From f6d32b979ea63e78011856a6ef5f2db120724aa6 Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Mon, 30 Oct 2023 15:19:28 +0000 Subject: [PATCH] build based on 90664bb --- dev/.documenter-siteinfo.json | 1 + dev/alt_features/index.html | 2 +- dev/alt_performance/index.html | 3 - dev/api/index.html | 6 +- dev/assets/documenter.js | 752 +++++++++++++++++++++---- dev/assets/search.js | 267 --------- dev/assets/themes/documenter-dark.css | 4 +- dev/assets/themes/documenter-light.css | 4 +- dev/assets/themeswap.js | 82 +-- dev/assets/warner.js | 87 +-- dev/background/index.html | 8 +- dev/debugging/index.html | 2 +- dev/formulas/index.html | 6 +- dev/index.html | 4 +- dev/roadmap/index.html | 2 +- dev/search/index.html | 2 - dev/search_index.js | 2 +- dev/tuto_builtin/index.html | 4 +- dev/tuto_custom/index.html | 8 +- 19 files changed, 777 insertions(+), 469 deletions(-) create mode 100644 dev/.documenter-siteinfo.json delete mode 100644 dev/alt_performance/index.html delete mode 100644 dev/assets/search.js delete mode 100644 dev/search/index.html diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json new file mode 100644 index 00000000..f835dd54 --- /dev/null +++ b/dev/.documenter-siteinfo.json @@ -0,0 +1 @@ +{"documenter":{"julia_version":"1.9.3","generation_timestamp":"2023-10-30T15:19:23","documenter_version":"1.1.2"}} \ No newline at end of file diff --git a/dev/alt_features/index.html b/dev/alt_features/index.html index 832c5863..5fefdc99 100644 --- a/dev/alt_features/index.html +++ b/dev/alt_features/index.html @@ -1,2 +1,2 @@ -Features · HiddenMarkovModels.jl
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Alternatives - features

We compare features among the following Julia packages:

We discard MarkovModels.jl because its focus is GPU computation. There are also more generic packages for probabilistic programming, which are able to perform MCMC or variational inference (eg. Turing.jl) but we leave those aside.

HMMs.jlHMMBase.jlHMMGradients.jl
AlgorithmsSim, FB, Vit, BWSim, FB, Vit, BWFB
Observation typesanythingNumber / Vectoranything
Observation distributionsDensityInterface.jlDistributions.jlmanual
Number typesanythingFloat64AbstractFloat
Priors / structurespossiblenopossible
Automatic differentiationyesnoyes
Multiple sequencesyesnoyes
Linear algebrayesyesno
Logarithmic probabilitieshalfwayhalfwayyes

Sim = Simulation, FB = Forward-Backward, Vit = Viterbi, BW = Baum-Welch

+Features · HiddenMarkovModels.jl
GitHub

Alternatives - features

We compare features among the following Julia packages:

We discard MarkovModels.jl because its focus is GPU computation. There are also more generic packages for probabilistic programming, which are able to perform MCMC or variational inference (eg. Turing.jl) but we leave those aside.

HMMs.jlHMMBase.jlHMMGradients.jl
AlgorithmsSim, FB, Vit, BWSim, FB, Vit, BWFB
Observation typesanythingNumber / Vectoranything
Observation distributionsDensityInterface.jlDistributions.jlmanual
Number typesanythingFloat64AbstractFloat
Priors / structurespossiblenopossible
Automatic differentiationyesnoyes
Multiple sequencesyesnoyes
Linear algebrayesyesno
Logarithmic probabilitieshalfwayhalfwayyes

Sim = Simulation, FB = Forward-Backward, Vit = Viterbi, BW = Baum-Welch

diff --git a/dev/alt_performance/index.html b/dev/alt_performance/index.html deleted file mode 100644 index ff17e0ba..00000000 --- a/dev/alt_performance/index.html +++ /dev/null @@ -1,3 +0,0 @@ - -Performance · HiddenMarkovModels.jl
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Alternatives - performance

We compare performance among the following packages:

The test case is an HMM with diagonal multivariate normal observations.

  • N: number of states
  • D: dimension of the observations
  • T: trajectory length
  • K: number of trajectories
  • I: number of Baum-Welch iterations
Why is this empty?

The benchmark suite is computationally expensive, and we only run it once for each new release. If the following section contains no plots and the links are broken, you're probably reading the development documentation or a local build of the website. Check out the stable documentation instead.

Reproducibility

These benchmarks were generated in the following environment: setup.txt.

If you want to run them on your machine:

  1. Clone the HiddenMarkovModels.jl repository

  2. Open a Julia REPL at the root

  3. Run the following commands

    include("benchmark/run_benchmarks.jl")
-include("benchmark/process_benchmarks.jl")

Remarks

Julia-to-Python overhead

Since the Python packages are called from Julia with PythonCall.jl, we pay a small overhead that is hard to quantify. On the plots, we compensate it by subtracting the runtime of the same algorithm for the smallest instance (N=1, D=1, T=2, K=1, I=1) from all Python-generated curves.

Allocations

A major bottleneck of performance in Julia is memory allocations. The benchmarks for HMMs.jl thus employ a custom implementation of diagonal multivariate normals, which is entirely allocation-free.

This partly explains the performance gap with HMMBase.jl as the dimension D grows beyond 1. Such a trick is also possible with HMMBase.jl, but more demanding since it requires subtyping Distribution from Distributions.jl, instead of just implementing DensityInterface.jl.

Parallelism

The packages we include have different approaches to parallelism, which can bias the evaluation in complex ways:

PackageStates NObservations DSequences K
HMMs.jlLinearAlgebra[2]depends[2]Threads[1]
HMMBase.jl-depends[2]-
hmmlearnNumPy[2]NumPy[2]NumPy[2]
pomegranatePyTorch[3]PyTorch[3]PyTorch[3]

We report each number of threads in setup.txt. Since OpenBLAS threads have negative interactions with Julia threads, we run the Julia benchmarks with a single OpenBLAS thread.

Numerical results

Low dimension

Full benchmark logs: low_dim.csv.

Here, pomegranate is not included because it is much slower on very small inputs.

High dimension

Full benchmark logs: high_dim.csv.

Here, HMMBase.jl is not included because it does not support multiple sequences.

  • 1affected by number of Julia threads
  • 2affected by number of OpenBLAS threads
  • 3affected by number of Pytorch threads
diff --git a/dev/api/index.html b/dev/api/index.html index edf6fc1b..1d89b324 100644 --- a/dev/api/index.html +++ b/dev/api/index.html @@ -1,8 +1,8 @@ -API reference · HiddenMarkovModels.jl
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API reference

HiddenMarkovModels — Module
HiddenMarkovModels

A Julia package for HMM modeling, simulation, inference and learning.

source

Types

Markov chains

HiddenMarkovModels.AbstractMarkovChain — Type
AbstractMarkovChain

Abstract supertype for a Markov chain amenable to simulation, inference and learning.

Required interface

  • initial_distribution(mc)
  • transition_matrix(mc)
  • fit!(mc, init_count, trans_count) (optional)

Applicable methods

  • rand([rng,] mc, T)
  • logdensityof(mc, state_seq)
  • fit(mc, state_seq_or_seqs) (if fit! is implemented)
source
HiddenMarkovModels.MarkovChain — Type
MarkovChain <: AbstractMarkovChain

Basic implementation of a discrete-state Markov chain.

Fields

  • init::AbstractVector: initial state probabilities
  • trans::AbstractMatrix: state transition matrix
source

Hidden Markov Models

HiddenMarkovModels.AbstractHiddenMarkovModel — Type
AbstractHiddenMarkovModel <: AbstractMarkovChain

Abstract supertype for an HMM amenable to simulation, inference and learning.

Required interface

  • initial_distribution(hmm)
  • transition_matrix(hmm)
  • obs_distribution(hmm, i)
  • fit!(hmm, init_count, trans_count, obs_seq, state_marginals) (optional)

Applicable methods

  • rand([rng,] hmm, T)
  • logdensityof(hmm, obs_seq) / logdensityof(hmm, obs_seqs, nb_seqs)
  • forward(hmm, obs_seq) / forward(hmm, obs_seqs, nb_seqs)
  • viterbi(hmm, obs_seq) / viterbi(hmm, obs_seqs, nb_seqs)
  • forward_backward(hmm, obs_seq) / forward_backward(hmm, obs_seqs, nb_seqs)
  • baum_welch(hmm, obs_seq) / baum_welch(hmm, obs_seqs, nb_seqs) if fit! is implemented
source
HiddenMarkovModels.HiddenMarkovModel — Type
HiddenMarkovModel{D} <: AbstractHiddenMarkovModel

Basic implementation of an HMM.

Fields

  • init::AbstractVector: initial state probabilities
  • trans::AbstractMatrix: state transition matrix
  • dists::AbstractVector{D}: observation distributions
source
HiddenMarkovModels.PermutedHMM — Type
PermutedHMM{H<:AbstractHMM}

Wrapper around an AbstractHMM that permutes its states.

This is computationally inefficient and mostly useful for evaluation.

Fields

  • hmm:H: the old HMM
  • perm::Vector{Int}: a permutation such that state i in the new HMM corresponds to state perm[i] in the old.
source

Basics

Base.rand — Function
rand([rng=default_rng(),] mc::AbstractMarkovChain, T)

Simulate mc for T time steps with a specified rng.

source
Base.length — Function
length(mc::AbstractMarkovChain)

Return the number of states of model.

source
HiddenMarkovModels.obs_distribution — Function
obs_distribution(hmm::AbstractHMM, i)

Return the observation distribution of hmm associated with state i.

The returned object dist must implement

  • rand(rng, dist)
  • DensityInterface.logdensityof(dist, x)
source

Inference

DensityInterface.logdensityof — Function
logdensityof(mc, state_seq)

Compute the loglikelihood of a single state sequence for a Markov chain.

source
logdensityof(hmm, obs_seq)

Apply the forward algorithm to compute the loglikelihood of a single observation sequence for an HMM.

Return a number.

source
logdensityof(hmm, obs_seqs, nb_seqs)

Apply the forward algorithm to compute the total loglikelihood of multiple observation sequences for an HMM.

Return a number.

Multithreading

This function is parallelized across sequences.

source
HiddenMarkovModels.forward — Function
forward(hmm, obs_seq)

Apply the forward algorithm to an HMM.

Return a tuple (α, logL) where

  • logL is the loglikelihood of the sequence
  • α[i] is the posterior probability of state i at the end of the sequence.
source
forward(hmm, obs_seqs, nb_seqs)

Apply the forward algorithm to an HMM, based on multiple observation sequences.

Return a vector of tuples (αₖ, logLₖ), where

  • logLₖ is the loglikelihood of sequence k
  • αₖ[i] is the posterior probability of state i at the end of sequence k
Multithreading

This function is parallelized across sequences.

source
HiddenMarkovModels.viterbi — Function
viterbi(hmm, obs_seq)

Apply the Viterbi algorithm to compute the most likely state sequence of an HMM.

Return a vector of integers.

source
viterbi(hmm, obs_seqs, nb_seqs)

Apply the Viterbi algorithm to compute the most likely state sequences of an HMM, based on multiple observation sequences.

Return a vector of vectors of integers.

Multithreading

This function is parallelized across sequences.

source
HiddenMarkovModels.forward_backward — Function
forward_backward(hmm, obs_seq)

Apply the forward-backward algorithm to estimate the posterior state marginals of an HMM.

Return a ForwardBackwardStorage.

source
forward_backward(hmm, obs_seqs, nb_seqs)

Apply the forward-backward algorithm to estimate the posterior state marginals of an HMM, based on multiple observation sequences.

Return a vector of ForwardBackwardStorage objects.

Multithreading

This function is parallelized across sequences.

source

Learning

StatsAPI.fit! — Function
fit!(mc::MC, init_count, trans_count)

Update mc in-place based on information generated from a state sequence.

source
fit!(hmm::HMM, init_count, trans_count, obs_seq, state_marginals)

Update hmm in-place based on information generated during forward-backward.

source
StatsAPI.fit — Function
fit(mc, state_seq_or_seqs)

Fit a Markov chain of the same type as mc to one or several state sequence(s).

Beware that mc must be an actual object of type MarkovChain, and not the type itself as is usually done eg. in Distributions.jl.

source
HiddenMarkovModels.baum_welch — Function
baum_welch(
+API reference · HiddenMarkovModels.jl
GitHub
API reference
HiddenMarkovModels — Module
HiddenMarkovModels
A Julia package for HMM modeling, simulation, inference and learning.
The alias HMMs is exported for the package name.
source
Types
Markov chains
HiddenMarkovModels.AbstractMarkovChain — Type
AbstractMarkovChain
Abstract supertype for a Markov chain amenable to simulation, inference and learning.
Required interface
  • initial_distribution(mc)
  • transition_matrix(mc)
  • fit!(mc, init_count, trans_count) (optional)
Applicable methods
  • rand([rng,] mc, T)
  • logdensityof(mc, state_seq)
  • fit(mc, state_seq_or_seqs) (if fit! is implemented)
source
HiddenMarkovModels.MarkovChain — Type
MarkovChain <: AbstractMarkovChain
Basic implementation of a discrete-state Markov chain.
Fields
  • init::AbstractVector: initial state probabilities
  • trans::AbstractMatrix: state transition matrix
source
Hidden Markov Models
HiddenMarkovModels.AbstractHiddenMarkovModel — Type
AbstractHiddenMarkovModel <: AbstractMarkovChain
Abstract supertype for an HMM amenable to simulation, inference and learning.
Required interface
  • initial_distribution(hmm)
  • transition_matrix(hmm)
  • obs_distribution(hmm, i)
  • fit!(hmm, init_count, trans_count, obs_seq, state_marginals) (optional)
Applicable methods
  • rand([rng,] hmm, T)
  • logdensityof(hmm, obs_seq) / logdensityof(hmm, obs_seqs, nb_seqs)
  • forward(hmm, obs_seq) / forward(hmm, obs_seqs, nb_seqs)
  • viterbi(hmm, obs_seq) / viterbi(hmm, obs_seqs, nb_seqs)
  • forward_backward(hmm, obs_seq) / forward_backward(hmm, obs_seqs, nb_seqs)
  • baum_welch(hmm, obs_seq) / baum_welch(hmm, obs_seqs, nb_seqs) if fit! is implemented
source
HiddenMarkovModels.HiddenMarkovModel — Type
HiddenMarkovModel{D} <: AbstractHiddenMarkovModel
Basic implementation of an HMM.
Fields
  • init::AbstractVector: initial state probabilities
  • trans::AbstractMatrix: state transition matrix
  • dists::AbstractVector{D}: observation distributions
source
HiddenMarkovModels.PermutedHMM — Type
PermutedHMM{H<:AbstractHMM}
Wrapper around an AbstractHMM that permutes its states.
This is computationally inefficient and mostly useful for evaluation.
Fields
  • hmm:H: the old HMM
  • perm::Vector{Int}: a permutation such that state i in the new HMM corresponds to state perm[i] in the old.
source
Basics
Base.rand — Function
rand([rng=default_rng(),] mc::AbstractMarkovChain, T)
Simulate mc for T time steps with a specified rng.
source
Base.length — Function
length(mc::AbstractMarkovChain)
Return the number of states of model.
source
HiddenMarkovModels.obs_distribution — Function
obs_distribution(hmm::AbstractHMM, i)
Return the observation distribution of hmm associated with state i.
The returned object dist must implement
  • rand(rng, dist)
  • DensityInterface.logdensityof(dist, x)
source
Inference
DensityInterface.logdensityof — Function
logdensityof(mc, state_seq)
Compute the loglikelihood of a single state sequence for a Markov chain.
source
logdensityof(hmm, obs_seq)
Apply the forward algorithm to compute the loglikelihood of a single observation sequence for an HMM.
Return a number.
source
logdensityof(hmm, obs_seqs, nb_seqs)
Apply the forward algorithm to compute the total loglikelihood of multiple observation sequences for an HMM.
Return a number.
Multithreading
This function is parallelized across sequences.
source
HiddenMarkovModels.forward — Function
forward(hmm, obs_seq)
Apply the forward algorithm to an HMM.
Return a tuple (α, logL) where
  • logL is the loglikelihood of the sequence
  • α[i] is the posterior probability of state i at the end of the sequence.
source
forward(hmm, obs_seqs, nb_seqs)
Apply the forward algorithm to an HMM, based on multiple observation sequences.
Return a vector of tuples (αₖ, logLₖ), where
  • logLₖ is the loglikelihood of sequence k
  • αₖ[i] is the posterior probability of state i at the end of sequence k
Multithreading
This function is parallelized across sequences.
source
HiddenMarkovModels.viterbi — Function
viterbi(hmm, obs_seq)
Apply the Viterbi algorithm to compute the most likely state sequence of an HMM.
Return a vector of integers.
source
viterbi(hmm, obs_seqs, nb_seqs)
Apply the Viterbi algorithm to compute the most likely state sequences of an HMM, based on multiple observation sequences.
Return a vector of vectors of integers.
Multithreading
This function is parallelized across sequences.
source
HiddenMarkovModels.forward_backward — Function
forward_backward(hmm, obs_seq)
Apply the forward-backward algorithm to estimate the posterior state marginals of an HMM.
Return a ForwardBackwardStorage.
source
forward_backward(hmm, obs_seqs, nb_seqs)
Apply the forward-backward algorithm to estimate the posterior state marginals of an HMM, based on multiple observation sequences.
Return a vector of ForwardBackwardStorage objects.
Multithreading
This function is parallelized across sequences.
source
Learning
StatsAPI.fit! — Function
fit!(mc::MC, init_count, trans_count)
Update mc in-place based on information generated from a state sequence.
source
fit!(hmm::HMM, init_count, trans_count, obs_seq, state_marginals)
Update hmm in-place based on information generated during forward-backward.
source
StatsAPI.fit — Function
fit(mc, state_seq_or_seqs)
Fit a Markov chain of the same type as mc to one or several state sequence(s).
Beware that mc must be an actual object of type MarkovChain, and not the type itself as is usually done eg. in Distributions.jl.
source
HiddenMarkovModels.baum_welch — Function
baum_welch(
     hmm_init, obs_seq;
     atol, max_iterations, check_loglikelihood_increasing
-)
Apply the Baum-Welch algorithm to estimate the parameters of an HMM starting from hmm_init.
Return a tuple (hmm_est, logL_evolution).
Keyword arguments
  • atol: Minimum loglikelihood increase at an iteration of the algorithm (otherwise the algorithm is deemed to have converged)
  • max_iterations: Maximum number of iterations of the algorithm
  • check_loglikelihood_increasing: Whether to throw an error if the loglikelihood decreases
source
baum_welch(
+)
Apply the Baum-Welch algorithm to estimate the parameters of an HMM starting from hmm_init.
Return a tuple (hmm_est, logL_evolution).
Keyword arguments
  • atol: Minimum loglikelihood increase at an iteration of the algorithm (otherwise the algorithm is deemed to have converged)
  • max_iterations: Maximum number of iterations of the algorithm
  • check_loglikelihood_increasing: Whether to throw an error if the loglikelihood decreases
source
baum_welch(
     hmm_init, obs_seqs, nb_seqs;
     atol, max_iterations, check_loglikelihood_increasing
-)
Apply the Baum-Welch algorithm to estimate the parameters of an HMM starting from hmm_init, based on nb_seqs observation sequences.
Return a tuple (hmm_est, logL_evolution).
Multithreading
This function is parallelized across sequences.
Keyword arguments
  • atol: Minimum loglikelihood increase at an iteration of the algorithm (otherwise the algorithm is deemed to have converged)
  • max_iterations: Maximum number of iterations of the algorithm
  • check_loglikelihood_increasing: Whether to throw an error if the loglikelihood decreases
source
Internals
HiddenMarkovModels.ForwardBackwardStorage — Type
ForwardBackwardStorage{R}
Store forward-backward quantities with element type R.
Fields
Let X denote the vector of hidden states and Y denote the vector of observations. The following fields are part of the API:
  • γ::Matrix{R}: posterior one-state marginals γ[i,t] = ℙ(X[t]=i | Y[1:T])
  • ξ::Array{R,3}: posterior two-state marginals ξ[i,j,t] = ℙ(X[t:t+1]=(i,j) | Y[1:T])
The following fields are internals and subject to change:
  • α::Matrix{R}: scaled forward variables α[i,t] proportional to ℙ(Y[1:t], X[t]=i) (up to a function of t)
  • β::Matrix{R}: scaled backward variables β[i,t] proportional to ℙ(Y[t+1:T] | X[t]=i) (up to a function of t)
  • c::Vector{R}: forward variable inverse normalizations c[t] = 1 / sum(α[:,t])
  • logm::Vector{R}: maximum of the observation loglikelihoods logm[t] = maximum(logB[:, t])
  • Bscaled::Matrix{R}: numerically stabilized observation likelihoods Bscaled[i,t] = exp.(logB[i,t] - logm[t])
  • Bβscaled::Matrix{R}: numerically stabilized product Bβscaled[i,t] = Bscaled[i,t] * β[i,t]
source
HiddenMarkovModels.fit_element_from_sequence! — Function
fit_element_from_sequence!(dists, i, x, w)
Modify the i-th element of dists by fitting it to an observation sequence x with associated weight sequence w.
The default behavior is a fallback on StatsAPI.fit!, which users are encouraged to implement if their observation distributions are mutable. If this is not possible, please override fit_element_from_sequence! directly.
source
HiddenMarkovModels.LightDiagNormal — Type
LightDiagNormal
An HMMs-compatible implementation of a multivariate normal distribution with diagonal covariance, enabling allocation-free estimation.
This is not part of the public API and is expected to change.
source
Notations
Integers
  • N: number of states
  • D: dimension of the observations
  • T: trajectory length
  • K: number of trajectories
Models and simulations
  • p or init: initial_distribution (vector of state probabilities)
  • A or trans: transition_matrix (matrix of transition probabilities)
  • dists: observation distribution (vector of rand-able and logdensityof-able objects)
  • state_seq: a sequence of states (vector of integers)
  • obs_seq: a sequence of observations (vector of individual observations)
  • obs_seqs: several sequences of observations
Forward backward
  • (log)b: vector of observation (log)likelihoods by state for an individual observation
  • (log)B: matrix of observation (log)likelihoods by state for a sequence of observations
  • α: scaled forward variables
  • β: scaled backward variables
  • γ: one-state marginals
  • ξ: two-state marginals
  • logL: loglikelihood of a sequence of observations
Index

+)

Apply the Baum-Welch algorithm to estimate the parameters of an HMM starting from hmm_init, based on nb_seqs observation sequences.

Return a tuple (hmm_est, logL_evolution).

Multithreading

This function is parallelized across sequences.

Keyword arguments

  • atol: Minimum loglikelihood increase at an iteration of the algorithm (otherwise the algorithm is deemed to have converged)
  • max_iterations: Maximum number of iterations of the algorithm
  • check_loglikelihood_increasing: Whether to throw an error if the loglikelihood decreases
source

Internals

HiddenMarkovModels.ForwardBackwardStorage — Type
ForwardBackwardStorage{R}

Store forward-backward quantities with element type R.

Fields

Let X denote the vector of hidden states and Y denote the vector of observations. The following fields are part of the API:

  • γ::Matrix{R}: posterior one-state marginals γ[i,t] = ℙ(X[t]=i | Y[1:T])
  • ξ::Array{R,3}: posterior two-state marginals ξ[i,j,t] = ℙ(X[t:t+1]=(i,j) | Y[1:T])

The following fields are internals and subject to change:

  • α::Matrix{R}: scaled forward variables α[i,t] proportional to ℙ(Y[1:t], X[t]=i) (up to a function of t)
  • β::Matrix{R}: scaled backward variables β[i,t] proportional to ℙ(Y[t+1:T] | X[t]=i) (up to a function of t)
  • c::Vector{R}: forward variable inverse normalizations c[t] = 1 / sum(α[:,t])
  • logm::Vector{R}: maximum of the observation loglikelihoods logm[t] = maximum(logB[:, t])
  • Bscaled::Matrix{R}: numerically stabilized observation likelihoods Bscaled[i,t] = exp.(logB[i,t] - logm[t])
  • Bβscaled::Matrix{R}: numerically stabilized product Bβscaled[i,t] = Bscaled[i,t] * β[i,t]
source
HiddenMarkovModels.fit_element_from_sequence! — Function
fit_element_from_sequence!(dists, i, x, w)

Modify the i-th element of dists by fitting it to an observation sequence x with associated weight sequence w.

The default behavior is a fallback on StatsAPI.fit!, which users are encouraged to implement if their observation distributions are mutable. If this is not possible, please override fit_element_from_sequence! directly.

source
HiddenMarkovModels.LightDiagNormal — Type
LightDiagNormal

An HMMs-compatible implementation of a multivariate normal distribution with diagonal covariance, enabling allocation-free estimation.

This is not part of the public API and is expected to change.

source

Notations

Integers

  • N: number of states
  • D: dimension of the observations
  • T: trajectory length
  • K: number of trajectories

Models and simulations

  • p or init: initial_distribution (vector of state probabilities)
  • A or trans: transition_matrix (matrix of transition probabilities)
  • dists: observation distribution (vector of rand-able and logdensityof-able objects)
  • state_seq: a sequence of states (vector of integers)
  • obs_seq: a sequence of observations (vector of individual observations)
  • obs_seqs: several sequences of observations

Forward backward

  • (log)b: vector of observation (log)likelihoods by state for an individual observation
  • (log)B: matrix of observation (log)likelihoods by state for a sequence of observations
  • α: scaled forward variables
  • β: scaled backward variables
  • γ: one-state marginals
  • ξ: two-state marginals
  • logL: loglikelihood of a sequence of observations

Index

diff --git a/dev/assets/documenter.js b/dev/assets/documenter.js index 6adfbbbf..f5311607 100644 --- a/dev/assets/documenter.js +++ b/dev/assets/documenter.js @@ -1,15 +1,16 @@ // Generated by Documenter.jl requirejs.config({ paths: { - 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.5.1/languages/julia.min', + 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia.min', 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', - 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.12.1/jquery-ui.min', - 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/contrib/auto-render.min', - 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.6.0/jquery.min', + 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.13.2/jquery-ui.min', + 'minisearch': 'https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min', + 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/contrib/auto-render.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.0/jquery.min', 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', - 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/katex.min', - 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.5.1/highlight.min', - 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.5.1/languages/julia-repl.min', + 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min', + 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min', + 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia-repl.min', }, shim: { "highlight-julia": { @@ -70,13 +71,90 @@ $(document).ready(function() { hljs.highlightAll(); }) +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +let timer = 0; +var isExpanded = true; + +$(document).on("click", ".docstring header", function () { + let articleToggleTitle = "Expand docstring"; + + debounce(() => { + if ($(this).siblings("section").is(":visible")) { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + } else { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + articleToggleTitle = "Collapse docstring"; + } + + $(this) + .find(".docstring-article-toggle-button") + .prop("title", articleToggleTitle); + $(this).siblings("section").slideToggle(); + }); +}); + +$(document).on("click", ".docs-article-toggle-button", function () { + let articleToggleTitle = "Expand docstring"; + let navArticleToggleTitle = "Expand all docstrings"; + + debounce(() => { + if (isExpanded) { + $(this).removeClass("fa-chevron-up").addClass("fa-chevron-down"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + + isExpanded = false; + + $(".docstring section").slideUp(); + } else { + $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + isExpanded = true; + articleToggleTitle = "Collapse docstring"; + navArticleToggleTitle = "Collapse all docstrings"; + + $(".docstring section").slideDown(); + } + + $(this).prop("title", navArticleToggleTitle); + $(".docstring-article-toggle-button").prop("title", articleToggleTitle); + }); +}); + +function debounce(callback, timeout = 300) { + if (Date.now() - timer > timeout) { + callback(); + } + + clearTimeout(timer); + + timer = Date.now(); +} + }) //////////////////////////////////////////////////////////////////////////////// require([], function() { function addCopyButtonCallbacks() { for (const el of document.getElementsByTagName("pre")) { const button = document.createElement("button"); - button.classList.add("copy-button", "fas", "fa-copy"); + button.classList.add("copy-button", "fa-solid", "fa-copy"); + button.setAttribute("aria-label", "Copy this code block"); + button.setAttribute("title", "Copy"); + el.appendChild(button); const success = function () { @@ -85,7 +163,7 @@ function addCopyButtonCallbacks() { }; const failure = function () { - button.classList.add("error", "fa-times"); + button.classList.add("error", "fa-xmark"); button.classList.remove("fa-copy"); }; @@ -94,7 +172,7 @@ function addCopyButtonCallbacks() { setTimeout(function () { button.classList.add("fa-copy"); - button.classList.remove("success", "fa-check", "fa-times"); + button.classList.remove("success", "fa-check", "fa-xmark"); }, 5000); }); } @@ -138,29 +216,418 @@ require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) { // Manages the top navigation bar (hides it when the user starts scrolling down on the // mobile). window.Headroom = Headroom; // work around buggy module loading? -$(document).ready(function() { - $('#documenter .docs-navbar').headroom({ - "tolerance": {"up": 10, "down": 10}, +$(document).ready(function () { + $("#documenter .docs-navbar").headroom({ + tolerance: { up: 10, down: 10 }, }); +}); + }) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'minisearch'], function($, minisearch) { + +// In general, most search related things will have "search" as a prefix. +// To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc + +let results = []; +let timer = undefined; + +let data = documenterSearchIndex["docs"].map((x, key) => { + x["id"] = key; // minisearch requires a unique for each object + return x; +}); + +// list below is the lunr 2.1.3 list minus the intersect with names(Base) +// (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) +// ideally we'd just filter the original list but it's not available as a variable +const stopWords = new Set([ + "a", + "able", + "about", + "across", + "after", + "almost", + "also", + "am", + "among", + "an", + "and", + "are", + "as", + "at", + "be", + "because", + "been", + "but", + "by", + "can", + "cannot", + "could", + "dear", + "did", + "does", + "either", + "ever", + "every", + "from", + "got", + "had", + "has", + "have", + "he", + "her", + "hers", + "him", + "his", + "how", + "however", + "i", + "if", + "into", + "it", + "its", + "just", + "least", + "like", + "likely", + "may", + "me", + "might", + "most", + "must", + "my", + "neither", + "no", + "nor", + "not", + "of", + "off", + "often", + "on", + "or", + "other", + "our", + "own", + "rather", + "said", + "say", + "says", + "she", + "should", + "since", + "so", + "some", + "than", + "that", + "the", + "their", + "them", + "then", + "there", + "these", + "they", + "this", + "tis", + "to", + "too", + "twas", + "us", + "wants", + "was", + "we", + "were", + "what", + "when", + "who", + "whom", + "why", + "will", + "would", + "yet", + "you", + "your", +]); + +let index = new minisearch({ + fields: ["title", "text"], // fields to index for full-text search + storeFields: ["location", "title", "text", "category", "page"], // fields to return with search results + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + } + + return word ?? null; + }, + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not find anything if searching for "add!", only for the entire qualification + tokenize: (string) => string.split(/[\s\-\.]+/), + // options which will be applied during the search + searchOptions: { + boost: { title: 100 }, + fuzzy: 2, + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + } + + return word ?? null; + }, + tokenize: (string) => string.split(/[\s\-\.]+/), + }, +}); + +index.addAll(data); + +let filters = [...new Set(data.map((x) => x.category))]; +var modal_filters = make_modal_body_filters(filters); +var filter_results = []; + +$(document).on("keyup", ".documenter-search-input", function (event) { + // Adding a debounce to prevent disruptions from super-speed typing! + debounce(() => update_search(filter_results), 300); +}); + +$(document).on("click", ".search-filter", function () { + if ($(this).hasClass("search-filter-selected")) { + $(this).removeClass("search-filter-selected"); + } else { + $(this).addClass("search-filter-selected"); + } + + // Adding a debounce to prevent disruptions from crazy clicking! + debounce(() => get_filters(), 300); +}); + +/** + * A debounce function, takes a function and an optional timeout in milliseconds + * + * @function callback + * @param {number} timeout + */ +function debounce(callback, timeout = 300) { + clearTimeout(timer); + timer = setTimeout(callback, timeout); +} + +/** + * Make/Update the search component + * + * @param {string[]} selected_filters + */ +function update_search(selected_filters = []) { + let initial_search_body = ` +
Type something to get started!
+ `; + + let querystring = $(".documenter-search-input").val(); + + if (querystring.trim()) { + results = index.search(querystring, { + filter: (result) => { + // Filtering results + if (selected_filters.length === 0) { + return result.score >= 1; + } else { + return ( + result.score >= 1 && selected_filters.includes(result.category) + ); + } + }, + }); + + let search_result_container = ``; + let search_divider = `
`; + + if (results.length) { + let links = []; + let count = 0; + let search_results = ""; + + results.forEach(function (result) { + if (result.location) { + // Checking for duplication of results for the same page + if (!links.includes(result.location)) { + search_results += make_search_result(result, querystring); + count++; + } + + links.push(result.location); + } + }); + + let result_count = `
${count} result(s)
`; + + search_result_container = ` +
+ ${modal_filters} + ${search_divider} + ${result_count} +
+ ${search_results} +
+
+ `; + } else { + search_result_container = ` +
+ ${modal_filters} + ${search_divider} +
0 result(s)
+
+
No result found!
+ `; + } + + if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").removeClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(search_result_container); + } else { + filter_results = []; + modal_filters = make_modal_body_filters(filters, filter_results); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(initial_search_body); + } +} + +/** + * Make the modal filter html + * + * @param {string[]} filters + * @param {string[]} selected_filters + * @returns string + */ +function make_modal_body_filters(filters, selected_filters = []) { + let str = ``; + + filters.forEach((val) => { + if (selected_filters.includes(val)) { + str += `${val}`; + } else { + str += `${val}`; + } + }); + + let filter_html = ` +
+ Filters: + ${str} +
+ `; + + return filter_html; +} + +/** + * Make the result component given a minisearch result data object and the value of the search input as queryString. + * To view the result object structure, refer: https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult + * + * @param {object} result + * @param {string} querystring + * @returns string + */ +function make_search_result(result, querystring) { + let search_divider = `
`; + let display_link = + result.location.slice(Math.max(0), Math.min(50, result.location.length)) + + (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div + + if (result.page !== "") { + display_link += ` (${result.page})`; + } + + let textindex = new RegExp(`\\b${querystring}\\b`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`\\b${querystring}\\b`, "i"), // For first occurrence + '$&' + ) + + "..." + : ""; // highlights the match + + let in_code = false; + if (!["page", "section"].includes(result.category.toLowerCase())) { + in_code = true; + } + + // We encode the full url to escape some special characters which can lead to broken links + let result_div = ` + +
+
${result.title}
+
${result.category}
+
+

+ ${display_result} +

+
+ ${display_link} +
+ + ${search_divider} + `; + + return result_div; +} + +/** + * Get selected filters, remake the filter html and lastly update the search modal + */ +function get_filters() { + let ele = $(".search-filters .search-filter-selected").get(); + filter_results = ele.map((x) => $(x).text().toLowerCase()); + modal_filters = make_modal_body_filters(filters, filter_results); + update_search(filter_results); +} }) //////////////////////////////////////////////////////////////////////////////// require(['jquery'], function($) { // Modal settings dialog -$(document).ready(function() { - var settings = $('#documenter-settings'); - $('#documenter-settings-button').click(function(){ - settings.toggleClass('is-active'); +$(document).ready(function () { + var settings = $("#documenter-settings"); + $("#documenter-settings-button").click(function () { + settings.toggleClass("is-active"); }); // Close the dialog if X is clicked - $('#documenter-settings button.delete').click(function(){ - settings.removeClass('is-active'); + $("#documenter-settings button.delete").click(function () { + settings.removeClass("is-active"); }); // Close dialog if ESC is pressed - $(document).keyup(function(e) { - if (e.keyCode == 27) settings.removeClass('is-active'); + $(document).keyup(function (e) { + if (e.keyCode == 27) settings.removeClass("is-active"); }); }); @@ -168,151 +635,242 @@ $(document).ready(function() { //////////////////////////////////////////////////////////////////////////////// require(['jquery'], function($) { +let search_modal_header = ` +
+
+

+ + + + +

+
+
+ +
+
+`; + +let initial_search_body = ` +
Type something to get started!
+`; + +let search_modal_footer = ` + +`; + +$(document.body).append( + ` + + ` +); + +document.querySelector(".docs-search-query").addEventListener("click", () => { + openModal(); +}); + +document.querySelector(".close-search-modal").addEventListener("click", () => { + closeModal(); +}); + +$(document).on("click", ".search-result-link", function () { + closeModal(); +}); + +document.addEventListener("keydown", (event) => { + if ((event.ctrlKey || event.metaKey) && event.key === "/") { + openModal(); + } else if (event.key === "Escape") { + closeModal(); + } + + return false; +}); + +// Functions to open and close a modal +function openModal() { + let searchModal = document.querySelector("#search-modal"); + + searchModal.classList.add("is-active"); + document.querySelector(".documenter-search-input").focus(); +} + +function closeModal() { + let searchModal = document.querySelector("#search-modal"); + let initial_search_body = ` +
Type something to get started!
+ `; + + searchModal.classList.remove("is-active"); + document.querySelector(".documenter-search-input").blur(); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".documenter-search-input").val(""); + $(".search-modal-card-body").html(initial_search_body); +} + +document + .querySelector("#search-modal .modal-background") + .addEventListener("click", () => { + closeModal(); + }); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + // Manages the showing and hiding of the sidebar. -$(document).ready(function() { +$(document).ready(function () { var sidebar = $("#documenter > .docs-sidebar"); - var sidebar_button = $("#documenter-sidebar-button") - sidebar_button.click(function(ev) { + var sidebar_button = $("#documenter-sidebar-button"); + sidebar_button.click(function (ev) { ev.preventDefault(); - sidebar.toggleClass('visible'); - if (sidebar.hasClass('visible')) { + sidebar.toggleClass("visible"); + if (sidebar.hasClass("visible")) { // Makes sure that the current menu item is visible in the sidebar. $("#documenter .docs-menu a.is-active").focus(); } }); - $("#documenter > .docs-main").bind('click', function(ev) { + $("#documenter > .docs-main").bind("click", function (ev) { if ($(ev.target).is(sidebar_button)) { return; } - if (sidebar.hasClass('visible')) { - sidebar.removeClass('visible'); + if (sidebar.hasClass("visible")) { + sidebar.removeClass("visible"); } }); -}) +}); // Resizes the package name / sitename in the sidebar if it is too wide. // Inspired by: https://github.com/davatron5000/FitText.js -$(document).ready(function() { +$(document).ready(function () { e = $("#documenter .docs-autofit"); function resize() { - var L = parseInt(e.css('max-width'), 10); + var L = parseInt(e.css("max-width"), 10); var L0 = e.width(); - if(L0 > L) { - var h0 = parseInt(e.css('font-size'), 10); - e.css('font-size', L * h0 / L0); + if (L0 > L) { + var h0 = parseInt(e.css("font-size"), 10); + e.css("font-size", (L * h0) / L0); // TODO: make sure it survives resizes? } } // call once and then register events resize(); $(window).resize(resize); - $(window).on('orientationchange', resize); + $(window).on("orientationchange", resize); }); // Scroll the navigation bar to the currently selected menu item -$(document).ready(function() { +$(document).ready(function () { var sidebar = $("#documenter .docs-menu").get(0); var active = $("#documenter .docs-menu .is-active").get(0); - if(typeof active !== 'undefined') { + if (typeof active !== "undefined") { sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; } -}) +}); }) //////////////////////////////////////////////////////////////////////////////// require(['jquery'], function($) { -function set_theme(theme) { - var active = null; - var disabled = []; - for (var i = 0; i < document.styleSheets.length; i++) { - var ss = document.styleSheets[i]; - var themename = ss.ownerNode.getAttribute("data-theme-name"); - if(themename === null) continue; // ignore non-theme stylesheets - // Find the active theme - if(themename === theme) active = ss; - else disabled.push(ss); - } - if(active !== null) { - active.disabled = false; - if(active.ownerNode.getAttribute("data-theme-primary") === null) { - document.getElementsByTagName('html')[0].className = "theme--" + theme; - } else { - document.getElementsByTagName('html')[0].className = ""; - } - disabled.forEach(function(ss){ - ss.disabled = true; - }); - } - - // Store the theme in localStorage - if(typeof(window.localStorage) !== "undefined") { - window.localStorage.setItem("documenter-theme", theme); - } else { - console.error("Browser does not support window.localStorage"); - } -} - // Theme picker setup -$(document).ready(function() { +$(document).ready(function () { // onchange callback - $('#documenter-themepicker').change(function themepick_callback(ev){ - var themename = $('#documenter-themepicker option:selected').attr('value'); - set_theme(themename); + $("#documenter-themepicker").change(function themepick_callback(ev) { + var themename = $("#documenter-themepicker option:selected").attr("value"); + if (themename === "auto") { + // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light'); + window.localStorage.removeItem("documenter-theme"); + } else { + // set_theme(themename); + window.localStorage.setItem("documenter-theme", themename); + } + // We re-use the global function from themeswap.js to actually do the swapping. + set_theme_from_local_storage(); }); // Make sure that the themepicker displays the correct theme when the theme is retrieved // from localStorage - if(typeof(window.localStorage) !== "undefined") { - var theme = window.localStorage.getItem("documenter-theme"); - if(theme !== null) { - $('#documenter-themepicker option').each(function(i,e) { - e.selected = (e.value === theme); - }) - } else { - $('#documenter-themepicker option').each(function(i,e) { - e.selected = $("html").hasClass(`theme--${e.value}`); - }) + if (typeof window.localStorage !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if (theme !== null) { + $("#documenter-themepicker option").each(function (i, e) { + e.selected = e.value === theme; + }); } } -}) +}); }) //////////////////////////////////////////////////////////////////////////////// require(['jquery'], function($) { // update the version selector with info from the siteinfo.js and ../versions.js files -$(document).ready(function() { +$(document).ready(function () { // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the // siteinfo.js file, we just return immediately and not display the version selector. - if (typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === 'boolean' && DOCUMENTER_VERSION_SELECTOR_DISABLED) { + if ( + typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" && + DOCUMENTER_VERSION_SELECTOR_DISABLED + ) { return; } var version_selector = $("#documenter .docs-version-selector"); var version_selector_select = $("#documenter .docs-version-selector select"); - version_selector_select.change(function(x) { - target_href = version_selector_select.children("option:selected").get(0).value; + version_selector_select.change(function (x) { + target_href = version_selector_select + .children("option:selected") + .get(0).value; window.location.href = target_href; }); // add the current version to the selector based on siteinfo.js, but only if the selector is empty - if (typeof DOCUMENTER_CURRENT_VERSION !== 'undefined' && $('#version-selector > option').length == 0) { - var option = $(""); + if ( + typeof DOCUMENTER_CURRENT_VERSION !== "undefined" && + $("#version-selector > option").length == 0 + ) { + var option = $( + "" + ); version_selector_select.append(option); } - if (typeof DOC_VERSIONS !== 'undefined') { + if (typeof DOC_VERSIONS !== "undefined") { var existing_versions = version_selector_select.children("option"); - var existing_versions_texts = existing_versions.map(function(i,x){return x.text}); - DOC_VERSIONS.forEach(function(each) { - var version_url = documenterBaseURL + "/../" + each; + var existing_versions_texts = existing_versions.map(function (i, x) { + return x.text; + }); + DOC_VERSIONS.forEach(function (each) { + var version_url = documenterBaseURL + "/../" + each + "/"; var existing_id = $.inArray(each, existing_versions_texts); // if not already in the version selector, add it as a new option, // otherwise update the old option with the URL and enable it if (existing_id == -1) { - var option = $(""); + var option = $( + "" + ); version_selector_select.append(option); } else { var option = existing_versions[existing_id]; @@ -326,6 +884,6 @@ $(document).ready(function() { if (version_selector_select.children("option").length > 0) { version_selector.toggleClass("visible"); } -}) +}); }) diff --git a/dev/assets/search.js b/dev/assets/search.js deleted file mode 100644 index c133f741..00000000 --- a/dev/assets/search.js +++ /dev/null @@ -1,267 +0,0 @@ -// Generated by Documenter.jl -requirejs.config({ - paths: { - 'lunr': 'https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min', - 'lodash': 'https://cdnjs.cloudflare.com/ajax/libs/lodash.js/4.17.21/lodash.min', - 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.6.0/jquery.min', - } -}); -//////////////////////////////////////////////////////////////////////////////// -require(['jquery', 'lunr', 'lodash'], function($, lunr, _) { - -$(document).ready(function() { - // parseUri 1.2.2 - // (c) Steven Levithan - // MIT License - function parseUri (str) { - var o = parseUri.options, - m = o.parser[o.strictMode ? "strict" : "loose"].exec(str), - uri = {}, - i = 14; - - while (i--) uri[o.key[i]] = m[i] || ""; - - uri[o.q.name] = {}; - uri[o.key[12]].replace(o.q.parser, function ($0, $1, $2) { - if ($1) uri[o.q.name][$1] = $2; - }); - - return uri; - }; - parseUri.options = { - strictMode: false, - key: ["source","protocol","authority","userInfo","user","password","host","port","relative","path","directory","file","query","anchor"], - q: { - name: "queryKey", - parser: /(?:^|&)([^&=]*)=?([^&]*)/g - }, - parser: { - strict: /^(?:([^:\/?#]+):)?(?:\/\/((?:(([^:@]*)(?::([^:@]*))?)?@)?([^:\/?#]*)(?::(\d*))?))?((((?:[^?#\/]*\/)*)([^?#]*))(?:\?([^#]*))?(?:#(.*))?)/, - loose: /^(?:(?![^:@]+:[^:@\/]*@)([^:\/?#.]+):)?(?:\/\/)?((?:(([^:@]*)(?::([^:@]*))?)?@)?([^:\/?#]*)(?::(\d*))?)(((\/(?:[^?#](?![^?#\/]*\.[^?#\/.]+(?:[?#]|$)))*\/?)?([^?#\/]*))(?:\?([^#]*))?(?:#(.*))?)/ - } - }; - - $("#search-form").submit(function(e) { - e.preventDefault() - }) - - // list below is the lunr 2.1.3 list minus the intersect with names(Base) - // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) - // ideally we'd just filter the original list but it's not available as a variable - lunr.stopWordFilter = lunr.generateStopWordFilter([ - 'a', - 'able', - 'about', - 'across', - 'after', - 'almost', - 'also', - 'am', - 'among', - 'an', - 'and', - 'are', - 'as', - 'at', - 'be', - 'because', - 'been', - 'but', - 'by', - 'can', - 'cannot', - 'could', - 'dear', - 'did', - 'does', - 'either', - 'ever', - 'every', - 'from', - 'got', - 'had', - 'has', - 'have', - 'he', - 'her', - 'hers', - 'him', - 'his', - 'how', - 'however', - 'i', - 'if', - 'into', - 'it', - 'its', - 'just', - 'least', - 'like', - 'likely', - 'may', - 'me', - 'might', - 'most', - 'must', - 'my', - 'neither', - 'no', - 'nor', - 'not', - 'of', - 'off', - 'often', - 'on', - 'or', - 'other', - 'our', - 'own', - 'rather', - 'said', - 'say', - 'says', - 'she', - 'should', - 'since', - 'so', - 'some', - 'than', - 'that', - 'the', - 'their', - 'them', - 'then', - 'there', - 'these', - 'they', - 'this', - 'tis', - 'to', - 'too', - 'twas', - 'us', - 'wants', - 'was', - 'we', - 'were', - 'what', - 'when', - 'who', - 'whom', - 'why', - 'will', - 'would', - 'yet', - 'you', - 'your' - ]) - - // add . as a separator, because otherwise "title": "Documenter.Anchors.add!" - // would not find anything if searching for "add!", only for the entire qualification - lunr.tokenizer.separator = /[\s\-\.]+/ - - // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names - lunr.trimmer = function (token) { - return token.update(function (s) { - return s.replace(/^[^a-zA-Z0-9@!]+/, '').replace(/[^a-zA-Z0-9@!]+$/, '') - }) - } - - lunr.Pipeline.registerFunction(lunr.stopWordFilter, 'juliaStopWordFilter') - lunr.Pipeline.registerFunction(lunr.trimmer, 'juliaTrimmer') - - var index = lunr(function () { - this.ref('location') - this.field('title',{boost: 100}) - this.field('text') - documenterSearchIndex['docs'].forEach(function(e) { - this.add(e) - }, this) - }) - var store = {} - - documenterSearchIndex['docs'].forEach(function(e) { - store[e.location] = {title: e.title, category: e.category, page: e.page} - }) - - $(function(){ - searchresults = $('#documenter-search-results'); - searchinfo = $('#documenter-search-info'); - searchbox = $('#documenter-search-query'); - searchform = $('.docs-search'); - sidebar = $('.docs-sidebar'); - function update_search(querystring) { - tokens = lunr.tokenizer(querystring) - results = index.query(function (q) { - tokens.forEach(function (t) { - q.term(t.toString(), { - fields: ["title"], - boost: 100, - usePipeline: true, - editDistance: 0, - wildcard: lunr.Query.wildcard.NONE - }) - q.term(t.toString(), { - fields: ["title"], - boost: 10, - usePipeline: true, - editDistance: 2, - wildcard: lunr.Query.wildcard.NONE - }) - q.term(t.toString(), { - fields: ["text"], - boost: 1, - usePipeline: true, - editDistance: 0, - wildcard: lunr.Query.wildcard.NONE - }) - }) - }) - searchinfo.text("Number of results: " + results.length) - searchresults.empty() - results.forEach(function(result) { - data = store[result.ref] - link = $(''+data.title+'') - link.attr('href', documenterBaseURL+'/'+result.ref) - if (data.category != "page"){ - cat = $('('+data.category+', '+data.page+')') - } else { - cat = $('('+data.category+')') - } - li = $('
  • ').append(link).append(" ").append(cat) - searchresults.append(li) - }) - } - - function update_search_box() { - querystring = searchbox.val() - update_search(querystring) - } - - searchbox.keyup(_.debounce(update_search_box, 250)) - searchbox.change(update_search_box) - - // Disable enter-key form submission for the searchbox on the search page - // and just re-run search rather than refresh the whole page. - searchform.keypress( - function(event){ - if (event.which == '13') { - if (sidebar.hasClass('visible')) { - sidebar.removeClass('visible'); - } - update_search_box(); - event.preventDefault(); - } - } - ); - - search_query_uri = parseUri(window.location).queryKey["q"] - if(search_query_uri !== undefined) { - search_query = decodeURIComponent(search_query_uri.replace(/\+/g, '%20')) - searchbox.val(search_query) - } - update_search_box(); - }) -}) - -}) diff --git a/dev/assets/themes/documenter-dark.css b/dev/assets/themes/documenter-dark.css index c94a294d..ec054ecc 100644 --- a/dev/assets/themes/documenter-dark.css +++ b/dev/assets/themes/documenter-dark.css @@ -1,7 +1,7 @@ -@keyframes spinAround{from{transform:rotate(0deg)}to{transform:rotate(359deg)}}html.theme--documenter-dark .tabs,html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark .pagination-next,html.theme--documenter-dark .pagination-link,html.theme--documenter-dark .pagination-ellipsis,html.theme--documenter-dark .breadcrumb,html.theme--documenter-dark .file,html.theme--documenter-dark .button,.is-unselectable,html.theme--documenter-dark .modal-close,html.theme--documenter-dark .delete{-webkit-touch-callout:none;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none}html.theme--documenter-dark .navbar-link:not(.is-arrowless)::after,html.theme--documenter-dark .select:not(.is-multiple):not(.is-loading)::after{border:3px solid rgba(0,0,0,0);border-radius:2px;border-right:0;border-top:0;content:" 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kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-dark.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content 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figure:not(:first-child){margin-top:2em}html.theme--documenter-dark .content figure:not(:last-child){margin-bottom:2em}html.theme--documenter-dark .content figure img{display:inline-block}html.theme--documenter-dark .content figure figcaption{font-style:italic}html.theme--documenter-dark .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--documenter-dark .content sup,html.theme--documenter-dark .content sub{font-size:75%}html.theme--documenter-dark .content table{width:100%}html.theme--documenter-dark .content table td,html.theme--documenter-dark .content table th{border:1px solid #5e6d6f;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--documenter-dark .content table th{color:#f2f2f2}html.theme--documenter-dark .content table th:not([align]){text-align:left}html.theme--documenter-dark .content table thead td,html.theme--documenter-dark .content table thead th{border-width:0 0 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.button.is-light.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-dark,html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:hover,html.theme--documenter-dark .content kbd.button:hover,html.theme--documenter-dark .button.is-dark.is-hovered,html.theme--documenter-dark .content kbd.button.is-hovered{background-color:#232829;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:focus,html.theme--documenter-dark 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kbd.button.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined,html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-outlined.is-focused{background-color:#282f2f;border-color:#282f2f;color:#fff}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading::after,html.theme--documenter-dark .content 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html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark 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.docstring>section>a.button.is-outlined.is-hovered.docs-sourcelink,html.theme--documenter-dark .button.is-primary.is-outlined:focus,html.theme--documenter-dark .docstring>section>a.button.is-outlined.docs-sourcelink:focus,html.theme--documenter-dark .button.is-primary.is-outlined.is-focused,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-focused.docs-sourcelink{background-color:#375a7f;border-color:#375a7f;color:#fff}html.theme--documenter-dark .button.is-primary.is-outlined.is-loading::after,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-loading.docs-sourcelink::after{border-color:transparent transparent #375a7f #375a7f !important}html.theme--documenter-dark .button.is-primary.is-outlined.is-loading:hover::after,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-loading.docs-sourcelink:hover::after,html.theme--documenter-dark .button.is-primary.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark 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.button.is-warning.is-active{background-color:#946e00;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-warning[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning{background-color:#ad8100;border-color:#ad8100;box-shadow:none}html.theme--documenter-dark .button.is-warning.is-inverted{background-color:#fff;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-inverted:hover,html.theme--documenter-dark .button.is-warning.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark 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.button.is-danger.is-light{background-color:#fdeeec;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#fce3e0;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#fcd8d5;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--documenter-dark .button.is-normal{font-size:1rem}html.theme--documenter-dark .button.is-medium{font-size:1.25rem}html.theme--documenter-dark .button.is-large{font-size:1.5rem}html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .button{background-color:#8c9b9d;border-color:#5e6d6f;box-shadow:none;opacity:.5}html.theme--documenter-dark .button.is-fullwidth{display:flex;width:100%}html.theme--documenter-dark .button.is-loading{color:transparent !important;pointer-events:none}html.theme--documenter-dark .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--documenter-dark .button.is-static{background-color:#282f2f;border-color:#5e6d6f;color:#dbdee0;box-shadow:none;pointer-events:none}html.theme--documenter-dark .button.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--documenter-dark .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--documenter-dark .buttons .button{margin-bottom:0.5rem}html.theme--documenter-dark .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--documenter-dark .buttons:last-child{margin-bottom:-0.5rem}html.theme--documenter-dark .buttons:not(:last-child){margin-bottom:1rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--documenter-dark .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--documenter-dark .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--documenter-dark .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--documenter-dark .buttons.has-addons 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#documenter .docs-sidebar ul.docs-menu ul.internal{margin:0 0.5rem 0.5rem;border-top:1px solid #5e6d6f}html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu ul.internal li{font-size:.85rem;border-left:none;margin-left:0;margin-top:0.5rem}html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu ul.internal .tocitem{width:100%;padding:0}html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu ul.internal .tocitem::before{content:"⚬";margin-right:0.4em}html.theme--documenter-dark #documenter .docs-sidebar form.docs-search{margin:auto;margin-top:0.5rem;margin-bottom:0.5rem}html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input{width:14.4rem}html.theme--documenter-dark #documenter .docs-sidebar #documenter-search-query{color:#868c98;width:14.4rem;box-shadow:inset 0 1px 2px rgba(10,10,10,0.1)}@media screen and (min-width: 1056px){html.theme--documenter-dark #documenter .docs-sidebar 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Gets put into the tag.. function set_theme_from_local_storage() { - // Intialize the theme to null, which means default + // Initialize the theme to null, which means default var theme = null; // If the browser supports the localstorage and is not disabled then try to get the // documenter theme - if(window.localStorage != null) { + if (window.localStorage != null) { // Get the user-picked theme from localStorage. May be `null`, which means the default // theme. - theme = window.localStorage.getItem("documenter-theme"); + theme = window.localStorage.getItem("documenter-theme"); } - // Check if the browser supports user color preference - var darkPreference = false; // Check if the users preference is for dark color scheme - if(window.matchMedia('(prefers-color-scheme: dark)').matches === true) { - darkPreference = true; - } + var darkPreference = + window.matchMedia("(prefers-color-scheme: dark)").matches === true; // Initialize a few variables for the loop: // // - active: will contain the index of the theme that should be active. Note that there @@ -24,43 +21,64 @@ function set_theme_from_local_storage() { // // - disabled: style sheets that should be disabled (i.e. all the theme style sheets // that are not the currently active theme) - var active = null; var disabled = []; var darkTheme = null; + var active = null; + var disabled = []; + var primaryLightTheme = null; + var primaryDarkTheme = null; for (var i = 0; i < document.styleSheets.length; i++) { var ss = document.styleSheets[i]; // The tag of each style sheet is expected to have a data-theme-name attribute // which must contain the name of the theme. The names in localStorage much match this. var themename = ss.ownerNode.getAttribute("data-theme-name"); // attribute not set => non-theme stylesheet => ignore - if(themename === null) continue; + if (themename === null) continue; // To distinguish the default (primary) theme, it needs to have the data-theme-primary // attribute set. - var isprimary = (ss.ownerNode.getAttribute("data-theme-primary") !== null); - // Check if the theme is primary dark theme - var isDarkTheme = (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null); - // If ss is for dark theme then set the value of darkTheme to the name of the theme - if(isDarkTheme) darkTheme = themename; + if (ss.ownerNode.getAttribute("data-theme-primary") !== null) { + primaryLightTheme = themename; + } + // Check if the theme is primary dark theme so that we could store its name in darkTheme + if (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null) { + primaryDarkTheme = themename; + } // If we find a matching theme (and it's not the default), we'll set active to non-null - if(themename === theme) active = i; + if (themename === theme) active = i; // Store the style sheets of inactive themes so that we could disable them - if(themename !== theme) disabled.push(ss); + if (themename !== theme) disabled.push(ss); } - if(active !== null) { + var activeTheme = null; + if (active !== null) { // If we did find an active theme, we'll (1) add the theme--$(theme) class to - document.getElementsByTagName('html')[0].className = "theme--" + theme; - // and (2) disable all the other theme stylesheets - disabled.forEach(function(ss){ - ss.disabled = true; - }); + document.getElementsByTagName("html")[0].className = "theme--" + theme; + activeTheme = theme; + } else { + // If we did _not_ find an active theme, then we need to fall back to the primary theme + // which can either be dark or light, depending on the user's OS preference. + var activeTheme = darkPreference ? primaryDarkTheme : primaryLightTheme; + // In case it somehow happens that the relevant primary theme was not found in the + // preceding loop, we abort without doing anything. + if (activeTheme === null) { + console.error("Unable to determine primary theme."); + return; + } + // When switching to the primary light theme, then we must not have a class name + // for the tag. That's only for non-primary or the primary dark theme. + if (darkPreference) { + document.getElementsByTagName("html")[0].className = + "theme--" + activeTheme; + } else { + document.getElementsByTagName("html")[0].className = ""; + } } - else if(darkTheme !== null && darkPreference === true) { - // If we did find an active theme, we'll (1) add the theme--$(theme) class to - document.getElementsByTagName('html')[0].className = "theme--" + darkTheme; - // and (2) disable all the other theme stylesheets - disabled.forEach(function(ss){ - if (ss.ownerNode.getAttribute("data-theme-name") !== darkTheme) { - ss.disabled = true; - } - }); + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // we'll disable all the stylesheets, except for the active one + ss.disabled = !(themename == activeTheme); } } set_theme_from_local_storage(); diff --git a/dev/assets/warner.js b/dev/assets/warner.js index 5531c885..3f6f5d00 100644 --- a/dev/assets/warner.js +++ b/dev/assets/warner.js @@ -1,49 +1,52 @@ -function maybeAddWarning () { - // DOCUMENTER_NEWEST is defined in versions.js, DOCUMENTER_CURRENT_VERSION and DOCUMENTER_STABLE - // in siteinfo.js. - // If either of these are undefined something went horribly wrong, so we abort. - if ( - window.DOCUMENTER_NEWEST === undefined || - window.DOCUMENTER_CURRENT_VERSION === undefined || - window.DOCUMENTER_STABLE === undefined - ) { - return - }; +function maybeAddWarning() { + // DOCUMENTER_NEWEST is defined in versions.js, DOCUMENTER_CURRENT_VERSION and DOCUMENTER_STABLE + // in siteinfo.js. + // If either of these are undefined something went horribly wrong, so we abort. + if ( + window.DOCUMENTER_NEWEST === undefined || + window.DOCUMENTER_CURRENT_VERSION === undefined || + window.DOCUMENTER_STABLE === undefined + ) { + return; + } - // Current version is not a version number, so we can't tell if it's the newest version. 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    Click here to go to the documentation for the latest stable release.'; - div.appendChild(closer); - document.body.appendChild(div); -}; + const div = document.createElement("div"); + div.classList.add("outdated-warning-overlay"); + const closer = document.createElement("button"); + closer.classList.add("outdated-warning-closer", "delete"); + closer.addEventListener("click", function () { + document.body.removeChild(div); + }); + const href = window.documenterBaseURL + "/../" + window.DOCUMENTER_STABLE; + div.innerHTML = + 'This documentation is not for the latest stable release, but for either the development version or an older release.
    Click here to go to the documentation for the latest stable release.'; + div.appendChild(closer); + document.body.appendChild(div); +} -if (document.readyState === 'loading') { - document.addEventListener('DOMContentLoaded', maybeAddWarning); +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", maybeAddWarning); } else { - maybeAddWarning(); -}; + maybeAddWarning(); +} diff --git a/dev/background/index.html b/dev/background/index.html index 8ace1811..14a79955 100644 --- a/dev/background/index.html +++ b/dev/background/index.html @@ -1,7 +1,7 @@ -Background · HiddenMarkovModels.jl
    Edit on GitHub

    Background

    What are HMMs?

    Hidden Markov Models (HMMs for short) are a statistical modeling framework that is ubiquitous in signal processing, bioinformatics and plenty of other fields. They capture the distribution of an observation sequence $(Y_t)$ by assuming the existence of a latent state sequence $(X_t)$ such that:

    • the sequence $(X_t)$ follows a (discrete time, discrete space) Markov chain
    • for each $t$, the distribution of $Y_t$ is entirely determined by the value of $X_t$

    What can we do with them?

    Imagine we are given an observation sequence $(Y_t)$ and a parametric family of HMMs $\{\mathbb{P}_\theta : \theta \in \Theta\}$. We can list several fundamental problems, each of which has a solution that relies on dynamic programming:

    ProblemGoalAlgorithm
    EvaluationLikelihood of the observation sequence $\mathbb{P}_\theta(Y_{1:T})$Forward
    InferenceState marginals $\mathbb{P}_\theta(X_t \vert Y_{1:T})$Forward-backward
    DecodingMost likely state sequence $\underset{X_{1:T}}{\mathrm{argmax}}~\mathbb{P}_\theta(X_{1:T} \vert Y_{1:T})$Viterbi
    LearningBest parameter $\underset{\theta}{\mathrm{argmax}}~\mathbb{P}_\theta(Y_{1:T})$Baum-Welch

    Our whole package is based on the tutorial by (Rabiner, 1989), you can refer to it for more details.

    Bibliography

    diff --git a/dev/debugging/index.html b/dev/debugging/index.html index 7e94a084..28691dd9 100644 --- a/dev/debugging/index.html +++ b/dev/debugging/index.html @@ -1,2 +1,2 @@ -Debugging · HiddenMarkovModels.jl
    Edit on GitHub

    Debugging

    Numerical overflow

    The most frequent error you will encounter is an OverflowError during forward-backward, telling you that "some values are infinite / NaN". This can happen for a variety of reasons, so here are a few leads worth investigating:

    • Increase the duration of the sequence / the number of sequences (to get more data)
    • Reduce the number of states (to make every one of them useful)
    • Add a prior to your transition matrix / observation distributions (to avoid degenerate behavior like zero variance in a Gaussian)
    • Pick a better initialization (to start closer to the supposed ground truth)
    • Use LogarithmicNumbers.jl in strategic places (to guarantee numerical stability). Note that these numbers don't play nicely with Distributions.jl, so you may have to roll out your own observation distribution.
    +Debugging · HiddenMarkovModels.jl
    GitHub

    Debugging

    Numerical overflow

    The most frequent error you will encounter is an OverflowError during forward-backward, telling you that "some values are infinite / NaN". This can happen for a variety of reasons, so here are a few leads worth investigating:

    • Increase the duration of the sequence / the number of sequences (to get more data)
    • Reduce the number of states (to make every one of them useful)
    • Add a prior to your transition matrix / observation distributions (to avoid degenerate behavior like zero variance in a Gaussian)
    • Pick a better initialization (to start closer to the supposed ground truth)
    • Use LogarithmicNumbers.jl in strategic places (to guarantee numerical stability). Note that these numbers don't play nicely with Distributions.jl, so you may have to roll out your own observation distribution.
    diff --git a/dev/formulas/index.html b/dev/formulas/index.html index ba4b4309..31d5951f 100644 --- a/dev/formulas/index.html +++ b/dev/formulas/index.html @@ -1,5 +1,5 @@ -Formulas · HiddenMarkovModels.jl
    Edit on GitHub

    Formulas

    Suppose we are given observations $Y_1, ..., Y_T$, with hidden states $X_1, ..., X_T$. Following (Rabiner, 1989), we use the following notations:

    • let $\pi \in \mathbb{R}^N$ be the initial state distribution $\pi_i = \mathbb{P}(X_1 = i)$
    • let $A \in \mathbb{R}^{N \times N}$ be the transition matrix $a_{i,j} = \mathbb{P}(X_{t+1}=j | X_t = i)$
    • let $B \in \mathbb{R}^{N \times T}$ be the matrix of statewise observation likelihoods $b_{i,t} = \mathbb{P}(Y_t | X_t = i)$

    Vanilla forward-backward

    Recursion

    The forward and backward variables are defined by

    \[\begin{align*} +Formulas · HiddenMarkovModels.jl

    GitHub

    Formulas

    Suppose we are given observations $Y_1, ..., Y_T$, with hidden states $X_1, ..., X_T$. Following (Rabiner, 1989), we use the following notations:

    • let $\pi \in \mathbb{R}^N$ be the initial state distribution $\pi_i = \mathbb{P}(X_1 = i)$
    • let $A \in \mathbb{R}^{N \times N}$ be the transition matrix $a_{i,j} = \mathbb{P}(X_{t+1}=j | X_t = i)$
    • let $B \in \mathbb{R}^{N \times T}$ be the matrix of statewise observation likelihoods $b_{i,t} = \mathbb{P}(Y_t | X_t = i)$

    Vanilla forward-backward

    Recursion

    The forward and backward variables are defined by

    \[\begin{align*} \alpha_{i,t} & = \mathbb{P}(Y_{1:t}, X_t=i) \\ \beta_{i,t} & = \mathbb{P}(Y_{t+1:T} | X_t=i) \end{align*}\]

    They are initialized with

    \[\begin{align*} @@ -14,7 +14,7 @@ \end{align*}\]

    Thus we deduce the one-state and two-state marginals

    \[\begin{align*} \gamma_{i,t} & = \mathbb{P}(X_t=i | Y_{1:T}) = \frac{1}{\mathcal{L}} \alpha_{i,t} \beta_{i,t} \\ \xi_{i,j,t} & = \mathbb{P}(X_t=i, X_{t+1}=j | Y_{1:T}) = \frac{1}{\mathcal{L}} \alpha_{i,t} a_{i,j} b_{j,t+1} \beta_{j,t+1} -\end{align*}\]

    Derivatives

    According to (Qin et al., 2000), derivatives of the likelihood can be obtained as follows:

    \[\begin{align*} +\end{align*}\]

    Derivatives

    According to (Qin *et al.*, 2000), derivatives of the likelihood can be obtained as follows:

    \[\begin{align*} \frac{\partial \mathcal{L}}{\partial \pi_i} &= \beta_{i,1} b_{i,1} \\ \frac{\partial \mathcal{L}}{\partial a_{i,j}} &= \sum_{t=1}^{T-1} \alpha_{i,t} b_{j,t+1} \beta_{j,t+1} \\ \frac{\partial \mathcal{L}}{\partial b_{j,1}} &= \pi_j \beta_{j,1} \\ @@ -56,4 +56,4 @@ \frac{\partial \log \mathcal{L}}{\partial a_{i,j}} &= \sum_{t=1}^{T-1} \bar{\alpha}_{i,t} \frac{b_{j,t+1}}{m_{t+1}} \bar{\beta}_{j,t+1} \\ \frac{\partial \log \mathcal{L}}{\partial \log b_{j,1}} &= \pi_j \frac{b_{j,1}}{m_1} \bar{\beta}_{j,1} = \frac{\bar{\alpha}_{j,1} \bar{\beta}_{j,1}}{c_1} = \gamma_{j,1} \\ \frac{\partial \log \mathcal{L}}{\partial \log b_{j,t}} &= \sum_{i=1}^N \bar{\alpha}_{i,t-1} a_{i,j} \frac{b_{j,t}}{m_t} \bar{\beta}_{j,t} = \frac{\bar{\alpha}_{j,t} \bar{\beta}_{j,t}}{c_t} = \gamma_{j,t} -\end{align*}\]

    +\end{align*}\]

    diff --git a/dev/index.html b/dev/index.html index 71bfdf70..f6ceed74 100644 --- a/dev/index.html +++ b/dev/index.html @@ -1,6 +1,6 @@ -Home · HiddenMarkovModels.jl
    Edit on GitHub

    HiddenMarkovModels.jl

    Stable Dev Build Status Coverage Code Style: Blue

    DOI Aqua QA JET

    A Julia package for HMM modeling, simulation, inference and learning.

    Getting started

    This package can be installed using Julia's package manager:

    pkg> add HiddenMarkovModels

    Then, you can create your first HMM as follows:

    using Distributions, HiddenMarkovModels
+Home · HiddenMarkovModels.jl
    GitHub
    HiddenMarkovModels.jl
    Stable Dev Build Status Coverage Code Style: Blue
    DOI Aqua QA JET
    A Julia package for HMM modeling, simulation, inference and learning.
    Getting started
    This package can be installed using Julia's package manager:
    pkg> add HiddenMarkovModels
    Then, you can create your first HMM as follows:
    using Distributions, HiddenMarkovModels
 init = [0.2, 0.8]
 trans = [0.1 0.9; 0.7 0.3]
 dists = [Normal(-1), Normal(1)]
-hmm = HMM(init, trans, dists)
    Take a look at the documentation to know what to do next!
    Main features
    This package is generic. Observations can be arbitrary Julia objects, not just scalars or arrays. Their distributions only need to implement rand(rng, dist) and logdensityof(dist, x) from DensityInterface.jl. Number types are not restricted to floating point, and automatic differentiation is supported in forward mode (ForwardDiff.jl) and reverse mode (ChainRules.jl).
    This package is fast. All the inference functions have allocation-free versions, which leverage efficient linear algebra subroutines. Multithreading is used to parallelize computations across sequences, and compatibility with various array types is ensured. We include extensive benchmarks against Julia and Python competitors thanks to BenchmarkTools.jl and PythonCall.jl.
    This package is reliable. It gives the same results as the previous reference package HMMBase.jl up to numerical accuracy. The test suite incorporates quality checks with Aqua.jl, as well as linting and type stability checks with JET.jl. A detailed documentation will help you find the functions you need.
    But this package is limited in scope. It is designed for HMMs with a small number of states, because memory and runtime scale quadratically (even if the transitions are sparse). It is also meant to perform best on a CPU, and not tested at all on GPUs.
    Contributing
    If you spot a bug or want to ask about a new feature, please open an issue on the GitHub repository. Once the issue receives positive feedback, feel free to try and fix it with a pull request that follows the BlueStyle guidelines.
    Acknowledgements
    A big thank you to Maxime Mouchet and Jacob Schreiber, the respective lead devs of HMMBase.jl and pomegranate, for their help and advice. Logo by Clément Mantoux based on a portrait of Andrey Markov.
    
+hmm = HMM(init, trans, dists)

    Take a look at the documentation to know what to do next!

    Main features

    This package is generic. Observations can be arbitrary Julia objects, not just scalars or arrays. Their distributions only need to implement rand(rng, dist) and logdensityof(dist, x) from DensityInterface.jl. Number types are not restricted to floating point, and automatic differentiation is supported in forward mode (ForwardDiff.jl) and reverse mode (ChainRules.jl).

    This package is fast. All the inference functions have allocation-free versions, which leverage efficient linear algebra subroutines. Multithreading is used to parallelize computations across sequences, and compatibility with various array types is ensured. We include extensive benchmarks against Julia and Python competitors thanks to BenchmarkTools.jl and PythonCall.jl.

    This package is reliable. It gives the same results as the previous reference package HMMBase.jl up to numerical accuracy. The test suite incorporates quality checks with Aqua.jl, as well as linting and type stability checks with JET.jl. A detailed documentation will help you find the functions you need.

    But this package is limited in scope. It is designed for HMMs with a small number of states, because memory and runtime scale quadratically (even if the transitions are sparse). It is also meant to perform best on a CPU, and not tested at all on GPUs.

    Contributing

    If you spot a bug or want to ask about a new feature, please open an issue on the GitHub repository. Once the issue receives positive feedback, feel free to try and fix it with a pull request that follows the BlueStyle guidelines.

    Acknowledgements

    A big thank you to Maxime Mouchet and Jacob Schreiber, the respective lead devs of HMMBase.jl and pomegranate, for their help and advice. Logo by Clément Mantoux based on a portrait of Andrey Markov.

    diff --git a/dev/roadmap/index.html b/dev/roadmap/index.html index 85d22479..cef15a47 100644 --- a/dev/roadmap/index.html +++ b/dev/roadmap/index.html @@ -1,2 +1,2 @@ -Roadmap · HiddenMarkovModels.jl
    Edit on GitHub
    +Roadmap · HiddenMarkovModels.jl
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    diff --git a/dev/search/index.html b/dev/search/index.html deleted file mode 100644 index 5ddb1f23..00000000 --- a/dev/search/index.html +++ /dev/null @@ -1,2 +0,0 @@ - -Search · HiddenMarkovModels.jl

    Loading search...

      diff --git a/dev/search_index.js b/dev/search_index.js index a6fdf20c..71c76b5d 100644 --- a/dev/search_index.js +++ b/dev/search_index.js @@ -1,3 +1,3 @@ var documenterSearchIndex = {"docs": -[{"location":"debugging/#Debugging","page":"Debugging","title":"Debugging","text":"","category":"section"},{"location":"debugging/#Numerical-overflow","page":"Debugging","title":"Numerical overflow","text":"","category":"section"},{"location":"debugging/","page":"Debugging","title":"Debugging","text":"The most frequent error you will encounter is an OverflowError during forward-backward, telling you that \"some values are infinite / NaN\". 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reference","text":"HMMs.ForwardBackwardStorage\nHMMs.fit_element_from_sequence!\nHMMs.LightDiagNormal","category":"page"},{"location":"api/#HiddenMarkovModels.ForwardBackwardStorage","page":"API reference","title":"HiddenMarkovModels.ForwardBackwardStorage","text":"ForwardBackwardStorage{R}\n\nStore forward-backward quantities with element type R.\n\nFields\n\nLet X denote the vector of hidden states and Y denote the vector of observations. The following fields are part of the API:\n\nγ::Matrix{R}: posterior one-state marginals γ[i,t] = ℙ(X[t]=i | Y[1:T])\nξ::Array{R,3}: posterior two-state marginals ξ[i,j,t] = ℙ(X[t:t+1]=(i,j) | Y[1:T])\n\nThe following fields are internals and subject to change:\n\nα::Matrix{R}: scaled forward variables α[i,t] proportional to ℙ(Y[1:t], X[t]=i) (up to a function of t)\nβ::Matrix{R}: scaled backward variables β[i,t] proportional to ℙ(Y[t+1:T] | X[t]=i) (up to a function of t)\nc::Vector{R}: forward variable inverse normalizations c[t] = 1 / sum(α[:,t])\nlogm::Vector{R}: maximum of the observation loglikelihoods logm[t] = maximum(logB[:, t])\nBscaled::Matrix{R}: numerically stabilized observation likelihoods Bscaled[i,t] = exp.(logB[i,t] - logm[t])\nBβscaled::Matrix{R}: numerically stabilized product Bβscaled[i,t] = Bscaled[i,t] * β[i,t]\n\n\n\n\n\n","category":"type"},{"location":"api/#HiddenMarkovModels.fit_element_from_sequence!","page":"API reference","title":"HiddenMarkovModels.fit_element_from_sequence!","text":"fit_element_from_sequence!(dists, i, x, w)\n\nModify the i-th element of dists by fitting it to an observation sequence x with associated weight sequence w.\n\nThe default behavior is a fallback on StatsAPI.fit!, which users are encouraged to implement if their observation distributions are mutable. If this is not possible, please override fit_element_from_sequence! directly.\n\n\n\n\n\n","category":"function"},{"location":"api/#HiddenMarkovModels.LightDiagNormal","page":"API reference","title":"HiddenMarkovModels.LightDiagNormal","text":"LightDiagNormal\n\nAn HMMs-compatible implementation of a multivariate normal distribution with diagonal covariance, enabling allocation-free estimation.\n\nThis is not part of the public API and is expected to change.\n\n\n\n\n\n","category":"type"},{"location":"api/#Notations","page":"API reference","title":"Notations","text":"","category":"section"},{"location":"api/#Integers","page":"API reference","title":"Integers","text":"","category":"section"},{"location":"api/","page":"API reference","title":"API reference","text":"N: number of states\nD: dimension of the observations\nT: trajectory length\nK: number of trajectories","category":"page"},{"location":"api/#Models-and-simulations","page":"API reference","title":"Models and simulations","text":"","category":"section"},{"location":"api/","page":"API reference","title":"API reference","text":"p or init: initial_distribution (vector of state probabilities)\nA or trans: transition_matrix (matrix of transition probabilities)\ndists: observation distribution (vector of rand-able and logdensityof-able objects)\nstate_seq: a sequence of states (vector of integers)\nobs_seq: a sequence of observations (vector of individual observations)\nobs_seqs: several sequences of observations","category":"page"},{"location":"api/#Forward-backward","page":"API reference","title":"Forward backward","text":"","category":"section"},{"location":"api/","page":"API reference","title":"API reference","text":"(log)b: vector of observation (log)likelihoods by state for an individual observation\n(log)B: matrix of observation (log)likelihoods by state for a sequence of observations\nα: scaled forward variables\nβ: scaled backward variables\nγ: one-state marginals\nξ: two-state marginals\nlogL: loglikelihood of a sequence of observations","category":"page"},{"location":"api/#Index","page":"API reference","title":"Index","text":"","category":"section"},{"location":"api/","page":"API reference","title":"API reference","text":"","category":"page"},{"location":"alt_performance/#Alternatives-performance","page":"Performance","title":"Alternatives - performance","text":"","category":"section"},{"location":"alt_performance/","page":"Performance","title":"Performance","text":"We compare performance among the following packages:","category":"page"},{"location":"alt_performance/","page":"Performance","title":"Performance","text":"HiddenMarkovModels.jl (abbreviated to HMMs.jl)\nHMMBase.jl\nhmmlearn\npomegranate","category":"page"},{"location":"alt_performance/","page":"Performance","title":"Performance","text":"The test case is an HMM with diagonal multivariate normal observations.","category":"page"},{"location":"alt_performance/","page":"Performance","title":"Performance","text":"N: number of states\nD: dimension of the observations\nT: trajectory length\nK: number of trajectories\nI: number of Baum-Welch iterations","category":"page"},{"location":"alt_performance/","page":"Performance","title":"Performance","text":"danger: Why is this empty?\nThe benchmark suite is computationally expensive, and we only run it once for each new release. If the following section contains no plots and the links are broken, you're probably reading the development documentation or a local build of the website. Check out the stable documentation instead.","category":"page"},{"location":"alt_performance/#Reproducibility","page":"Performance","title":"Reproducibility","text":"","category":"section"},{"location":"alt_performance/","page":"Performance","title":"Performance","text":"These benchmarks were generated in the following environment: setup.txt.","category":"page"},{"location":"alt_performance/","page":"Performance","title":"Performance","text":"If you want to run them on your machine:","category":"page"},{"location":"alt_performance/","page":"Performance","title":"Performance","text":"Clone the HiddenMarkovModels.jl repository\nOpen a Julia REPL at the root\nRun the following commands\ninclude(\"benchmark/run_benchmarks.jl\")\ninclude(\"benchmark/process_benchmarks.jl\")","category":"page"},{"location":"alt_performance/#Remarks","page":"Performance","title":"Remarks","text":"","category":"section"},{"location":"alt_performance/#Julia-to-Python-overhead","page":"Performance","title":"Julia-to-Python overhead","text":"","category":"section"},{"location":"alt_performance/","page":"Performance","title":"Performance","text":"Since the Python packages are called from Julia with PythonCall.jl, we pay a small overhead that is hard to quantify. On the plots, we compensate it by subtracting the runtime of the same algorithm for the smallest instance (N=1, D=1, T=2, K=1, I=1) from all Python-generated curves.","category":"page"},{"location":"alt_performance/#Allocations","page":"Performance","title":"Allocations","text":"","category":"section"},{"location":"alt_performance/","page":"Performance","title":"Performance","text":"A major bottleneck of performance in Julia is memory allocations. The benchmarks for HMMs.jl thus employ a custom implementation of diagonal multivariate normals, which is entirely allocation-free.","category":"page"},{"location":"alt_performance/","page":"Performance","title":"Performance","text":"This partly explains the performance gap with HMMBase.jl as the dimension D grows beyond 1. Such a trick is also possible with HMMBase.jl, but more demanding since it requires subtyping Distribution from Distributions.jl, instead of just implementing DensityInterface.jl.","category":"page"},{"location":"alt_performance/#Parallelism","page":"Performance","title":"Parallelism","text":"","category":"section"},{"location":"alt_performance/","page":"Performance","title":"Performance","text":"The packages we include have different approaches to parallelism, which can bias the evaluation in complex ways:","category":"page"},{"location":"alt_performance/","page":"Performance","title":"Performance","text":"Package States N Observations D Sequences K\nHMMs.jl LinearAlgebra[2] depends[2] Threads[1]\nHMMBase.jl - depends[2] -\nhmmlearn NumPy[2] NumPy[2] NumPy[2]\npomegranate PyTorch[3] PyTorch[3] PyTorch[3]","category":"page"},{"location":"alt_performance/","page":"Performance","title":"Performance","text":"[1]: affected by number of Julia threads","category":"page"},{"location":"alt_performance/","page":"Performance","title":"Performance","text":"[2]: affected by number of OpenBLAS threads","category":"page"},{"location":"alt_performance/","page":"Performance","title":"Performance","text":"[3]: affected by number of Pytorch threads","category":"page"},{"location":"alt_performance/","page":"Performance","title":"Performance","text":"We report each number of threads in setup.txt. Since OpenBLAS threads have negative interactions with Julia threads, we run the Julia benchmarks with a single OpenBLAS thread.","category":"page"},{"location":"alt_performance/#Numerical-results","page":"Performance","title":"Numerical results","text":"","category":"section"},{"location":"alt_performance/#Low-dimension","page":"Performance","title":"Low dimension","text":"","category":"section"},{"location":"alt_performance/","page":"Performance","title":"Performance","text":"Full benchmark logs: low_dim.csv.","category":"page"},{"location":"alt_performance/","page":"Performance","title":"Performance","text":"(Image: ) (Image: ) (Image: ) (Image: )","category":"page"},{"location":"alt_performance/","page":"Performance","title":"Performance","text":"Here, pomegranate is not included because it is much slower on very small inputs.","category":"page"},{"location":"alt_performance/#High-dimension","page":"Performance","title":"High dimension","text":"","category":"section"},{"location":"alt_performance/","page":"Performance","title":"Performance","text":"Full benchmark logs: high_dim.csv.","category":"page"},{"location":"alt_performance/","page":"Performance","title":"Performance","text":"(Image: ) (Image: ) (Image: ) (Image: )","category":"page"},{"location":"alt_performance/","page":"Performance","title":"Performance","text":"Here, HMMBase.jl is not included because it does not support multiple sequences.","category":"page"},{"location":"tuto_builtin/#Tutorial-built-in-HMM","page":"Built-in HMM","title":"Tutorial - built-in HMM","text":"","category":"section"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"using Distributions\nusing HiddenMarkovModels\n\nusing Random; Random.seed!(63)","category":"page"},{"location":"tuto_builtin/#Construction","page":"Built-in HMM","title":"Construction","text":"","category":"section"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"Creating a model:","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"function gaussian_hmm(N; noise=0)\n p = ones(N) / N # initial distribution\n A = rand_trans_mat(N) # transition matrix\n dists = [Normal(i + noise * randn(), 0.5) for i in 1:N] # observation distributions\n return HMM(p, A, dists)\nend","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"Checking its contents:","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"N = 3\nhmm = gaussian_hmm(N)\ntransition_matrix(hmm)","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"[obs_distribution(hmm, i) for i in 1:N]","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"Simulating a sequence:","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"T = 1000\nstate_seq, obs_seq = rand(hmm, T);\nfirst(state_seq, 10)'","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"first(obs_seq, 10)'","category":"page"},{"location":"tuto_builtin/#Inference","page":"Built-in HMM","title":"Inference","text":"","category":"section"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"Computing the loglikelihood of an observation sequence:","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"logdensityof(hmm, obs_seq)","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"Inferring the most likely state sequence:","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"most_likely_state_seq = viterbi(hmm, obs_seq);\nfirst(most_likely_state_seq, 10)'","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"Learning the parameters based on an observation sequence:","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"hmm_init = gaussian_hmm(N, noise=1)\nhmm_est, logL_evolution = baum_welch(hmm_init, obs_seq);\nfirst(logL_evolution), last(logL_evolution)","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"Correcting state order because we know observation means are increasing in the true model:","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"[obs_distribution(hmm_est, i) for i in 1:N]","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"perm = sortperm(1:3, by=i->obs_distribution(hmm_est, i).μ)","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"hmm_est = PermutedHMM(hmm_est, perm)","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"Evaluating errors:","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"cat(transition_matrix(hmm_est), transition_matrix(hmm), dims=3)","category":"page"},{"location":"roadmap/#Roadmap","page":"Roadmap","title":"Roadmap","text":"","category":"section"},{"location":"roadmap/","page":"Roadmap","title":"Roadmap","text":"Here are some of the things that I would like to work on soon-ish:","category":"page"},{"location":"roadmap/","page":"Roadmap","title":"Roadmap","text":"numerical stability in large-dimensional settings with sparse transitions\nSIMD optimization with LoopVectorization.jl or Tullio.jl\nspectral estimation methods\ninput-output HMMs in my other package ControlledMarkovModels.jl","category":"page"},{"location":"roadmap/","page":"Roadmap","title":"Roadmap","text":"Contributors are welcome!","category":"page"},{"location":"alt_features/#Alternatives-features","page":"Features","title":"Alternatives - features","text":"","category":"section"},{"location":"alt_features/","page":"Features","title":"Features","text":"We compare features among the following Julia packages:","category":"page"},{"location":"alt_features/","page":"Features","title":"Features","text":"HiddenMarkovModels.jl (abbreviated to HMMs.jl)\nHMMBase.jl\nHMMGradients.jl","category":"page"},{"location":"alt_features/","page":"Features","title":"Features","text":"We discard MarkovModels.jl because its focus is GPU computation. There are also more generic packages for probabilistic programming, which are able to perform MCMC or variational inference (eg. Turing.jl) but we leave those aside.","category":"page"},{"location":"alt_features/","page":"Features","title":"Features","text":" HMMs.jl HMMBase.jl HMMGradients.jl\nAlgorithms Sim, FB, Vit, BW Sim, FB, Vit, BW FB\nObservation types anything Number / Vector anything\nObservation distributions DensityInterface.jl Distributions.jl manual\nNumber types anything Float64 AbstractFloat\nPriors / structures possible no possible\nAutomatic differentiation yes no yes\nMultiple sequences yes no yes\nLinear algebra yes yes no\nLogarithmic probabilities halfway halfway yes","category":"page"},{"location":"alt_features/","page":"Features","title":"Features","text":"Sim = Simulation, FB = Forward-Backward, Vit = Viterbi, BW = Baum-Welch","category":"page"},{"location":"background/#Background","page":"Background","title":"Background","text":"","category":"section"},{"location":"background/#What-are-HMMs?","page":"Background","title":"What are HMMs?","text":"","category":"section"},{"location":"background/","page":"Background","title":"Background","text":"Hidden Markov Models (HMMs for short) are a statistical modeling framework that is ubiquitous in signal processing, bioinformatics and plenty of other fields. They capture the distribution of an observation sequence (Y_t) by assuming the existence of a latent state sequence (X_t) such that:","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"the sequence (X_t) follows a (discrete time, discrete space) Markov chain\nfor each t, the distribution of Y_t is entirely determined by the value of X_t","category":"page"},{"location":"background/#What-can-we-do-with-them?","page":"Background","title":"What can we do with them?","text":"","category":"section"},{"location":"background/","page":"Background","title":"Background","text":"Imagine we are given an observation sequence (Y_t) and a parametric family of HMMs mathbbP_theta theta in Theta. We can list several fundamental problems, each of which has a solution that relies on dynamic programming:","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"Problem Goal Algorithm\nEvaluation Likelihood of the observation sequence mathbbP_theta(Y_1T) Forward\nInference State marginals mathbbP_theta(X_t vert Y_1T) Forward-backward\nDecoding Most likely state sequence undersetX_1TmathrmargmaxmathbbP_theta(X_1T vert Y_1T) Viterbi\nLearning Best parameter undersetthetamathrmargmaxmathbbP_theta(Y_1T) Baum-Welch","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"Our whole package is based on the tutorial by (Rabiner, 1989), you can refer to it for more details.","category":"page"},{"location":"background/#Bibliography","page":"Background","title":"Bibliography","text":"","category":"section"},{"location":"background/","page":"Background","title":"Background","text":"","category":"page"},{"location":"tuto_custom/#Tutorial-custom-HMM","page":"Custom HMM","title":"Tutorial - custom HMM","text":"","category":"section"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"The built-in HMM is perfect when the initial state distribution, transition matrix and emission distributions can be updated independently with Maximum Likelihood Estimation. But some of these parameters might be correlated, or fixed. Or they might come with a prior, which forces you to use Maximum A Posteriori instead.","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"In such cases, it is necessary to implement a new subtype of AbstractHMM with all its required methods.","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"using Distributions\nusing HiddenMarkovModels\nusing LinearAlgebra\nusing RequiredInterfaces\nusing StatsAPI\n\nusing Random; Random.seed!(63)","category":"page"},{"location":"tuto_custom/#Interface","page":"Custom HMM","title":"Interface","text":"","category":"section"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"To ascertain that a type indeed satisfies the interface, you can use RequiredInterfaces.jl.","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"RequiredInterfaces.check_interface_implemented(AbstractHMM, HMM)","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"If your implementation is insufficient, the test will list missing methods.","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"struct EmptyHMM end\nRequiredInterfaces.check_interface_implemented(AbstractHMM, EmptyHMM)","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"Note that this test does not check the StatsAPI.fit! method. Since it is only used in the Baum-Welch algorithm, it is an optional part of the AbstractHMM interface.","category":"page"},{"location":"tuto_custom/#Example","page":"Custom HMM","title":"Example","text":"","category":"section"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"We show how to implement an HMM whose initial distribution is always the equilibrium distribution of the underlying Markov chain. The code that follows is not efficient (it leads to a lot of allocations), but it would be fairly easy to optimize if needed.","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"The equilibrium distribution of a Markov chain is the (only) left eigenvector associated with the left eigenvalue 1.","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"function markov_equilibrium(A)\n p = real.(eigvecs(A')[:, end])\n return p ./ sum(p)\nend","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"We now define our custom HMM by taking inspiration from src/types/hmm.jl and making a few modifications:","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"struct EquilibriumHMM{R,D} <: AbstractHMM\n trans::Matrix{R}\n dists::Vector{D}\nend","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"The interface is only different as far as the initialization is concerned.","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"Base.length(hmm::EquilibriumHMM) = length(hmm.dists)\nHMMs.initial_distribution(hmm::EquilibriumHMM) = markov_equilibrium(hmm.trans) # this is new\nHMMs.transition_matrix(hmm::EquilibriumHMM) = hmm.trans\nHMMs.obs_distribution(hmm::EquilibriumHMM, i::Integer) = hmm.dists[i]","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"As for fitting, we simply ignore the initialization count and copy the rest of the original code (with a few simplifications):","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"function StatsAPI.fit!(\n hmm::EquilibriumHMM{R,D}, init_count, trans_count, obs_seq, state_marginals\n) where {R,D}\n hmm.trans .= trans_count ./ sum(trans_count, dims=2)\n for i in 1:N\n hmm.dists[i] = fit(D, obs_seq, state_marginals[i, :])\n end\nend","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"Let's take our new model for a spin:","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"function gaussian_equilibrium_hmm(N; noise=0)\n A = rand_trans_mat(N)\n dists = [Normal(i + noise * randn(), 0.5) for i in 1:N]\n return EquilibriumHMM(A, dists)\nend;","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"N = 3\nhmm = gaussian_equilibrium_hmm(N);\ntransition_matrix(hmm)","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"[obs_distribution(hmm, i) for i in 1:N]","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"We can estimate parameters based on several observation sequences. Note that as soon as we tamper with the re-estimation procedure, the loglikelihood is no longer guaranteed to increase during Baum-Welch, which is why we turn off the corresponding check.","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"T = 1000\nnb_seqs = 10\nobs_seqs = [rand(hmm, T).obs_seq for _ in 1:nb_seqs]\n\nhmm_init = gaussian_equilibrium_hmm(N; noise=1)\nhmm_est, logL_evolution = baum_welch(\n hmm_init, obs_seqs, nb_seqs; check_loglikelihood_increasing=false\n);\nfirst(logL_evolution), last(logL_evolution)","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"Let's correct the state order:","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"[obs_distribution(hmm_est, i) for i in 1:N]","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"perm = sortperm(1:3, by=i->obs_distribution(hmm_est, i).μ)","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"hmm_est = PermutedHMM(hmm_est, perm)","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"And finally evaluate the errors:","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"cat(transition_matrix(hmm_est), transition_matrix(hmm), dims=3)","category":"page"},{"location":"formulas/#Formulas","page":"Formulas","title":"Formulas","text":"","category":"section"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"Suppose we are given observations Y_1 Y_T, with hidden states X_1 X_T. Following (Rabiner, 1989), we use the following notations:","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"let pi in mathbbR^N be the initial state distribution pi_i = mathbbP(X_1 = i)\nlet A in mathbbR^N times N be the transition matrix a_ij = mathbbP(X_t+1=j X_t = i)\nlet B in mathbbR^N times T be the matrix of statewise observation likelihoods b_it = mathbbP(Y_t X_t = i)","category":"page"},{"location":"formulas/#Vanilla-forward-backward","page":"Formulas","title":"Vanilla forward-backward","text":"","category":"section"},{"location":"formulas/#Recursion","page":"Formulas","title":"Recursion","text":"","category":"section"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"The forward and backward variables are defined by","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nalpha_it = mathbbP(Y_1t X_t=i) \nbeta_it = mathbbP(Y_t+1T X_t=i)\nendalign*","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"They are initialized with","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nalpha_i1 = pi_i b_i1 \nbeta_iT = 1\nendalign*","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"and satisfy the dynamic programming equations","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nalpha_jt+1 = left(sum_i=1^N alpha_it a_ijright) b_jt+1 \nbeta_it = sum_j=1^N a_ij b_jt+1 beta_jt+1\nendalign*","category":"page"},{"location":"formulas/#Likelihood","page":"Formulas","title":"Likelihood","text":"","category":"section"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"The likelihood of the whole sequence of observations is given by","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"mathcalL = mathbbP(Y_1T) = sum_i=1^N alpha_iT","category":"page"},{"location":"formulas/#Marginals","page":"Formulas","title":"Marginals","text":"","category":"section"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"We notice that","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nalpha_it beta_it = mathbbP(Y_1T X_t=i) \nalpha_it a_ij b_jt+1 beta_jt+1 = mathbbP(Y_1T X_t=i X_t+1=j)\nendalign*","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"Thus we deduce the one-state and two-state marginals","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\ngamma_it = mathbbP(X_t=i Y_1T) = frac1mathcalL alpha_it beta_it \nxi_ijt = mathbbP(X_t=i X_t+1=j Y_1T) = frac1mathcalL alpha_it a_ij b_jt+1 beta_jt+1\nendalign*","category":"page"},{"location":"formulas/#Derivatives","page":"Formulas","title":"Derivatives","text":"","category":"section"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"According to (Qin et al., 2000), derivatives of the likelihood can be obtained as follows:","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nfracpartial mathcalLpartial pi_i = beta_i1 b_i1 \nfracpartial mathcalLpartial a_ij = sum_t=1^T-1 alpha_it b_jt+1 beta_jt+1 \nfracpartial mathcalLpartial b_j1 = pi_j beta_j1 \nfracpartial mathcalLpartial b_jt = left(sum_i=1^N alpha_it-1 a_ijright) beta_jt \nendalign*","category":"page"},{"location":"formulas/#Scaled-forward-backward","page":"Formulas","title":"Scaled forward-backward","text":"","category":"section"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"In this package, we use a slightly different version of the algorithm, including both the traditional scaling of (Rabiner, 1989) and a normalization of B using m_t = max_i b_it.","category":"page"},{"location":"formulas/#Recursion-2","page":"Formulas","title":"Recursion","text":"","category":"section"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"The variables are initialized with","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nhatalpha_i1 = pi_i fracb_i1m_1 c_1 = frac1sum_i hatalpha_i1 baralpha_i1 = c_1 hatalpha_i1 \nhatbeta_iT = 1 barbeta_1T = c_T hatbeta_1T\nendalign*","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"and satisfy the dynamic programming equations","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nhatalpha_jt+1 = left(sum_i=1^N baralpha_it a_ijright) fracb_jt+1m_t+1 c_t+1 = frac1sum_j hatalpha_jt+1 baralpha_jt+1 = c_t+1 hatalpha_jt+1 \nhatbeta_it = sum_j=1^N a_ij fracb_jt+1m_t+1 barbeta_jt+1 barbeta_jt = c_t hatbeta_jt\nendalign*","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"In terms of the original variables, we find","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nbaralpha_it = alpha_it left(prod_s=1^t fracc_sm_sright) \nbarbeta_it = beta_it left(c_t prod_s=t+1^T fracc_sm_sright)\nendalign*","category":"page"},{"location":"formulas/#Likelihood-2","page":"Formulas","title":"Likelihood","text":"","category":"section"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"Since we normalized baralpha at each time step,","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"1 = sum_i=1^N baralpha_iT = left(sum_i=1^N alpha_iTright) left(prod_s=1^T fracc_sm_sright) ","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"which means","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"mathcalL = sum_i=1^N alpha_iT = prod_s=1^T fracm_sc_s","category":"page"},{"location":"formulas/#Marginals-2","page":"Formulas","title":"Marginals","text":"","category":"section"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"We can now express the marginals using scaled variables:","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\ngamma_it = frac1mathcalL alpha_it beta_it = frac1mathcalL left(baralpha_it prod_s=1^t fracm_sc_sright) left(barbeta_it frac1c_t prod_s=t+1^T fracm_sc_sright) \n= frac1mathcalL fracbaralpha_it barbeta_itc_t left(prod_s=1^T fracm_sc_sright) = fracbaralpha_it barbeta_itc_t\nendalign*","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nxi_ijt = frac1mathcalL alpha_it a_ij b_jt+1 beta_jt+1 \n= frac1mathcalL left(baralpha_it prod_s=1^t fracm_sc_sright) a_ij b_jt+1 left(barbeta_jt+1 frac1c_t+1 prod_s=t+2^T fracm_sc_sright) \n= frac1mathcalL baralpha_it a_ij fracb_jt+1m_t+1 barbeta_jt+1 left(prod_s=1^T fracm_sc_sright) \n= baralpha_it a_ij fracb_jt+1m_t+1 barbeta_jt+1\nendalign*","category":"page"},{"location":"formulas/#Derivatives-2","page":"Formulas","title":"Derivatives","text":"","category":"section"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"And we also need to adapt the derivatives. For the initial distribution,","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nfracpartial mathcalLpartial pi_i = beta_i1 b_i1 = left(barbeta_i1 frac1c_1 prod_s=2^T fracm_sc_s right) b_i1 \n= left(prod_s=1^T fracm_sc_sright) barbeta_i1 fracb_i1m_1 = mathcalL barbeta_i1 fracb_i1m_1 \nendalign*","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"For the transition matrix,","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nfracpartial mathcalLpartial a_ij = sum_t=1^T-1 alpha_it b_jt+1 beta_jt+1 \n= sum_t=1^T-1 left(baralpha_it prod_s=1^t fracm_sc_s right) b_jt+1 left(barbeta_jt+1 frac1c_t+1 prod_s=t+2^T fracm_sc_s right) \n= sum_t=1^T-1 baralpha_it fracb_jt+1m_t+1 barbeta_jt+1 left(prod_s=1^T fracm_sc_s right) \n= mathcalL sum_t=1^T-1 baralpha_it fracb_jt+1m_t+1 barbeta_jt+1 \nendalign*","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"And for the statewise observation likelihoods,","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nfracpartial mathcalLpartial b_j1 = pi_j beta_j1 = pi_j barbeta_j1 frac1c_1 prod_s=2^T fracm_sc_s = mathcalL pi_j barbeta_j1 frac1m_1\nendalign*","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nfracpartial mathcalLpartial b_jt = left(sum_i=1^N alpha_it-1 a_ijright) beta_jt \n= sum_i=1^N left(baralpha_it-1 prod_s=1^t-1 fracm_sc_sright) a_ij left(barbeta_jt frac1c_t prod_s=t+1^T fracm_sc_s right) \n= sum_i=1^N baralpha_it-1 a_ij barbeta_jt frac1m_t left(prod_s=1^T fracm_sc_sright) \n= mathcalL sum_i=1^N baralpha_it-1 a_ij barbeta_jt frac1m_t \nendalign*","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"Finally, we note that","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"fracpartial log mathcalLpartial log b_jt = fracpartial log mathcalLpartial b_jt b_jt","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"To sum up,","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nfracpartial log mathcalLpartial pi_i = fracb_i1m_1 barbeta_i1 \nfracpartial log mathcalLpartial a_ij = sum_t=1^T-1 baralpha_it fracb_jt+1m_t+1 barbeta_jt+1 \nfracpartial log mathcalLpartial log b_j1 = pi_j fracb_j1m_1 barbeta_j1 = fracbaralpha_j1 barbeta_j1c_1 = gamma_j1 \nfracpartial log mathcalLpartial log b_jt = sum_i=1^N baralpha_it-1 a_ij fracb_jtm_t barbeta_jt = fracbaralpha_jt barbeta_jtc_t = gamma_jt\nendalign*","category":"page"},{"location":"","page":"Home","title":"Home","text":"EditURL = \"https://github.com/gdalle/HiddenMarkovModels.jl/blob/main/README.md\"","category":"page"},{"location":"#HiddenMarkovModels.jl","page":"Home","title":"HiddenMarkovModels.jl","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"(Image: Stable) (Image: Dev) (Image: Build Status) (Image: Coverage) (Image: Code Style: Blue)","category":"page"},{"location":"","page":"Home","title":"Home","text":"(Image: DOI) (Image: Aqua QA) (Image: JET)","category":"page"},{"location":"","page":"Home","title":"Home","text":"A Julia package for HMM modeling, simulation, inference and learning.","category":"page"},{"location":"#Getting-started","page":"Home","title":"Getting started","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This package can be installed using Julia's package manager:","category":"page"},{"location":"","page":"Home","title":"Home","text":"pkg> add HiddenMarkovModels","category":"page"},{"location":"","page":"Home","title":"Home","text":"Then, you can create your first HMM as follows:","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Distributions, HiddenMarkovModels\ninit = [0.2, 0.8]\ntrans = [0.1 0.9; 0.7 0.3]\ndists = [Normal(-1), Normal(1)]\nhmm = HMM(init, trans, dists)","category":"page"},{"location":"","page":"Home","title":"Home","text":"Take a look at the documentation to know what to do next!","category":"page"},{"location":"#Main-features","page":"Home","title":"Main features","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This package is generic. Observations can be arbitrary Julia objects, not just scalars or arrays. Their distributions only need to implement rand(rng, dist) and logdensityof(dist, x) from DensityInterface.jl. Number types are not restricted to floating point, and automatic differentiation is supported in forward mode (ForwardDiff.jl) and reverse mode (ChainRules.jl).","category":"page"},{"location":"","page":"Home","title":"Home","text":"This package is fast. All the inference functions have allocation-free versions, which leverage efficient linear algebra subroutines. Multithreading is used to parallelize computations across sequences, and compatibility with various array types is ensured. We include extensive benchmarks against Julia and Python competitors thanks to BenchmarkTools.jl and PythonCall.jl.","category":"page"},{"location":"","page":"Home","title":"Home","text":"This package is reliable. It gives the same results as the previous reference package HMMBase.jl up to numerical accuracy. The test suite incorporates quality checks with Aqua.jl, as well as linting and type stability checks with JET.jl. A detailed documentation will help you find the functions you need.","category":"page"},{"location":"","page":"Home","title":"Home","text":"But this package is limited in scope. It is designed for HMMs with a small number of states, because memory and runtime scale quadratically (even if the transitions are sparse). It is also meant to perform best on a CPU, and not tested at all on GPUs.","category":"page"},{"location":"#Contributing","page":"Home","title":"Contributing","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"If you spot a bug or want to ask about a new feature, please open an issue on the GitHub repository. Once the issue receives positive feedback, feel free to try and fix it with a pull request that follows the BlueStyle guidelines.","category":"page"},{"location":"#Acknowledgements","page":"Home","title":"Acknowledgements","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"A big thank you to Maxime Mouchet and Jacob Schreiber, the respective lead devs of HMMBase.jl and pomegranate, for their help and advice. Logo by Clément Mantoux based on a portrait of Andrey Markov.","category":"page"}] +[{"location":"debugging/#Debugging","page":"Debugging","title":"Debugging","text":"","category":"section"},{"location":"debugging/#Numerical-overflow","page":"Debugging","title":"Numerical overflow","text":"","category":"section"},{"location":"debugging/","page":"Debugging","title":"Debugging","text":"The most frequent error you will encounter is an OverflowError during forward-backward, telling you that \"some values are infinite / NaN\". This can happen for a variety of reasons, so here are a few leads worth investigating:","category":"page"},{"location":"debugging/","page":"Debugging","title":"Debugging","text":"Increase the duration of the sequence / the number of sequences (to get more data)\nReduce the number of states (to make every one of them useful)\nAdd a prior to your transition matrix / observation distributions (to avoid degenerate behavior like zero variance in a Gaussian)\nPick a better initialization (to start closer to the supposed ground truth)\nUse LogarithmicNumbers.jl in strategic places (to guarantee numerical stability). Note that these numbers don't play nicely with Distributions.jl, so you may have to roll out your own observation distribution.","category":"page"},{"location":"api/#API-reference","page":"API reference","title":"API reference","text":"","category":"section"},{"location":"api/","page":"API reference","title":"API reference","text":"HiddenMarkovModels","category":"page"},{"location":"api/#HiddenMarkovModels","page":"API reference","title":"HiddenMarkovModels","text":"HiddenMarkovModels\n\nA Julia package for HMM modeling, simulation, inference and learning.\n\nThe alias HMMs is exported for the package name.\n\n\n\n\n\n","category":"module"},{"location":"api/#Types","page":"API reference","title":"Types","text":"","category":"section"},{"location":"api/#Markov-chains","page":"API reference","title":"Markov chains","text":"","category":"section"},{"location":"api/","page":"API reference","title":"API reference","text":"AbstractMarkovChain\nMarkovChain\nAbstractMC\nMC","category":"page"},{"location":"api/#HiddenMarkovModels.AbstractMarkovChain","page":"API reference","title":"HiddenMarkovModels.AbstractMarkovChain","text":"AbstractMarkovChain\n\nAbstract supertype for a Markov chain amenable to simulation, inference and learning.\n\nRequired interface\n\ninitial_distribution(mc)\ntransition_matrix(mc)\nfit!(mc, init_count, trans_count) (optional)\n\nApplicable methods\n\nrand([rng,] mc, T)\nlogdensityof(mc, state_seq)\nfit(mc, state_seq_or_seqs) (if fit! is implemented)\n\n\n\n\n\n","category":"type"},{"location":"api/#HiddenMarkovModels.MarkovChain","page":"API reference","title":"HiddenMarkovModels.MarkovChain","text":"MarkovChain <: AbstractMarkovChain\n\nBasic implementation of a discrete-state Markov chain.\n\nFields\n\ninit::AbstractVector: initial state probabilities\ntrans::AbstractMatrix: state transition matrix\n\n\n\n\n\n","category":"type"},{"location":"api/#HiddenMarkovModels.AbstractMC","page":"API reference","title":"HiddenMarkovModels.AbstractMC","text":"AbstractMC\n\nAlias for the type AbstractMarkovChain.\n\n\n\n\n\n","category":"type"},{"location":"api/#HiddenMarkovModels.MC","page":"API reference","title":"HiddenMarkovModels.MC","text":"MC\n\nAlias for the type MarkovChain.\n\n\n\n\n\n","category":"type"},{"location":"api/#Hidden-Markov-Models","page":"API reference","title":"Hidden Markov Models","text":"","category":"section"},{"location":"api/","page":"API reference","title":"API reference","text":"AbstractHiddenMarkovModel\nHiddenMarkovModel\nAbstractHMM\nHiddenMarkovModels.PermutedHMM\nHMM","category":"page"},{"location":"api/#HiddenMarkovModels.AbstractHiddenMarkovModel","page":"API reference","title":"HiddenMarkovModels.AbstractHiddenMarkovModel","text":"AbstractHiddenMarkovModel <: AbstractMarkovChain\n\nAbstract supertype for an HMM amenable to simulation, inference and learning.\n\nRequired interface\n\ninitial_distribution(hmm)\ntransition_matrix(hmm)\nobs_distribution(hmm, i)\nfit!(hmm, init_count, trans_count, obs_seq, state_marginals) (optional)\n\nApplicable methods\n\nrand([rng,] hmm, T)\nlogdensityof(hmm, obs_seq) / logdensityof(hmm, obs_seqs, nb_seqs)\nforward(hmm, obs_seq) / forward(hmm, obs_seqs, nb_seqs)\nviterbi(hmm, obs_seq) / viterbi(hmm, obs_seqs, nb_seqs)\nforward_backward(hmm, obs_seq) / forward_backward(hmm, obs_seqs, nb_seqs)\nbaum_welch(hmm, obs_seq) / baum_welch(hmm, obs_seqs, nb_seqs) if fit! is implemented\n\n\n\n\n\n","category":"type"},{"location":"api/#HiddenMarkovModels.HiddenMarkovModel","page":"API reference","title":"HiddenMarkovModels.HiddenMarkovModel","text":"HiddenMarkovModel{D} <: AbstractHiddenMarkovModel\n\nBasic implementation of an HMM.\n\nFields\n\ninit::AbstractVector: initial state probabilities\ntrans::AbstractMatrix: state transition matrix\ndists::AbstractVector{D}: observation distributions\n\n\n\n\n\n","category":"type"},{"location":"api/#HiddenMarkovModels.AbstractHMM","page":"API reference","title":"HiddenMarkovModels.AbstractHMM","text":"AbstractHMM\n\nAlias for the type AbstractHiddenMarkovModel.\n\n\n\n\n\n","category":"type"},{"location":"api/#HiddenMarkovModels.PermutedHMM","page":"API reference","title":"HiddenMarkovModels.PermutedHMM","text":"PermutedHMM{H<:AbstractHMM}\n\nWrapper around an AbstractHMM that permutes its states.\n\nThis is computationally inefficient and mostly useful for evaluation.\n\nFields\n\nhmm:H: the old HMM\nperm::Vector{Int}: a permutation such that state i in the new HMM corresponds to state perm[i] in the old.\n\n\n\n\n\n","category":"type"},{"location":"api/#HiddenMarkovModels.HMM","page":"API reference","title":"HiddenMarkovModels.HMM","text":"HMM\n\nAlias for the type HiddenMarkovModel.\n\n\n\n\n\n","category":"type"},{"location":"api/#Basics","page":"API reference","title":"Basics","text":"","category":"section"},{"location":"api/","page":"API reference","title":"API reference","text":"rand\nlength\ninitial_distribution\ntransition_matrix\nobs_distribution","category":"page"},{"location":"api/#Base.rand","page":"API reference","title":"Base.rand","text":"rand([rng=default_rng(),] mc::AbstractMarkovChain, T)\n\nSimulate mc for T time steps with a specified rng.\n\n\n\n\n\n","category":"function"},{"location":"api/#Base.length","page":"API reference","title":"Base.length","text":"length(mc::AbstractMarkovChain)\n\nReturn the number of states of model.\n\n\n\n\n\n","category":"function"},{"location":"api/#HiddenMarkovModels.initial_distribution","page":"API reference","title":"HiddenMarkovModels.initial_distribution","text":"initial_distribution(mc::AbstractMarkovChain)\n\nReturn the initial state probabilities of mc.\n\n\n\n\n\n","category":"function"},{"location":"api/#HiddenMarkovModels.transition_matrix","page":"API reference","title":"HiddenMarkovModels.transition_matrix","text":"transition_matrix(mc::AbstractMarkovChain)\n\nReturn the state transition probabilities of mc.\n\n\n\n\n\n","category":"function"},{"location":"api/#HiddenMarkovModels.obs_distribution","page":"API reference","title":"HiddenMarkovModels.obs_distribution","text":"obs_distribution(hmm::AbstractHMM, i)\n\nReturn the observation distribution of hmm associated with state i.\n\nThe returned object dist must implement\n\nrand(rng, dist)\nDensityInterface.logdensityof(dist, x)\n\n\n\n\n\n","category":"function"},{"location":"api/#Inference","page":"API reference","title":"Inference","text":"","category":"section"},{"location":"api/","page":"API reference","title":"API reference","text":"logdensityof\nforward\nviterbi\nforward_backward","category":"page"},{"location":"api/#DensityInterface.logdensityof","page":"API reference","title":"DensityInterface.logdensityof","text":"logdensityof(mc, state_seq)\n\nCompute the loglikelihood of a single state sequence for a Markov chain.\n\n\n\n\n\nlogdensityof(hmm, obs_seq)\n\nApply the forward algorithm to compute the loglikelihood of a single observation sequence for an HMM.\n\nReturn a number.\n\n\n\n\n\nlogdensityof(hmm, obs_seqs, nb_seqs)\n\nApply the forward algorithm to compute the total loglikelihood of multiple observation sequences for an HMM.\n\nReturn a number.\n\nwarning: Multithreading\nThis function is parallelized across sequences.\n\n\n\n\n\n","category":"function"},{"location":"api/#HiddenMarkovModels.forward","page":"API reference","title":"HiddenMarkovModels.forward","text":"forward(hmm, obs_seq)\n\nApply the forward algorithm to an HMM.\n\nReturn a tuple (α, logL) where\n\nlogL is the loglikelihood of the sequence\nα[i] is the posterior probability of state i at the end of the sequence.\n\n\n\n\n\nforward(hmm, obs_seqs, nb_seqs)\n\nApply the forward algorithm to an HMM, based on multiple observation sequences.\n\nReturn a vector of tuples (αₖ, logLₖ), where\n\nlogLₖ is the loglikelihood of sequence k\nαₖ[i] is the posterior probability of state i at the end of sequence k\n\nwarning: Multithreading\nThis function is parallelized across sequences.\n\n\n\n\n\n","category":"function"},{"location":"api/#HiddenMarkovModels.viterbi","page":"API reference","title":"HiddenMarkovModels.viterbi","text":"viterbi(hmm, obs_seq)\n\nApply the Viterbi algorithm to compute the most likely state sequence of an HMM.\n\nReturn a vector of integers.\n\n\n\n\n\nviterbi(hmm, obs_seqs, nb_seqs)\n\nApply the Viterbi algorithm to compute the most likely state sequences of an HMM, based on multiple observation sequences.\n\nReturn a vector of vectors of integers.\n\nwarning: Multithreading\nThis function is parallelized across sequences.\n\n\n\n\n\n","category":"function"},{"location":"api/#HiddenMarkovModels.forward_backward","page":"API reference","title":"HiddenMarkovModels.forward_backward","text":"forward_backward(hmm, obs_seq)\n\nApply the forward-backward algorithm to estimate the posterior state marginals of an HMM.\n\nReturn a ForwardBackwardStorage.\n\n\n\n\n\nforward_backward(hmm, obs_seqs, nb_seqs)\n\nApply the forward-backward algorithm to estimate the posterior state marginals of an HMM, based on multiple observation sequences.\n\nReturn a vector of ForwardBackwardStorage objects.\n\nwarning: Multithreading\nThis function is parallelized across sequences.\n\n\n\n\n\n","category":"function"},{"location":"api/#Learning","page":"API reference","title":"Learning","text":"","category":"section"},{"location":"api/","page":"API reference","title":"API reference","text":"fit!\nfit\nbaum_welch","category":"page"},{"location":"api/#StatsAPI.fit!","page":"API reference","title":"StatsAPI.fit!","text":"fit!(mc::MC, init_count, trans_count)\n\nUpdate mc in-place based on information generated from a state sequence.\n\n\n\n\n\nfit!(hmm::HMM, init_count, trans_count, obs_seq, state_marginals)\n\nUpdate hmm in-place based on information generated during forward-backward.\n\n\n\n\n\n","category":"function"},{"location":"api/#StatsAPI.fit","page":"API reference","title":"StatsAPI.fit","text":"fit(mc, state_seq_or_seqs)\n\nFit a Markov chain of the same type as mc to one or several state sequence(s).\n\nBeware that mc must be an actual object of type MarkovChain, and not the type itself as is usually done eg. in Distributions.jl.\n\n\n\n\n\n","category":"function"},{"location":"api/#HiddenMarkovModels.baum_welch","page":"API reference","title":"HiddenMarkovModels.baum_welch","text":"baum_welch(\n hmm_init, obs_seq;\n atol, max_iterations, check_loglikelihood_increasing\n)\n\nApply the Baum-Welch algorithm to estimate the parameters of an HMM starting from hmm_init.\n\nReturn a tuple (hmm_est, logL_evolution).\n\nKeyword arguments\n\natol: Minimum loglikelihood increase at an iteration of the algorithm (otherwise the algorithm is deemed to have converged)\nmax_iterations: Maximum number of iterations of the algorithm\ncheck_loglikelihood_increasing: Whether to throw an error if the loglikelihood decreases\n\n\n\n\n\nbaum_welch(\n hmm_init, obs_seqs, nb_seqs;\n atol, max_iterations, check_loglikelihood_increasing\n)\n\nApply the Baum-Welch algorithm to estimate the parameters of an HMM starting from hmm_init, based on nb_seqs observation sequences.\n\nReturn a tuple (hmm_est, logL_evolution).\n\nwarning: Multithreading\nThis function is parallelized across sequences.\n\nKeyword arguments\n\natol: Minimum loglikelihood increase at an iteration of the algorithm (otherwise the algorithm is deemed to have converged)\nmax_iterations: Maximum number of iterations of the algorithm\ncheck_loglikelihood_increasing: Whether to throw an error if the loglikelihood decreases\n\n\n\n\n\n","category":"function"},{"location":"api/#Internals","page":"API reference","title":"Internals","text":"","category":"section"},{"location":"api/","page":"API reference","title":"API reference","text":"HMMs.ForwardBackwardStorage\nHMMs.fit_element_from_sequence!\nHMMs.LightDiagNormal","category":"page"},{"location":"api/#HiddenMarkovModels.ForwardBackwardStorage","page":"API reference","title":"HiddenMarkovModels.ForwardBackwardStorage","text":"ForwardBackwardStorage{R}\n\nStore forward-backward quantities with element type R.\n\nFields\n\nLet X denote the vector of hidden states and Y denote the vector of observations. The following fields are part of the API:\n\nγ::Matrix{R}: posterior one-state marginals γ[i,t] = ℙ(X[t]=i | Y[1:T])\nξ::Array{R,3}: posterior two-state marginals ξ[i,j,t] = ℙ(X[t:t+1]=(i,j) | Y[1:T])\n\nThe following fields are internals and subject to change:\n\nα::Matrix{R}: scaled forward variables α[i,t] proportional to ℙ(Y[1:t], X[t]=i) (up to a function of t)\nβ::Matrix{R}: scaled backward variables β[i,t] proportional to ℙ(Y[t+1:T] | X[t]=i) (up to a function of t)\nc::Vector{R}: forward variable inverse normalizations c[t] = 1 / sum(α[:,t])\nlogm::Vector{R}: maximum of the observation loglikelihoods logm[t] = maximum(logB[:, t])\nBscaled::Matrix{R}: numerically stabilized observation likelihoods Bscaled[i,t] = exp.(logB[i,t] - logm[t])\nBβscaled::Matrix{R}: numerically stabilized product Bβscaled[i,t] = Bscaled[i,t] * β[i,t]\n\n\n\n\n\n","category":"type"},{"location":"api/#HiddenMarkovModels.fit_element_from_sequence!","page":"API reference","title":"HiddenMarkovModels.fit_element_from_sequence!","text":"fit_element_from_sequence!(dists, i, x, w)\n\nModify the i-th element of dists by fitting it to an observation sequence x with associated weight sequence w.\n\nThe default behavior is a fallback on StatsAPI.fit!, which users are encouraged to implement if their observation distributions are mutable. If this is not possible, please override fit_element_from_sequence! directly.\n\n\n\n\n\n","category":"function"},{"location":"api/#HiddenMarkovModels.LightDiagNormal","page":"API reference","title":"HiddenMarkovModels.LightDiagNormal","text":"LightDiagNormal\n\nAn HMMs-compatible implementation of a multivariate normal distribution with diagonal covariance, enabling allocation-free estimation.\n\nThis is not part of the public API and is expected to change.\n\n\n\n\n\n","category":"type"},{"location":"api/#Notations","page":"API reference","title":"Notations","text":"","category":"section"},{"location":"api/#Integers","page":"API reference","title":"Integers","text":"","category":"section"},{"location":"api/","page":"API reference","title":"API reference","text":"N: number of states\nD: dimension of the observations\nT: trajectory length\nK: number of trajectories","category":"page"},{"location":"api/#Models-and-simulations","page":"API reference","title":"Models and simulations","text":"","category":"section"},{"location":"api/","page":"API reference","title":"API reference","text":"p or init: initial_distribution (vector of state probabilities)\nA or trans: transition_matrix (matrix of transition probabilities)\ndists: observation distribution (vector of rand-able and logdensityof-able objects)\nstate_seq: a sequence of states (vector of integers)\nobs_seq: a sequence of observations (vector of individual observations)\nobs_seqs: several sequences of observations","category":"page"},{"location":"api/#Forward-backward","page":"API reference","title":"Forward backward","text":"","category":"section"},{"location":"api/","page":"API reference","title":"API reference","text":"(log)b: vector of observation (log)likelihoods by state for an individual observation\n(log)B: matrix of observation (log)likelihoods by state for a sequence of observations\nα: scaled forward variables\nβ: scaled backward variables\nγ: one-state marginals\nξ: two-state marginals\nlogL: loglikelihood of a sequence of observations","category":"page"},{"location":"api/#Index","page":"API reference","title":"Index","text":"","category":"section"},{"location":"api/","page":"API reference","title":"API reference","text":"","category":"page"},{"location":"tuto_builtin/#Tutorial-built-in-HMM","page":"Built-in HMM","title":"Tutorial - built-in HMM","text":"","category":"section"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"using Distributions\nusing HiddenMarkovModels\n\nusing Random; Random.seed!(63)","category":"page"},{"location":"tuto_builtin/#Construction","page":"Built-in HMM","title":"Construction","text":"","category":"section"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"Creating a model:","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"function gaussian_hmm(N; noise=0)\n p = ones(N) / N # initial distribution\n A = rand_trans_mat(N) # transition matrix\n dists = [Normal(i + noise * randn(), 0.5) for i in 1:N] # observation distributions\n return HMM(p, A, dists)\nend","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"Checking its contents:","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"N = 3\nhmm = gaussian_hmm(N)\ntransition_matrix(hmm)","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"[obs_distribution(hmm, i) for i in 1:N]","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"Simulating a sequence:","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"T = 1000\nstate_seq, obs_seq = rand(hmm, T);\nfirst(state_seq, 10)'","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"first(obs_seq, 10)'","category":"page"},{"location":"tuto_builtin/#Inference","page":"Built-in HMM","title":"Inference","text":"","category":"section"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"Computing the loglikelihood of an observation sequence:","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"logdensityof(hmm, obs_seq)","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"Inferring the most likely state sequence:","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"most_likely_state_seq = viterbi(hmm, obs_seq);\nfirst(most_likely_state_seq, 10)'","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"Learning the parameters based on an observation sequence:","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"hmm_init = gaussian_hmm(N, noise=1)\nhmm_est, logL_evolution = baum_welch(hmm_init, obs_seq);\nfirst(logL_evolution), last(logL_evolution)","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"Correcting state order because we know observation means are increasing in the true model:","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"[obs_distribution(hmm_est, i) for i in 1:N]","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"perm = sortperm(1:3, by=i->obs_distribution(hmm_est, i).μ)","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"hmm_est = PermutedHMM(hmm_est, perm)","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"Evaluating errors:","category":"page"},{"location":"tuto_builtin/","page":"Built-in HMM","title":"Built-in HMM","text":"cat(transition_matrix(hmm_est), transition_matrix(hmm), dims=3)","category":"page"},{"location":"roadmap/#Roadmap","page":"Roadmap","title":"Roadmap","text":"","category":"section"},{"location":"roadmap/","page":"Roadmap","title":"Roadmap","text":"Here are some of the things that I would like to work on soon-ish:","category":"page"},{"location":"roadmap/","page":"Roadmap","title":"Roadmap","text":"numerical stability in large-dimensional settings with sparse transitions\nSIMD optimization with LoopVectorization.jl or Tullio.jl\nspectral estimation methods\ninput-output HMMs in my other package ControlledMarkovModels.jl","category":"page"},{"location":"roadmap/","page":"Roadmap","title":"Roadmap","text":"Contributors are welcome!","category":"page"},{"location":"alt_features/#Alternatives-features","page":"Features","title":"Alternatives - features","text":"","category":"section"},{"location":"alt_features/","page":"Features","title":"Features","text":"We compare features among the following Julia packages:","category":"page"},{"location":"alt_features/","page":"Features","title":"Features","text":"HiddenMarkovModels.jl (abbreviated to HMMs.jl)\nHMMBase.jl\nHMMGradients.jl","category":"page"},{"location":"alt_features/","page":"Features","title":"Features","text":"We discard MarkovModels.jl because its focus is GPU computation. There are also more generic packages for probabilistic programming, which are able to perform MCMC or variational inference (eg. Turing.jl) but we leave those aside.","category":"page"},{"location":"alt_features/","page":"Features","title":"Features","text":" HMMs.jl HMMBase.jl HMMGradients.jl\nAlgorithms Sim, FB, Vit, BW Sim, FB, Vit, BW FB\nObservation types anything Number / Vector anything\nObservation distributions DensityInterface.jl Distributions.jl manual\nNumber types anything Float64 AbstractFloat\nPriors / structures possible no possible\nAutomatic differentiation yes no yes\nMultiple sequences yes no yes\nLinear algebra yes yes no\nLogarithmic probabilities halfway halfway yes","category":"page"},{"location":"alt_features/","page":"Features","title":"Features","text":"Sim = Simulation, FB = Forward-Backward, Vit = Viterbi, BW = Baum-Welch","category":"page"},{"location":"background/#Background","page":"Background","title":"Background","text":"","category":"section"},{"location":"background/#What-are-HMMs?","page":"Background","title":"What are HMMs?","text":"","category":"section"},{"location":"background/","page":"Background","title":"Background","text":"Hidden Markov Models (HMMs for short) are a statistical modeling framework that is ubiquitous in signal processing, bioinformatics and plenty of other fields. They capture the distribution of an observation sequence (Y_t) by assuming the existence of a latent state sequence (X_t) such that:","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"the sequence (X_t) follows a (discrete time, discrete space) Markov chain\nfor each t, the distribution of Y_t is entirely determined by the value of X_t","category":"page"},{"location":"background/#What-can-we-do-with-them?","page":"Background","title":"What can we do with them?","text":"","category":"section"},{"location":"background/","page":"Background","title":"Background","text":"Imagine we are given an observation sequence (Y_t) and a parametric family of HMMs mathbbP_theta theta in Theta. We can list several fundamental problems, each of which has a solution that relies on dynamic programming:","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"Problem Goal Algorithm\nEvaluation Likelihood of the observation sequence mathbbP_theta(Y_1T) Forward\nInference State marginals mathbbP_theta(X_t vert Y_1T) Forward-backward\nDecoding Most likely state sequence undersetX_1TmathrmargmaxmathbbP_theta(X_1T vert Y_1T) Viterbi\nLearning Best parameter undersetthetamathrmargmaxmathbbP_theta(Y_1T) Baum-Welch","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"Our whole package is based on the tutorial by (Rabiner, 1989), you can refer to it for more details.","category":"page"},{"location":"background/#Bibliography","page":"Background","title":"Bibliography","text":"","category":"section"},{"location":"background/","page":"Background","title":"Background","text":"
      ","category":"page"},{"location":"tuto_custom/#Tutorial-custom-HMM","page":"Custom HMM","title":"Tutorial - custom HMM","text":"","category":"section"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"The built-in HMM is perfect when the initial state distribution, transition matrix and emission distributions can be updated independently with Maximum Likelihood Estimation. But some of these parameters might be correlated, or fixed. Or they might come with a prior, which forces you to use Maximum A Posteriori instead.","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"In such cases, it is necessary to implement a new subtype of AbstractHMM with all its required methods.","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"using Distributions\nusing HiddenMarkovModels\nusing LinearAlgebra\nusing RequiredInterfaces\nusing StatsAPI\n\nusing Random; Random.seed!(63)","category":"page"},{"location":"tuto_custom/#Interface","page":"Custom HMM","title":"Interface","text":"","category":"section"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"To ascertain that a type indeed satisfies the interface, you can use RequiredInterfaces.jl.","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"RequiredInterfaces.check_interface_implemented(AbstractHMM, HMM)","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"If your implementation is insufficient, the test will list missing methods.","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"struct EmptyHMM end\nRequiredInterfaces.check_interface_implemented(AbstractHMM, EmptyHMM)","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"Note that this test does not check the StatsAPI.fit! method. Since it is only used in the Baum-Welch algorithm, it is an optional part of the AbstractHMM interface.","category":"page"},{"location":"tuto_custom/#Example","page":"Custom HMM","title":"Example","text":"","category":"section"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"We show how to implement an HMM whose initial distribution is always the equilibrium distribution of the underlying Markov chain. The code that follows is not efficient (it leads to a lot of allocations), but it would be fairly easy to optimize if needed.","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"The equilibrium distribution of a Markov chain is the (only) left eigenvector associated with the left eigenvalue 1.","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"function markov_equilibrium(A)\n p = real.(eigvecs(A')[:, end])\n return p ./ sum(p)\nend","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"We now define our custom HMM by taking inspiration from src/types/hmm.jl and making a few modifications:","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"struct EquilibriumHMM{R,D} <: AbstractHMM\n trans::Matrix{R}\n dists::Vector{D}\nend","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"The interface is only different as far as the initialization is concerned.","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"Base.length(hmm::EquilibriumHMM) = length(hmm.dists)\nHMMs.initial_distribution(hmm::EquilibriumHMM) = markov_equilibrium(hmm.trans) # this is new\nHMMs.transition_matrix(hmm::EquilibriumHMM) = hmm.trans\nHMMs.obs_distribution(hmm::EquilibriumHMM, i::Integer) = hmm.dists[i]","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"As for fitting, we simply ignore the initialization count and copy the rest of the original code (with a few simplifications):","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"function StatsAPI.fit!(\n hmm::EquilibriumHMM{R,D}, init_count, trans_count, obs_seq, state_marginals\n) where {R,D}\n hmm.trans .= trans_count ./ sum(trans_count, dims=2)\n for i in 1:N\n hmm.dists[i] = fit(D, obs_seq, state_marginals[i, :])\n end\nend","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"Let's take our new model for a spin:","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"function gaussian_equilibrium_hmm(N; noise=0)\n A = rand_trans_mat(N)\n dists = [Normal(i + noise * randn(), 0.5) for i in 1:N]\n return EquilibriumHMM(A, dists)\nend;","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"N = 3\nhmm = gaussian_equilibrium_hmm(N);\ntransition_matrix(hmm)","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"[obs_distribution(hmm, i) for i in 1:N]","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"We can estimate parameters based on several observation sequences. Note that as soon as we tamper with the re-estimation procedure, the loglikelihood is no longer guaranteed to increase during Baum-Welch, which is why we turn off the corresponding check.","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"T = 1000\nnb_seqs = 10\nobs_seqs = [rand(hmm, T).obs_seq for _ in 1:nb_seqs]\n\nhmm_init = gaussian_equilibrium_hmm(N; noise=1)\nhmm_est, logL_evolution = baum_welch(\n hmm_init, obs_seqs, nb_seqs; check_loglikelihood_increasing=false\n);\nfirst(logL_evolution), last(logL_evolution)","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"Let's correct the state order:","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"[obs_distribution(hmm_est, i) for i in 1:N]","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"perm = sortperm(1:3, by=i->obs_distribution(hmm_est, i).μ)","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"hmm_est = PermutedHMM(hmm_est, perm)","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"And finally evaluate the errors:","category":"page"},{"location":"tuto_custom/","page":"Custom HMM","title":"Custom HMM","text":"cat(transition_matrix(hmm_est), transition_matrix(hmm), dims=3)","category":"page"},{"location":"formulas/#Formulas","page":"Formulas","title":"Formulas","text":"","category":"section"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"Suppose we are given observations Y_1 Y_T, with hidden states X_1 X_T. Following (Rabiner, 1989), we use the following notations:","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"let pi in mathbbR^N be the initial state distribution pi_i = mathbbP(X_1 = i)\nlet A in mathbbR^N times N be the transition matrix a_ij = mathbbP(X_t+1=j X_t = i)\nlet B in mathbbR^N times T be the matrix of statewise observation likelihoods b_it = mathbbP(Y_t X_t = i)","category":"page"},{"location":"formulas/#Vanilla-forward-backward","page":"Formulas","title":"Vanilla forward-backward","text":"","category":"section"},{"location":"formulas/#Recursion","page":"Formulas","title":"Recursion","text":"","category":"section"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"The forward and backward variables are defined by","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nalpha_it = mathbbP(Y_1t X_t=i) \nbeta_it = mathbbP(Y_t+1T X_t=i)\nendalign*","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"They are initialized with","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nalpha_i1 = pi_i b_i1 \nbeta_iT = 1\nendalign*","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"and satisfy the dynamic programming equations","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nalpha_jt+1 = left(sum_i=1^N alpha_it a_ijright) b_jt+1 \nbeta_it = sum_j=1^N a_ij b_jt+1 beta_jt+1\nendalign*","category":"page"},{"location":"formulas/#Likelihood","page":"Formulas","title":"Likelihood","text":"","category":"section"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"The likelihood of the whole sequence of observations is given by","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"mathcalL = mathbbP(Y_1T) = sum_i=1^N alpha_iT","category":"page"},{"location":"formulas/#Marginals","page":"Formulas","title":"Marginals","text":"","category":"section"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"We notice that","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nalpha_it beta_it = mathbbP(Y_1T X_t=i) \nalpha_it a_ij b_jt+1 beta_jt+1 = mathbbP(Y_1T X_t=i X_t+1=j)\nendalign*","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"Thus we deduce the one-state and two-state marginals","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\ngamma_it = mathbbP(X_t=i Y_1T) = frac1mathcalL alpha_it beta_it \nxi_ijt = mathbbP(X_t=i X_t+1=j Y_1T) = frac1mathcalL alpha_it a_ij b_jt+1 beta_jt+1\nendalign*","category":"page"},{"location":"formulas/#Derivatives","page":"Formulas","title":"Derivatives","text":"","category":"section"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"According to (Qin *et al.*, 2000), derivatives of the likelihood can be obtained as follows:","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nfracpartial mathcalLpartial pi_i = beta_i1 b_i1 \nfracpartial mathcalLpartial a_ij = sum_t=1^T-1 alpha_it b_jt+1 beta_jt+1 \nfracpartial mathcalLpartial b_j1 = pi_j beta_j1 \nfracpartial mathcalLpartial b_jt = left(sum_i=1^N alpha_it-1 a_ijright) beta_jt \nendalign*","category":"page"},{"location":"formulas/#Scaled-forward-backward","page":"Formulas","title":"Scaled forward-backward","text":"","category":"section"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"In this package, we use a slightly different version of the algorithm, including both the traditional scaling of (Rabiner, 1989) and a normalization of B using m_t = max_i b_it.","category":"page"},{"location":"formulas/#Recursion-2","page":"Formulas","title":"Recursion","text":"","category":"section"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"The variables are initialized with","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nhatalpha_i1 = pi_i fracb_i1m_1 c_1 = frac1sum_i hatalpha_i1 baralpha_i1 = c_1 hatalpha_i1 \nhatbeta_iT = 1 barbeta_1T = c_T hatbeta_1T\nendalign*","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"and satisfy the dynamic programming equations","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nhatalpha_jt+1 = left(sum_i=1^N baralpha_it a_ijright) fracb_jt+1m_t+1 c_t+1 = frac1sum_j hatalpha_jt+1 baralpha_jt+1 = c_t+1 hatalpha_jt+1 \nhatbeta_it = sum_j=1^N a_ij fracb_jt+1m_t+1 barbeta_jt+1 barbeta_jt = c_t hatbeta_jt\nendalign*","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"In terms of the original variables, we find","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nbaralpha_it = alpha_it left(prod_s=1^t fracc_sm_sright) \nbarbeta_it = beta_it left(c_t prod_s=t+1^T fracc_sm_sright)\nendalign*","category":"page"},{"location":"formulas/#Likelihood-2","page":"Formulas","title":"Likelihood","text":"","category":"section"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"Since we normalized baralpha at each time step,","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"1 = sum_i=1^N baralpha_iT = left(sum_i=1^N alpha_iTright) left(prod_s=1^T fracc_sm_sright) ","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"which means","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"mathcalL = sum_i=1^N alpha_iT = prod_s=1^T fracm_sc_s","category":"page"},{"location":"formulas/#Marginals-2","page":"Formulas","title":"Marginals","text":"","category":"section"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"We can now express the marginals using scaled variables:","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\ngamma_it = frac1mathcalL alpha_it beta_it = frac1mathcalL left(baralpha_it prod_s=1^t fracm_sc_sright) left(barbeta_it frac1c_t prod_s=t+1^T fracm_sc_sright) \n= frac1mathcalL fracbaralpha_it barbeta_itc_t left(prod_s=1^T fracm_sc_sright) = fracbaralpha_it barbeta_itc_t\nendalign*","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nxi_ijt = frac1mathcalL alpha_it a_ij b_jt+1 beta_jt+1 \n= frac1mathcalL left(baralpha_it prod_s=1^t fracm_sc_sright) a_ij b_jt+1 left(barbeta_jt+1 frac1c_t+1 prod_s=t+2^T fracm_sc_sright) \n= frac1mathcalL baralpha_it a_ij fracb_jt+1m_t+1 barbeta_jt+1 left(prod_s=1^T fracm_sc_sright) \n= baralpha_it a_ij fracb_jt+1m_t+1 barbeta_jt+1\nendalign*","category":"page"},{"location":"formulas/#Derivatives-2","page":"Formulas","title":"Derivatives","text":"","category":"section"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"And we also need to adapt the derivatives. For the initial distribution,","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nfracpartial mathcalLpartial pi_i = beta_i1 b_i1 = left(barbeta_i1 frac1c_1 prod_s=2^T fracm_sc_s right) b_i1 \n= left(prod_s=1^T fracm_sc_sright) barbeta_i1 fracb_i1m_1 = mathcalL barbeta_i1 fracb_i1m_1 \nendalign*","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"For the transition matrix,","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nfracpartial mathcalLpartial a_ij = sum_t=1^T-1 alpha_it b_jt+1 beta_jt+1 \n= sum_t=1^T-1 left(baralpha_it prod_s=1^t fracm_sc_s right) b_jt+1 left(barbeta_jt+1 frac1c_t+1 prod_s=t+2^T fracm_sc_s right) \n= sum_t=1^T-1 baralpha_it fracb_jt+1m_t+1 barbeta_jt+1 left(prod_s=1^T fracm_sc_s right) \n= mathcalL sum_t=1^T-1 baralpha_it fracb_jt+1m_t+1 barbeta_jt+1 \nendalign*","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"And for the statewise observation likelihoods,","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nfracpartial mathcalLpartial b_j1 = pi_j beta_j1 = pi_j barbeta_j1 frac1c_1 prod_s=2^T fracm_sc_s = mathcalL pi_j barbeta_j1 frac1m_1\nendalign*","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nfracpartial mathcalLpartial b_jt = left(sum_i=1^N alpha_it-1 a_ijright) beta_jt \n= sum_i=1^N left(baralpha_it-1 prod_s=1^t-1 fracm_sc_sright) a_ij left(barbeta_jt frac1c_t prod_s=t+1^T fracm_sc_s right) \n= sum_i=1^N baralpha_it-1 a_ij barbeta_jt frac1m_t left(prod_s=1^T fracm_sc_sright) \n= mathcalL sum_i=1^N baralpha_it-1 a_ij barbeta_jt frac1m_t \nendalign*","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"Finally, we note that","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"fracpartial log mathcalLpartial log b_jt = fracpartial log mathcalLpartial b_jt b_jt","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"To sum up,","category":"page"},{"location":"formulas/","page":"Formulas","title":"Formulas","text":"beginalign*\nfracpartial log mathcalLpartial pi_i = fracb_i1m_1 barbeta_i1 \nfracpartial log mathcalLpartial a_ij = sum_t=1^T-1 baralpha_it fracb_jt+1m_t+1 barbeta_jt+1 \nfracpartial log mathcalLpartial log b_j1 = pi_j fracb_j1m_1 barbeta_j1 = fracbaralpha_j1 barbeta_j1c_1 = gamma_j1 \nfracpartial log mathcalLpartial log b_jt = sum_i=1^N baralpha_it-1 a_ij fracb_jtm_t barbeta_jt = fracbaralpha_jt barbeta_jtc_t = gamma_jt\nendalign*","category":"page"},{"location":"","page":"Home","title":"Home","text":"EditURL = \"https://github.com/gdalle/HiddenMarkovModels.jl/blob/main/README.md\"","category":"page"},{"location":"#HiddenMarkovModels.jl","page":"Home","title":"HiddenMarkovModels.jl","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"(Image: Stable) (Image: Dev) (Image: Build Status) (Image: Coverage) (Image: Code Style: Blue)","category":"page"},{"location":"","page":"Home","title":"Home","text":"(Image: DOI) (Image: Aqua QA) (Image: JET)","category":"page"},{"location":"","page":"Home","title":"Home","text":"A Julia package for HMM modeling, simulation, inference and learning.","category":"page"},{"location":"#Getting-started","page":"Home","title":"Getting started","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This package can be installed using Julia's package manager:","category":"page"},{"location":"","page":"Home","title":"Home","text":"pkg> add HiddenMarkovModels","category":"page"},{"location":"","page":"Home","title":"Home","text":"Then, you can create your first HMM as follows:","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Distributions, HiddenMarkovModels\ninit = [0.2, 0.8]\ntrans = [0.1 0.9; 0.7 0.3]\ndists = [Normal(-1), Normal(1)]\nhmm = HMM(init, trans, dists)","category":"page"},{"location":"","page":"Home","title":"Home","text":"Take a look at the documentation to know what to do next!","category":"page"},{"location":"#Main-features","page":"Home","title":"Main features","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This package is generic. Observations can be arbitrary Julia objects, not just scalars or arrays. Their distributions only need to implement rand(rng, dist) and logdensityof(dist, x) from DensityInterface.jl. Number types are not restricted to floating point, and automatic differentiation is supported in forward mode (ForwardDiff.jl) and reverse mode (ChainRules.jl).","category":"page"},{"location":"","page":"Home","title":"Home","text":"This package is fast. All the inference functions have allocation-free versions, which leverage efficient linear algebra subroutines. Multithreading is used to parallelize computations across sequences, and compatibility with various array types is ensured. We include extensive benchmarks against Julia and Python competitors thanks to BenchmarkTools.jl and PythonCall.jl.","category":"page"},{"location":"","page":"Home","title":"Home","text":"This package is reliable. It gives the same results as the previous reference package HMMBase.jl up to numerical accuracy. The test suite incorporates quality checks with Aqua.jl, as well as linting and type stability checks with JET.jl. A detailed documentation will help you find the functions you need.","category":"page"},{"location":"","page":"Home","title":"Home","text":"But this package is limited in scope. It is designed for HMMs with a small number of states, because memory and runtime scale quadratically (even if the transitions are sparse). It is also meant to perform best on a CPU, and not tested at all on GPUs.","category":"page"},{"location":"#Contributing","page":"Home","title":"Contributing","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"If you spot a bug or want to ask about a new feature, please open an issue on the GitHub repository. Once the issue receives positive feedback, feel free to try and fix it with a pull request that follows the BlueStyle guidelines.","category":"page"},{"location":"#Acknowledgements","page":"Home","title":"Acknowledgements","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"A big thank you to Maxime Mouchet and Jacob Schreiber, the respective lead devs of HMMBase.jl and pomegranate, for their help and advice. Logo by Clément Mantoux based on a portrait of Andrey Markov.","category":"page"}] } diff --git a/dev/tuto_builtin/index.html b/dev/tuto_builtin/index.html index 8026d3f4..81c50025 100644 --- a/dev/tuto_builtin/index.html +++ b/dev/tuto_builtin/index.html @@ -1,5 +1,5 @@ -Built-in HMM · HiddenMarkovModels.jl
      Edit on GitHub

      Tutorial - built-in HMM

      using Distributions
+Built-in HMM · HiddenMarkovModels.jl
      GitHub
      Tutorial - built-in HMM
      using Distributions
 using HiddenMarkovModels
 
 using Random; Random.seed!(63)
      Random.TaskLocalRNG()
      Construction
      Creating a model:
      function gaussian_hmm(N; noise=0)
@@ -38,4 +38,4 @@
 [:, :, 2] =
  0.323233  0.512336   0.164431
  0.337336  0.0936302  0.569033
- 0.474491  0.211947   0.313561
      
+ 0.474491  0.211947   0.313561
      diff --git a/dev/tuto_custom/index.html b/dev/tuto_custom/index.html index dc0c5e0a..983fb410 100644 --- a/dev/tuto_custom/index.html +++ b/dev/tuto_custom/index.html @@ -1,5 +1,5 @@ -Custom HMM · HiddenMarkovModels.jl
      Edit on GitHub

      Tutorial - custom HMM

      The built-in HMM is perfect when the initial state distribution, transition matrix and emission distributions can be updated independently with Maximum Likelihood Estimation. But some of these parameters might be correlated, or fixed. Or they might come with a prior, which forces you to use Maximum A Posteriori instead.

      In such cases, it is necessary to implement a new subtype of AbstractHMM with all its required methods.

      using Distributions
+Custom HMM · HiddenMarkovModels.jl
      GitHub
      Tutorial - custom HMM
      The built-in HMM is perfect when the initial state distribution, transition matrix and emission distributions can be updated independently with Maximum Likelihood Estimation. But some of these parameters might be correlated, or fixed. Or they might come with a prior, which forces you to use Maximum A Posteriori instead.
      In such cases, it is necessary to implement a new subtype of AbstractHMM with all its required methods.
      using Distributions
 using HiddenMarkovModels
 using LinearAlgebra
 using RequiredInterfaces
@@ -7,10 +7,10 @@
 
 using Random; Random.seed!(63)
      Random.TaskLocalRNG()
      Interface
      To ascertain that a type indeed satisfies the interface, you can use RequiredInterfaces.jl.
      RequiredInterfaces.check_interface_implemented(AbstractHMM, HMM)
      true
      If your implementation is insufficient, the test will list missing methods.
      struct EmptyHMM end
 RequiredInterfaces.check_interface_implemented(AbstractHMM, EmptyHMM)
      4-element Vector{Tuple{Any, Tuple}}:
- (length, (Main.EmptyHMM,))
  (HiddenMarkovModels.initial_distribution, (Main.EmptyHMM,))
  (HiddenMarkovModels.transition_matrix, (Main.EmptyHMM,))
- (HiddenMarkovModels.obs_distribution, (Main.EmptyHMM, Integer))
      Note that this test does not check the StatsAPI.fit! method. Since it is only used in the Baum-Welch algorithm, it is an optional part of the AbstractHMM interface.
      Example
      We show how to implement an HMM whose initial distribution is always the equilibrium distribution of the underlying Markov chain. The code that follows is not efficient (it leads to a lot of allocations), but it would be fairly easy to optimize if needed.
      The equilibrium distribution of a Markov chain is the (only) left eigenvector associated with the left eigenvalue $1$.
      function markov_equilibrium(A)
+ (HiddenMarkovModels.obs_distribution, (Main.EmptyHMM, Integer))
+ (length, (Main.EmptyHMM,))
      Note that this test does not check the StatsAPI.fit! method. Since it is only used in the Baum-Welch algorithm, it is an optional part of the AbstractHMM interface.
      Example
      We show how to implement an HMM whose initial distribution is always the equilibrium distribution of the underlying Markov chain. The code that follows is not efficient (it leads to a lot of allocations), but it would be fairly easy to optimize if needed.
      The equilibrium distribution of a Markov chain is the (only) left eigenvector associated with the left eigenvalue $1$.
      function markov_equilibrium(A)
     p = real.(eigvecs(A')[:, end])
     return p ./ sum(p)
 end
      markov_equilibrium (generic function with 1 method)
      We now define our custom HMM by taking inspiration from src/types/hmm.jl and making a few modifications:
      struct EquilibriumHMM{R,D} <: AbstractHMM
@@ -61,4 +61,4 @@
 [:, :, 2] =
  0.323233  0.512336   0.164431
  0.337336  0.0936302  0.569033
- 0.474491  0.211947   0.313561
      
+ 0.474491  0.211947   0.313561