Algorithm类介绍(core)

本文详细解析了人工智能算法在数据处理领域的关键应用,包括数据挖掘、机器学习、深度学习等技术,旨在帮助读者理解如何利用这些技术解决实际问题。

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参考:http://blog.youkuaiyun.com/yang_xian521/article/details/7533922

转载于:https://www.cnblogs.com/573177885qq/p/4749561.html

### Viterbi Algorithm Explanation The **Viterbi algorithm** is a dynamic programming technique used primarily for finding the most likely sequence of hidden states—called the *Viterbi path*—in models such as Hidden Markov Models (HMMs). It operates based on two key probabilities: 1. The **transition probability**, which defines the likelihood of moving from one state to another. 2. The **emission probability**, which represents the likelihood of observing a particular symbol given a specific state. Mathematically, the goal is to maximize the joint probability \( P(L|O) \), where \( L \) denotes the label sequence and \( O \) represents the observation sequence. This can be expressed using Bayes' theorem: \[ P(L|O) = \frac{P(O|L)P(L)}{P(O)} \] Here, \( P(O|L) \) refers to the emission probability, while \( P(L) \) corresponds to the transition probability between labels[^4]. To compute this efficiently, the Viterbi algorithm employs recursion over time steps, maintaining both the maximum probability at each step and the corresponding optimal predecessor state. Below is an outline of its core components: #### Key Components - At any point in time \( t \), let \( v_t(j) \) denote the highest probability along a single path ending with state \( j \). - Let \( p_t(j) \) store the index of the previous state leading optimally to state \( j \). These values are updated iteratively according to the following equations: \[ v_{t}(j)=\max _{1 \leq i \leq N}\left[v_{t-1}(i)a(i,j)\right]b(j,o_{t}) \] Where: - \( a(i,j) \): Transition probability from state \( i \) to state \( j \), - \( b(j,o_t) \): Emission probability of emitting observation \( o_t \) when in state \( j \)[^4]. Once all sequences have been processed through the final stage, traceback begins starting from the terminal node having maximal value among possible end-states until reaching initial conditions. ### Python Implementation Example Below demonstrates a basic implementation utilizing numpy arrays for clarity purposes only; actual applications may require optimizations depending upon use cases. ```python import numpy as np def viterbi(obs_seq, trans_prob, emit_prob, init_prob): T = len(obs_seq) num_states = len(trans_prob) # Initialize matrices prob_matrix = np.zeros((T, num_states)) prev_state = np.zeros((T, num_states)) # Initialization Step for s in range(num_states): prob_matrix[0][s] = init_prob[s] * emit_prob[s][obs_seq[0]] # Recursion Steps for t in range(1, T): for s in range(num_states): max_val = -np.inf best_prev = None for sprev in range(num_states): val = prob_matrix[t-1][sprev]*trans_prob[sprev][s] if val > max_val: max_val = val best_prev = sprev prob_matrix[t][s] = max_val * emit_prob[s][obs_seq[t]] prev_state[t][s] = best_prev # Termination & Traceback last_best = np.argmax(prob_matrix[-1]) seq = [last_best] for t in reversed(range(T-1)): last_best = int(prev_state[t+1][last_best]) seq.append(last_best) return list(reversed(seq)), prob_matrix[-1][seq[-1]] # Sample Data Setup 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}, } observation_map = {act:i for i, act in enumerate(observations)} init_p = np.array([start_probability[state] for state in states]) trn_p = np.array([[transition_probability[a][b] for b in states] for a in states]) emit_p = np.array([[emission_probability[a][b] for b in observations] for a in states]) sequence_indices = [observation_map[o] for o in ["walk", "shop"]] result_sequence, result_prob = viterbi(sequence_indices, trn_p, emit_p, init_p) print("Most probable sequence:", [states[i] for i in result_sequence], "\nProbability:", result_prob) ``` This script initializes parameters representing weather transitions alongside activities performed under certain circumstances before executing `viterbi` function call providing results accordingly.
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