Syntax

C#
[SerializableAttribute] public class Covariances

Visual Basic (Declaration)
<SerializableAttribute> _ Public Class Covariances

Visual C++
[SerializableAttribute] public ref class Covariances

Remarks

Class Covariances computes estimates of correlations, covariances, or sums of squares and crossproducts for a data matrix x. Weights and frequencies are allowed but not required.

The means, (corrected) sums of squares, and (corrected) sums of crossproducts are computed using the method of provisional means. Let $x_{ki}$ denote the mean based on i observations for the k-th variable, denote the frequency of the i-th observation, denote the weight of the i-th observations, and $c_{jki}$ denote the sum of crossproducts (or sum of squares if j = k) based on i observations. Then the method of provisional means finds new means and sums of crossproducts as shown in the example below.

The means and crossproducts are initialized as follows:

$x_{k0} = 0.0\,\,\,\,\,for\,\,k = 1,\, \ldots ,\,p$

$c_{jk0} = 0.0\,\,\,for\,\,j,\,k = 1,\, \ldots ,\,p$

where p denotes the number of variables. Letting $x_{k,i+1}$ denote the k-th variable of observation i + 1, each new observation leads to the following updates for $x_{ki}$ and $c_{jki}$ using the update constant $r_{i+1}$ :

$r_{i + 1} = \frac{{f_{i + 1} w_{i + 1} }}{{\sum\limits_{l = 1}^{i + 1} {f_l w_l } }}$

$\bar x_{k,\;i + 1} = \bar x_{ki} + \left( {x_{k,\;i + 1} - \bar x_{ki} } \right)r_{i + 1}$

$c_{jk,\;i + 1} = c_{jki} + f_{i + 1} w_{i + 1} \left( {x_{j,\;i + 1} - \bar x_{ji} } \right)\left( {x_{k,\;i + 1} - \bar x_{ki} } \right)\left( {1 - r_{i + 1} } \right)$

The default value for weights and frequencies is 1. Means and variances are computed based on the valid data for each variable or, if required, based on all the valid data for each pair of variables.

Inheritance Hierarchy

System..::.Object
Imsl.Stat..::.Covariances

Syntax

Remarks

Inheritance Hierarchy

See Also