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DanieleGregori`ArXivExplore`
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The **Viterbi algorithm** is a dynamic programming algorithm used to find the most likely sequence of hidden states in a hidden Markov model (HMM). It is widely used in various fields such as signal processing, speech recognition, bioinformatics, and gravitational wave (GW) data analysis.### Key Concepts:1. **Hidden Markov Model (HMM)**: - An HMM is a statistical model where the system being modeled is assumed to follow a Markov process with hidden (unobservable) states. In an HMM, you have: - **Hidden states**: The actual states of the system, which are not directly observable. - **Observations**: The data or signals that are observed, which are related to the hidden states. - **Transition probabilities**: The probabilities of moving from one hidden state to another. - **Emission probabilities**: The probabilities of observing a particular output given a hidden state.2. **Viterbi Algorithm**: - The Viterbi algorithm is used to find the most probable sequence of hidden states that could have produced a given sequence of observations. It does this by maximizing the joint probability of the hidden states and the observations. - The algorithm works by breaking the problem into smaller subproblems and solving them recursively, which makes it computationally efficient.### Steps of the Viterbi Algorithm:1. **Initialization**: - Start by initializing the probabilities of the first observation being generated by each possible hidden state. 2. **Recursion**: - For each subsequent observation, calculate the probability of each hidden state being the most likely state, given the previous states and the current observation. This is done by considering all possible transitions from the previous states and selecting the one with the highest probability.3. **Termination**: - Once all observations have been processed, the algorithm identifies the final state with the highest probability.4. **Backtracking**: - After determining the most likely final state, the algorithm traces back through the sequence of states to reconstruct the most likely sequence of hidden states.### Application in Gravitational Wave Searches:In the context of gravitational wave (GW) searches, the Viterbi algorithm is used to detect signals that evolve over time, such as continuous waves (CWs) or inspiral signals from binary black hole mergers. The algorithm is particularly useful when the signal's frequency evolution is not well-modeled by simple power laws, as it can detect arbitrary curves in the time-frequency domain.- **Advantages**: - **Model-agnostic**: Unlike other methods (e.g., Frequency-Hough), the Viterbi algorithm does not assume a specific form for the signal's frequency evolution. This makes it flexible and capable of detecting signals with complex time-frequency behavior. - **Computational efficiency**: The algorithm is computationally efficient, making it suitable for large-scale searches where computational resources are a limiting factor.- **Disadvantages**: - **Sensitivity**: The Viterbi algorithm may be less sensitive than other methods (e.g., Frequency-Hough) that are tailored to specific signal models. However, its flexibility and efficiency often make it a preferred choice in certain parameter spaces.### Summary:The Viterbi algorithm is a powerful tool for detecting the most likely sequence of hidden states in a hidden Markov model. In gravitational wave searches, it is used to identify signals with complex frequency evolution, offering a flexible and computationally efficient approach to detecting signals that may not follow simple models.
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