### Install HiddenMarkovModels.jl

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable

Use Julia's package manager to add the HiddenMarkovModels package.

```julia
pkg> add HiddenMarkovModels
```

--------------------------------

### Get Initial State Probabilities

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Retrieves the initial state probabilities for a given HMM.

```julia
initialization(hmm)
```

--------------------------------

### Initialize PeriodicHMM Guess

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/temporal

Sets up an initial guess for a PeriodicHMM, including initial probabilities, transition matrices per period, and emission distributions per period. This is used as a starting point for the Baum-Welch algorithm.

```julia
init_guess = [0.4, 0.2, 0.3]
trans_per_guess = ntuple(_ -> [
    0.4 0.3 0.3
    0.3 0.4 0.3
    0.3 0.3 0.4
], Val(3))
dists_per_guess = (
    [Normal(1.5), Normal(2.2), Normal(2.5)],
    [Normal(3.5), Normal(4.2), Normal(4.5)],
    [Normal(5.5), Normal(6.2), Normal(6.5)],
)
hmm_guess = PeriodicHMM(init_guess, trans_per_guess, dists_per_guess);
```

--------------------------------

### Get Log Initial State Probabilities

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Returns the vector of initial state log-probabilities for an HMM. Falls back on the `initialization` function if direct log-probabilities are not available.

```julia
log_initialization(hmm)

```

--------------------------------

### Get State Transition Probabilities

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Retrieves the state transition probabilities for an HMM, optionally considering a control input.

```julia
transition_matrix(hmm)
transition_matrix(hmm, control)
```

--------------------------------

### Get Log Transition Matrix

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Returns the matrix of state transition log-probabilities for an HMM. Can optionally consider a control input, which influences transitions from the previous time step.

```julia
log_transition_matrix(hmm)

```

```julia
log_transition_matrix(hmm, control)

```

--------------------------------

### Baum-Welch with Sparse Transition Guess

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/types

Apply the Baum-Welch algorithm to estimate HMM parameters from observed sequences, starting with an initial guess that maintains the correct sparsity pattern.

```julia
init_guess = [0.3, 0.4, 0.3]
trans_guess = sparse([
    0.6 0.4 0
    0 0.6 0.4
    0.4 0 0.6
])
dists_guess = [Normal(1.1), Normal(2.1), Normal(3.1)]
hmm_guess = HMM(init_guess, trans_guess, dists_guess)
```

```julia
hmm_est, loglikelihood_evolution = baum_welch(hmm_guess, obs_seq)
first(loglikelihood_evolution), last(loglikelihood_evolution)
```

--------------------------------

### Get Number of States

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Returns the total number of states in the HMM.

```julia
length(hmm)
```

--------------------------------

### Estimate HMM Parameters with Baum-Welch Algorithm

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Use `baum_welch` to estimate HMM parameters from observation sequences, starting with an initial guess. It returns the estimated HMM and the log-likelihood evolution. Convergence criteria like `atol` and `max_iterations` can be specified.

```julia
baum_welch(hmm_guess, obs_seq; ...)

```

```julia
baum_welch(
    hmm_guess,
    obs_seq,
    control_seq;
    seq_ends,
    atol,
    max_iterations,
    loglikelihood_increasing
)

```

--------------------------------

### Generate Multiple Observation Sequences

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/basics

Generates a specified number of long observation sequences with random lengths. This is a setup step for processing multiple sequences.

```julia
nb_seqs = 1000
long_obs_seqs = [last(rand(rng, hmm, rand(rng, 100:200))) for k in 1:nb_seqs];
typ_of(long_obs_seqs)
```

```julia
Vector{Vector{Vector{Float64}}} (alias for Array{Array{Array{Float64, 1}, 1}, 1})
```

--------------------------------

### Get Observation Distributions

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Retrieves the observation distributions for each state of an HMM, optionally considering a control input. These distributions must support sampling and log-density calculation.

```julia
obs_distributions(hmm)
obs_distributions(hmm, control)
```

--------------------------------

### Create a Basic Hidden Markov Model (HMM)

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable

Initialize a Hidden Markov Model with given initial probabilities, transition matrix, and observation distributions. Requires Distributions and HiddenMarkovModels packages.

```julia
using Distributions, HiddenMarkovModels
init = [0.6, 0.4]
trans = [0.7 0.3; 0.2 0.8]
dists = [Normal(-1.0), Normal(1.0)]
hmm = HMM(init, trans, dists)
```

--------------------------------

### Get Subsequence Indices

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Calculates the start and end indices for a specific subsequence within a concatenated sequence, given the sequence end markers.

```julia
seq_limits(seq_ends, k)
```

--------------------------------

### Initialize Guess for HMM Parameters

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/interfaces

Sets up initial guesses for the HMM parameters, including initial state probabilities, transition matrix, and custom observation distributions, to be used in the Baum-Welch algorithm.

```julia
init_guess = [0.5, 0.5]
trans_guess = [0.6 0.4; 0.3 0.7]
dists_guess = [StuffDist(-1.1), StuffDist(+1.1)]
hmm_guess = HMM(init_guess, trans_guess, dists_guess);
```

--------------------------------

### Create Periodic HMM Instance

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/temporal

Sets up the parameters for a periodic HMM, including initial state probabilities, transition matrices for each period, and observation distributions for each period. Then, it instantiates the `PeriodicHMM`.

```julia
init = [0.6, 0.3, 0.1]
trans_per = (
    [ # l = 1 -> mostly switch to next state
        0.2 0.8 0.0
        0.0 0.2 0.8
        0.8 0.0 0.2
    ],
    [ # l = 2 -> mostly switch to previous state
        0.2 0.0 0.8
        0.8 0.2 0.0
        0.0 0.8 0.2
    ],
    [ # l = 3 -> mostly stay in current state
        0.8 0.1 0.1
        0.1 0.8 0.1
        0.1 0.1 0.8
    ],
)
dists_per = (
    [Normal(1.0), Normal(2.0), Normal(3.0)],
    [Normal(3.0), Normal(4.0), Normal(5.0)],
    [Normal(5.0), Normal(6.0), Normal(7.0)],
)
hmm = PeriodicHMM(init, trans_per, dists_per);
```

--------------------------------

### Initialize Random Number Generator

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/controlled

Sets up a stable random number generator for reproducible simulations.

```julia
rng = StableRNG(63);
```

--------------------------------

### Initialize Guess for Learning

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/controlled

Sets up an initial guess for the HMM parameters to be used in the Baum-Welch learning algorithm.

```julia
init_guess = [0.5, 0.5]
trans_guess = [0.6 0.4; 0.3 0.7]
dist_coeffs_guess = [-2 * ones(d), 2 * ones(d)]
hmm_guess = ControlledGaussianHMM(init_guess, trans_guess, dist_coeffs_guess);
```

--------------------------------

### seq_limits Function

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Calculates the start and end indices for a specific subsequence within a set of sequences defined by their end points.

```APIDOC
## Function: `seq_limits`

```julia
seq_limits(seq_ends, k)
```

Return a tuple `(t1, t2)` giving the begin and end indices of subsequence `k` within a set of sequences ending at `seq_ends`.
```

--------------------------------

### Instantiate and Fit Custom PriorHMM

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/interfaces

Demonstrates how to instantiate a `PriorHMM` with initial guesses and then fit it to observation sequences using the `baum_welch` function. This shows the practical application of the custom HMM structure.

```julia
trans_prior_count = 10
prior_hmm_guess = PriorHMM(init_guess, trans_guess, dists_guess, trans_prior_count);

prior_hmm_est, prior_logl_evolution = baum_welch(prior_hmm_guess, obs_seq)
first(prior_logl_evolution), last(prior_logl_evolution)
```

--------------------------------

### Initialize Forward Algorithm Storage

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

The `initialize_forward` function prepares the storage for the forward algorithm, taking the HMM, observation sequence, and optional control sequence and sequence ends as input.

```julia
initialize_forward(hmm, obs_seq, control_seq; seq_ends)

```

--------------------------------

### Initialize Forward-Backward Storage

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Initializes the storage for forward-backward algorithm computations. Requires the HMM, observation sequence, and control sequence.

```julia
initialize_forward_backward(
    hmm,
    obs_seq,
    control_seq;
    seq_ends,
    transition_marginals
)

```

--------------------------------

### Implement AbstractHMM Interface Methods

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/interfaces

Implement the core methods required by the AbstractHMM interface: `initialization`, `transition_matrix`, and `obs_distributions`. These methods provide access to the HMM's parameters.

```julia
HiddenMarkovModels.initialization(hmm::PriorHMM) = hmm.init
HiddenMarkovModels.transition_matrix(hmm::PriorHMM) = hmm.trans
HiddenMarkovModels.obs_distributions(hmm::PriorHMM) = hmm.dists
```

--------------------------------

### Compare Estimated and Original HMM Parameters

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/basics

Compares the transition matrix, observation distributions, and initializations of the estimated HMM with the original HMM. This helps assess the accuracy of the parameter estimation.

```julia
cat(transition_matrix(hmm_est_concat), transition_matrix(hmm); dims=3)
```

```julia
2×2×2 Array{Float64, 3}:
[:, :, 1] =
 0.700421  0.299579
 0.203473  0.796527

[:, :, 2] =
 0.7  0.3
 0.2  0.8
```

```julia
map(mean, hcat(obs_distributions(hmm_est_concat), obs_distributions(hmm)))
```

```julia
2×2 Matrix{Vector{Float64}}:
 [-0.502073, -0.800874]  [-0.5, -0.8]
 [0.504019, 0.801399]    [0.5, 0.8]
```

```julia
hcat(initialization(hmm_est_concat), initialization(hmm))
```

```julia
2×2 Matrix{Float64}:
 0.605645  0.6
 0.394355  0.4
```

--------------------------------

### Calculate Prior Log-Likelihood of HMM Parameters

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Use `logdensityof` to get the prior log-likelihood of the HMM parameters. It can be called with or without control sequences and sequence end indicators.

```julia
logdensityof(hmm)

```

```julia
logdensityof(hmm, obs_seq; ...)

```

```julia
logdensityof(hmm, obs_seq, control_seq; seq_ends)

```

--------------------------------

### HiddenMarkovModels.baum_welch

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Applies the Baum-Welch algorithm to estimate the parameters of an HMM given an observation sequence, starting from an initial guess. Returns the estimated HMM and the evolution of the log-likelihood over iterations.

```APIDOC
## baum_welch

### Description
Apply the Baum-Welch algorithm to estimate the parameters of an HMM on `obs_seq`, starting from `hmm_guess`.

### Signature
```julia
baum_welch(hmm_guess, obs_seq; ...)
baum_welch(
    hmm_guess,
    obs_seq,
    control_seq;
    seq_ends,
    atol,
    max_iterations,
    loglikelihood_increasing
)
```

### Returns
Return a tuple `(hmm_est, loglikelihood_evolution)` where `hmm_est` is the estimated HMM and `loglikelihood_evolution` is a vector of loglikelihood values, one per iteration of the algorithm.

### 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
* `loglikelihood_increasing`: whether to throw an error if the loglikelihood decreases
```

--------------------------------

### Import necessary packages

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/basics

Import the required packages for using Hidden Markov Models, including distributions, linear algebra, and random number generation.

```julia
using Distributions
using HiddenMarkovModels
using LinearAlgebra
using Random
using StableRNGs
```

--------------------------------

### HiddenMarkovModels.initialize_viterbi

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Initializes the storage for the Viterbi algorithm.

```APIDOC
## initialize_viterbi

### Description
Initialize the storage for the Viterbi algorithm.

### Signature
```julia
initialize_viterbi(hmm, obs_seq, control_seq; seq_ends)
```
```

--------------------------------

### Retrieve Sequence Limits

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/basics

Uses `seq_limits` to find the start and end indices of a specific sequence within the concatenated observation vector. This is useful for analyzing individual sequence results.

```julia
start2, stop2 = seq_limits(seq_ends, 2)
```

```julia
(175, 279)
```

```julia
best_state_seq_concat[start2:stop2] == first(viterbi(hmm, long_obs_seqs[2]))
```

```julia
true
```

--------------------------------

### Instantiate DiffusionHMM and generate sequences

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/autodiff

Creates an instance of DiffusionHMM and generates observation and control sequences.

```julia
init = [0.6, 0.4]
trans = [0.7 0.3; 0.3 0.7]
means = [-1.0, 1.0]
hmm = DiffusionHMM(init, trans, means);
```

```julia
control_seqs = [rand(rng, 3), rand(rng, 5)];
obs_seqs = [rand(rng, hmm, control_seqs[k]).obs_seq for k in 1:2];

control_seq = reduce(vcat, control_seqs)
obs_seq = reduce(vcat, obs_seqs)
seq_ends = cumsum(length.(obs_seqs));
```

--------------------------------

### Check Possible and Impossible Transitions

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/types

Verify that transitions defined in the sparse matrix occur during simulation, while transitions that are structurally forbidden (zero in the sparse matrix) do not.

```julia
count(isequal((2, 2)), state_transitions)
```

```julia
count(isequal((2, 1)), state_transitions)
```

--------------------------------

### Display Log-Likelihood Evolution

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/interfaces

Shows the initial and final log-likelihood values obtained during the Baum-Welch estimation.

```julia
first(loglikelihood_evolution), last(loglikelihood_evolution)
```

--------------------------------

### initialization Function

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Retrieves the initial state probabilities for a given HMM.

```APIDOC
## Function: `initialization`

```julia
initialization(hmm)
```

Return the vector of initial state probabilities for `hmm`.
```

--------------------------------

### Generate Data from a Vanilla HMM

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/types

Define a standard HMM with initial probabilities, transition matrix, and normal distributions, then generate a state and observation sequence.

```julia
init = [0.6, 0.4]
trans = [0.7 0.3; 0.2 0.8]
dists = [Normal(-1.0), Normal(1.0)]
hmm = HMM(init, trans, dists)
state_seq, obs_seq = rand(rng, hmm, 100);
```

--------------------------------

### Simulate HMM Sequence with Sparse Transitions

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/types

Simulate a sequence of states and observations from an HMM defined with a sparse transition matrix. Transitions outside the non-zero coefficients are impossible.

```julia
state_seq, obs_seq = rand(rng, hmm, 1000)
state_transitions = collect(zip(state_seq[1:(end - 1)], state_seq[2:end]))
```

--------------------------------

### HiddenMarkovModels.initialize_forward

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Initializes the storage for the forward algorithm.

```APIDOC
## initialize_forward

### Description
Initialize the storage for the forward algorithm.

### Signature
```julia
initialize_forward(hmm, obs_seq, control_seq; seq_ends)
```
```

--------------------------------

### Initialize Viterbi Algorithm Storage

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

The `initialize_viterbi` function sets up the storage for the Viterbi algorithm, requiring the HMM, observation sequence, and optional control sequence and sequence end indicators.

```julia
initialize_viterbi(hmm, obs_seq, control_seq; seq_ends)

```

--------------------------------

### Forward-Backward Algorithm Initialization

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Initializes the storage for the forward-backward algorithm. This function prepares the necessary data structures to hold intermediate results.

```APIDOC
## Function: initialize_forward_backward

### Description
Initializes the storage for the forward-backward algorithm.

### Signature
```julia
initialize_forward_backward(hmm, obs_seq, control_seq; seq_ends, transition_marginals)
```

### Parameters
* `hmm`: The Hidden Markov Model object.
* `obs_seq`: The sequence of observations.
* `control_seq`: The sequence of control inputs.
* `seq_ends` (Optional): Specifies the end of sequences.
* `transition_marginals` (Optional): Specifies transition marginals.
```

--------------------------------

### Compare Estimated and True Transition Matrices

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/controlled

Compares the transition matrix estimated by the Baum-Welch algorithm with the true transition matrix used for simulation.

```julia
cat(hmm_est.trans, hmm.trans; dims=3)
```

--------------------------------

### Sample Observation Sequence from HMM

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/interfaces

Generates a sequence of states and observations from the HMM. The observations will be of the custom `Stuff` type.

```julia
state_seq, obs_seq = rand(rng, hmm, 100)
```

--------------------------------

### Create HMM with Custom Distributions

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/interfaces

Initializes an HMM object using custom initial state probabilities, transition matrix, and custom observation distributions.

```julia
init = [0.6, 0.4]
trans = [0.7 0.3; 0.2 0.8]
dists = [StuffDist(-1.0), StuffDist(+1.0)]
hmm = HMM(init, trans, dists);
```

--------------------------------

### Log-likelihood Evolution

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/temporal

Displays the initial and final log-likelihood values obtained during the Baum-Welch estimation. This indicates the convergence of the algorithm.

```julia
(-255695.48690011754, -243058.7258113099)
```

--------------------------------

### Instantiate ControlledGaussianHMM Model

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/controlled

Creates an instance of the ControlledGaussianHMM with specified dimensions, initial probabilities, transition matrix, and distribution coefficients.

```julia
d = 3
init = [0.6, 0.4]
trans = [0.7 0.3; 0.2 0.8]
dist_coeffs = [-ones(d), ones(d)]
hmm = ControlledGaussianHMM(init, trans, dist_coeffs);
```

--------------------------------

### Compare Estimated and True Distribution Coefficients (State 1)

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/controlled

Compares the estimated distribution coefficients for the first state with the true coefficients.

```julia
hcat(hmm_est.dist_coeffs[1], hmm.dist_coeffs[1])
```

--------------------------------

### Compare Transition Matrices of HMMs

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/interfaces

Compares the transition matrices of a standard HMM and a custom `PriorHMM` after estimation. This visualization helps understand the effect of the prior on the transition probabilities.

```julia
cat(transition_matrix(hmm_est), transition_matrix(prior_hmm_est); dims=3)
```

--------------------------------

### Run Baum-Welch Algorithm with Custom Observations

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/interfaces

Executes the Baum-Welch algorithm to estimate HMM parameters from the observed sequence of custom `Stuff` objects. It returns the estimated HMM and the evolution of the log-likelihood.

```julia
hmm_est, loglikelihood_evolution = baum_welch(hmm, obs_seq)
```

--------------------------------

### Preallocate gradient storage for Enzyme.jl

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/autodiff

Enzyme.jl requires preallocated storage for gradients. Use `Enzyme.make_zero` to create zero-initialized arrays matching the types and shapes of the parameters to be differentiated.

```julia
∇parameters_enzyme = Enzyme.make_zero(parameters)
∇obs_enzyme = Enzyme.make_zero(obs_seq)
∇control_enzyme = Enzyme.make_zero(control_seq);
```

--------------------------------

### Display first three states of the sequence

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/basics

Show the first three elements of the simulated state sequence. The state sequence is a vector of integers.

```julia
state_seq[1:3]
```

--------------------------------

### Implement HMM Interface for ControlledGaussianHMM

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/controlled

Provides the necessary methods for the AbstractHMM interface: initialization, transition matrix, and observation distributions. The observation distributions depend on the provided control vector.

```julia
function HMMs.initialization(hmm::ControlledGaussianHMM)
    return hmm.init
end

function HMMs.transition_matrix(hmm::ControlledGaussianHMM, control::AbstractVector)
    return hmm.trans
end

function HMMs.obs_distributions(hmm::ControlledGaussianHMM, control::AbstractVector)
    return [Normal(dot(hmm.dist_coeffs[i], control), 1.0) for i in 1:length(hmm)]
end
```

--------------------------------

### Run Baum-Welch Learning and Check Likelihood

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/controlled

Executes the Baum-Welch algorithm to estimate the HMM parameters from the data and reports the initial and final log-likelihood values.

```julia
hmm_est, loglikelihood_evolution = baum_welch(hmm_guess, obs_seq, control_seq; seq_ends)
first(loglikelihood_evolution), last(loglikelihood_evolution)
```

--------------------------------

### Run Baum-Welch Algorithm

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/temporal

Executes the Baum-Welch algorithm to estimate HMM parameters. Requires an initial guess of the HMM, the observation sequence, and the control sequence. Returns the estimated HMM and the evolution of the log-likelihood.

```julia
hmm_est, loglikelihood_evolution = baum_welch(hmm_guess, obs_seq, control_seq; seq_ends);
first(loglikelihood_evolution), last(loglikelihood_evolution)
```

--------------------------------

### Run Baum-Welch Algorithm

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Executes the Baum-Welch algorithm for parameter estimation. Updates the forward-backward storage and log-likelihood evolution in-place. Supports custom convergence criteria.

```julia
baum_welch!(
    fb_storage,
    logL_evolution,
    hmm,
    obs_seq,
    control_seq;
    seq_ends,
    atol,
    max_iterations,
    loglikelihood_increasing
)

```

--------------------------------

### Import Necessary Packages

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/temporal

Imports the required Julia packages for working with distributions, HiddenMarkovModels, random number generation, and statistics.

```julia
using Distributions
using HiddenMarkovModels
import HiddenMarkovModels as HMMs
using Random
using StableRNGs
using StatsAPI
```

--------------------------------

### Sequence Formatting Utilities

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Helper functions and explanations for formatting observation and control sequences, especially when dealing with multiple sequences concatenated into a single vector.

```APIDOC
## Sequence Formatting

Most algorithms ingest data with `obs_seq` (mandatory), `control_seq` (optional), and `seq_ends` (optional).

For single sequences, `obs_seq` and `control_seq` are vectors, and `seq_ends` is not needed.

For multiple sequences:
```julia
obs_seq = reduce(vcat, obs_seqs)
control_seq = reduce(vcat, control_seqs)
seq_ends = cumsum(length.(obs_seqs))
```
```

--------------------------------

### Compare Estimated vs True Transition Matrices (Period 1)

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/temporal

Compares the estimated transition matrix for the first period of the HMM with the true transition matrix. This helps in evaluating the accuracy of the learned parameters.

```julia
cat(transition_matrix(hmm_est, 1), transition_matrix(hmm, 1); dims=3)
```

--------------------------------

### Implement In-Place Fitting for Custom Distribution

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/interfaces

Implements the `StatsAPI.fit!` method for in-place fitting of the custom distribution using weighted observation sequences. This method updates the distribution's parameters.

```julia
function StatsAPI.fit!( 
    dist::StuffDist, obs_seq::AbstractVector{<:Stuff}, weight_seq::AbstractVector{<:Real}
)
    dist.quantity_mean =
        sum(weight_seq[k] * obs_seq[k].quantity for k in eachindex(obs_seq, weight_seq)) /
        sum(weight_seq)
    return nothing
end
```

--------------------------------

### Perform Forward-Backward Algorithm In-Place

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Performs the forward-backward algorithm and updates the provided storage object in-place. Requires the storage, HMM, observation sequence, and control sequence.

```julia
forward_backward!(
    storage,
    hmm,
    obs_seq,
    control_seq;
    seq_ends,
    transition_marginals
)

```

--------------------------------

### Log Initial State Probabilities

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Returns the vector of initial state log-probabilities for a given HMM.

```APIDOC
## Function: log_initialization

### Description
Return the vector of initial state log-probabilities for `hmm`.

### Signature
```julia
log_initialization(hmm)
```

### Parameters
* `hmm`: The Hidden Markov Model object.
```

--------------------------------

### Inspect Estimated Observation Distributions

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/interfaces

Retrieves and displays the observation distributions from the HMM estimated by the Baum-Welch algorithm, showing the updated parameters of the custom `StuffDist`.

```julia
obs_distributions(hmm_est)
```

--------------------------------

### HiddenMarkovModels.forward!

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Performs the forward algorithm in-place, updating the provided storage with alpha values and log-likelihoods.

```APIDOC
## forward!

### Description
Perform the forward algorithm in-place.

### Signature
```julia
forward!(
    storage,
    hmm,
    obs_seq,
    control_seq;
    seq_ends,
    error_if_not_finite
)
```
```

--------------------------------

### Compare Estimated and True Distribution Coefficients (State 2)

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/controlled

Compares the estimated distribution coefficients for the second state with the true coefficients.

```julia
hcat(hmm_est.dist_coeffs[2], hmm.dist_coeffs[2])
```

--------------------------------

### Import Necessary Packages

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/types

Import required packages for using HiddenMarkovModels.jl, including distributions, linear algebra, and specialized number types.

```julia
using Distributions
using HiddenMarkovModels
using LinearAlgebra
using LogarithmicNumbers
using Measurements
using Random
using SparseArrays
using StableRNGs
```

--------------------------------

### Import Necessary Packages

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/interfaces

Imports required packages for working with custom distributions and HMMs.

```julia
using DensityInterface
using Distributions
using HiddenMarkovModels
import HiddenMarkovModels as HMMs
using LinearAlgebra
using Random: Random, AbstractRNG
using StableRNGs
using StatsAPI
```

--------------------------------

### Log Transition Matrix

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Returns the matrix of state transition log-probabilities for an HMM, optionally considering control inputs.

```APIDOC
## Function: log_transition_matrix

### Description
Return the matrix of state transition log-probabilities for `hmm` (possibly when `control` is applied).

### Signature
```julia
log_transition_matrix(hmm)
log_transition_matrix(hmm, control)
```

### Parameters
* `hmm`: The Hidden Markov Model object.
* `control` (Optional): The control input.
```

--------------------------------

### Construct HMM with Uncertain Parameters

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/types

Create a new HMM using Measurements.jl to introduce uncertainty in the observation means. Note that uncertainty in transition parameters is not supported.

```julia
dists_guess = [Normal(-1.0 ± 0.1), Normal(1.0 ± 0.2)]
hmm_uncertain = HMM(init, trans, dists_guess)
```

--------------------------------

### Run Forward Algorithm for Log-Likelihood

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

The `forward` function computes the log-likelihood of an observation sequence using the forward algorithm. It can optionally use control sequences and specify error handling for non-finite values.

```julia
forward(hmm, obs_seq; ...)

```

```julia
forward(
    hmm,
    obs_seq,
    control_seq;
    seq_ends,
    error_if_not_finite
)

```

--------------------------------

### Implement HMM interface methods for DiffusionHMM

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/autodiff

Implements initialization, transition matrix, and observation distribution methods for the DiffusionHMM, incorporating a control parameter λ.

```julia
HMMs.initialization(hmm::DiffusionHMM) = hmm.init

function HMMs.transition_matrix(hmm::DiffusionHMM, λ::Number)
    N = length(hmm)
    return (1 - λ) * hmm.trans + λ * ones(N, N) / N
end

function HMMs.obs_distributions(hmm::DiffusionHMM, λ::Number)
    return [Normal((1 - λ) * hmm.means[i] + λ * 0) for i in 1:length(hmm)]
end
```

--------------------------------

### Prepare Data for Inference

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/temporal

Generates a collection of observation sequences (`obs_seqs`) and their corresponding control sequences (`control_seqs`) of variable lengths. These are then concatenated into single `obs_seq` and `control_seq` vectors, with `seq_ends` marking the boundaries of the original sequences.

```julia
control_seqs = [1:rand(rng, 100:200) for k in 1:1000]
obs_seqs = [rand(rng, hmm, control_seqs[k]).obs_seq for k in eachindex(control_seqs)]

obs_seq = reduce(vcat, obs_seqs)
control_seq = reduce(vcat, control_seqs)
seq_ends = cumsum(length.(obs_seqs));
```

--------------------------------

### Baum-Welch Algorithm for Parameter Estimation

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/basics

Estimates HMM parameters (initialization, transition matrix, observation distributions) using the Expectation-Maximization algorithm. Requires an initial guess.

```julia
init_guess = [0.5, 0.5]
trans_guess = [0.6 0.4; 0.3 0.7]
dists_guess = [MvNormal([-0.4, -0.7], I), MvNormal([0.4, 0.7], I)]
hmm_guess = HMM(init_guess, trans_guess, dists_guess);
```

```julia
_, long_obs_seq = rand(rng, hmm, 200)
hmm_est, loglikelihood_evolution = baum_welch(hmm_guess, long_obs_seq);
```

```julia
first(loglikelihood_evolution), last(loglikelihood_evolution)
```

```text
(-613.0225125416881, -605.3027257918021)
```

```julia
cat(transition_matrix(hmm_est), transition_matrix(hmm); dims=3)
```

```text
2×2×2 Array{Float64, 3}:
[:, :, 1] =
 0.676764  0.323236
 0.197387  0.802613

[:, :, 2] =
 0.7  0.3
 0.2  0.8
```

```julia
map(mean, hcat(obs_distributions(hmm_est), obs_distributions(hmm)))
```

```text
2×2 Matrix{Vector{Float64}}:
 [-0.256414, -0.563875]  [-0.5, -0.8]
 [0.652218, 0.841332]    [0.5, 0.8]
```

```julia
hcat(initialization(hmm_est), initialization(hmm))
```

```text
2×2 Matrix{Float64}:
 1.91765e-19  0.6
 1.0          0.4
```

--------------------------------

### Generate Random Transition Matrix

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Generates a random transition matrix of a specified size with normalized uniform entries.

```APIDOC
## Function: rand_trans_mat

### Description
Generate a random transition matrix of size `(N, N)` with normalized uniform entries.

### Signature
```julia
rand_trans_mat([rng, ::Type{R},] N)
```

### Parameters
* `rng` (Optional): Random number generator.
* `R` (Optional): Type of the elements.
* `N`: The size of the transition matrix (N x N).
```

--------------------------------

### Implement Log-Density Calculation for Custom Distribution

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/interfaces

Implements the `DensityInterface.logdensityof` method to compute the log-probability density of an observation given the custom distribution. The calculation can be up to an additive constant.

```julia
function DensityInterface.logdensityof(dist::StuffDist, obs::Stuff)
    return -abs2(obs.quantity - dist.quantity_mean) / 2
end
```

--------------------------------

### Perform Forward Algorithm In-Place

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

The `forward!` function executes the forward algorithm in-place, updating the provided `storage` object. It requires the HMM, observation sequence, and optional control sequence and sequence end indicators.

```julia
forward!(
    storage,
    hmm,
    obs_seq,
    control_seq;
    seq_ends,
    error_if_not_finite
)

```

--------------------------------

### Import necessary packages

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/autodiff

Imports required Julia packages for automatic differentiation, component arrays, distributions, and HiddenMarkovModels.

```julia
using ComponentArrays
using DensityInterface
using Distributions
using Enzyme: Enzyme
using ForwardDiff: ForwardDiff
using HiddenMarkovModels
import HiddenMarkovModels as HMMs
using LinearAlgebra
using Random: Random, AbstractRNG
using StableRNGs
using StatsAPI
using Zygote: Zygote
```

--------------------------------

### Simulate Sequence with Control Sequence

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/temporal

Simulates a state and observation sequence from the `PeriodicHMM`. Unlike standard HMMs, this requires providing a `control_seq` which dictates the time-dependent parameters used at each step.

```julia
control_seq = 1:10
state_seq, obs_seq = rand(rng, hmm, control_seq);
```

--------------------------------

### Compare Estimated vs True Transition Matrices (Period 2)

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/temporal

Compares the estimated transition matrix for the second period of the HMM with the true transition matrix. This helps in evaluating the accuracy of the learned parameters.

```julia
cat(transition_matrix(hmm_est, 2), transition_matrix(hmm, 2); dims=3)
```

--------------------------------

### Simulate state and observation sequences

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/basics

Simulate a pair of state and observation sequences from a defined HMM for a specified duration T.

```julia
T = 20
state_seq, obs_seq = rand(rng, hmm, T);
```

--------------------------------

### Import Dependencies for Controlled HMM

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/controlled

Imports necessary Julia packages for working with distributions, HiddenMarkovModels, linear algebra, random number generation, and statistics.

```julia
using Distributions
using HiddenMarkovModels
import HiddenMarkovModels as HMMs
using LinearAlgebra
using Random
using StableRNGs
using StatsAPI
```

--------------------------------

### Generate Control Sequences and Observation Sequences

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/controlled

Generates multiple sequences of control inputs and corresponding observation sequences using the defined HMM. Each sequence has a random length.

```julia
control_seqs = [[randn(rng, d) for t in 1:rand(100:200)] for k in 1:1000];
obs_seqs = [rand(rng, hmm, control_seq).obs_seq for control_seq in control_seqs];

obs_seq = reduce(vcat, obs_seqs)
control_seq = reduce(vcat, control_seqs)
seq_ends = cumsum(length.(obs_seqs));
```

--------------------------------

### Verify Uncertainty Propagation

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/types

Check if the computed log-density for the HMM with uncertainties is approximately equal to the log-density of the vanilla HMM.

```julia
Measurements.value(logdensityof(hmm_uncertain, obs_seq)) ≈ logdensityof(hmm, obs_seq)
```

--------------------------------

### Implement Learning Function for ControlledGaussianHMM

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/controlled

Overrides the `fit!` function to implement the learning algorithm for the ControlledGaussianHMM. It updates initial probabilities, transition matrix, and observation coefficients using forward-backward results and weighted least squares.

```julia
function StatsAPI.fit!( 
    hmm::ControlledGaussianHMM{T}, 
    fb_storage::HMMs.ForwardBackwardStorage, 
    obs_seq::AbstractVector, 
    control_seq::AbstractVector; 
    seq_ends, 
) where {T}
    (; γ, ξ) = fb_storage
    N = length(hmm)

    hmm.init .= 0
    hmm.trans .= 0
    for k in eachindex(seq_ends)
        t1, t2 = HMMs.seq_limits(seq_ends, k)
        hmm.init .+= γ[:, t1]
        hmm.trans .+= sum(ξ[t1:t2])
    end
    hmm.init ./= sum(hmm.init)
    for row in eachrow(hmm.trans)
        row ./= sum(row)
    end

    U = reduce(hcat, control_seq)'
    y = obs_seq
    for i in 1:N
        W = sqrt.(Diagonal(γ[i, :]))
        hmm.dist_coeffs[i] = (W * U) \ (W * y)
    end
end
```

--------------------------------

### Compare Estimated vs True Transition Matrices (Period 3)

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/temporal

Compares the estimated transition matrix for the third period of the HMM with the true transition matrix. This helps in evaluating the accuracy of the learned parameters.

```julia
cat(transition_matrix(hmm_est, 3), transition_matrix(hmm, 3); dims=3)
```

--------------------------------

### Simulate HMM

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Generates a state and observation sequence by simulating the HMM for a specified number of time steps or based on a control sequence.

```julia
rand([rng,] hmm, T)
rand([rng,] hmm, control_seq)
```

--------------------------------

### Implement Periodic HMM Interface

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/temporal

Implements the necessary interface functions for `PeriodicHMM`. This includes defining the period, initialization, and time-dependent transition matrices and observation distributions by using the modulo operator on the time step.

```julia
period(::PeriodicHMM{T,D,L}) where {T,D,L} = L

function HMMs.initialization(hmm::PeriodicHMM)
    return hmm.init
end

function HMMs.transition_matrix(hmm::PeriodicHMM, t::Integer)
    l = (t - 1) % period(hmm) + 1
    return hmm.trans_per[l]
end

function HMMs.obs_distributions(hmm::PeriodicHMM, t::Integer)
    l = (t - 1) % period(hmm) + 1
    return hmm.dists_per[l]
end
```

--------------------------------

### Log-likelihood of Observation Sequence

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/basics

A wrapper around the forward algorithm to directly compute the log-likelihood of an observation sequence P(Y1:T).

```julia
logdensityof(hmm, obs_seq)
```

```text
-54.214899298272634
```

--------------------------------

### Update HMM In-Place using Forward-Backward Storage

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

The `StatsAPI.fit!` function updates an HMM in-place using pre-computed forward-backward statistics. The `fb_storage` is used as scratch space and its contents are not guaranteed after the call.

```julia
StatsAPI.fit!(
    hmm, fb_storage::ForwardBackwardStorage,
    obs_seq, [control_seq]; seq_ends,
)

```

--------------------------------

### Base Overloads

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Overloads for Base package functions.

```APIDOC
## Base Overloads

Functions from the Base package that are extended for HiddenMarkovModels.

### `Base.eltype(hmm)`
Returns the element type of the HMM.

### `Base.length(hmm)`
Returns the number of states in the HMM.
```

--------------------------------

### Forward-Backward Algorithm Execution

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Executes the forward-backward algorithm in-place, updating the provided storage with computed values.

```APIDOC
## Function: forward_backward!

### Description
Executes the forward-backward algorithm in-place.

### Signature
```julia
forward_backward!(storage, hmm, obs_seq, control_seq; seq_ends, transition_marginals)
```

### Parameters
* `storage`: The storage object to be updated with results.
* `hmm`: The Hidden Markov Model object.
* `obs_seq`: The sequence of observations.
* `control_seq`: The sequence of control inputs.
* `seq_ends` (Optional): Specifies the end of sequences.
* `transition_marginals` (Optional): Specifies transition marginals.
```

--------------------------------

### Retrieve Estimated Sparse Transition Matrix

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/types

Inspect the transition matrix of the HMM estimated by the Baum-Welch algorithm. The sparsity pattern from the initial guess is preserved in the estimated model.

```julia
transition_matrix(hmm_est)
```

--------------------------------

### Forward-Backward Algorithm for Smoothed State Marginals

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/basics

Computes smoothed state marginals (probability of current state given all observations) and the observation sequence log-likelihood. Useful for learning.

```julia
smoothed_state_marginals, obs_seq_loglikelihood_fb = forward_backward(hmm, obs_seq);
only(obs_seq_loglikelihood_fb)
```

```text
-54.214899298272634
```

```julia
filtered_state_marginals[:, T - 1] ≈ smoothed_state_marginals[:, T - 1]
```

```text
false
```

```julia
filtered_state_marginals[:, T] ≈ smoothed_state_marginals[:, T]
```

```text
true
```

--------------------------------

### StatsAPI.fit!

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Updates an HMM in-place using information computed during the forward-backward algorithm. The provided storage object may be reused as scratch space and its contents should not be trusted afterwards.

```APIDOC
## StatsAPI.fit!

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

### Signature
```julia
StatsAPI.fit!(
    hmm, fb_storage::ForwardBackwardStorage,
    obs_seq, [control_seq]; seq_ends,
)
```

### Details
This function is allowed to reuse `fb_storage` as a scratch space, so its contents should not be trusted afterwards.
```

--------------------------------

### Verify gradient results with forward diff

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/autodiff

Compare the computed gradients from Enzyme.jl (reverse mode) with those obtained from forward-mode differentiation to ensure correctness.

```julia
∇parameters_enzyme ≈ ∇parameters_forwarddiff
```

```julia
∇obs_enzyme ≈ ∇obs_forwarddiff
```

```julia
∇control_enzyme ≈ ∇control_forwarddiff
```

--------------------------------

### HMM Type

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

A concrete implementation of a basic Hidden Markov Model, storing initial probabilities, transition matrices, and observation distributions.

```APIDOC
## Type: `HMM`

Basic implementation of an HMM.

### Fields
  * `init::AbstractVector` : initial state probabilities
  * `trans::AbstractMatrix`: state transition probabilities
  * `dists::AbstractVector`: observation distributions
  * `loginit::AbstractVector`: logarithms of initial state probabilities
  * `logtrans::AbstractMatrix`: logarithms of state transition probabilities
```

--------------------------------

### HiddenMarkovModels.forward_backward

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Applies the forward-backward algorithm to compute posterior state and transition marginals for an HMM given an observation sequence. Returns the posterior marginals (gamma) and the log-likelihood.

```APIDOC
## forward_backward

### Description
Apply the forward-backward algorithm to infer the posterior state and transition marginals during sequence `obs_seq` for `hmm`.

### Signature
```julia
forward_backward(hmm, obs_seq; ...)
forward_backward(hmm, obs_seq, control_seq; seq_ends)
```

### Returns
Return a tuple `(storage.γ, storage.logL)` where `storage` is of type `ForwardBackwardStorage`.
```

--------------------------------

### Calculate Log-Density of Observations

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/types

Compute the log-density of the observation sequence using both the original HMM and the HMM with uncertain parameters.

```julia
logdensityof(hmm, obs_seq)
```

```julia
logdensityof(hmm_uncertain, obs_seq)
```

--------------------------------

### Construct Sparse Transition Matrix for HMM

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/types

Define an HMM with a sparse transition matrix to represent structurally forbidden transitions. This is useful for large models where not all state transitions are possible.

```julia
trans = sparse([
    0.7 0.3 0
    0 0.7 0.3
    0.3 0 0.7
])
```

```julia
init = [0.2, 0.6, 0.2]
dists = [Normal(1.0), Normal(2.0), Normal(3.0)]
hmm = HMM(init, trans, dists)
```

--------------------------------

### Define a Hidden Markov Model

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/basics

Define an HMM with initial state probabilities, transition probabilities, and observation distributions. Any scalar- or vector-valued distribution from Distributions.jl can be used.

```julia
init = [0.6, 0.4]
trans = [0.7 0.3; 0.2 0.8]
dists = [MvNormal([-0.5, -0.8], I), MvNormal([0.5, 0.8], I)]
hmm = HMM(init, trans, dists)
```

--------------------------------

### Validate Zygote gradients against ForwardDiff gradients

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/autodiff

Compares the gradients computed by Zygote.jl (reverse mode) with those computed by ForwardDiff.jl (forward mode) to ensure correctness.

```julia
∇parameters_zygote ≈ ∇parameters_forwarddiff
```

```julia
∇obs_zygote ≈ ∇obs_forwarddiff
```

```julia
∇control_zygote ≈ ∇control_forwarddiff
```

--------------------------------

### Generate Random Transition Matrix

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Generates a random transition matrix of size (N, N) with normalized uniform entries. Optionally accepts a random number generator and element type.

```julia
rand_trans_mat([rng, ::Type{R},] N)

```

--------------------------------

### HiddenMarkovModels.viterbi!

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/api

Performs the Viterbi algorithm in-place, updating the provided storage with the most likely state sequence and its log-likelihood.

```APIDOC
## viterbi!

### Description
Perform the Viterbi algorithm in-place.

### Signature
```julia
viterbi!(storage, hmm, obs_seq, control_seq; seq_ends)
```
```

--------------------------------

### Display Simulated Observations

Source: https://gdalle.github.io/HiddenMarkovModels.jl/stable/examples/temporal

Displays the transpose of the simulated observation sequence. The output shows a sequence of floating-point numbers representing the observed values.

```julia
obs_seq'
```