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covarmtk

Calculate the covariance of two strided arrays provided known means and using a one-pass textbook algorithm.

The population covariance of two finite size populations of size N is given by

$$\mathop{\mathrm{cov_N}} = \frac{1}{N} \sum_{i=0}^{N-1} (x_i - \mu_x)(y_i - \mu_y)$$

where the population means are given by

$$\mu_x = \frac{1}{N} \sum_{i=0}^{N-1} x_i$$

and

$$\mu_y = \frac{1}{N} \sum_{i=0}^{N-1} y_i$$

Often in the analysis of data, the true population covariance is not known a priori and must be estimated from samples drawn from population distributions. If one attempts to use the formula for the population covariance, the result is biased and yields a biased sample covariance. To compute an unbiased sample covariance for samples of size n,

$$\mathop{\mathrm{cov_n}} = \frac{1}{n-1} \sum_{i=0}^{n-1} (x_i - \bar{x}_n)(y_i - \bar{y}_n)$$

where sample means are given by

$$\bar{x} = \frac{1}{n} \sum_{i=0}^{n-1} x_i$$

and

$$\bar{y} = \frac{1}{n} \sum_{i=0}^{n-1} y_i$$

The use of the term n-1 is commonly referred to as Bessel's correction. Depending on the characteristics of the population distributions, other correction factors (e.g., n-1.5, n+1, etc) can yield better estimators.

Installation

npm install @stdlib/stats-strided-covarmtk

Alternatively,

• To load the package in a website via a script tag without installation and bundlers, use the ES Module available on the esm branch (see README).
• If you are using Deno, visit the deno branch (see README for usage intructions).
• For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

Usage

var covarmtk = require( '@stdlib/stats-strided-covarmtk' );

covarmtk( N, correction, meanx, x, strideX, meany, y, strideY )

Computes the covariance of two strided arrays provided known means and using a one-pass textbook algorithm.

var x = [ 1.0, -2.0, 2.0 ];
var y = [ 2.0, -2.0, 1.0 ];

var v = covarmtk( x.length, 1, 1.0/3.0, x, 1, 1.0/3.0, y, 1 );
// returns ~3.8333

The function has the following parameters:

• N: number of indexed elements.
• correction: degrees of freedom adjustment. Setting this parameter to a value other than 0 has the effect of adjusting the divisor during the calculation of the covariance according to N-c where c corresponds to the provided degrees of freedom adjustment. When computing the population covariance, setting this parameter to 0 is the standard choice (i.e., the provided arrays contain data constituting entire populations). When computing the unbiased sample covariance, setting this parameter to 1 is the standard choice (i.e., the provided arrays contain data sampled from larger populations; this is commonly referred to as Bessel's correction).
• meanx: mean of x.
• x: first input Array or typed array.
• strideX: stride length for x.
• meany: mean of y.
• y: second input Array or typed array.
• strideY: stride length for y.

The N and stride parameters determine which elements in the strided arrays are accessed at runtime. For example, to compute the covariance of every other element in x and y,

var x = [ 1.0, 2.0, 2.0, -7.0, -2.0, 3.0, 4.0, 2.0 ];
var y = [ 2.0, 1.0, 2.0, 1.0, -2.0, 2.0, 3.0, 4.0 ];

var v = covarmtk( 4, 1, 1.25, x, 2, 1.25, y, 2 );
// returns 5.25

Note that indexing is relative to the first index. To introduce an offset, use typed array views.

var Float64Array = require( '@stdlib/array-float64' );

var x0 = new Float64Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] );
var y0 = new Float64Array( [ 2.0, -2.0, 2.0, 1.0, -2.0, 4.0, 3.0, 2.0 ] );

var x1 = new Float64Array( x0.buffer, x0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
var y1 = new Float64Array( y0.buffer, y0.BYTES_PER_ELEMENT*1 ); // start at 2nd element

var v = covarmtk( 4, 1, 1.25, x1, 2, 1.25, y1, 2 );
// returns ~1.9167

covarmtk.ndarray( N, correction, meanx, x, strideX, offsetX, meany, y, strideY, offsetY )

Computes the covariance of two strided arrays provided known means and using a one-pass textbook algorithm and alternative indexing semantics.

var x = [ 1.0, -2.0, 2.0 ];
var y = [ 2.0, -2.0, 1.0 ];

var v = covarmtk.ndarray( x.length, 1, 1.0/3.0, x, 1, 0, 1.0/3.0, y, 1, 0 );
// returns ~3.8333

The function has the following additional parameters:

• offsetX: starting index for x.
• offsetY: starting index for y.

While typed array views mandate a view offset based on the underlying buffer, the offset parameters support indexing semantics based on starting indices. For example, to calculate the covariance for every other element in x and y starting from the second element

var x = [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ];
var y = [ -7.0, 2.0, 2.0, 1.0, -2.0, 2.0, 3.0, 4.0 ];

var v = covarmtk.ndarray( 4, 1, 1.25, x, 2, 1, 1.25, y, 2, 1 );
// returns 6.0

Notes

• If N <= 0, both functions return NaN.
• If N - c is less than or equal to 0 (where c corresponds to the provided degrees of freedom adjustment), both functions return NaN.
• Both functions support array-like objects having getter and setter accessors for array element access (e.g., @stdlib/array-base/accessor).
• Depending on the environment, the typed versions (dcovarmtk, scovarmtk, etc.) are likely to be significantly more performant.

Examples

var discreteUniform = require( '@stdlib/random-array-discrete-uniform' );
var covarmtk = require( '@stdlib/stats-strided-covarmtk' );

var opts = {
    'dtype': 'generic'
};
var x = discreteUniform( 10, -50, 50, opts );
console.log( x );

var y = discreteUniform( 10, -50, 50, opts );
console.log( y );

var v = covarmtk( x.length, 1, 0.0, x, 1, 0.0, y, 1 );
console.log( v );

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Notice

This package is part of stdlib, a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop stdlib, see the main project repository.

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License

See LICENSE.

Copyright

Copyright © 2016-2025. The Stdlib Authors.