我正在测量一个PCA空间和一个’特征空间’的质心，这两个空间涵盖了大约20种处理和3个组。如果我正确理解了我数学老师的讲解，这两个空间之间的距离应该是相同的。然而，我计算的结果显示它们并不相同，我想知道我的计算方法是否有误。
我使用著名的葡萄酒数据集作为我的方法/最小工作示例的说明：
library(ggbiplot)data(wine)treatments <- 1:2 #treatments to be considerd for this calculationwine.pca <- prcomp(wine[treatments], scale. = TRUE)#calculate the centroids for the feature/treatment space and the pca spacedf.wine.x <- as.data.frame(wine.pca$x)df.wine.x$groups <- wine.classwine$groups <- wine.classfeature.centroids <- aggregate(wine[treatments], list(Type = wine$groups), mean)pca.centroids <- aggregate(df.wine.x[treatments], list(Type = df.wine.x$groups), mean)pca.centroidsfeature.centroids#calculate distance between the centroids of barolo and grignolinodist(rbind(feature.centroids[feature.centroids$Type == "barolo",][-1],feature.centroids[feature.centroids$Type == "grignolino",][-1]), method = "euclidean")dist(rbind(pca.centroids[pca.centroids$Type == "barolo",][-1],pca.centroids[pca.centroids$Type == "grignolino",][-1]), method = "euclidean")最后两行返回的特征空间距离为1.468087，而PCA空间内的距离为1.80717，这表明其中可能存在问题…
回答：
这是因为缩放和中心化的问题，如果你不进行缩放和中心化，原始特征空间和PCA特征空间的距离将完全相同。
wine.pca <- prcomp(wine[treatments], scale=FALSE, center=FALSE)dist(rbind(feature.centroids[feature.centroids$Type == "barolo",][-1],feature.centroids[feature.centroids$Type == "grignolino",][-1]), method = "euclidean")# 1# 2 1.468087dist(rbind(pca.centroids[pca.centroids$Type == "barolo",][-1],pca.centroids[pca.centroids$Type == "grignolino",][-1]), method = "euclidean")# 1# 2 1.468087另一种获得相同结果的方法是先对原始数据进行缩放/中心化处理，然后再应用带有缩放/中心化的PCA，如下所示：
wine[treatments] <- scale(wine[treatments], center = TRUE)wine.pca <- prcomp(wine[treatments], scale = TRUE)dist(rbind(feature.centroids[feature.centroids$Type == "barolo",][-1],feature.centroids[feature.centroids$Type == "grignolino",][-1]), method = "euclidean")# 1# 2 1.80717dist(rbind(pca.centroids[pca.centroids$Type == "barolo",][-1],pca.centroids[pca.centroids$Type == "grignolino",][-1]), method = "euclidean")# 1# 2 1.80717