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# sampleCovarianceMatrix -- sample covariance matrix of observation vectors

- Usage:
    `sampleCovarianceMatrix U`
- Inputs:
  - U, a matrix, or List of sample data
- Outputs:
  - a matrix, sample covariance matrix of the sample data

## Description

The sample covariance matrix is $S = \frac{1}{n} \sum_{i=1}^{n} (X^{(i)}-\bar{X}) (X^{(i)}-\bar{X})^T$. Note that for normally distributed random variables, $S$ is the maximum likelihood estimator (MLE) for the covariance matrix. This is different from the unbiased estimator, which uses a denominator of $n-1$ instead of $n$.

Sample data is input as a matrix or a list. The rows of the matrix or the elements of the list are observation vectors.

    i1 : L= {{1,2,1,-1},{2,1,3,0},{-1, 0, 1, 1},{-5, 3, 4, -6}};

    i2 : sampleCovarianceMatrix(L)

    o2 = | 115/16 -13/8 -29/16 47/8  |
         | -13/8  5/4   7/8    -11/4 |
         | -29/16 7/8   27/16  -21/8 |
         | 47/8   -11/4 -21/8  29/4  |

                  4       4
    o2 : Matrix QQ  <-- QQ

    i3 : U= matrix{{1,2,1,-1},{2,1,3,0},{-1, 0, 1, 1},{-5, 3, 4, -6}};

                  4       4
    o3 : Matrix ZZ  <-- ZZ

    i4 : sampleCovarianceMatrix(U)

    o4 = | 115/16 -13/8 -29/16 47/8  |
         | -13/8  5/4   7/8    -11/4 |
         | -29/16 7/8   27/16  -21/8 |
         | 47/8   -11/4 -21/8  29/4  |

                  4       4
    o4 : Matrix QQ  <-- QQ

## Ways to use sampleCovarianceMatrix:

- `sampleCovarianceMatrix(List)`
- `sampleCovarianceMatrix(Matrix)`

## For the programmer

The object sampleCovarianceMatrix is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/GraphicalModelsMLE.m2:953:0.