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
# 7. 傅立叶变换
# 先通过快速傅立叶变换把时域信号转成频域信号, 然后把其中一个频域和另外一个频域的复共轭做乘积, 再把结果做快速傅立叶逆变换, 得到在时域信号的互相关.
def compare_multiple_arrays_with_max_shift(reference_array, other_arrays, max_shift):
start_ns = time.time_ns()
fft_reference = np.fft.rfft(reference_array)
reference_norm = np.linalg.norm(reference_array)
results = []
for array in other_arrays:
fft_array = np.fft.rfft(array)
cross_correlation = np.fft.irfft(fft_reference * np.conj(fft_array))
max_shift_arr = np.concatenate((cross_correlation[-max_shift:], cross_correlation[:max_shift + 1]))
max_shift_abs_arr = np.abs(max_shift_arr)
lag_index = np.argmax(max_shift_abs_arr)
max_correlation = max_shift_abs_arr[lag_index]
lag = lag_index - max_shift
array_norm = np.linalg.norm(array)
norm = reference_norm * array_norm
correlation_coefficient = 0 if norm == 0 else max_correlation / norm
results.append((correlation_coefficient, lag))
print("method 0 used: ", time.time_ns() - start_ns)
return results
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:
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:
