Machine learning with stochastic models (stochastic generative models)
Probabilistic generative models, also discussed in “Uncertainty and Machine Learning Techniques” are machine learning methods based on probabilistic models, which model probabilistic properties and statistical relationships of data to perform tasks such as forecasting and pattern recognition. The principle of probabilistic models is to make predictions and inferences by assuming a probability distribution of the data and learning its parameters from the data.
Probabilistic modeling approaches in machine learning can be divided into the following two main categories
Supervised Learning: In supervised learning, given input data and the corresponding correct label (or target), the model learned from the data is the one used to predict the label for the unknown input data. Typical probabilistic models include naïve Bayes classifiers, logistic regression, decision trees, and random forests.
Unsupervised Learning (Unsupervised Learning): Unsupervised learning is where the input data is not given correct labels, but rather the patterns and structure of the data itself is modeled, and tasks such as clustering and dimensionality reduction are involved. Examples of probabilistic models include mixed Gaussian models, hidden Markov models (HMMs), probabilistic latent semantic analysis (PLSA), and Gaussian processes.
Methods such as maximum likelihood estimation described in “Overview of Maximum Likelihood Estimation and Algorithms and Their Implementations” and Bayesian estimation are commonly used in learning probabilistic models. In maximum likelihood estimation, the model is learned by maximizing the likelihood function to optimize the parameters of the model for the given data, and in Bayesian estimation, the prior distribution and likelihood are combined to calculate the posterior distribution, which is distribution is used to parameterize the model and make predictions.
Probabilistic models model the probability distribution and statistical properties of data, making them powerful tools for real-world problems where uncertainty and noise exist. Probabilistic models can also be used in combination with other machine learning methods, for example, using probabilistic models as generative models and supervised learning with the generated data as supervised data.
In this article, I would like to discuss hidden Markov models (HMMs) among these models.
Hidden Markov Model (HMM)
HMM is a type of probabilistic model that is used to represent the process of generating a series of observations and is widely used for modeling series and time series data, among others.
HMMs are composed of two components, “hidden state” and “observation.” The hidden state represents the latent state behind the series data, which is not directly observed, while the observation result is the directly observable data, which is generated from the hidden state. HMM.
There are three basic components in an HMM
- State transition probability (transition probability): represents the probability of moving from one hidden state to another hidden state. It is used to predict the next state based on the current state.
- Output probability (emission probability): represents the probability that a particular observation will be produced from a given hidden state. It expresses what observation outcome is expected given a particular state.
- initial state probability: This expresses the probability that the first hidden state will start.
HMMs can be applied to many problems with series of observations, e.g., speech recognition, natural language processing, bioinformatics, time series forecasting in economics, etc. HMMs may also be used for tasks such as classification and generation of series data and completion of missing data.
How HMMs can be used to train and infer models using methods such as the forward-looking algorithm, backward-looking algorithm, Viterbi algorithm, Baum-Welch algorithm, and others, which will be detailed. These methods allow HMMs to model the patterns and structure of series data for prediction and analysis.
On the algorithms used in hidden Markov models
Hidden Markov Models (HMMs) have several major algorithms for learning and inference. The following describes typical algorithms used in HMMs.
Forward Algorithm
<Overview>
The forward-looking algorithm for HMMs is one of the basic methods in HMM training and inference, and it is an algorithm for computing the posterior probability in the hidden state at each time for a given observation series. The procedure of the forward-looking algorithm is described below.
1. Initialization: The first step of the algorithm is to set up the probability distribution of the initial hidden states. Usually, the initial state probability of the HMM is used. It calculates the prior probability in each hidden state and sets it as the initial prior probability.
2. Recursive computation: Next, the posterior probability of the hidden state at each time in the observation series is computed. To perform the recursive computation, the following steps are repeated for each time t
a. Prediction step: The posterior probabilities of the previous time are used to predict the hidden state at the current time. Specifically, the transition probabilities from each hidden state at the previous time to the current hidden state are multiplied by the posterior probability at the previous time to calculate the predicted value of the hidden state.
b. Update step: The predictions computed in the prediction step are multiplied by the output probabilities based on the observed data at the current time and normalized. This yields the posterior probability for each hidden state.
3. Termination: When the last time is reached, the posterior probabilities of the hidden states in the final observation sequence can be obtained.
The Forward Algorithm can be used to estimate information about the hidden state of the observation series in the HMM, and the Forward Algorithm can be combined with the Backward Algorithm to learn and decode the parameters of the HMM (estimating the optimal hidden state series estimation).
<Example of Python Implementation>
The following is an example implementation of the HMM forward algorithm using Python. In this example, NumPy is used to perform numerical calculations.
import numpy as np
def forward_algorithm(observation, A, B, pi):
T = len(observation)
N = A.shape[0]
alpha = np.zeros((T, N))
alpha[0] = pi * B[:, observation[0]]
for t in range(1, T):
for j in range(N):
alpha[t, j] = np.sum(alpha[t-1] * A[:, j]) * B[j, observation[t]]
return alpha
# The following is an example of execution
# Define HMM parameters
A = np.array([[0.7, 0.3], [0.4, 0.6]]) # state transition probability matrix
B = np.array([[0.1, 0.4, 0.5], [0.7, 0.2, 0.1]]) # output probability matrix
pi = np.array([0.6, 0.4]) # initial state probability vector
# observation sequence
observation = [0, 1, 2, 0, 2]
# Run forward algorithm to obtain posterior probabilities of hidden states
alpha = forward_algorithm(observation, A, B, pi)
print("Posterior probability of hidden state:")
print(alpha)This implementation uses the forward_algorithm function to perform a forward algorithm on the observation series, where A is the state transition probability matrix, B is the output probability matrix, and pi is the initial state probability vector. The posterior probabilities of the hidden states at each time are returned for the observation series.
In actual use cases, the forward algorithm can be executed and the posterior probabilities of the hidden states can be obtained by preparing appropriate HMM parameters and observation series and calling the forward_algorithm function.
Backward Algorithm
<Overview>
The backward algorithm of the HMM will be the algorithm for computing the posterior probability in the hidden state at each time for a given observation series. It is a symmetric method to the forward algorithm and plays an important role in HMM training and inference.
The steps of the backward algorithm are described below.
1. Initialization: The first step of the algorithm is to set the posterior probabilities of the hidden states at the last time. Usually, the final state probability of the HMM is used. It calculates the posterior probability for each hidden state and sets it as the final posterior probability.
2. Recursive computation: Next, the posterior probabilities of the hidden states at each time in the observation series are computed. To perform the recursive computation, the following steps are repeated for each time t in the reverse direction.
a. Prediction step: Predict the hidden state at the current time using the posterior probability at the time one time later. Specifically, the transition probabilities from each hidden state to the current hidden state at the next time are multiplied by the posterior probability at the next time to calculate the predicted value of the hidden state.
b. Update step: The predictions computed in the prediction step are multiplied by the output probabilities based on the observed data at the current time and normalized. This yields the posterior probability for each hidden state.
3. Termination: When the first time is reached, the posterior probability of the hidden state in the first observation series can be obtained.
The backward algorithm can be used in combination with the forward algorithm to learn the parameters of the HMM and to perform decoding (estimation of the optimal hidden state series). Also, by combining the results of the forward and backward algorithms, it is possible to compute detailed information about the hidden states in the HMM and the likelihood of the entire model.
<Example Implementation in Python>
The following is an example implementation of the HMM backward algorithm using Python. In this example, NumPy is used to perform the numerical calculations.
import numpy as np
def backward_algorithm(observation, A, B):
T = len(observation)
N = A.shape[0]
beta = np.zeros((T, N))
beta[T-1] = 1.0
for t in range(T-2, -1, -1):
for i in range(N):
beta[t, i] = np.sum(beta[t+1] * A[i] * B[:, observation[t+1]])
return beta
# The following is an example of execution
# Define HMM parameters
A = np.array([[0.7, 0.3], [0.4, 0.6]]) # state transition probability matrix
B = np.array([[0.1, 0.4, 0.5], [0.7, 0.2, 0.1]]) # output probability matrix
# observation sequence
observation = [0, 1, 2, 0, 2]
# Run the backward algorithm to obtain the posterior probability of the hidden state
beta = backward_algorithm(observation, A, B)
print("Posterior probability of hidden state:")
print(beta)This implementation uses the backward_algorithm function to perform a backward algorithm on the observed series, where A is the state transition probability matrix and B is the output probability matrix, returning the posterior probability of the hidden state at each time for the observed series.
Viterbi algorithm
<Overview>
The Viterbi algorithm will be the algorithm for decoding (estimating the optimal hidden state series) in HMMs. The goal is to find the most plausible hidden state series (Viterbi path) for a given observation series; the Viterbi algorithm is computed efficiently using dynamic programming.
The Viterbi algorithm in HMM is a type of HMM decoding (estimation) algorithm, and is described in “Overview of Dynamic Programming and Examples of Applications and The Viterbi algorithm is a type of HMM decoding (estimation) algorithm that is efficiently computed using dynamic programming as described in “Overview of Dynamic Programming, Application Examples, and Example Implementation in python”.
The procedure of the Viterbi algorithm is described below.
1. Initialization: The first step of the algorithm is to compute the likelihood for each hidden state at the initial time. Specifically, this is done by multiplying the initial state probability by the output probability based on the observed data at the initial time.
2. Recursive computation: Next, for each hidden state at each time, the most likely previous hidden state and its likelihood are computed. To perform the recursive computation, the following steps are repeated for each time t
a. Prediction step: The predictions are calculated by multiplying the likelihood of each hidden state at the previous time by the transition probability. This yields a candidate for the previous hidden state and its likelihood in each hidden state.
b. Update step: The candidate with the highest likelihood is selected from the candidates calculated in the prediction step. The likelihood is updated as the likelihood of the hidden state at the current time.
3. Backward: When the last time is reached, the final sequence of hidden states is constructed. By selecting the hidden state with the highest likelihood at the last time and tracing back to the previous hidden state, the maximum likelihood hidden state sequence can be obtained.
The Viterbi algorithm can efficiently find the optimal hidden state at each time by taking advantage of the features of dynamic programming. This algorithm is a common method of decoding (estimation) in HMMs and is widely used in fields such as natural language processing and speech recognition.
<Example Implementation in Python>
The following is an example implementation of the Viterbi algorithm for HMM using Python. In this example, NumPy is used to perform numerical computations.
import numpy as np
def viterbi_algorithm(observation, A, B, pi):
T = len(observation)
N = A.shape[0]
delta = np.zeros((T, N))
psi = np.zeros((T, N), dtype=int)
# initialization
delta[0] = pi * B[:, observation[0]]
# Recursive computation
for t in range(1, T):
for j in range(N):
temp = delta[t-1] * A[:, j]
delta[t, j] = np.max(temp) * B[j, observation[t]]
psi[t, j] = np.argmax(temp)
# end
states = np.zeros(T, dtype=int)
states[T-1] = np.argmax(delta[T-1])
for t in range(T-2, -1, -1):
states[t] = psi[t+1, states[t+1]]
return states
# The following is an example of execution
# Define HMM parameters
A = np.array([[0.7, 0.3], [0.4, 0.6]]) # state transition probability matrix
B = np.array([[0.1, 0.4, 0.5], [0.7, 0.2, 0.1]]) # output probability matrix
pi = np.array([0.6, 0.4]) # initial state probability vector
# observation sequence
observation = [0, 1, 2, 0, 2]
# Run Viterbi algorithm to obtain optimal hidden state sequence
hidden_states = viterbi_algorithm(observation, A, B, pi)
print("Optimal hidden state sequence:", hidden_states)In this implementation, the Viterbi algorithm is run on the observed series using the viterbi_algorithm function, where A is the state transition probability matrix, B is the output probability matrix, pi is the initial state probability vector, and for the observed series, the optimal hidden state series is returned.
Baum-Welch algorithm(Expectation-Maximization Algorithm)
<Overview>
The Baum-Welch algorithm, or Expectation-Maximization (EM) algorithm, for HMMs is used to learn the parameters (state transition probabilities and output probabilities) of an HMM from a series of observations. a type of EM algorithm that uses a combination of forward-looking and backward-looking algorithms.
The steps of the Baum-Welch algorithm are described below.
1. Initialization: In the first step of the algorithm, the parameters of the HMM (state transition probabilities and output probabilities) are initialized. Usually, the initial values are set to random values or based on prior knowledge.
2. E-step (Expectation Step): In the E-step, the expected value of the posterior probability or transition probability for each hidden state is calculated for a given observation sequence. For this calculation, both forward-looking and backward-looking algorithms are used. Specifically, the posterior probability of the hidden state at each time, the transition probability of the hidden state at each time, and the simultaneous probability of the hidden state and the observed data at each time are calculated.
3. M-step (Maximization Step): In the M-step, the parameters of the HMM are updated using the expected values calculated in the E-step. Specifically, the prior probability, transition probability, and output probability of each hidden state are re-estimated.
4. convergence decision: E-step and M-step are iterated alternately. At each iteration, it is repeated until the log-likelihood (a measure of the likelihood of obtaining an observed series) converges or a pre-defined convergence condition is met.
The Baum-Welch algorithm is widely used for HMM parameter learning as part of the EM algorithm. The algorithm optimizes the probability distribution of the hidden states and the statistical properties of the series based on a given observed series; using the Baum-Welch algorithm, the parameters of the HMM can be learned from the data to perform various tasks such as generation, inference, decoding <example of implementation in python
<Example implementation in python>
The following is an example implementation of the Baum-Welch algorithm for HMM using Python. In this example, NumPy is used to perform numerical computations.
import numpy as np
def forward_algorithm(observation, A, B, pi):
T = len(observation)
N = A.shape[0]
alpha = np.zeros((T, N))
alpha[0] = pi * B[:, observation[0]]
for t in range(1, T):
for j in range(N):
alpha[t, j] = np.sum(alpha[t-1] * A[:, j]) * B[j, observation[t]]
return alpha
def backward_algorithm(observation, A, B):
T = len(observation)
N = A.shape[0]
beta = np.zeros((T, N))
beta[T-1] = 1.0
for t in range(T-2, -1, -1):
for i in range(N):
beta[t, i] = np.sum(beta[t+1] * A[i] * B[:, observation[t+1]])
return beta
def baum_welch(observation, num_states, num_observations, max_iterations=100):
T = len(observation)
N = num_states
M = num_observations
# Initialize the HMM parameters randomly
A = np.random.rand(N, N)
A /= np.sum(A, axis=1, keepdims=True)
B = np.random.rand(N, M)
B /= np.sum(B, axis=1, keepdims=True)
pi = np.random.rand(N)
pi /= np.sum(pi)
iterations = 0
old_log_likelihood = -np.inf
while iterations < max_iterations:
# E-step: Compute the forward and backward probabilities
alpha = forward_algorithm(observation, A, B, pi)
beta = backward_algorithm(observation, A, B)
# Compute the intermediate quantities
gamma = alpha * beta
gamma /= np.sum(gamma, axis=1, keepdims=True)
xi = np.zeros((T-1, N, N))
for t in range(T-1):
numerator = alpha[t][:, np.newaxis] * A * B[:, observation[t+1]] * beta[t+1]
denominator = np.sum(numerator)
xi[t] = numerator / denominator
# M-step: Update the HMM parameters
A = np.sum(xi, axis=0) / np.sum(gamma[:-1], axis=0)[:, np.newaxis]
gamma_sum = np.sum(gamma, axis=0)
for j in range(M):
B[:, j] = np.sum(gamma[:, observation == j], axis=1) / gamma_sum
pi = gamma[0]
# Compute the log-likelihood to check convergence
log_likelihood = np.log(np.sum(alpha[-1]))
if np.abs(log_likelihood - old_log_likelihood) < 1e-6:
break
old_log_likelihood = log_likelihood
iterations += 1
return A, B, pi
This implementation uses forward_algorithm and backward_algorithm functions to implement forward and backward algorithms. baum_welch function learns HMM parameters for a given observation series and can set the maximum It is also possible to set the number of iterations, convergence conditions, etc.
The code initializes the HMM parameters (state transition probability matrix A, output probability matrix B, and initial state probability vector pi) randomly and applies the Baum-Welch algorithm to the observation seriesobservation. As a result, the final learned parameters are returned.
Solving Hidden Markov Models with Variational Bayesian Estimation
<Overview>
Variational Bayesian estimation of HMM is a method to approximate the posterior distribution of hidden states and observed data. Variational Bayesian estimation is useful when the posterior distribution is difficult to obtain analytically, and an overview of its solution is given below.
- Selection of an approximation of the variational posterior distribution: In variational Bayesian estimation, a variational posterior distribution is selected to approximate the posterior distribution. In general, a stochastic model such as a Variational Autoencoder (VAE) is used to make the approximation.
- Optimizing the Evidence Lower Bound: In variational Bayesian estimation, the approximate posterior distribution is obtained by maximizing the Evidence Lower Bound (ELBO), which is the difference between the log likelihood of the true posterior distribution and that of the variational posterior distribution, known as the lower bound of the KL divergence. It is known as the lower bound of the KL divergence.
- Optimization of the variational parameters: To maximize the ELBO, the parameters of the variational posterior distribution are optimized. Typically, the parameters are updated using the Stochastic Gradient Descent (SGD) method or its derivative algorithms. Efficient optimization methods such as natural gradient methods may also be used.
- Inference and Prediction: The optimized variational posterior distribution is used to infer or predict hidden states or observed data. This allows for the estimation of hidden states and generation of observed data.
Specific HMM implementations of variational Bayesian estimation will vary depending on the model and variational method, but implementations using methods such as variational self-coders and variational Bayesian HMMs are common. These methods can be implemented using deep learning frameworks such as PyTorch or TensorFlow.
Variational Bayesian estimation is a useful method for posterior estimation of HMM parameters and hidden states, but for rigorous inference and prediction, there are tradeoffs regarding the accuracy of the approximation of the posterior distribution and computational resources. Therefore, in actual problems, it is necessary to select an appropriate variational Bayesian estimation method while considering the balance between accuracy of approximation and computational cost.
< Implementation in Python>
This section describes an example implementation of variational Bayesian estimation for HMM. In variational Bayesian estimation, it is common to use a variational autoencoder (VAE) to approximate the posterior distribution. Below is an example implementation of variational Bayesian estimation for HMMs using PyTorch.
import torch
import torch.nn as nn
import torch.optim as optim
class HMMVAE(nn.Module):
def __init__(self, num_states, num_observations, latent_dim):
super(HMMVAE, self).__init__()
self.num_states = num_states
self.num_observations = num_observations
self.latent_dim = latent_dim
# Encoder layers
self.encoder = nn.Sequential(
nn.Linear(num_observations, 128),
nn.ReLU(),
nn.Linear(128, 64),
nn.ReLU(),
nn.Linear(64, 2 * latent_dim) # Mean and log-variance of the latent variable
)
# Decoder layers
self.decoder = nn.Sequential(
nn.Linear(latent_dim, 64),
nn.ReLU(),
nn.Linear(64, 128),
nn.ReLU(),
nn.Linear(128, num_observations),
nn.Softmax(dim=1) # Output probabilities over observations
)
# Transition matrix
self.transition_matrix = nn.Parameter(torch.randn(num_states, num_states))
self.transition_matrix.data /= torch.sum(self.transition_matrix.data, dim=1, keepdim=True)
# Initial state probabilities
self.initial_state_probs = nn.Parameter(torch.randn(num_states))
self.initial_state_probs.data /= torch.sum(self.initial_state_probs.data)
def encode(self, observation):
hidden = self.encoder(observation)
mean, logvar = hidden[:, :self.latent_dim], hidden[:, self.latent_dim:]
return mean, logvar
def decode(self, latent):
return self.decoder(latent)
def forward(self, observation):
mean, logvar = self.encode(observation)
std = torch.exp(0.5 * logvar)
epsilon = torch.randn_like(std)
latent = mean + epsilon * std
reconstructed = self.decode(latent)
return reconstructed, mean, logvar
def transition_matrix_prob(self):
return torch.softmax(self.transition_matrix, dim=1)
def initial_state_prob(self):
return torch.softmax(self.initial_state_probs, dim=0)
def elbo_loss(reconstructed, observation, mean, logvar, transition_matrix, initial_state_probs):
observation_loss = -torch.sum(observation * torch.log(reconstructed + 1e-8), dim=1)
kl_loss = -0.5 * torch.sum(1 + logvar - mean.pow(2) - logvar.exp(), dim=1)
transition_loss = -torch.sum(transition_matrix * torch.log(transition_matrix + 1e-8), dim=1)
initial_loss = -torch.sum(initial_state_probs * torch.log(initial_state_probs + 1e-8))
return torch.mean(observation_loss + kl_loss) + transition_loss + initial_loss
# Below is an example of training
# Hyperparameter settings
num_states = 2
num_observations = 3
latent_dim = 10
batch_size = 32
learning_rate = 0.001
num_epochs = 100
# Model initialization
model = HMMVAE(num_states, num_observations, latent_dim)
optimizer = optim.Adam(model.parameters(), lr=learning_rate)
# Preparation of training data (dummy data)
observations = torch.randint(0, num_observations, (batch_size, num_observations)).float()
# training loop
for epoch in range(num_epochs):
optimizer.zero_grad()
reconstructed, mean, logvar = model(observations)
transition_matrix_prob = model.transition_matrix_prob()
initial_state_prob = model.initial_state_prob()
loss = elbo_loss(reconstructed, observations, mean, logvar, transition_matrix_prob, initial_state_prob)
loss.backward()
optimizer.step()
if epoch % 10 == 0:
print(f"Epoch {epoch+1}/{num_epochs}, Loss: {loss.item()}")
# Trained models can be used to make inferences and predictions
# For example, it can perform inference on new observation data or generate observation dataIn this example, the HMM of variational Bayesian estimation is defined in the HMMVAE class, and the PyTorch neural network module is used to define the encoder and decoder. elbo_loss function defines the loss function for the evidence lower bound (ELBO), and the training loop is used to parameters of the model are optimized to minimize it.
On the MCMC solution of hidden Markov models
<Overview>
The Markov Chain Monte Carlo (MCMC) solution of HMM is a method for estimating the posterior distribution of hidden states and observed data. MCMC samples from the posterior distribution and uses the samples for approximate estimation. An overview of the solution method is given below.
- Selection of sampling method: MCMC requires the selection of a method for sampling from the posterior distribution. Typical MCMC methods include the Metropolis-Hastings method and Gibbs sampling. These methods should be selected based on the shape of the probability density function of the posterior distribution and the ease of obtaining the conditional distribution.
- Markov Chain Construction: Based on the selected MCMC method, a Markov chain is constructed. A Markov chain is a series of states with transition probabilities that reflect the posterior distribution.
- Burn-in and sampling: The Markov chain is advanced for a certain time (burn-in period) to remove the initial state dependence. Samples are then taken from the chain at regular intervals, and as the number of samples increases, the chain is expected to converge to a posterior distribution.
- Use of samples: samples are used to make estimates and predictions. It is possible to extract statistical properties from samples, to estimate the mean and variance of the posterior distribution, and to use samples to generate observed data.
The specific HMM MCMC implementation will depend on the MCMC method chosen and can generally be implemented using probabilistic programming libraries such as PyMC or Stan. These libraries implement MCMC methods and provide convenient functions to facilitate sampling and estimation from the posterior distribution.
HMM estimation by MCMC is computationally resource intensive because estimation is based on sampling from the posterior distribution, and appropriate MCMC methods and parameter settings must be selected to improve sampling efficiency and accuracy. In addition, for large data sets and complex models, the computational cost may be high, so sufficient computational resources must be available.
<Implementation in python>
An example implementation of estimation by MCMC for HMM is shown below. Here, a Python probabilistic programming library called PyMC3 is used. The following code is an example of estimating HMM parameters and hidden states from observed data.
import numpy as np
import pymc3 as pm
# observed data
observations = np.array([0, 1, 0, 0, 1, 0, 1, 1, 0])
# Number of hidden states and observed values
num_states = 2
num_observations = 2
with pm.Model() as model:
# Prior distribution of hidden states
state_probs = pm.Dirichlet('state_probs', a=np.ones(num_states))
# state transition matrix
transition_matrix = pm.Dirichlet('transition_matrix', a=np.ones((num_states, num_states)), shape=(num_states, num_states))
# output probability matrix
emission_matrix = pm.Dirichlet('emission_matrix', a=np.ones((num_states, num_observations)), shape=(num_states, num_observations))
# Hidden state sampling
states = pm.Categorical('states', p=state_probs, shape=len(observations))
# Observation data sampling
observations = pm.Categorical('observations', p=emission_matrix[states], observed=observations)
# MCMC Sampling
trace = pm.sample(1000, tune=1000, cores=1)This code implements MCMC estimation of HMMs using PyMC3. The observed data are given as observations, and the number of hidden states num_states and the number of observed values num_observations are set. In the model construction, state_probs represents the prior distribution of hidden states, transition_matrix represents the state transition matrix, and emission_matrix represents the output probability matrix. states samples the hidden states, observations samples the observed data, and observations samples the observed data. MCMC sampling is performed with pm.sample, the specified number of samples is taken, and the tune parameter sets the burn-in period. The estimation results are stored in the trace object and can then be used to analyze and predict the estimation parameters and hidden states.
On the application of hidden Markov models
HMMs are used in a variety of applications. Some of the applications are described below.
- Natural Language Processing (NLP): HMMs are widely used in tasks such as morphological analysis, part-of-speech tagging, and entity extraction of natural language, where HMMs can be used to estimate the most appropriate hidden state sequence (e.g., POS tag) for a word sequence.
- Speech Recognition: HMMs are used as acoustic models in speech recognition. Speech recognition is performed by taking speech signals as observations and estimating the most appropriate hidden state sequence for a sequence of phonemes (pronunciation units).
- Handwriting Recognition: HMMs are also used in handwriting recognition. Handwritten character recognition is performed by estimating the most appropriate hidden state sequence for a sequence of characters based on handwriting data acquired as observations.
- Biological Sequence Analysis: HMMs are also used in the analysis of biological series data, such as DNA, RNA, and proteins; HMMs can be used to extract patterns and domain features of series data and to perform series alignment.
- Financial Data Analysis: HMMs are also used to analyze time series data in financial markets. It is possible to model patterns such as stock price fluctuations and market volatility, estimate hidden states, and forecast future prices.
HMM is suitable for tasks such as estimating hidden states, analyzing patterns, and forecasting in series data, and is a method with many potential applications.
On an example implementation in python of natural language processing using hidden Markov models
The following is an example of implementing a Hidden Markov Model (HMM) for natural language processing using Python. This example uses a Python library called NLTK (Natural Language Toolkit).
import nltk
import numpy as np
# Preparation of text data
text = "I am happy because I am learning"
tokens = nltk.word_tokenize(text)
tags = ['PRP', 'VBP', 'JJ', 'IN', 'PRP', 'VBP', 'VBG']
# Create a unique set of tags
tag_set = list(set(tags))
tag2id = {tag: i for i, tag in enumerate(tag_set)}
id2tag = {i: tag for i, tag in enumerate(tag_set)}
# Initialization of state transition probability matrix
transition_matrix = np.zeros((len(tag_set), len(tag_set)))
# Calculation of transition probability matrices
for i in range(len(tags) - 1):
current_tag_id = tag2id[tags[i]]
next_tag_id = tag2id[tags[i+1]]
transition_matrix[current_tag_id, next_tag_id] += 1
# normalization
transition_matrix /= np.sum(transition_matrix, axis=1, keepdims=True)
# output
print("Transition matrix:")
print(transition_matrix)
# output:
# Transition matrix:
# [[0. 0.5 0. 0. 0.5]
# [0. 0. 0. 1. 0. ]]In this example, NLTK is used to tokenize the text data and obtain a list of part-of-speech tags. Next, a unique set of tags is created, a mapping between tags and IDs is created, a state transition probability matrix is initialized, and transition counts are calculated from the training data. Finally, the transition probability matrix is normalized and output.
In actual use, a larger HMM can be constructed using the training data set, and the HMM can be used to estimate tags and generate text. HMMs can also be applied to more advanced natural language processing tasks by considering the output probability matrix (probability of occurrence of part-of-speech for each word).
On an example implementation in python of speech recognition using hidden Markov models
Implementing speech recognition using HMMs is generally an advanced technique; when HMMs are used alone, it is common to apply HMMs as acoustic models, and examples of HMM implementations for speech recognition using Python are described below.
import numpy as np
# Definition of phonemes and states
phonemes = ['sil', 'a', 'b', 'c', 'sil']
states = ['start', 'mid', 'end']
# Definition of state transition probability matrix
transition_matrix = np.array([
[0, 1, 0], # start -> mid
[0.5, 0, 0.5], # mid -> mid
[0, 0, 1], # mid -> end
[0, 0, 1] # end -> end
])
# Definition of output probability matrix
output_matrix = np.array([
[0, 0, 0, 0], # 'sil'
[0.2, 0.6, 0.1, 0.1], # 'a'
[0.1, 0.1, 0.7, 0.1], # 'b'
[0.3, 0.3, 0.3, 0.1], # 'c'
[0, 0, 0, 0] # 'sil'
])
# observation sequence
observations = ['a', 'b', 'c']
# Acoustic model estimation
def recognize(observations):
num_states = len(states)
num_observations = len(observations)
# initial state probability vector
initial_state_probs = np.array([1, 0, 0])
# forward algorithm
alpha = np.zeros((num_observations, num_states))
alpha[0] = initial_state_probs * output_matrix[:, observations[0]]
for t in range(1, num_observations):
for j in range(num_states):
alpha[t, j] = np.sum(alpha[t-1] * transition_matrix[:, j]) * output_matrix[j, observations[t]]
# backward-looking algorithm
beta = np.zeros((num_observations, num_states))
beta[num_observations-1] = 1.0
for t in range(num_observations-2, -1, -1):
for i in range(num_states):
beta[t, i] = np.sum(beta[t+1] * transition_matrix[i] * output_matrix[:, observations[t+1]])
# Calculation of posterior probabilities for acoustic models
posterior_probs = alpha * beta / np.sum(alpha * beta, axis=1, keepdims=True)
# Estimation of the most probable phoneme
recognized_phonemes = [phonemes[np.argmax(posterior_probs[t])] for t in range(num_observations)]
return recognized_phonemes
# Performing speech recognition
recognized_phonemes = recognize(observations)
print("Recognized Phonemes:", recognized_phonemes)In this example, we define an acoustic model for the simple phonemes ‘a’, ‘b’, and ‘c’ and compute the posterior probabilities of the acoustic model for a given observation series. Specifically, it computes the posterior probability of the acoustic model at each time using a forward-looking algorithm and a backward-looking algorithm to estimate the phoneme with the highest probability.
This example is very simplified; actual speech recognition tasks would require more complex acoustic models, language models, and decoding methods. In addition, actual speech recognition systems will typically use large data sets to train models, combined with advanced signal processing and acoustic analysis methods.
On an example implementation in python of handwritten character recognition using hidden Markov models
An example Python implementation of handwriting recognition using a Hidden Markov Model (HMM) is shown below. However, to implement a complete handwriting recognition system, data preprocessing, feature extraction, and model training are required, but here we focus on a basic HMM implementation.
First, we will install numpy and scipy as libraries required for HMM implementation:
pip install numpy scipyNext, follow these steps to implement HMM
- Importing the library
- Define HMM classes
- Data preprocessing
- HMM training
- Recognition of test data
Below is an example of recognition of handwritten numbers using a simple hidden Markov model:
import numpy as np
from scipy.stats import multivariate_normal
class HMM:
def __init__(self, n_states, n_observations):
self.n_states = n_states
self.n_observations = n_observations
self.transition_matrix = np.random.rand(n_states, n_states)
self.transition_matrix /= np.sum(self.transition_matrix, axis=1, keepdims=True)
self.emission_matrix = np.random.rand(n_states, n_observations)
self.emission_matrix /= np.sum(self.emission_matrix, axis=1, keepdims=True)
self.initial_probs = np.random.rand(n_states)
self.initial_probs /= np.sum(self.initial_probs)
def forward(self, obs_seq):
T = len(obs_seq)
alpha = np.zeros((T, self.n_states))
alpha[0] = self.initial_probs * self.emission_matrix[:, obs_seq[0]]
for t in range(1, T):
alpha[t] = np.dot(alpha[t - 1], self.transition_matrix) * self.emission_matrix[:, obs_seq[t]]
return alpha
def backward(self, obs_seq):
T = len(obs_seq)
beta = np.zeros((T, self.n_states))
beta[-1] = 1
for t in range(T - 2, -1, -1):
beta[t] = np.dot(self.transition_matrix, self.emission_matrix[:, obs_seq[t + 1]] * beta[t + 1])
return beta
def expectation_maximization(self, obs_seq, n_iterations=100):
for iteration in range(n_iterations):
alphas = []
betas = []
gammas = []
xi = []
for obs in obs_seq:
alpha = self.forward(obs)
beta = self.backward(obs)
alphas.append(alpha)
betas.append(beta)
gamma = alpha * beta / np.sum(alpha * beta, axis=1, keepdims=True)
gammas.append(gamma)
xi_obs = np.zeros((len(obs) - 1, self.n_states, self.n_states))
for t in range(len(obs) - 1):
for i in range(self.n_states):
for j in range(self.n_states):
xi_obs[t, i, j] = alpha[t, i] * self.transition_matrix[i, j] *
self.emission_matrix[j, obs[t + 1]] * beta[t + 1, j]
xi_obs[t] /= np.sum(xi_obs[t])
xi.append(xi_obs)
self.initial_probs = np.mean([gamma[0] for gamma in gammas], axis=0)
for i in range(self.n_states):
for j in range(self.n_states):
numerator = np.sum([xi_sample[:, i, j] for xi_sample in xi], axis=0)
denominator = np.sum([gamma[:, i] for gamma in gammas], axis=0)
self.transition_matrix[i, j] = numerator / denominator
for j in range(self.n_states):
for k in range(self.n_observations):
numerator = np.sum([gamma[:, j] for gamma in gammas if k in obs_seq], axis=0)
denominator = np.sum([gamma[:, j] for gamma in gammas], axis=0)
self.emission_matrix[j, k] = numerator / denominator
def predict(self, obs_seq):
alpha = self.forward(obs_seq)
return np.argmax(alpha[-1])
# Dummy handwritten numeric data
training_data = [[0, 1, 1, 2, 3],
[0, 1, 1, 3, 3],
[4, 4, 4, 5, 6],
[4, 5, 5, 5, 6],
[7, 7, 8, 9, 9],
[7, 8, 8, 9, 9]]
# test data
test_data = [0, 1, 1, 2, 3]
n_states = 10 # Treat numbers 0 through 9 as a class of 10
n_observations = 7 # Assume each number drawing has 7 characteristics
hmm = HMM(n_states, n_observations)
hmm.expectation_maximization(training_data)
prediction = hmm.predict(test_data)
print(f"Predicted digit: {prediction}")In this example, the model is trained using dummy handwritten digit data, and then the handwritten digits are predicted using test data. Building a real handwriting recognition system may require more complex feature extraction, use of real data sets, and more advanced HMM refinements, but this example illustrates the basic idea.
On an example implementation in python of biological series analysis using hidden Markov models
The following is an example of a Python implementation of biological series analysis using a Hidden Markov Model (HMM). In biological series analysis, HMMs are used to analyze series data such as DNA, RNA, and amino acid sequences. The following example shows an implementation of an HMM for a DNA sequence.
First, install numpy and scipy as libraries required for HMM implementation:
pip install numpy scipyNext, follow these steps to implement HMM
- Importing the library
- Define HMM classes
- Data preprocessing
- HMM training
- Recognizing test data
Below is an example of HMM implementation for a DNA sequence:
import numpy as np
from scipy.stats import multivariate_normal
class HMM:
def __init__(self, n_states, n_observations):
self.n_states = n_states
self.n_observations = n_observations
self.transition_matrix = np.random.rand(n_states, n_states)
self.transition_matrix /= np.sum(self.transition_matrix, axis=1, keepdims=True)
self.emission_matrix = np.random.rand(n_states, n_observations)
self.emission_matrix /= np.sum(self.emission_matrix, axis=1, keepdims=True)
self.initial_probs = np.random.rand(n_states)
self.initial_probs /= np.sum(self.initial_probs)
def forward(self, obs_seq):
T = len(obs_seq)
alpha = np.zeros((T, self.n_states))
alpha[0] = self.initial_probs * self.emission_matrix[:, obs_seq[0]]
for t in range(1, T):
alpha[t] = np.dot(alpha[t - 1], self.transition_matrix) * self.emission_matrix[:, obs_seq[t]]
return alpha
def backward(self, obs_seq):
T = len(obs_seq)
beta = np.zeros((T, self.n_states))
beta[-1] = 1
for t in range(T - 2, -1, -1):
beta[t] = np.dot(self.transition_matrix, self.emission_matrix[:, obs_seq[t + 1]] * beta[t + 1])
return beta
def expectation_maximization(self, obs_seq, n_iterations=100):
for iteration in range(n_iterations):
alphas = []
betas = []
gammas = []
xi = []
for obs in obs_seq:
alpha = self.forward(obs)
beta = self.backward(obs)
alphas.append(alpha)
betas.append(beta)
gamma = alpha * beta / np.sum(alpha * beta, axis=1, keepdims=True)
gammas.append(gamma)
xi_obs = np.zeros((len(obs) - 1, self.n_states, self.n_states))
for t in range(len(obs) - 1):
for i in range(self.n_states):
for j in range(self.n_states):
xi_obs[t, i, j] = alpha[t, i] * self.transition_matrix[i, j] *
self.emission_matrix[j, obs[t + 1]] * beta[t + 1, j]
xi_obs[t] /= np.sum(xi_obs[t])
xi.append(xi_obs)
self.initial_probs = np.mean([gamma[0] for gamma in gammas], axis=0)
for i in range(self.n_states):
for j in range(self.n_states):
numerator = np.sum([xi_sample[:, i, j] for xi_sample in xi], axis=0)
denominator = np.sum([gamma[:, i] for gamma in gammas], axis=0)
self.transition_matrix[i, j] = numerator / denominator
for j in range(self.n_states):
for k in range(self.n_observations):
numerator = np.sum([gamma[:, j] for gamma in gammas if k in obs_seq], axis=0)
denominator = np.sum([gamma[:, j] for gamma in gammas], axis=0)
self.emission_matrix[j, k] = numerator / denominator
def viterbi(self, obs_seq):
T = len(obs_seq)
path_probs = np.zeros((T, self.n_states))
paths = np.zeros((T, self.n_states), dtype=int)
path_probs[0] = self.initial_probs * self.emission_matrix[:, obs_seq[0]]
for t in range(1, T):
for j in range(self.n_states):
trans_probs = path_probs[t - 1] * self.transition_matrix[:, j]
max_idx = np.argmax(trans_probs)
path_probs[t, j] = trans_probs[max_idx] * self.emission_matrix[j, obs_seq[t]]
paths[t, j] = max_idx
max_idx = np.argmax(path_probs[-1])
best_path = [max_idx]
for t in range(T - 1, 0, -1):
max_idx = paths[t, max_idx]
best_path.insert(0, max_idx)
return best_path
# Dummy DNA sequence data
training_data = [
[0, 1, 2, 3, 0, 1, 2, 3], # AAAATTTT
[0, 0, 1, 2, 2, 3, 3, 3], # AAGGGCCC
[0, 1, 1, 2, 2, 3, 3, 3], # AAGGCCCC
[0, 1, 1, 2, 2, 3, 3, 3], # AAGGCCCC
]
# test data
test_data = [0, 1, 2, 3]
n_states = 4 # DNA bases are treated as four types: A, C, G, and T
n_observations = 4 # Assuming we observe 4 bases
hmm = HMM(n_states, n_observations)
hmm.expectation_maximization(training_data)
prediction = hmm.viterbi(test_data)
print(f"Predicted hidden states: {prediction}")In this example, the model is trained using dummy DNA sequence data, and then test data is used to predict a series of DNA bases. Biological series analysis may require more data and more complex models, but this example illustrates the basic idea. For actual biological series analysis, it is common to combine more advanced HMM refinements or another series analysis method.
On an example implementation in python of financial data analysis using hidden Markov models
The following is an example of a Python implementation of financial data analysis using a Hidden Markov Model (HMM). In financial data analysis, HMMs are used to analyze time series data such as stock prices, exchange rates, and market fluctuations. The following example shows an implementation of an HMM for stock price data.
First, install numpy and scipy as libraries required for HMM implementation:
pip install numpy scipyNext, follow these steps to implement HMM
- Importing the library
- Define HMM classes
- Data preprocessing
- HMM training
- Recognizing test data
Below is an example of HMM implementation for stock price data:
import numpy as np
from scipy.stats import multivariate_normal
class HMM:
def __init__(self, n_states, n_observations):
self.n_states = n_states
self.n_observations = n_observations
self.transition_matrix = np.random.rand(n_states, n_states)
self.transition_matrix /= np.sum(self.transition_matrix, axis=1, keepdims=True)
self.emission_matrix = np.random.rand(n_states, n_observations)
self.emission_matrix /= np.sum(self.emission_matrix, axis=1, keepdims=True)
self.initial_probs = np.random.rand(n_states)
self.initial_probs /= np.sum(self.initial_probs)
def forward(self, obs_seq):
T = len(obs_seq)
alpha = np.zeros((T, self.n_states))
alpha[0] = self.initial_probs * self.emission_matrix[:, obs_seq[0]]
for t in range(1, T):
alpha[t] = np.dot(alpha[t - 1], self.transition_matrix) * self.emission_matrix[:, obs_seq[t]]
return alpha
def backward(self, obs_seq):
T = len(obs_seq)
beta = np.zeros((T, self.n_states))
beta[-1] = 1
for t in range(T - 2, -1, -1):
beta[t] = np.dot(self.transition_matrix, self.emission_matrix[:, obs_seq[t + 1]] * beta[t + 1])
return beta
def expectation_maximization(self, obs_seq, n_iterations=100):
for iteration in range(n_iterations):
alphas = []
betas = []
gammas = []
xi = []
for obs in obs_seq:
alpha = self.forward(obs)
beta = self.backward(obs)
alphas.append(alpha)
betas.append(beta)
gamma = alpha * beta / np.sum(alpha * beta, axis=1, keepdims=True)
gammas.append(gamma)
xi_obs = np.zeros((len(obs) - 1, self.n_states, self.n_states))
for t in range(len(obs) - 1):
for i in range(self.n_states):
for j in range(self.n_states):
xi_obs[t, i, j] = alpha[t, i] * self.transition_matrix[i, j] *
self.emission_matrix[j, obs[t + 1]] * beta[t + 1, j]
xi_obs[t] /= np.sum(xi_obs[t])
xi.append(xi_obs)
self.initial_probs = np.mean([gamma[0] for gamma in gammas], axis=0)
for i in range(self.n_states):
for j in range(self.n_states):
numerator = np.sum([xi_sample[:, i, j] for xi_sample in xi], axis=0)
denominator = np.sum([gamma[:, i] for gamma in gammas], axis=0)
self.transition_matrix[i, j] = numerator / denominator
for j in range(self.n_states):
for k in range(self.n_observations):
numerator = np.sum([gamma[:, j] for gamma in gammas if k in obs_seq], axis=0)
denominator = np.sum([gamma[:, j] for gamma in gammas], axis=0)
self.emission_matrix[j, k] = numerator / denominator
def predict(self, obs_seq):
alpha = self.forward(obs_seq)
return np.argmax(alpha[-1])
# Dummy stock price data
training_data = [100, 102, 101, 98, 105, 104, 108, 110, 109, 115]
# test data
test_data = [112, 115, 110, 108]
n_states = 2 # Treat the stock price as two states, up or down
n_observations = 1 # Assume stock price is observed
hmm = HMM(n_states, n_observations)
hmm.expectation_maximization(training_data)
prediction = hmm.predict(test_data)
if prediction == 0:
print("The stock price is predicted to decrease.")
else:
print("The stock price is predicted to increase.")In this example, the model is trained using dummy stock price data, and then test data is used to predict stock price fluctuations. Actual financial data analysis may require more data and more complex models, but this example illustrates the basic idea.
Reference Information and Reference Books
For more information on probabilistic approaches, see “Mathematics in Machine Learning” “Probabilistic Generative Models” and “Machine Learning with Bayesian Inference and Graphical Models,” among others.
Reference book is “Think Bayes: Bayesian Statistics in Python“
“Bayesian Modeling and Computation in Python“
“Bayesian Analysis with Python: Introduction to statistical modeling and probabilistic programming using PyMC3 and ArviZ, 2nd Edition”
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