Last Updated : 06 Jun, 2024
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The Viterbi algorithm is a dynamic programming algorithm for finding the most likely sequence of hidden states in a Hidden Markov Model (HMM). It is widely used in various applications such as speech recognition, bioinformatics, and natural language processing. This article delves into the fundamentals of the Viterbi algorithm, its applications, and a step-by-step guide to its implementation.
Table of Content
- Understanding Hidden Markov Models (HMMs)
- The Viterbi Algorithm
- Initialization in Viterbi Algorithm
- The Forward Algorithm
- The Backward Algorithm
- Decoding with Viterbi Algorithm
- Optimizing Viterbi Algorithm
- Example: Viterbi Algorithm in Action
- Applications of Viterbi in HMM
- Conclusion
A Hidden Markov Model (HMM) is a statistical model that represents systems with hidden states and observable events. It consists of:
- States: Hidden conditions that are not directly observable.
- Observations: Observable events influenced by the hidden states.
- Transition Probabilities: Probabilities of moving from one state to another.
- Emission Probabilities: Probabilities of observing a particular event given a state.
- Initial State Probabilities: Probabilities of the system starting in a particular state.
Understanding HMM Parameters
To fully understand the Viterbi algorithm, it is essential to grasp the parameters of an HMM:
- States (π): The set of hidden states in the model, denoted as [Tex]π ={π _1, π _2, β¦, π _N}[/Tex].
- Observations (π): The sequence of observed events, denoted as [Tex]π={π_1, π_2, β¦, π_T}[/Tex].
- Transition Probabilities (π΄): The probability of transitioning from one state to another, denoted as [Tex]π΄ = {π_{ij}}[/Tex] where [Tex]π_{ij} = π(π _j β£ π _i)[/Tex] .
- Emission Probabilities (π΅): The probability of observing a particular event from a state, denoted as [Tex]π΅ = {π_j(π_π‘)}[/Tex] where [Tex]π_j(π_t) = π(π_t β£ π _j)[/Tex].
- Initial Probabilities (π): The probability of starting in a particular state, denoted as [Tex]\pi = \{ \pi_i \}[/Tex]where [Tex]\pi_i = P(s_i \text{ at } t=1)[/Tex].
The Viterbi Algorithm
The Viterbi algorithm is a fundamental dynamic programming technique widely used in the context of Hidden Markov Models (HMMs) to uncover the most likely sequence of hidden states given a sequence of observed events.
Given a series of observed events, the Viterbi algorithm determines the most likely order of hidden states in an HMM. Andrew Viterbi, who first proposed the algorithm in 1967, is honored by the algorithmβs name. The most likely sequence of states is computed efficiently by the Viterbi method, which divides the issue into smaller subproblems and solves them recursively.
The Viterbi algorithm operates by iteratively computing the highest probability paths to each state at each time step, storing these probabilities, and backtracking to determine the most probable sequence of hidden states. This method ensures an efficient and accurate decoding of hidden state sequences, making it indispensable for applications that require precise sequence analysis and pattern recognition.
Initialization in Viterbi Algorithm
Initialization is the first step of the Viterbi algorithm. It sets up the initial probabilities for the starting states based on the initial state probabilities and the emission probabilities for the first observation.
Mathematically it can be represented as:
[Tex]V_1 (j) = \pi_j. b_j(o_1) \forall j \epsilon \{1, β¦,N\} [/Tex]
[Tex]\text{Path}_j(1) = [j] \forall j \epsilon \{ 1, β¦, N \}[/Tex]
The Forward Algorithm
The Forward Algorithm is used to compute the probability of observing a sequence of observations given an HMM. It is a dynamic programming algorithm that recursively calculates the probabilities of partial observation sequences.
Step 1: Initialization
[Tex]\alpha_1(j) = \pi_j. b_j (o_1)[/Tex]
Step 2: Recursion
[Tex]\alpha_t (j) = \sum_{i=1}^{N} \alpha_{t-1}(i).a_{ij}.b_j(o_t)[/Tex]
Step 3: Termination
[Tex]P(Oβ£Ξ»)= \sum_{j=1}^{N} \alpha_T (j)[/Tex]
The Backward Algorithm
The Backward Algorithm complements the Forward Algorithm by computing the probability of the ending part of the observation sequence, starting from a given state.
Step 1: Initialization
[Tex]\beta_T (j) = 1[/Tex]
Step 2: Recursion
[Tex]Ξ²_t (i)= \sum_{j=1}^{N} \beta_{t+1}(j).a_{ij}.b_j (o_{t+1})[/Tex]
Step 3: Termination
[Tex]P(O|\lambda) = \sum_{j=1}^{N} \pi_j. b_j(o_1). \beta_1(j)[/Tex]
Decoding with Viterbi Algorithm
Decoding in the context of HMMs refers to determining the most likely sequence of hidden states given an observation sequence. The Viterbi algorithm achieves this by maximizing the probability of the hidden state sequence.
Recursion
[Tex]V_t (j) = \max_i [V_{t-1}(i).a_{ij}. b_j (o_t)] \forall j \epsilon \{ 1,β¦, N\}[/Tex]
[Tex]\text{Path}_j(t) = [\argmax_i [V_{t-1}(i). a_{ij}], j][/Tex]
Termination Step
[Tex]P* = \max_j V_T (j)[/Tex]
[Tex]\text{Best Path} = \argmax_j V_T(j)[/Tex]
Optimizing Viterbi Algorithm
To optimize the Viterbi algorithm, consider the following:
- Pruning: Reducing the state space by eliminating states with very low probabilities.
- Logarithms: Using log probabilities to avoid underflow issues.
- Beam Search: Keeping track of only the top kkk most likely paths at each step to reduce computational complexity.
Example: Viterbi Algorithm in Action
Consider an HMM with two states (Rainy, Sunny) and three observations (Walk, Shop, Clean). The following matrices define the model:
- States: S={Rainy,Sunny}
- Observations: O={Walk,Shop,Clean}
- Transition Probabilities (A): [Tex]A= \begin{pmatrix} 0.7 && 0.3 \\ 0.4 && 0.6\end{pmatrix}[/Tex]
- Emission Probabilities (B): [Tex]B = \begin{pmatrix} 0.1 && 0.4 && 0.5 \\0.6 && 0.3 && 0.1\end{pmatrix}[/Tex]
- Initial Probabilities [Tex](\pi)[/Tex]: [Tex]\pi = \begin{pmatrix} 0.6 && 0.4 \end{pmatrix} [/Tex]
Given the observation sequence (Walk, Shop, Clean), the Viterbi algorithm computes the most probable state sequence.
import numpy as npdef viterbi(obs, states, start_p, trans_p, emit_p): V = [{}] path = {} for y in states: V[0][y] = start_p[y] * emit_p[y][obs[0]] path[y] = [y] for t in range(1, len(obs)): V.append({}) newpath = {} for y in states: (prob, state) = max( [(V[t-1][y0] * trans_p[y0][y] * emit_p[y][obs[t]], y0) for y0 in states] ) V[t][y] = prob newpath[y] = path[state] + [y] path = newpath (prob, state) = max([(V[-1][y], y) for y in states]) return (prob, path[state])states = ('Rainy', 'Sunny')observations = ('Walk', 'Shop', 'Clean')start_probability = {'Rainy': 0.6, 'Sunny': 0.4}transition_probability = { 'Rainy' : {'Rainy': 0.7, 'Sunny': 0.3}, 'Sunny' : {'Rainy': 0.4, 'Sunny': 0.6}, }emission_probability = { 'Rainy' : {'Walk': 0.1, 'Shop': 0.4, 'Clean': 0.5}, 'Sunny' : {'Walk': 0.6, 'Shop': 0.3, 'Clean': 0.1}, }print(viterbi(observations, states, start_probability, transition_probability, emission_probability))
Output:
(0.01344, ['Sunny', 'Rainy', 'Rainy'])
The output of the Viterbi algorithm implementation is a tuple: (0.01344, [βSunnyβ, βRainyβ, βRainyβ]).
Explanation of the Output
- Probability (0.01344):
- This value represents the highest probability of the most likely sequence of states that can produce the given observation sequence (Walk, Shop, Clean).
- In other words, it is the probability of the observation sequence occurring given the most probable path of hidden states.
- Most Probable State Sequence ([βSunnyβ, βRainyβ, βRainyβ]):
- This list represents the sequence of hidden states that has the highest probability of resulting in the observed sequence (Walk, Shop, Clean).
- Each element in the list corresponds to the most probable state at each time step:
- Sunny for the first observation (Walk)
- Rainy for the second observation (Shop)
- Rainy for the third observation (Clean)
Applications of Viterbi in HMM
The Viterbi algorithm is widely used in various applications:
- Speech Recognition: The Viterbi algorithm in speech recognition converts spoken words into text by determining the most likely word or phoneme sequence that corresponds to the detected acoustic signals. HMMs simulate the speech process by using observations to represent acoustic features and hidden states to represent phonemes.
- Bioinformatics: The Viterbi algorithm is used in bioinformatics to align DNA sequences, predict protein structures, and locate gene sequences. Biological sequences can be modeled using HMMs, where observations correspond to sequences of nucleotides or amino acids, and hidden states represent biological functions.
- Natural Language Processing: In NLP, the Viterbi algorithm is used for tasks such as part-of-speech tagging, named entity recognition, and machine translation. HMMs can model linguistic sequences, with hidden states representing grammatical categories and observations representing words.
- Error Correction: In digital communications, the Viterbi algorithm is used for error correction to decode transmitted messages that may have been corrupted by noise. HMMs can model the transmission process, with hidden states representing the transmitted symbols and observations representing the received symbols.
Conclusion
A key component of sequence analysis in HMMs is the Viterbi algorithm, which provides a reliable way to infer the most likely order of hidden states from events that have been seen. Through comprehension of its principles and uses, one can effectively utilize the algorithm to address complicated issues in a variety of domains. Because of its versatility and effectiveness, the Viterbi algorithm is a vital tool in contemporary computational analysis.
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