.. _matrixStats:

Statistics on Matrices
==========================

.. include:: ../replace.txt


Column Statistics
------------------

When a matrix is passed to a statistical function, the default is
to work on each column of the matrix independently. The result is a row
vector, which may in turn be used with other statistical functions if
desired.

.. figure:: columnMean.png
   :align: center
   :width: 15%

Some functions return multiple values for each column. For example, the
``diff`` function calculates the difference between adjacent elements in
each column of the input matrix.  The ``cumsum`` function also returns a
matrix rather than a row vector.

.. figure:: columnDiff.png
   :align: center
   :width: 15%

Changing Dimension
-------------------

Many statistical functions accept an optional dimension argument that
specifies whether the operation should be applied to the columns
independently (the default) or to the rows.

If matrix :math:`\bf{A}` has size :math:`10{\times}20`, then ``mean(A,2)``
returns a 10 element column vector; whereas, ``mean(A)`` returns a 20 element
row vector.

::

    >> A = randi(25,10,20);
    >> mC = mean(A);
    >> mr = mean(A,2);
    >> whos
    Name       Size            Bytes  Class     Attributes

    A         10x20             1600  double              
    mC         1x20              160  double              
    mr        10x1                80  double    

Some functions, such as ``min``, ``max``, and ``diff``, use the second argument
for other purposes, which makes dimension the third argument. To
skip the second argument, use a pair of empty square brackets for an empty
vector, ``[]``.

::

    >> Amin = min(A,[],2);

.. _covariance:

Covariance and Correlation
---------------------------

.. index:: covariance, correlation

**Covariance** shows how distinct variables relate to each
other.  It is calculated in the same manner that variance is calculated for a
single variable. Variance (square of the standard deviation) is the expected
value of the squared difference between each sample and the mean of the
variable. Note that covariance and correlation can be defined either
in terms of populations or samples. We mostly use the notion for sampled data. 

.. math:: \sigma_x^2 = E(X - \mu_x)^2

.. math:: s_x^2 = \frac{1}{n - 1} \sum_{i = 1}^n (x_i - \bar{x})^2

Similarly, the covariance between two variables is the product of the
differences between samples and their respective means.

.. math:: \sigma_{xy} = E[(X - \mu_x)(Y - \mu_y)] 

.. math:: S_{xy} = 
        \frac{1}{n - 1} \sum_{i = 1}^n (x_i - \bar{x})(y_i - \bar{y})

Thus, the covariance between a variable and itself is its variance,
:math:`s_{xx} = s_x^2`. Covariance is represented with a
symmetric matrix because :math:`s_{xy} = s_{yx}`.
The variances of each variable will be on the diagonal of the matrix.

For example, consider taking a sampling of the age, height, and weight of 
:math:`n` children. We could construct a covariance matrix as follows.

.. math:: \mathbf{Covariance}(a, h, w) = 
                   \mat{s_{aa}, s_{ah}, s_{aw};
                        s_{ha}, s_{hh}, s_{hw};
                        s_{wa}, s_{wh}, s_{ww}}

The **correlation coefficient** of two variables is a measure of their
linear dependence. 

.. math:: r_{xy} = \frac{s_{xy}}{s_x s_y}

A matrix of correlation coefficients has ones on the diagonal since variables
are directly correlated to themselves. Correlation values near zero indicate
that the variables are mostly independent of each other, while correlation
values near one or negative one indicate positive or negative correlation
relationships. Correlation coefficients are generally more useful than
covariance values because they are scaled to always have the same range
(:math:`-1 \leq r \leq 1`).

.. math:: \mathbf{R}(a, h, w) = \mat{1, r_{ah}, r_{aw};
                        r_{ha}, 1, r_{hw};
                        r_{wa}, r_{wh}, 1}

The |M| functions to compute the covariance and correlation coefficient
matrices are ``cov`` and ``corrcoef``.  In the following example, matrix
:math:`\bf{A}` has 100 random numbers in each of two columns. Half of the value
of the second column come from the first column and half come from another
random number generator.  The variance of each column is on the diagonal of the
covariance matrix. The covariance between the two columns is on the
off--diagonal.  The off--diagonal of the matrix of correlation coefficients
shows that the two columns have a positive correlation.

::

    >> A = 10*randn(100, 1);
    >> A(:,2) = 0.5*A(:,1) + 5*randn(100, 1);

    >> Acov = cov(A)
    Acov =
       94.1505   50.0808
       50.0808   50.6121

    >> Acorr = corrcoef(A)
    Acorr =
        1.0000    0.7255
        0.7255    1.0000