时序数据相似性检测

看到一篇关于时序相似性检测的文章 How can we quantify similarity between time series?, 里面涉及了几种算法的对比, 非常有启发性. 于是问 ChatGPT 要了几个关于相似性的算法, 也做了相关的对比.

这里记录一下相关的代码和步骤. 这个内容是在 Jupyter Notebook 写的. 主要分为4部分.

定义几种算法

import numpy as np
import pywt

# 1. 欧拉距离 1.长度相同
def euclidean_similarity(actual, predic):
    return np.sqrt(np.sum((actual - predic) ** 2))

# 2. mean absolute percentage error (MAPE) 1.长度相同, 值不能为0
def mape_similarity(actual, predic):
    return np.mean(np.abs((actual - predic) / actual))

# 3. Pearson correlation coefficient 1.长度相同, 1.0 表示完全相同
def correlation_similarity(actual, predic):
    a_diff = actual - np.mean(actual)
    p_diff = predic - np.mean(predic)
    numerator = np.sum(a_diff * p_diff)
    denominator = np.sqrt(np.sum(a_diff ** 2)) * np.sqrt(np.sum(p_diff ** 2))
    return numerator / denominator

# 4. 余弦相似性, 1.长度相同, 1.0 表示完全相同
def cosine_similarity(x, y):
    numerator = sum([a * b for a,b in zip(x,y)])
    denominator = np.sqrt(sum([a**2 for a in x])) * np.sqrt(sum([b**2 for b in y]))
    return numerator / denominator

# 5. 基于傅立叶变换 再使用余弦相似
def fft_similarity(x, y):
    # 对两个时序数据进行傅里叶变换
    fft_x = np.fft.fft(x)
    fft_y = np.fft.fft(y)

    # 对频谱数据进行归一化处理
    norm_x = np.abs(fft_x) / len(x)
    norm_y = np.abs(fft_y) / len(y)

    # 计算余弦相似度
    numerator = np.dot(norm_x, norm_y)
    denominator = np.linalg.norm(norm_x) * np.linalg.norm(norm_y)
    similarity = numerator / denominator

    return similarity

# 6. 小波变换 -
def wavelet_similarity(x, y, wavelet='db4', level=3, threshold='soft', threshold_value=0.1):
    # 对两个时序数据进行小波变换
    coeffs_x = pywt.wavedec(x, wavelet=wavelet, level=level)
    coeffs_y = pywt.wavedec(y, wavelet=wavelet, level=level)

    # 对小波系数进行阈值处理
    if threshold == 'hard':
        thresh_coeffs_x = [pywt.threshold(c, threshold_value, mode='hard') for c in coeffs_x]
        thresh_coeffs_y = [pywt.threshold(c, threshold_value, mode='hard') for c in coeffs_y]
    elif threshold == 'soft':
        thresh_coeffs_x = [pywt.threshold(c, threshold_value, mode='soft') for c in coeffs_x]
        thresh_coeffs_y = [pywt.threshold(c, threshold_value, mode='soft') for c in coeffs_y]
    else:
        raise ValueError("Invalid threshold mode. Must be 'hard' or 'soft'.")

    # 计算余弦相似度
    flat_x = np.concatenate(thresh_coeffs_x)
    flat_y = np.concatenate(thresh_coeffs_y)
    numerator = np.dot(flat_x, flat_y)
    denominator = np.linalg.norm(flat_x) * np.linalg.norm(flat_y)
    similarity = numerator / denominator

    return similarity

定义一些辅助函数

# 一些辅助函数
def __normalize(vals):
    min_v = min(vals)
    max_v = max(vals)
    return [(x - min_v) / (max_v - min_v) * 100 for x in vals]

def __fill_0_for_empty_ts(x_values, y_values):
    """
    x_values & y_values are 2 dimension array, the 1st column is the millisecond, the 2nd
    column is the value of that time, ex: [[], [], []]
    """
    start_time = round(min(min(x_values)[0], min(y_values)[0]) / 60000)  # the min minute
    end_time = round(max(max(x_values)[0], max(y_values)[0]) / 60000)  # the max minute

    count = int(end_time - start_time) + 1
    x_vals = np.zeros(count)
    y_vals = np.zeros(count)

    for element in x_values:
        minute = round(element[0] / 60000)
        x_vals[minute - start_time] = float(element[1])

    for element in y_values:
        minute = round(element[0] / 60000)
        y_vals[minute - start_time] = float(element[1])

    return x_vals, y_vals

定义对比和测试函数

import matplotlib.pyplot as plt
import pandas as pd
from IPython.display import display

def __prepare(ts_x, ts_y):
    x = ts_x[:,1]
    y = ts_y[:,1]

    fig, (ax1, ax2, ax3) = plt.subplots(3, 1, figsize=(30, 8))

    ax1.plot(np.array(ts_x[:,0], dtype='datetime64[ms]'), x, color='blue', label='ts_x')
    ax2.plot(np.array(ts_y[:,0], dtype='datetime64[ms]'), y, color='red', label='ts_y')


    nx = np.array(__normalize(x))
    ny = np.array(__normalize(y))
    ax3.plot(np.array(ts_y[:,0], dtype='datetime64[ms]'), nx, color='blue', label='normalized_x')
    ax3.plot(np.array(ts_y[:,0], dtype='datetime64[ms]'), ny, color='red', label='normalized_y')
    ax3.set_title('normalized')
    plt.tight_layout()
    
    return x, y, nx, ny

def test_similarity(x, y, similarity_func):
    return similarity_func(x, y)

# calc shift 
def test_similarity_with_move(ts_x, ts_y, similarity_func):
    steps = 8
    length = len(ts_x)
    best_score = similarity_func(ts_x, ts_y)
    best_shift = 0
    for move in range(1, steps):
        score = similarity_func(ts_x[move:], ts_y[:(length - move)])
        # print(f"{best_score} vs {score} with {similarity_func.__name__} shift: {move}")
        best_score, best_shift = (score, move) if score > best_score else (best_score, best_shift)
        
        score = similarity_func(ts_y[move:], ts_x[:(length - move)])
        # print(f"{best_score} vs {score} with {similarity_func.__name__} shift: {-move}")
        best_score, best_shift = (score, move) if score > best_score else (best_score, best_shift)
    
    return best_score, best_shift

def compare(ts_x, ts_y):
    algorithms = [correlation_similarity, cosine_similarity, fft_similarity, wavelet_similarity]
    x, y, nx, ny = __prepare(ts_x, ts_y)
    result = {"algorithm":[], "raw":[], "normalized":[], "rawWithShift":[], "normalizedWithShift":[]}
    for algorithm in algorithms:
        result["algorithm"].append(algorithm.__name__)
        result["raw"].append(test_similarity(x, y, algorithm))
        result["normalized"].append(test_similarity(nx, ny, algorithm))
        result["rawWithShift"].append(test_similarity_with_move(x, y, algorithm))
        result["normalizedWithShift"].append(test_similarity_with_move(nx, ny, algorithm))

    df = pd.DataFrame(result)
    display(df)
    

对比测试

明显区分度测试

# 测试1 明显区分度 原始数据
ts_x = 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ts_y = 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compare(ts_x, ts_y)

结果:
verySimi.png

不明显区分度

# 测试2 一般区分度 
ts_x = 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compare(ts_x, ts_y)

结果:
lowSimi.png

标签: none

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