# Statistics notes

From raju

#### Using svd to get principal components

Here are some useful Matlab commands to experiment.

```>> x = rand(100,4); c=x'*x, [u, s, v] = svd(x); sum(diag(c)), pca=u*s; c2 = (pca)'* (pca), sum(diag(c2))

c =
32.3085   25.3927   24.2248   22.5342
25.3927   36.5004   25.6572   24.9433
24.2248   25.6572   31.2583   23.5037
22.5342   24.9433   23.5037   29.5961
ans =
129.6633
c2 =
105.7363    0.0000   -0.0000    0.0000
0.0000    9.0590   -0.0000    0.0000
-0.0000   -0.0000    8.1320   -0.0000
0.0000    0.0000   -0.0000    6.7361
ans =
129.6633

>> subplot(2,2,1); plot(pca(:,1), 'x-'); subplot(2,2,2); plot(pca(:,2), 'x-'); subplot(2,2,3); plot(pca(:,3), 'x-'); subplot(2,2,4); plot(pca(:,4), 'x-')
```

Notice how sum of the variances of x-variables is same as sum of variances of the pca variables. While the original x-variables have covariances, the pca are mutually orthogonal (i.e. pca covariance matrix is diagonal).

#### Correlation coefficient for multiple variables

The multiple correlation coefficient generalizes the standard coefficient of correlation for multiple variables. For details see [1]. This article can also be found in the "Multiple Correlation Coefficient" section in [2].

Ref:

1. https://www.utd.edu/~herve/Abdi-MCC2007-pretty.pdf
2. Encyclopedia of Measurement and Statistics edited by Neil J. Salkind, 2007.

#### correlation coefficient

• Pearson correlation coefficient
measures linear correlation between two variables
https://en.wikipedia.org/wiki/Pearson_correlation_coefficient
• Spearman's rank correlation coefficient
Pearson correlation coefficient between the rank variables
measures monotonic correlation between two variables
https://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient
• Kendall rank correlation coefficient
https://en.wikipedia.org/wiki/Kendall_rank_correlation_coefficient

#### Type I and Type II errors

Action ${\displaystyle H_{0}}$ is actually ...
True False
Do not reject ${\displaystyle H_{0}}$ Correct outcome Type II error
Reject ${\displaystyle H_{0}}$ Type I error Correct outcome