# Statistics notes

From raju

## Contents

#### 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:

- https://www.utd.edu/~herve/Abdi-MCC2007-pretty.pdf
- 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

#### Type I and Type II errors

Action | [math]H_0[/math] is actually | ... |
---|---|---|

True | False | |

Do not reject [math]H_0[/math] | Correct outcome | Type II error |

Reject [math]H_0[/math] | Type I error | Correct outcome |