聚類分析 (Cluster Analysis)
香港中文大學數學系
2026年6月27日
1 聚類分析 (Cluster Analysis, CA)
聚類分析（Cluster analysis）是一種數據分析技術，旨在識別和聚合具同質性的觀測值或數據點——即表現出可比較特徵和內在屬性的實體。作為一種數據探勘（data mining）方法，它構成了一種無監督學習（unsupervised learning）演算法，能揭示隱藏在數據集中的潛在模式和內在結構關係。
基於質心的聚類（Centroid-based clustering，例如 K-means）和階層分群（hierarchical clustering）是聚類分析中廣泛採用的兩種不同方法。
2 Python 程式碼
2.1 步驟 1：載入數據
隨機抽取 20 個觀測樣本，並設定隨機種子以確保結果可重現
描述性統計
## === Descriptive Statistics ===## Aroma Flavor ... Quakers Category.Two.Defects
## count 20.000000 20.000000 ... 20.0 20.000000
## mean 7.576000 7.605000 ... 0.0 4.050000
## std 0.266959 0.367058 ... 0.0 6.021671
## min 7.170000 6.920000 ... 0.0 0.000000
## 25% 7.420000 7.420000 ... 0.0 0.000000
## 50% 7.540000 7.625000 ... 0.0 2.500000
## 75% 7.750000 7.750000 ... 0.0 4.000000
## max 8.080000 8.670000 ... 0.0 20.000000
##
## [8 rows x 15 columns]移除標準差為零的欄位（Uniformity, Clean.Cup, Sweetness, Category.Two.Defects）
drop_cols = ["Uniformity", "Clean.Cup", "Sweetness", "Category.Two.Defects"]
df = df.drop(columns=drop_cols)移除最後一個欄位（字元型類別變數）
2.3 步驟 3：特徵縮放 (Scaling)
對數據集進行縮放（Scaling）至關重要，因為各個特徵通常具有差異極大的測量範圍。如果不進行縮放，數值較大的變數將不成比例地扭曲聚類結果。
聚類分析中兩種廣泛使用的縮放方法是：標準化（standardization），它透過減去每個特徵的平均值並除以其標準差來轉換數據；以及正規化（normalization），它將所有數值重新縮放至一個固定的有界範圍內。
\[ X_{stand} = \frac{X - \mu_x}{\sigma_x} \\ X_{norm} = \frac{X - X_{min}}{X_{max}-X_{min}} \]
import pandas as pd
from sklearn.preprocessing import StandardScaler, MinMaxScaler
# ----------------------
# 標準化 (Z-score)
# ----------------------
scaler_standard = StandardScaler()
df_scaled_standard = pd.DataFrame(
scaler_standard.fit_transform(df_com),
columns=df_com.columns
)
# ----------------------
# 最大最小正規化 (0~1 範圍)
# ----------------------
scaler_minmax = MinMaxScaler()
df_scaled_norm = pd.DataFrame(
scaler_minmax.fit_transform(df_com),
columns=df_com.columns
)
# 如果您只需要標準化後的數據：
df_scaled = df_scaled_standard 2.4 步驟 4：最佳聚類數量 (Optimal Number of Clusters)
最佳的聚類數量取決於數據集及其預期應用。儘管如此，目前有幾種方法可以幫助找出基於質心的聚類的最佳分群數。
2.4.1 肘部法則 (Elbow Method)
這種方法會繪製群內誤差平方和（Within-cluster sum of squares, WSS）與聚類數量的關係圖。WSS 量化了數據點在其各自的聚類中聚集的緊密程度。肘部法則透過找出 WSS 曲線上遞減速率開始趨於平緩的點（即拐點）來確定最佳聚類數量；這個拐點通常被選為理想的聚類數。
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.cluster import KMeans
import numpy as np
# 計算群內誤差平方和 (WSS) 以用於肘部法則
wss = []
max_k = 10 # 您可以調整要測試的最大聚類數量
for k in range(1, max_k + 1):
kmeans = KMeans(n_clusters=k, random_state=100)
kmeans.fit(df_scaled)
wss.append(kmeans.inertia_) # inertia = WSSKMeans(n_clusters=10, random_state=100)In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook.
On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.
Parameters
Fitted attributes
10 features
| kmeans0 |
| kmeans1 |
| kmeans2 |
| kmeans3 |
| kmeans4 |
| kmeans5 |
| kmeans6 |
| kmeans7 |
| kmeans8 |
| kmeans9 |
# 繪製肘部曲線
plt.figure(figsize=(8, 5))
plt.plot(range(1, max_k + 1), wss, marker='o')
plt.xlabel("Number of clusters k")
plt.ylabel("Within-cluster sum of squares (WSS)")
plt.title("Elbow method")
plt.axvline(x=2, linestyle='--', color='red') # 在 k=2 處畫紅色垂直虛線
plt.show()
2.4.2 輪廓係數/寬度 (Silhouette Score/Width)
輪廓係數（Silhouette score）量化了每個數據點與其分配到的聚類（相對於相鄰聚類）的匹配程度。其數值範圍從 -1 到 1：得分為 1 表示該點完美契合其聚類，得分為 0 表示該點恰好位於兩個聚類的邊界上，而得分為 -1 則表示該點更適合分配到其他聚類。最佳的聚類數量通常是能產生最高平均輪廓係數的那個值。
import matplotlib.pyplot as plt
import pandas as pd
from sklearn.cluster import KMeans
from sklearn.metrics import silhouette_score, silhouette_samples
import numpy as np
# ----------------------
# 相當於 fviz_nbclust(..., method = "silhouette")
# 繪製不同 k 值的平均輪廓係數
# ----------------------
max_k = 10
sil_avg_scores = []
k_range = range(2, max_k + 1) # 當 k=1 時輪廓係數未定義
for k in k_range:
km = KMeans(n_clusters=k, random_state=100)
cluster_labels = km.fit_predict(df_scaled)
avg_sil = silhouette_score(df_scaled, cluster_labels)
sil_avg_scores.append(avg_sil)
plt.figure(figsize=(8, 5))
plt.plot(k_range, sil_avg_scores, marker="o")
plt.xlabel("Number of clusters k")
plt.ylabel("Average Silhouette Score")
plt.title("Silhouette method")
plt.show()
# ----------------------
# 相當於 silhouette() + plot + fviz_silhouette，針對 k=2
# ----------------------
# 擬合 k=2 的 KMeans
km_res = KMeans(n_clusters=2, random_state=100)
labels = km_res.fit_predict(df_scaled)
# 計算每一個樣本的輪廓值
sil_vals = silhouette_samples(df_scaled, labels)
# 簡單的輪廓圖 (對應基礎 R 語言的 plot(sil))
fig, ax = plt.subplots(figsize=(8, 5))
y_lower = 10
for cluster_id in np.unique(labels):
# 提取目前聚類的輪廓係數並進行排序
cluster_sil = sil_vals[labels == cluster_id]
cluster_sil.sort()
y_upper = y_lower + len(cluster_sil)
ax.fill_betweenx(np.arange(y_lower, y_upper), 0, cluster_sil, alpha=0.7)
y_lower = y_upper + 10
ax.axvline(x=np.mean(sil_vals), color="red", linestyle="--")
ax.set_xlabel("Silhouette Coefficient")
ax.set_ylabel("Sample Index")
ax.set_title("Silhouette Plot (k=2)")
plt.show()
輪廓係數為正數表示觀測值很適合其分配的聚類。
2.4.3 間隙統計量方法 (Gap Statistic Method)
間隙統計量（Gap statistic）透過比較觀測到的群內離散度與參考分佈來協助確定最佳聚類數。其核心概念是選擇能使間隙統計量最大化的聚類數量。較大的間隙表示偏離隨機性的程度越顯著，意味著聚類結構定義得越好。透過比較不同聚類數量的間隙，我們可以找出能提供最鮮明且具意義之聚類模式的分群數。
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.cluster import KMeans
from sklearn.metrics import pairwise_distances_argmin_min
def calculate_gap_statistic(data, max_k, n_references=10):
"""
重製 factoextra::fviz_nbclust(..., method = "gap_stat")
計算每個 k 的間隙統計量與標準誤
"""
# 將 dataframe 轉換為 numpy 陣列
X = data.values if hasattr(data, "values") else np.array(data)
n_samples, n_features = X.shape
k_range = np.arange(1, max_k + 1)
gap_scores = []
gap_standard_errors = []
for k in k_range:
# 1. 計算實際數據的群內離散度 Wk
kmeans = KMeans(n_clusters=k, random_state=100)
labels = kmeans.fit_predict(X)
dists, _ = pairwise_distances_argmin_min(X, kmeans.cluster_centers_)
wk = np.sum(dists ** 2)
# 2. 產生隨機參考數據集並計算參考的 Wkb
ref_wk_log = []
for _ in range(n_references):
ref_data = np.random.uniform(
low=X.min(axis=0),
high=X.max(axis=0),
size=X.shape
)
ref_kmeans = KMeans(n_clusters=k, random_state=100)
ref_labels = ref_kmeans.fit_predict(ref_data)
ref_dists, _ = pairwise_distances_argmin_min(ref_data, ref_kmeans.cluster_centers_)
ref_wk = np.sum(ref_dists ** 2)
ref_wk_log.append(np.log(ref_wk))
# 間隙統計量公式
log_wk = np.log(wk)
mean_log_ref = np.mean(ref_wk_log)
gap = mean_log_ref - log_wk
gap_scores.append(gap)
# 誤差線的標準誤
sd_ref = np.std(ref_wk_log)
se = sd_ref * np.sqrt(1 + 1 / n_references)
gap_standard_errors.append(se)
return k_range, np.array(gap_scores), np.array(gap_standard_errors)
# ---------------------- 執行間隙統計量圖 ----------------------
# 請將 df_scaled 替換為您實際的標準化 dataframe 變數
max_k_test = 10
k_values, gap_values, gap_error = calculate_gap_statistic(df_scaled, max_k=max_k_test)
# 繪製相當於 R fviz_nbclust gap_stat 的圖表
plt.figure(figsize=(8, 5))
plt.errorbar(k_values, gap_values, yerr=gap_error, marker="o", capsize=4)
plt.xlabel("Number of clusters k")
plt.ylabel("Gap statistic")
plt.title("Gap statistic method")
plt.xticks(k_values)## ([<matplotlib.axis.XTick object at 0x000001FA46F48920>, <matplotlib.axis.XTick object at 0x000001FA4A371B20>, <matplotlib.axis.XTick object at 0x000001FA4CA879B0>, <matplotlib.axis.XTick object at 0x000001FA4A3FB680>, <matplotlib.axis.XTick object at 0x000001FA4A3FA9C0>, <matplotlib.axis.XTick object at 0x000001FA4A3F92E0>, <matplotlib.axis.XTick object at 0x000001FA4A440DD0>, <matplotlib.axis.XTick object at 0x000001FA4A4412B0>, <matplotlib.axis.XTick object at 0x000001FA4A3FA4B0>, <matplotlib.axis.XTick object at 0x000001FA4A440DA0>], [Text(1, 0, '1'), Text(2, 0, '2'), Text(3, 0, '3'), Text(4, 0, '4'), Text(5, 0, '5'), Text(6, 0, '6'), Text(7, 0, '7'), Text(8, 0, '8'), Text(9, 0, '9'), Text(10, 0, '10')])
2.4.4 綜合以上所有方法 (Combination of All)
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.cluster import KMeans
from sklearn.metrics import silhouette_score, pairwise_distances_argmin_min
# --------------------------
# 1. 輔助函數: 肘部法則 (WSS)
# --------------------------
def elbow_curve(data, max_k=10, seed=100):
X = data.values
wss = []
for k in range(1, max_k+1):
km = KMeans(n_clusters=k, random_state=seed)
km.fit(X)
wss.append(km.inertia_)
return np.arange(1, max_k+1), np.array(wss)
# --------------------------
# 2. 輔助函數: 輪廓係數
# --------------------------
def silhouette_curve(data, max_k=10, seed=100):
X = data.values
k_range = np.arange(2, max_k+1)
sil_scores = []
for k in k_range:
km = KMeans(n_clusters=k, random_state=seed)
labels = km.fit_predict(X)
sil_scores.append(silhouette_score(X, labels))
return k_range, np.array(sil_scores)
# --------------------------
# 3. 輔助函數: 間隙統計量
# --------------------------
def gap_stat_curve(data, max_k=10, n_ref=10, seed=100):
X = data.values
n, p = X.shape
k_list = np.arange(1, max_k+1)
gaps = []
np.random.seed(seed)
for k in k_list:
# 實際數據 Wk
km = KMeans(n_clusters=k, random_state=seed)
labels = km.fit_predict(X)
dists, _ = pairwise_distances_argmin_min(X, km.cluster_centers_)
wk = np.sum(dists ** 2)
log_wk = np.log(wk)
# 參考均勻數據集
ref_log = []
for _ in range(n_ref):
ref = np.random.uniform(X.min(axis=0), X.max(axis=0), X.shape)
km_r = KMeans(n_clusters=k, random_state=seed)
r_labels = km_r.fit_predict(ref)
r_dists, _ = pairwise_distances_argmin_min(ref, km_r.cluster_centers_)
ref_log.append(np.log(np.sum(r_dists ** 2)))
gap = np.mean(ref_log) - log_wk
gaps.append(gap)
return k_list, np.array(gaps)
# --------------------------
# 4. 對縮放後的數據執行這三種方法
# --------------------------
max_cluster = 10
k_elbow, wss_vals = elbow_curve(df_scaled, max_k=max_cluster)
k_sil, sil_vals = silhouette_curve(df_scaled, max_k=max_cluster)
k_gap, gap_vals = gap_stat_curve(df_scaled, max_k=max_cluster)
# 為每種方法找出最佳的 k
# 肘部法則: 挑選肘部拐點（簡單的啟發式方法：最大下降率）
wss_drop = np.diff(wss_vals)
opt_elbow = np.argmin(wss_drop) + 2
# 輪廓係數: 具有最高輪廓係數的 k
opt_sil = k_sil[np.argmax(sil_vals)]
# 間隙統計量: 具有最大間隙值的 k
opt_gap = k_gap[np.argmax(gap_vals)]
# 匯總投票結果 (與 R n_clust 的多數決邏輯相符)
votes = [opt_elbow, opt_sil, opt_gap]
from collections import Counter
vote_count = Counter(votes)
majority_k = vote_count.most_common(1)[0][0]
# 印出摘要 (相當於 R 中的 print(n_clust))
print("=== Optimal Cluster Count Votes from Multiple Methods ===")## === Optimal Cluster Count Votes from Multiple Methods ===## Elbow Method optimal k: 2## Silhouette Method optimal k: 2## Gap Statistic optimal k: 1##
## Vote tally: {np.int64(2): 2, np.int64(1): 1}## Majority-supported optimal number of clusters: 2# --------------------------
# 繪製組合結果 (相當於 R 中的 plot(n_clust))
# --------------------------
fig, axes = plt.subplots(3, 1, figsize=(8, 10))
# 子圖 1: 肘部法則
axes[0].plot(k_elbow, wss_vals, marker="o")
axes[0].axvline(opt_elbow, color="red", ls="--", label=f"Opt k={opt_elbow}")
axes[0].set_title("Elbow Method (WSS)")
axes[0].set_xlabel("Number of clusters k")
axes[0].set_ylabel("Within-cluster Sum of Squares")
axes[0].legend()
axes[0].grid(alpha=0.3)
# 子圖 2: 輪廓係數
axes[1].plot(k_sil, sil_vals, marker="o", color="green")
axes[1].axvline(opt_sil, color="red", ls="--", label=f"Opt k={opt_sil}")
axes[1].set_title("Silhouette Score Method")
axes[1].set_xlabel("Number of clusters k")
axes[1].set_ylabel("Average Silhouette Score")
axes[1].legend()
axes[1].grid(alpha=0.3)
# 子圖 3: 間隙統計量
axes[2].plot(k_gap, gap_vals, marker="o", color="orange")
axes[2].axvline(opt_gap, color="red", ls="--", label=f"Opt k={opt_gap}")
axes[2].set_title("Gap Statistic Method")
axes[2].set_xlabel("Number of clusters k")
axes[2].set_ylabel("Gap Statistic Value")
axes[2].legend()
axes[2].grid(alpha=0.3)
plt.tight_layout()
plt.show()
2.5 步驟 5：階層分群 (Hierarchical Clustering)
2.5.1 步驟 5.1：距離 (Distance)
2.5.1.1 數值數據 (Numeric Data)
使用函數 dist()：
import pandas as pd
import numpy as np
from scipy.spatial.distance import pdist
# 將縮放後的 dataframe 轉換為 numpy 陣列
X = df_scaled.values
# ----------------------
# 數值數據距離矩陣 (相當於 R dist())
# pdist 返回與 R dist 物件相同的壓縮距離矩陣
# ----------------------
# 歐氏距離 (R method = 'euclidian' / euclidean)
df_dist_euc = pdist(X, metric="euclidean")
# 最大值 / 柴比雪夫距離
# df_dist_max = pdist(X, metric="chebyshev")
# 曼哈頓 / L1 距離
# df_dist_man = pdist(X, metric="cityblock")
# 坎培拉距離
# df_dist_can = pdist(X, metric="canberra")
# 閔可夫斯基距離 (預設 p=2 即為歐氏距離)
# df_dist_mink = pdist(X, metric="minkowski", p=3)
# ----------------------
# 類別二元距離 (R method = "binary")
# 用於虛擬化後的二元類別數據矩陣 dummified_data
# ----------------------
# dummified_data_np = dummified_data.values
# dist_binary = pdist(dummified_data_np, metric="jaccard")2.5.2 步驟 5.2：連結準則 (Linkage Criteria)
連結（Linkage）定義了在階層分群過程中用於合併聚類的規則。目前存在多種連結準則，而所選的準則可能會大幅改變聚類的結果：
- 完全連結（Complete linkage）：使用兩個聚類之間的最大距離
- 單一連結（Single linkage）：使用兩個聚類之間的最小距離
- 平均連結（Average linkage）：使用兩個聚類之間的平均成對距離
- 華德法（Ward’s linkage）：最小化群內總方差
以下是使用歐氏距離（Euclidean distance）以及上述各項連結方法，從數值數據生成階層分群的程式碼。
from scipy.cluster.hierarchy import linkage
# df_dist_euc 是 pdist() 產生的壓縮歐氏距離矩陣
# linkage() 將壓縮距離矩陣作為輸入，相當於 R hclust()
hclust_single = linkage(df_dist_euc, method="single")
hclust_complete = linkage(df_dist_euc, method="complete")
hclust_average = linkage(df_dist_euc, method="average")
hclust_centroid = linkage(df_dist_euc, method="centroid")
hclust_ward = linkage(df_dist_euc, method="ward")獲取聚類資訊：
from scipy.cluster.hierarchy import fcluster
# 將樹切成恰好 2 個聚類 (相當於 cutree(..., k = 2))
clusters_k2 = fcluster(hclust_ward, t=2, criterion="maxclust")
# 在高度 h = 0.7 處切割樹 (相當於 cutree(..., h = 0.7))
clusters_h07 = fcluster(hclust_ward, t=0.7, criterion="distance")import pandas as pd
from scipy.cluster.hierarchy import fcluster
# 透過將樹狀圖切成 2 個聚類來分配聚類標籤
cluster_labels = fcluster(hclust_ward, t=2, criterion="maxclust")
# 加上 cluster 欄位並計算每個聚類的平均摘要
df_num["cluster"] = cluster_labels
cluster_summary = df_num.groupby("cluster").mean().add_suffix("_Average")
print(cluster_summary)## Aroma_Average ... Category.One.Defects_Average
## cluster ...
## 1 7.425455 ... 0.636364
## 2 7.760000 ... 0.222222
##
## [2 rows x 10 columns]樹狀圖（Dendrogram）有助於視覺化，並能幫助決定要使用的聚類數量，或是應該在哪個高度進行切割。
import numpy as np
import matplotlib.pyplot as plt
from scipy.cluster.hierarchy import dendrogram, fcluster
# 總樣本數
n_samples = hclust_ward.shape[0] + 1
# ====================== 1. 基礎樹狀圖 + 安全的藍色切割線 ======================
plt.figure(figsize=(10, 6))
dendrogram(hclust_ward)## {'icoord': [[35.0, 35.0, 45.0, 45.0], [25.0, 25.0, 40.0, 40.0], [65.0, 65.0, 75.0, 75.0], [95.0, 95.0, 105.0, 105.0], [85.0, 85.0, 100.0, 100.0], [70.0, 70.0, 92.5, 92.5], [55.0, 55.0, 81.25, 81.25], [32.5, 32.5, 68.125, 68.125], [15.0, 15.0, 50.3125, 50.3125], [5.0, 5.0, 32.65625, 32.65625], [115.0, 115.0, 125.0, 125.0], [135.0, 135.0, 145.0, 145.0], [185.0, 185.0, 195.0, 195.0], [175.0, 175.0, 190.0, 190.0], [165.0, 165.0, 182.5, 182.5], [155.0, 155.0, 173.75, 173.75], [140.0, 140.0, 164.375, 164.375], [120.0, 120.0, 152.1875, 152.1875], [18.828125, 18.828125, 136.09375, 136.09375]], 'dcoord': [[0.0, np.float64(2.8211994490582724), np.float64(2.8211994490582724), 0.0], [0.0, np.float64(3.3416529959187184), np.float64(3.3416529959187184), np.float64(2.8211994490582724)], [0.0, np.float64(1.3541797122858186), np.float64(1.3541797122858186), 0.0], [0.0, np.float64(1.2640743886952885), np.float64(1.2640743886952885), 0.0], [0.0, np.float64(2.428994380285572), np.float64(2.428994380285572), np.float64(1.2640743886952885)], [np.float64(1.3541797122858186), np.float64(2.7691404691045887), np.float64(2.7691404691045887), np.float64(2.428994380285572)], [0.0, np.float64(3.559154299785624), np.float64(3.559154299785624), np.float64(2.7691404691045887)], [np.float64(3.3416529959187184), np.float64(4.403923017090526), np.float64(4.403923017090526), np.float64(3.559154299785624)], [0.0, np.float64(6.027631053839452), np.float64(6.027631053839452), np.float64(4.403923017090526)], [0.0, np.float64(6.590497475713074), np.float64(6.590497475713074), np.float64(6.027631053839452)], [0.0, np.float64(2.2713867810103223), np.float64(2.2713867810103223), 0.0], [0.0, np.float64(1.41825729301792), np.float64(1.41825729301792), 0.0], [0.0, np.float64(1.3122498495396393), np.float64(1.3122498495396393), 0.0], [0.0, np.float64(1.6038764997622879), np.float64(1.6038764997622879), np.float64(1.3122498495396393)], [0.0, np.float64(2.2724840629843372), np.float64(2.2724840629843372), np.float64(1.6038764997622879)], [0.0, np.float64(3.137028030598917), np.float64(3.137028030598917), np.float64(2.2724840629843372)], [np.float64(1.41825729301792), np.float64(4.843050954713168), np.float64(4.843050954713168), np.float64(3.137028030598917)], [np.float64(2.2713867810103223), np.float64(7.30855846442715), np.float64(7.30855846442715), np.float64(4.843050954713168)], [np.float64(6.590497475713074), np.float64(12.194460625748167), np.float64(12.194460625748167), np.float64(7.30855846442715)]], 'ivl': ['8', '16', '19', '2', '3', '6', '5', '15', '1', '11', '17', '4', '13', '0', '9', '14', '12', '7', '10', '18'], 'leaves': [8, 16, 19, 2, 3, 6, 5, 15, 1, 11, 17, 4, 13, 0, 9, 14, 12, 7, 10, 18], 'color_list': ['C1', 'C1', 'C1', 'C1', 'C1', 'C1', 'C1', 'C1', 'C1', 'C1', 'C2', 'C2', 'C2', 'C2', 'C2', 'C2', 'C2', 'C2', 'C0'], 'leaves_color_list': ['C1', 'C1', 'C1', 'C1', 'C1', 'C1', 'C1', 'C1', 'C1', 'C1', 'C1', 'C2', 'C2', 'C2', 'C2', 'C2', 'C2', 'C2', 'C2', 'C2']}# 安全計算 k=2 的切割高度
clusters_k2 = fcluster(hclust_ward, t=2, criterion="maxclust")
dist_vals = np.unique(hclust_ward[:, 2])
dist_vals_sorted = np.sort(dist_vals)
if len(dist_vals_sorted) >= 2:
cut_height = dist_vals_sorted[-2]
else:
cut_height = dist_vals_sorted[-1]
plt.axhline(y=cut_height, c="blue", lw=2, linestyle="--")
plt.title("Ward Dendrogram with Cut Line k=2")
plt.xlabel("Samples")
plt.ylabel("Distance")
plt.xticks(rotation=45, ha="right")## (array([ 5, 15, 25, 35, 45, 55, 65, 75, 85, 95, 105, 115, 125,
## 135, 145, 155, 165, 175, 185, 195]), [Text(5, 0, '8'), Text(15, 0, '16'), Text(25, 0, '19'), Text(35, 0, '2'), Text(45, 0, '3'), Text(55, 0, '6'), Text(65, 0, '5'), Text(75, 0, '15'), Text(85, 0, '1'), Text(95, 0, '11'), Text(105, 0, '17'), Text(115, 0, '4'), Text(125, 0, '13'), Text(135, 0, '0'), Text(145, 0, '9'), Text(155, 0, '14'), Text(165, 0, '12'), Text(175, 0, '7'), Text(185, 0, '10'), Text(195, 0, '18')])
# ====================== 2. 簡單樹狀圖 (移除損壞的分支顏色函數) ======================
plt.figure(figsize=(10, 6))
dendrogram(hclust_ward)## {'icoord': [[35.0, 35.0, 45.0, 45.0], [25.0, 25.0, 40.0, 40.0], [65.0, 65.0, 75.0, 75.0], [95.0, 95.0, 105.0, 105.0], [85.0, 85.0, 100.0, 100.0], [70.0, 70.0, 92.5, 92.5], [55.0, 55.0, 81.25, 81.25], [32.5, 32.5, 68.125, 68.125], [15.0, 15.0, 50.3125, 50.3125], [5.0, 5.0, 32.65625, 32.65625], [115.0, 115.0, 125.0, 125.0], [135.0, 135.0, 145.0, 145.0], [185.0, 185.0, 195.0, 195.0], [175.0, 175.0, 190.0, 190.0], [165.0, 165.0, 182.5, 182.5], [155.0, 155.0, 173.75, 173.75], [140.0, 140.0, 164.375, 164.375], [120.0, 120.0, 152.1875, 152.1875], [18.828125, 18.828125, 136.09375, 136.09375]], 'dcoord': [[0.0, np.float64(2.8211994490582724), np.float64(2.8211994490582724), 0.0], [0.0, np.float64(3.3416529959187184), np.float64(3.3416529959187184), np.float64(2.8211994490582724)], [0.0, np.float64(1.3541797122858186), np.float64(1.3541797122858186), 0.0], [0.0, np.float64(1.2640743886952885), np.float64(1.2640743886952885), 0.0], [0.0, np.float64(2.428994380285572), np.float64(2.428994380285572), np.float64(1.2640743886952885)], [np.float64(1.3541797122858186), np.float64(2.7691404691045887), np.float64(2.7691404691045887), np.float64(2.428994380285572)], [0.0, np.float64(3.559154299785624), np.float64(3.559154299785624), np.float64(2.7691404691045887)], [np.float64(3.3416529959187184), np.float64(4.403923017090526), np.float64(4.403923017090526), np.float64(3.559154299785624)], [0.0, np.float64(6.027631053839452), np.float64(6.027631053839452), np.float64(4.403923017090526)], [0.0, np.float64(6.590497475713074), np.float64(6.590497475713074), np.float64(6.027631053839452)], [0.0, np.float64(2.2713867810103223), np.float64(2.2713867810103223), 0.0], [0.0, np.float64(1.41825729301792), np.float64(1.41825729301792), 0.0], [0.0, np.float64(1.3122498495396393), np.float64(1.3122498495396393), 0.0], [0.0, np.float64(1.6038764997622879), np.float64(1.6038764997622879), np.float64(1.3122498495396393)], [0.0, np.float64(2.2724840629843372), np.float64(2.2724840629843372), np.float64(1.6038764997622879)], [0.0, np.float64(3.137028030598917), np.float64(3.137028030598917), np.float64(2.2724840629843372)], [np.float64(1.41825729301792), np.float64(4.843050954713168), np.float64(4.843050954713168), np.float64(3.137028030598917)], [np.float64(2.2713867810103223), np.float64(7.30855846442715), np.float64(7.30855846442715), np.float64(4.843050954713168)], [np.float64(6.590497475713074), np.float64(12.194460625748167), np.float64(12.194460625748167), np.float64(7.30855846442715)]], 'ivl': ['8', '16', '19', '2', '3', '6', '5', '15', '1', '11', '17', '4', '13', '0', '9', '14', '12', '7', '10', '18'], 'leaves': [8, 16, 19, 2, 3, 6, 5, 15, 1, 11, 17, 4, 13, 0, 9, 14, 12, 7, 10, 18], 'color_list': ['C1', 'C1', 'C1', 'C1', 'C1', 'C1', 'C1', 'C1', 'C1', 'C1', 'C2', 'C2', 'C2', 'C2', 'C2', 'C2', 'C2', 'C2', 'C0'], 'leaves_color_list': ['C1', 'C1', 'C1', 'C1', 'C1', 'C1', 'C1', 'C1', 'C1', 'C1', 'C1', 'C2', 'C2', 'C2', 'C2', 'C2', 'C2', 'C2', 'C2', 'C2']}plt.title("Ward Dendrogram")
plt.xlabel("Samples")
plt.ylabel("Distance")
plt.xticks(rotation=45, ha="right")## (array([ 5, 15, 25, 35, 45, 55, 65, 75, 85, 95, 105, 115, 125,
## 135, 145, 155, 165, 175, 185, 195]), [Text(5, 0, '8'), Text(15, 0, '16'), Text(25, 0, '19'), Text(35, 0, '2'), Text(45, 0, '3'), Text(55, 0, '6'), Text(65, 0, '5'), Text(75, 0, '15'), Text(85, 0, '1'), Text(95, 0, '11'), Text(105, 0, '17'), Text(115, 0, '4'), Text(125, 0, '13'), Text(135, 0, '0'), Text(145, 0, '9'), Text(155, 0, '14'), Text(165, 0, '12'), Text(175, 0, '7'), Text(185, 0, '10'), Text(195, 0, '18')])
2.6 步驟 5：K-means 聚類 (K-means Clustering)
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
from sklearn.cluster import KMeans
from sklearn.decomposition import PCA
# 1. 執行 K-Means 聚類 (相當於 kmeans(df_scaled, centers = 2))
km_model = KMeans(n_clusters=2, random_state=100)
cluster_assign = km_model.fit_predict(df_scaled)
# 印出模型資訊，同等於在 R 中印出 km 物件
print("K-Means Model Output")## K-Means Model Output## Cluster centers:
## [[ 0.55342061 0.65939883 0.68248618 0.61146386 0.62287806 0.42464149
## 0.64082913 0.67537713 -0.15838297 -0.23102817]
## [-0.67640297 -0.80593191 -0.83414977 -0.74734472 -0.76129541 -0.51900626
## -0.7832356 -0.82546094 0.19357919 0.28236777]]## Inertia (total within-cluster sum of squares): 124.3613## Cluster labels for each sample:
## [0 1 1 0 0 0 1 0 1 0 0 1 0 0 0 1 1 1 0 1]# 2. 相當於 fviz_cluster: PCA 2D 投影用於聚類視覺化
# fviz_cluster 會自動使用 PCA 來降維至 2 維
pca = PCA(n_components=2)
pca_result = pca.fit_transform(df_scaled)
pca_df = pd.DataFrame(pca_result, columns=["PC1", "PC2"])
pca_df["cluster"] = cluster_assign
plt.figure(figsize=(8, 5))
sns.scatterplot(data=pca_df, x="PC1", y="PC2", hue="cluster", palette="Set1", s=60)
# 將聚類中心投影至 PCA 空間並繪製
centers_pca = pca.transform(km_model.cluster_centers_)
plt.scatter(centers_pca[:, 0], centers_pca[:, 1], marker="X", c="black", s=200, label="Centers")
plt.legend(title="Clusters")
plt.title("K-Means Clusters (PCA Projection)")
sns.set_style("white") # match theme_minimal()
plt.tight_layout()
plt.show()
# 3. 原始變數 Aroma 與 Aftertaste 的散佈圖，並依聚類著色
df_km = df_num.copy()
df_km["cluster"] = cluster_assign
plt.figure(figsize=(8, 5))
sns.scatterplot(data=df_km, x="Aroma", y="Aftertaste", hue="cluster", palette="Set1", s=60)
plt.legend(title="Clusters")
plt.title("Clusters: Aroma vs Aftertaste")
sns.set_style("white")
plt.tight_layout()
plt.show()
# 4. 聚類平均值摘要表 (dplyr group_by + summarise 所有平均值)
cluster_summary = df_km.groupby("cluster").mean().add_suffix("_Average")
print("\nCluster Variable Average Summary:")##
## Cluster Variable Average Summary:## Aroma_Average ... Category.One.Defects_Average
## cluster ...
## 0 7.72 ... 0.181818
## 1 7.40 ... 0.777778
##
## [2 rows x 10 columns]