R做的主成分分析散点图(biplot)如何解释？

实例
某中学随机抽取某年级30名学生，测量其身高，体重，胸围，坐高，针对这30名中学生身体四项指标数据做主成分分析。
student<-data.frame(
X1=c(148, 139, 160, 149, 159, 142, 153, 150, 151, 139,
140, 161, 158, 140, 137, 152, 149, 145, 160, 156,
151, 147, 157, 147, 157, 151, 144, 141, 139, 148),
X2=c(41, 34, 49, 36, 45, 31, 43, 43, 42, 31,
29, 47, 49, 33, 31, 35, 47, 35, 47, 44,
42, 38, 39, 30, 48, 36, 36, 30, 32, 38),
X3=c(72, 71, 77, 67, 80, 66, 76, 77, 77, 68,
64, 78, 78, 67, 66, 73, 82, 70, 74, 78,
73, 73, 68, 65, 80, 74, 68, 67, 68, 70),
X4=c(78, 76, 86, 79, 86, 76, 83, 79, 80, 74,
74, 84, 83, 77, 73, 79, 79, 77, 87, 85,
82, 78, 80, 75, 88, 80, 76, 76, 73, 78)
)
#主成分分析
student.pr <- princomp(student, cor = TRUE)
#显示结果
summary(student.pr, loadings=TRUE)
#预测，显示各样本主成分的值
pre<-predict(student.pr)
#显示碎石图
screeplot(student.pr,type="lines")
# 主成分分析散点图
biplot(student.pr)

碎石图可以帮助选择主成分的个数，，跟上面的累积方差贡献率基本上是一个意思，从你画的碎石土，两个主成分就可以了
双重信息图biplot，表明各个变量在3，5,25,12,13等学生上面表现的强烈，

which(X1==max(X1))
which(X2==max(X2))
which(X3==max(X3))
which(X4==max(X4))
可以查看最大的那个指标是哪个学生

吉林小王子 发表于 2012-3-6 21:00
which(X1==max(X1))
which(X2==max(X2))
which(X3==max(X3))

真是好兄弟！

吉林小王子 发表于 2012-3-6 20:28
碎石图可以帮助选择主成分的个数，，跟上面的累积方差贡献率基本上是一个意思，从你画的碎石土，两个主成分 ...

我反复看，还是看不太懂。
是不是箭头长短代表着主成分矩阵系数的大小？

如上图，在第一主成分轴（横轴）上，四个变量都是负值，所以箭头都在0的左边。而且箭头长短与四个值相一致。
在纵轴（主成分2）上呢，则x1和x4是正，x2和x3是负，这也吻合了。

这个我到没有研究过，你说的有道理，我看了一下，我自己书上的例子也符合你的说法，你观察的很细致啊，治学严谨，佩服啊，佩服。学习了

吉林小王子 发表于 2012-3-6 23:06
这个我到没有研究过，你说的有道理，我看了一下，我自己书上的例子也符合你的说法，你观察的很细致啊，治学 ...

兄弟过奖了，我是百思不得其解，在你的启发下才有所突破的。
但是还有未解的，左下横纵坐标明白了，但右边和上边仍有两个坐标轴，我还没弄明白。

值得好好学习呀

A biplot is constructed by using the singular value decomposition (SVD) to obtain a low-rank approximation to a transformed version of the data matrix X, whose n rows are the samples (also called the cases, or objects), and whose p columns are the variables. The transformed data matrix Y is obtained from the original matrix X by centering and optionally standardizing the columns (the variables). Using the SVD, we can write Y = ∑k=1,...pdkukvkT;, where the uk are n-dimensional column vectors, the vk are p-dimensional column vectors, and the dk are a non-increasing sequence of non-negative scalars. The biplot is formed from two scatterplots that share a common set of axes and have a between-set scalar product interpretation. The first scatterplot is formed from the points (d1αu1i,  d2αu2i), for i = 1,...,n. The second plot is formed from the points (d11-αv1j, d21-αv2j), for j = 1,...,p. This is the biplot formed by the dominant two terms of the SVD, which can then be represented in a two-dimensional display. Typical choices of α are 1 (to give a distance interpretation to the row display) and 0 (to give a distance interpretation to the column display), and in some rare cases α=1/2 to obtain a symmetrically scaled biplot (which gives no distance interpretation to the rows or the columns, but only the scalar product interpretation). The set of points depicting the variables can be drawn as arrows from the origin to reinforce the idea that they represent biplot axes onto which the samples can be projected to approximate the original data.
按照上面的试了一下，跟这个不一样，你再琢磨一下

Read more: http://www.answers.com/topic/biplot#ixzz1oOSmXsFe

吉林小王子 发表于 2012-3-7 10:20
A biplot is constructed by using the singular value decomposition (SVD) to obtain a low-rank approxi ...

多谢小王子！