Widely used in fields like speech recognition and bioinformatics, HMM predicts sequences by considering the probabilities of transitioning between hidden states. In this blog, we will explore Markov chains, the workings of Hidden Markov Models, and the practical implementation of HMM. Let’s get started.
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What are Markov Chains?
To understand Hidden Markov Models, it is essential to first familiarize oneself with Markov Chains, as they serve as the building blocks for Hidden Markov Models.
Markov chains are mathematical models of a series of events where the possibility of going to any given state depends only on the state at that point in time and the state that has been reached. This feature represents the Markov property.
A set of states and the probabilities of changing between them make up a Markov chain. A matrix known as the transition matrix is frequently used to depict these transitions. Given the current state, the Markov property suggests that the system’s future behavior will be independent of its past behavior.
Applications of Markov chains can be found in computer science, biology, economics, and physics, among other disciplines. They are employed in the modeling of systems that display stochastic (random) behavior over time, including molecular motion, financial transactions, or random walks. Markov chains are useful for forecasting a system’s future state probability distribution given its present state and transition probabilities.
The statistical Hidden Markov Model (HMM) is employed in machine learning for modeling systems that consist of a series of observable events, but the underlying state of the system is not readily visible (‘hidden’). The model suggests the existence of an underlying Markov process with hidden states, each of which has a probability associated with its production of observable symbols.
Three different sets of parameters define an HMM:
- State Transition Probabilities (A): The probabilities of changing from one hidden state to another are shown by the state transition probabilities (A). The probability of transitioning to a particular state depends only on the current hidden state and not on the sequence of events that preceded it.
- Emission Probabilities (B): These describe the probabilities of observing certain symbols (emissions) given the current hidden state. A probability is linked to each hidden state to emit distinct observable symbols.
- Initial State Probabilities (π): The probability of beginning in each hidden state is represented by the initial state probabilities (π). It gives the first hidden state’s probability distribution definition.
Speech recognition, natural language processing, bioinformatics (for gene prediction), and finance (for stock price modeling) are just a few of the fields in which HMMs find usage. HMMs are frequently trained using the Baum-Welch algorithm and decoded using the Viterbi technique.
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The Hidden Markov Model (HMM) is a probabilistic framework used to model systems that produce observable symbols from hidden states. According to the model, there is an underlying Markov process that has a number of hidden states and that, with a given probability, emits observable symbols in each state. An HMM’s Algorithm can be implemented in detail as follows:
- Specify the group of hidden states
- Provide the initial probability of each hidden condition
- Describe the collection of observable symbols
- Indicate the probability of each hidden state emitting a specific observable sign
- State Transitions
Give an explanation of the probabilities of changing from one hidden state to another. The state transition matrix captures this.
A series of observable symbols is produced by considering the probability of emission and hidden states. In accordance with its emission probabilities, every hidden state emits an observable symbol.
- Training (Learning)
Modify the model’s parameters (transition, emission, and initial state probabilities) according to the implications of the observed data if it hasn’t already been trained. For this, the expectation-maximization Baum-Welch method is frequently employed.
The Viterbi method is frequently used to determine the most likely sequence of hidden states that might have produced the observed symbols, given a sequence of observable symbols.
Observable symbols or hidden states can be predicted by HMMs based on observed historical data and the current state.
When the underlying system is thought to have a Markovian property—that is, when the future state depends simply on the current state and is unaffected by the preceding series of events—the model becomes particularly helpful.
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Implementing a Hidden Markov Model (HMM) from scratch can be complex due to the various mathematical computations involved. However, here is a basic example using the hmmlearn library in Python. Before running this code, make sure to install the library by running:
pip install hmmlearn
Now, here’s a simple example of a 2-state HMM with discrete observations:
from hmmlearn import hmm
import numpy as np
# Define the HMM model
model = hmm.MultinomialHMM(n_components=2, n_iter=100)
# Training data (sequence of observations)
X = np.array([[0, 1, 0, 1, 0, 0, 1, 1, 1, 0]]).reshape(-1, 1)
# Fit the model to the training data
# Predict the hidden states for the training data
hidden_states = model.predict(X)
# Print the model parameters
print("\nInitial state probabilities:")
print("\nPredicted hidden states:")
The model is trained on a series of observations (0s and 1s) in this example. The model parameters (transition matrix, emission matrix, initial state probabilities) and the predicted hidden states are printed. This will give us the output:
In this Python code, a Hidden Markov Model (HMM) is implemented using the `hmmlearn` library. The HMM is trained on a sequence of observations denoted by the variable `X`, which represents a binary sequence `[0, 1, 0, 1, 0, 0, 1, 1, 1, 0]`.
The model is configured with two hidden states (`n_components=2`) and iterates 100 times during training (`n_iter=100`). After training, the code predicts the hidden states for the input sequence using the `predict` method. The output includes the transition matrix, emission matrix, initial state probabilities, and the predicted hidden states.
The transition matrix (`transmat_`) shows the probabilities of transitioning between hidden states; the emission matrix (`emissionprob_`) represents the probabilities of observing certain outcomes in each state; and the initial state probabilities (`startprob_`) indicate the likelihood of starting in each hidden state.
The predicted hidden states are printed as a result of the model’s inference from the input sequence.
The ability of Hidden Markov Models (HMMs) to simulate systems with hidden states and observable outcomes makes them useful in a variety of applications. These are a few notable applications:
- Speech Recognition: HMMs are used to recognize spoken words by modeling the sequence of acoustic features extracted from the speech signal. The hidden states of the HMM represent the different sound units in the language, while the observations are the acoustic features. The transition probabilities between hidden states represent the likelihood of one phoneme transitioning to another, and the emission probabilities represent the likelihood of observing a particular acoustic feature given a hidden state.
- Finance: HMMs are applied in finance for modeling stock price movements and identifying different market states. The hidden states can represent market conditions (bull, bear, or stagnant), and the observed data is the historical stock prices.
- Robotics: HMMs are used in robotics to perform tasks such as mapping and localizing robots. The visible data consists of sensor measurements, while the hidden states can represent various locations or features on a map.
- Natural Language Processing (NLP): Hidden Markov Models play a crucial role in natural language processing (NLP), where they are applied to tasks like part-of-speech tagging, parsing, and machine translation. In part-of-speech tagging, HMMs help assign specific parts of speech to each word within a sentence. For parsing, these models are utilized to build a tree structure that represents the grammatical arrangement of a sentence. Additionally, in the world of machine translation, HMMs are employed to facilitate the translation of sentences from one language to another.
- Gesture Recognition: In computer vision, hand gestures are identified by the use of HMMs in video clips. The sequence of picture frames represents the observed data, and the hidden states correlate to various gesture types.
- Bioinformatics: HMMs are used in bioinformatics tasks such as gene prediction, protein structure prediction, and DNA sequence analysis. In gene prediction, HMMs are used to identify the location of genes in a DNA sequence. In protein structure prediction, HMMs are used to predict the secondary structure of a protein from its amino acid sequence. In DNA sequence analysis, HMMs are used to identify patterns in DNA sequences that may indicate the presence of genes or other regulatory elements.
- Healthcare: HMMs can be used in the field of healthcare to perform tasks such as modeling the course of diseases, interpreting medical pictures, and forecasting patient outcomes based on time-series data.
Hidden Markov Models (HMMs) serve as powerful tools across a diverse range of applications, showcasing their versatility in modeling sequential data with hidden states, particularly within the field of machine learning. HMMs offer a strong foundation for comprehending and forecasting sequential patterns, which is useful for understanding spoken language in voice recognition, interpreting grammatical structures in natural language processing, or negotiating the complex dynamics of financial markets. As we continue to explore and advance in various domains, HMMs stand as reliable and adaptable instruments, contributing significantly to the understanding and manipulation of sequential data in a myriad of real-world scenarios, making them indispensable in the field of machine learning.
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