我們將使用輪廓分?jǐn)?shù)和一些距離指標(biāo)來執(zhí)行時(shí)間序列聚類實(shí)驗(yàn)，并且進(jìn)行可視化

讓我們看看下面的時(shí)間序列:

如果沿著y軸移動(dòng)序列添加隨機(jī)噪聲，并隨機(jī)化這些序列，那么它們幾乎無法分辨，如下圖所示-現(xiàn)在很難將時(shí)間序列列分組為簇:

上面的圖表是使用以下腳本創(chuàng)建的:

# Import necessary libraries
 import os
 import pandas as pd
 import numpy as np
 
 # Import random module with an alias 'rand'
 import random as rand
 from scipy import signal
 
 # Import the matplotlib library for plotting
 import matplotlib.pyplot as plt
 
 # Generate an array 'x' ranging from 0 to 5*pi with a step of 0.1
 x = np.arange(0, 5*np.pi, 0.1)
 
 # Generate square, sawtooth, sin, and cos waves based on 'x'
 y_square = signal.square(np.pi * x)
 y_sawtooth = signal.sawtooth(np.pi * x)
 y_sin = np.sin(x)
 y_cos = np.cos(x)
 
 # Create a DataFrame 'df_waves' to store the waveforms
 df_waves = pd.DataFrame([x, y_sawtooth, y_square, y_sin, y_cos]).transpose()
 
 # Rename the columns of the DataFrame for clarity
 df_waves = df_waves.rename(columns={0: 'time',
                                     1: 'sawtooth',
                                     2: 'square',
                                     3: 'sin',
                                     4: 'cos'})
 
 # Plot the original waveforms against time
 df_waves.plot(x='time', legend=False)
 plt.show()
 
 # Add noise to the waveforms and plot them again
 for col in df_waves.columns:
     if col != 'time':
         for i in range(1, 10):
             # Add noise to each waveform based on 'i' and a random value
             df_waves['{}_{}'.format(col, i)] = df_waves[col].apply(lambda x: x + i + rand.random() * 0.25 * i)
 
 # Plot the waveforms with added noise against time
 df_waves.plot(x='time', legend=False)
 plt.show()

現(xiàn)在我們需要確定聚類的基礎(chǔ)。這里有兩種方法:

把接近于一組的波形分組——較低歐幾里得距離的波形將聚在一起。

把看起來相似的波形分組——它們有相似的形狀，但歐幾里得距離可能不低

距離度量

一般來說，我們希望根據(jù)形狀對(duì)時(shí)間序列進(jìn)行分組，對(duì)于這樣的聚類-可能希望使用距離度量，如相關(guān)性，這些度量或多或少與波形的線性移位無關(guān)。

讓我們看看上面定義的帶有噪聲的波形對(duì)之間的歐幾里得距離和相關(guān)性的熱圖:

可以看到歐幾里得距離對(duì)波形進(jìn)行分組是很困難的，因?yàn)槿魏我唤M波形對(duì)的模式都是相似的。例如，除了對(duì)角線元素外，square & cos之間的相關(guān)形狀與square和square之間的相關(guān)形狀非常相似

所有的形狀都可以很容易地使用相關(guān)熱圖組合在一起——因?yàn)轭愃频牟ㄐ尉哂蟹浅８叩南嚓P(guān)性(sin-sin對(duì))，而像sin和cos這樣的波形幾乎沒有相關(guān)性。

輪廓分?jǐn)?shù)

通過上面熱圖和分析，根據(jù)高相關(guān)性分配組看起來是一個(gè)好主意，但是我們?nèi)绾味x相關(guān)閾值呢？看起來像一個(gè)迭代過程，容易出現(xiàn)不準(zhǔn)確和大量的人工工作。

在這種情況下，我們可以使用輪廓分?jǐn)?shù)（Silhouette score），它為執(zhí)行的聚類分配一個(gè)分?jǐn)?shù)。我們的目標(biāo)是使輪廓分?jǐn)?shù)最大化。

輪廓分?jǐn)?shù)（Silhouette Score）是一種用于評(píng)估聚類質(zhì)量的指標(biāo)，它可以幫助你確定數(shù)據(jù)點(diǎn)是否被正確地分配到它們的簇中。較高的輪廓分?jǐn)?shù)表示簇內(nèi)數(shù)據(jù)點(diǎn)相互之間更加相似，而不同簇之間的數(shù)據(jù)點(diǎn)差異更大，這通常是良好的聚類結(jié)果。

輪廓分?jǐn)?shù)的計(jì)算方法如下：

1. 對(duì)于每個(gè)數(shù)據(jù)點(diǎn) i，計(jì)算以下兩個(gè)值：- a(i)：數(shù)據(jù)點(diǎn) i 到同一簇中所有其他點(diǎn)的平均距離（簇內(nèi)平均距離）。- b(i)：數(shù)據(jù)點(diǎn) i 到與其不同簇中的所有簇的平均距離，取最小值（最近簇的平均距離）。
2. 然后，計(jì)算每個(gè)數(shù)據(jù)點(diǎn)的輪廓系數(shù) s(i)，它定義為：s(i) = frac{b(i) - a(i)}{max{a(i), b(i)}}
3. 最后，計(jì)算整個(gè)數(shù)據(jù)集的輪廓分?jǐn)?shù)，它是所有數(shù)據(jù)點(diǎn)的輪廓系數(shù)的平均值：text{輪廓分?jǐn)?shù)} = frac{1}{N} sum_{i=1}^{N} s(i)

其中，N 是數(shù)據(jù)點(diǎn)的總數(shù)。

輪廓分?jǐn)?shù)的取值范圍在 -1 到 1 之間，具體含義如下：

• 輪廓分?jǐn)?shù)接近1：表示簇內(nèi)數(shù)據(jù)點(diǎn)相似度高，不同簇之間的差異很大，是一個(gè)好的聚類結(jié)果。
• 輪廓分?jǐn)?shù)接近0：表示數(shù)據(jù)點(diǎn)在簇內(nèi)的相似度與簇間的差異相當(dāng)，可能是重疊的聚類或者不明顯的聚類。
• 輪廓分?jǐn)?shù)接近-1：表示數(shù)據(jù)點(diǎn)更適合分配到其他簇，不同簇之間的差異相比簇內(nèi)差異更小，通常是一個(gè)糟糕的聚類結(jié)果。

一些重要的知識(shí)點(diǎn):

在所有點(diǎn)上的高平均輪廓分?jǐn)?shù)(接近1)表明簇的定義良好且明顯。

低或負(fù)的平均輪廓分?jǐn)?shù)(接近-1)表明重疊或形成不良的集群。

0左右的分?jǐn)?shù)表示該點(diǎn)位于兩個(gè)簇的邊界上。

聚類

現(xiàn)在讓我們嘗試對(duì)時(shí)間序列進(jìn)行分組。我們已經(jīng)知道存在四種不同的波形，因此理想情況下應(yīng)該有四個(gè)簇。

歐氏距離

pca = decomposition.PCA(n_components=2)
 pca.fit(df_man_dist_euc)
 df_fc_cleaned_reduced_euc = pd.DataFrame(pca.transform(df_man_dist_euc).transpose(), 
                                               index = ['PC_1','PC_2'],
                                               columns = df_man_dist_euc.transpose().columns)
 
 index = 0
 range_n_clusters = [2, 3, 4, 5, 6, 7, 8]
 
 # Iterate over different cluster numbers
 for n_clusters in range_n_clusters:
     # Create a subplot with silhouette plot and cluster visualization
     fig, (ax1, ax2) = plt.subplots(1, 2)
     fig.set_size_inches(15, 7)
 
     # Set the x and y axis limits for the silhouette plot
     ax1.set_xlim([-0.1, 1])
     ax1.set_ylim([0, len(df_man_dist_euc) + (n_clusters + 1) * 10])
 
     # Initialize the KMeans clusterer with n_clusters and random seed
     clusterer = KMeans(n_clusters=n_clusters, n_init="auto", random_state=10)
     cluster_labels = clusterer.fit_predict(df_man_dist_euc)
 
     # Calculate silhouette score for the current cluster configuration
     silhouette_avg = silhouette_score(df_man_dist_euc, cluster_labels)
     print("For n_clusters =", n_clusters, "The average silhouette_score is :", silhouette_avg)
     sil_score_results.loc[index, ['number_of_clusters', 'Euclidean']] = [n_clusters, silhouette_avg]
     index += 1
     
     # Calculate silhouette values for each sample
     sample_silhouette_values = silhouette_samples(df_man_dist_euc, cluster_labels)
     
     y_lower = 10
 
     # Plot the silhouette plot
     for i in range(n_clusters):
         # Aggregate silhouette scores for samples in the cluster and sort them
         ith_cluster_silhouette_values = sample_silhouette_values[cluster_labels == i]
         ith_cluster_silhouette_values.sort()
 
         # Set the y_upper value for the silhouette plot
         size_cluster_i = ith_cluster_silhouette_values.shape[0]
         y_upper = y_lower + size_cluster_i
 
         color = cm.nipy_spectral(float(i) / n_clusters)
 
         # Fill silhouette plot for the current cluster
         ax1.fill_betweenx(np.arange(y_lower, y_upper), 0, ith_cluster_silhouette_values, facecolor=color, edgecolor=color, alpha=0.7)
 
         # Label the silhouette plot with cluster numbers
         ax1.text(-0.05, y_lower + 0.5 * size_cluster_i, str(i))
         y_lower = y_upper + 10  # Update y_lower for the next plot
 
     # Set labels and title for the silhouette plot
     ax1.set_title("The silhouette plot for the various clusters.")
     ax1.set_xlabel("The silhouette coefficient values")
     ax1.set_ylabel("Cluster label")
 
     # Add vertical line for the average silhouette score
     ax1.axvline(x=silhouette_avg, color="red", linestyle="--")
     ax1.set_yticks([])  # Clear the yaxis labels / ticks
     ax1.set_xticks([-0.1, 0, 0.2, 0.4, 0.6, 0.8, 1])
 
     # Plot the actual clusters
     colors = cm.nipy_spectral(cluster_labels.astype(float) / n_clusters)
     ax2.scatter(df_fc_cleaned_reduced_euc.transpose().iloc[:, 0], df_fc_cleaned_reduced_euc.transpose().iloc[:, 1],
                 marker=".", s=30, lw=0, alpha=0.7, c=colors, edgecolor="k")
 
     # Label the clusters and cluster centers
     centers = clusterer.cluster_centers_
     ax2.scatter(centers[:, 0], centers[:, 1], marker="o", c="white", alpha=1, s=200, edgecolor="k")
 
     for i, c in enumerate(centers):
         ax2.scatter(c[0], c[1], marker="$%d$" % i, alpha=1, s=50, edgecolor="k")
 
     # Set labels and title for the cluster visualization
     ax2.set_title("The visualization of the clustered data.")
     ax2.set_xlabel("Feature space for the 1st feature")
     ax2.set_ylabel("Feature space for the 2nd feature")
 
     # Set the super title for the whole plot
     plt.suptitle("Silhouette analysis for KMeans clustering on sample data with n_clusters = %d" % n_clusters,
                  fontsize=14, fontweight="bold")
 
 plt.savefig('sil_score_eucl.png')
 plt.show()

可以看到無論分成多少簇，數(shù)據(jù)都是混合的，并不能為任何數(shù)量的簇提供良好的輪廓分?jǐn)?shù)。這與我們基于歐幾里得距離熱圖的初步評(píng)估的預(yù)期一致

相關(guān)性

pca = decomposition.PCA(n_components=2)
 pca.fit(df_man_dist_corr)
 df_fc_cleaned_reduced_corr = pd.DataFrame(pca.transform(df_man_dist_corr).transpose(), 
                                               index = ['PC_1','PC_2'],
                                               columns = df_man_dist_corr.transpose().columns)
 
 index=0
 range_n_clusters = [2,3,4,5,6,7,8]
 for n_clusters in range_n_clusters:
     # Create a subplot with 1 row and 2 columns
     fig, (ax1, ax2) = plt.subplots(1, 2)
     fig.set_size_inches(15, 7)
 
     # The 1st subplot is the silhouette plot
     # The silhouette coefficient can range from -1, 1 but in this example all
     # lie within [-0.1, 1]
     ax1.set_xlim([-0.1, 1])
     # The (n_clusters+1)*10 is for inserting blank space between silhouette
     # plots of individual clusters, to demarcate them clearly.
     ax1.set_ylim([0, len(df_man_dist_corr) + (n_clusters + 1) * 10])
 
     # Initialize the clusterer with n_clusters value and a random generator
     # seed of 10 for reproducibility.
     clusterer = KMeans(n_clusters=n_clusters, n_init="auto", random_state=10)
     cluster_labels = clusterer.fit_predict(df_man_dist_corr)
 
     # The silhouette_score gives the average value for all the samples.
     # This gives a perspective into the density and separation of the formed
     # clusters
     silhouette_avg = silhouette_score(df_man_dist_corr, cluster_labels)
     print(
         "For n_clusters =",
         n_clusters,
         "The average silhouette_score is :",
         silhouette_avg,
     )
     sil_score_results.loc[index,['number_of_clusters','corrlidean']] = [n_clusters,silhouette_avg]
     index=index+1
     
     sample_silhouette_values = silhouette_samples(df_man_dist_corr, cluster_labels)
     
     y_lower = 10
     for i in range(n_clusters):
         # Aggregate the silhouette scores for samples belonging to
         # cluster i, and sort them
         ith_cluster_silhouette_values = sample_silhouette_values[cluster_labels == i]
 
         ith_cluster_silhouette_values.sort()
 
         size_cluster_i = ith_cluster_silhouette_values.shape[0]
         y_upper = y_lower + size_cluster_i
 
         color = cm.nipy_spectral(float(i) / n_clusters)
         ax1.fill_betweenx(
             np.arange(y_lower, y_upper),
             0,
             ith_cluster_silhouette_values,
             facecolor=color,
             edgecolor=color,
             alpha=0.7,
         )
 
         # Label the silhouette plots with their cluster numbers at the middle
         ax1.text(-0.05, y_lower + 0.5 * size_cluster_i, str(i))
 
         # Compute the new y_lower for next plot
         y_lower = y_upper + 10  # 10 for the 0 samples
 
     ax1.set_title("The silhouette plot for the various clusters.")
     ax1.set_xlabel("The silhouette coefficient values")
     ax1.set_ylabel("Cluster label")
 
     # The vertical line for average silhouette score of all the values
     ax1.axvline(x=silhouette_avg, color="red", linestyle="--")
 
     ax1.set_yticks([])  # Clear the yaxis labels / ticks
     ax1.set_xticks([-0.1, 0, 0.2, 0.4, 0.6, 0.8, 1])
 
     # 2nd Plot showing the actual clusters formed
     colors = cm.nipy_spectral(cluster_labels.astype(float) / n_clusters)
     
     ax2.scatter(
         df_fc_cleaned_reduced_corr.transpose().iloc[:, 0], 
         df_fc_cleaned_reduced_corr.transpose().iloc[:, 1], marker=".", s=30, lw=0, alpha=0.7, c=colors, edgecolor="k"
     )
     
 #     for i in range(len(df_fc_cleaned_cleaned_reduced.transpose().iloc[:, 0])):
 #                         ax2.annotate(list(df_fc_cleaned_cleaned_reduced.transpose().index)[i], 
 #                                      (df_fc_cleaned_cleaned_reduced.transpose().iloc[:, 0][i], 
 #                                       df_fc_cleaned_cleaned_reduced.transpose().iloc[:, 1][i] + 0.2))
         
     # Labeling the clusters
     centers = clusterer.cluster_centers_
     # Draw white circles at cluster centers
     ax2.scatter(
         centers[:, 0],
         centers[:, 1],
         marker="o",
         c="white",
         alpha=1,
         s=200,
         edgecolor="k",
     )
 
     for i, c in enumerate(centers):
         ax2.scatter(c[0], c[1], marker="$%d$" % i, alpha=1, s=50, edgecolor="k")
 
     ax2.set_title("The visualization of the clustered data.")
     ax2.set_xlabel("Feature space for the 1st feature")
     ax2.set_ylabel("Feature space for the 2nd feature")
 
     plt.suptitle(
         "Silhouette analysis for KMeans clustering on sample data with n_clusters = %d"
         % n_clusters,
         fontsize=14,
         fontweight="bold",
     )
 
 plt.show()

當(dāng)選擇的簇?cái)?shù)為4時(shí)，我們可以清楚地看到分離的簇，其他結(jié)果通常比歐氏距離要好得多。

歐幾里得距離與相關(guān)廓形評(píng)分的比較

輪廓分?jǐn)?shù)表明基于相關(guān)性的距離矩陣在簇?cái)?shù)為4時(shí)效果最好，而在歐氏距離的情況下效果就不那么明顯了結(jié)論

總結(jié)

在本文中，我們研究了如何使用歐幾里得距離和相關(guān)度量執(zhí)行時(shí)間序列聚類，并觀察了這兩種情況下的結(jié)果如何變化。如果我們?cè)谠u(píng)估聚類時(shí)結(jié)合Silhouette，我們可以使聚類步驟更加客觀，因?yàn)樗峁┝艘环N很好的直觀方式來查看聚類的分離情況。