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I have a problem trying to find a combination (or weighted average) of variables (statistics) that best explains the sample statistics. A – n x p matrix (n: observations p: variables, here are statistics for each observation)

The steps used are:

  1. Use correlation matrix C of A to get eigenvectors $V (V1, V2,\cdots,Vp)$ and eigenvalues $D(D1,D2,\cdots,Dp)$.

  2. Then pick up top 3 eigenvalues and eigenvectors $(V1, V2, V3)$ and $(D1,D2,D3)$ which explain 85% of the variance for the example. Transform $V1$ to $V1^*$ as: $V1^* = (\frac{v11}{\sum_1^pvp1}, \frac{v21}{\sum_1^pvp1}, \frac{vp1}{\sum_1^pvp1})$

  3. Then form a vector using these 3 as: $V*=\frac{D1}{(D1+D2+D3)}\cdot V1^*+\frac{D2}{(D1+D2+D3)}\cdot V2^*+\frac{D3}{(D1+D2+D3)}\cdot V3^*$

So $V^*$ is a $p \times 1$ vector and used as weights for these variables (statistics) assuming this will give the best explanation of the variance.

Please see attached image if the formula is not clear. Thanksenter image description here

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    $\begingroup$ What's the question? I don't see a question mark anywhere $\endgroup$
    – user20160
    Jun 15, 2018 at 9:51
  • $\begingroup$ At the end of title, maybe? $\endgroup$ Jun 15, 2018 at 10:56
  • $\begingroup$ Please format the question properly. Use bold or slant style for text to highlight the important point to be visible at one sight. $\endgroup$
    – ironman
    Jun 15, 2018 at 10:58

1 Answer 1

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Problem of finding linear combination of variables that best explains them (= has the largest possible variance) is well known and solution for it is known as Principal Component Analysis (PCA).

It can be shown (by Lagrange multipliers, for instance) that eigenvector coresponding to largest eigenvalue gives coeficients of desired linear combination.

So, any other linear combination (including $V^*$, you described) has variance lower or equal to that given by eigenvector coresponding to largest eigenvalue.

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