使用ANOVA检验比较含KPCA预处理的聚类算法性能，如何选输入数据？

Great question! When using ANOVA to compare your four clustering configurations (k-means++ with/without KPCA, hierarchical agglomerative clustering with/without KPCA), you need two key components: a numerical performance metric (the dependent variable) and group labels (the independent variable). Here's a step-by-step breakdown of what to extract and how to structure your data:

Step 1: Choose a Clustering Performance Metric #

First, pick one (or more, run ANOVA separately for each) metric from scikit-learn that quantifies how well each clustering run performed. Your choice depends on whether you have ground truth labels for your dataset:

If you have ground truth labels (external metrics):

• Adjusted Rand Index (adjusted_rand_score): Measures similarity between cluster assignments and true labels, adjusted for random chance. Ranges from -1 (worst) to 1 (perfect).
• Normalized Mutual Information (normalized_mutual_info_score): Quantifies shared information between clusters and true labels, normalized to a 0-1 scale. Higher values mean better alignment.
• Adjusted Mutual Information (adjusted_mutual_info_score): Similar to NMI but adjusted for chance, making it more reliable for small datasets.

If you don't have ground truth labels (internal metrics):

• Silhouette Score (silhouette_score): Evaluates how similar each data point is to its own cluster versus other clusters. Ranges from -1 (poor clustering) to 1 (excellent clustering).
• Calinski-Harabasz Index (calinski_harabasz_score): Ratio of between-cluster variance to within-cluster variance. Higher scores indicate more distinct clusters.
• Davies-Bouldin Index (davies_bouldin_score): Average similarity between each cluster and its most similar neighbor. Lower values mean better clustering (you might invert this metric to make higher scores better for easier ANOVA interpretation).

Step 2: Generate Replicate Scores for Each Group #

ANOVA needs multiple data points per group to assess variance within groups. Here's how to get them for each of your four configurations:

For k-means++ (stochastic algorithm):

Since k-means++ uses random centroid initialization, run it 10-30 times with different random_state values. For each run:

1. Apply KPCA (if part of the group) or use raw data.
2. Fit the k-means++ model and get cluster labels.
3. Compute your chosen performance metric.
   Store all these scores as a list for the group.

For Hierarchical Agglomerative Clustering (deterministic algorithm):

This algorithm doesn't have random initialization, so use bootstrapping to generate replicate scores:

1. For each replicate, create a bootstrap sample (random sample with replacement from your original dataset).
2. Apply KPCA (if part of the group) to the bootstrap sample.
3. Fit hierarchical clustering (keep parameters like linkage method and cluster count fixed) and get labels.
4. Compute the performance metric using the bootstrap sample's data/labels (or ground truth if available).
   Repeat this 10-30 times to build a list of scores for the group.

Step 3: Structure Data for ANOVA #

Once you have four lists of scores (one per group), format them for ANOVA analysis in Python. Here are two common approaches:

Using scipy.stats.f_oneway (simple one-way ANOVA):

from scipy.stats import f_oneway

# Replace these with your actual score lists
kmeans_nokpca = [0.72, 0.75, 0.71, 0.73, 0.74]
kmeans_kpca = [0.81, 0.79, 0.83, 0.80, 0.82]
hier_nokpca = [0.68, 0.70, 0.67, 0.69, 0.66]
hier_kpca = [0.76, 0.74, 0.78, 0.75, 0.77]

# Run ANOVA
f_stat, p_val = f_oneway(kmeans_nokpca, kmeans_kpca, hier_nokpca, hier_kpca)
print(f"F-statistic: {f_stat:.2f}, p-value: {p_val:.4f}")

Using statsmodels (detailed ANOVA table):

If you want more context (like sum of squares, degrees of freedom), use pandas and statsmodels to structure your data into a DataFrame:

import pandas as pd
import statsmodels.api as sm
from statsmodels.formula.api import ols

# Combine scores and group labels into a DataFrame
data = {
    'score': kmeans_nokpca + kmeans_kpca + hier_nokpca + hier_kpca,
    'group': ['kmeans_nokpca']*5 + ['kmeans_kpca']*5 + ['hier_nokpca']*5 + ['hier_kpca']*5
}
df = pd.DataFrame(data)

# Fit ANOVA model
model = ols('score ~ C(group)', data=df).fit()
anova_table = sm.stats.anova_lm(model, typ=2)
print(anova_table)

Quick Notes: #

• Check ANOVA Assumptions: ANOVA assumes your scores are normally distributed within each group and have homogeneous variances. Test normality with the Shapiro-Wilk test and variance homogeneity with Levene's test. If assumptions are violated, use the non-parametric Kruskal-Wallis test instead.
• Post-Hoc Tests: If ANOVA returns a significant p-value (p < 0.05), use post-hoc tests like Tukey's HSD to find which specific groups differ significantly.

内容的提问来源于stack exchange，提问作者Beg