请教一个Princomp过程中的问题

shiyiming

应用sas进行主成分分析的时候，有几个参数我不明白，请大家帮忙解答一下：
1、NOINT 解释为模型中不使用截距。主成分分析仅仅是求一个原始变量线性组 合，与截距何干？
2、singual p 为奇异检验标准。何为奇异检验，何用？
3、在sas网上帮助文件princomp过程的overview中提到一个Y = XB + E模型，我不大懂，能解释一下吗？主成分分析不是PLS，与拟和有什么关系呢？

shiyiming

Without going into SAS specific interpretation, I would like to to make a heuristic explaination.

1、NOINT 解释为模型中不使用截距。主成分分析仅仅是求一个原始变量线性组 合，与截距何干？

Without intersection, COMP=ax+by
With intersection, COMP=ax+by+c

2、singual p 为奇异检验标准。何为奇异检验，何用？

The principal components are supposed to be independent to each other, the covariace matrix is supposed to be of full rank.  But, the concept of full rank is a qualitative measurement, not quantitive.  Singular value test test the hypotheses that some of the singular values to be zero.  In the other words, test the independence of these principal components.  Quite often, the test is performed on the ratio of the smallest singular value to the largest.  As for difference between sigular value to eigenvalue, please check the textbook of the Linear Algebra.

3、在sas网上帮助文件princomp过程的overview中提到一个Y = XB + E模型，我不大懂，能解释一下吗？主成分分析不是PLS，与拟和有什么关系呢？

I think it is related to the first question.

shiyiming

但是，princomp的结果中没有截距的数值，怎样显示呢？加入截距对数据的处理又有什么好处了？是如何推导含截距的主成分分析的公式吗？
多谢！

shiyiming

I checked the SAS online documentation, it says that
-------------------------
NOINT
omits the intercept from the model. In other words, the NOINT option requests that the covariance or correlation matrix not be corrected for the mean. When you use the PRINCOMP procedure with the NOINT option, the covariance matrix and, hence, the standard deviations are not corrected for the mean. If you are interested in the standard deviations corrected for the mean, you can get them by using a procedure such as the MEANS procedure.
--------------------------
As matter of the fact, intercept in the formula is in fact the correction of the mean referred here.  There is no advantage or disadvantage to include or to exclude the intercept, it has impact on how do you interpret your results.  I thinkk NOINT is rarely used in analysis.

As for the third question, I also checked the documentation, it might be a different problem.  It says that a subset of principal components can form a least square solution to  the euqation Y=XB+E.  It has a relationship to the Moore-Penrose Generlized Inverse, if you are interested in it.

shiyiming

xic，感谢的热心解答，但我还想同你继续讨论：
应用Princomp过程进行主成分分析时并没有使用noint这个参数
如：proc princomp data=***;
      val  ***;
      run;
所以是不是可以理解为“With intersection, COMP=ax+by+c”，但我的手工进行计算(并没有加入截距c）的结果且与sas结果相同。 这又说明什么呢？
帮助文件中的解释“The NOINT option omits the intercept from the model”，其中所指的模型（model）是不是主成分回归模型哟？
对于同一数据集我应用noint参数和不使用noint参数的结果是不同的（这是当然），可以确定上程序中的确是应用了截距的，真不知截距到底指什么？

shiyiming

I think when you did it by hand, you have already substract the mean to calculate the covariance matrix, it is equivalent to add an intercept term.  Substract the mean is required in most of statistical textbook, I think.

shiyiming

我的确是减了均值的，也许正如你所说的，其实这样已经加入了截距！
我想再去验证一下没有截距的情况，再同你讨论。

shiyiming

主成分分析的第一步就是将 原始变量 进行标准化！

shiyiming

不加截距的主成分分析是如何做的？