PCA空间和“特征空间”分歧中的质心距离计算

Question

我正在测量跨PCA空间和'特征空间'的〜20种治疗和3组的质心。如果我正确理解我的数学老师，他们之间的距离应该是相同的。然而，按照我计算的方式，他们不是，我想知道如果我做数学的方式，他们中的任何一个都是错误的。

我使用的是臭名昭著的葡萄酒的数据集作为说明我的方法/ MWE：

library(ggbiplot) 
data(wine) 
treatments <- 1:2 #treatments to be considerd for this calculation 
wine.pca <- prcomp(wine[treatments], scale. = TRUE) 
#calculate the centroids for the feature/treatment space and the pca space 
df.wine.x <- as.data.frame(wine.pca$x) 
df.wine.x$groups <- wine.class 
wine$groups <- wine.class 
feature.centroids <- aggregate(wine[treatments], list(Type = wine$groups), mean) 
pca.centroids <- aggregate(df.wine.x[treatments], list(Type = df.wine.x$groups), mean) 
pca.centroids 
feature.centroids 
#calculate distance between the centroids of barolo and grignolino 
dist(rbind(feature.centroids[feature.centroids$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") 

的最后两行中的PCA空间内的功能空间和1.80717的距离返回1.468087，表明有美中不足...

Answer 1

这是因为缩放和居中，如果你不做缩放和居中的距离将原来和PCA特征空间完全相同。

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 1.468087 
dist(rbind(pca.centroids[pca.centroids$Type == "barolo",][-1],pca.centroids[pca.centroids$Type == "grignolino",][-1]), method = "euclidean") 
#   1 
# 2 1.468087 

另一种方法是，以获得相同的结果是缩放/中心的原始数据，然后用缩放应用PCA /定心类似如下：

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 1.80717 
dist(rbind(pca.centroids[pca.centroids$Type == "barolo",][-1],pca.centroids[pca.centroids$Type == "grignolino",][-1]), method = "euclidean") 
#  1 
# 2 1.80717