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covarmtk

Calculate the covariance of two one-dimensional ndarrays 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-base-ndarray-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-base-ndarray-covarmtk' );

covarmtk( arrays )

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

var scalar2ndarray = require( '@stdlib/ndarray-from-scalar' );
var ndarray = require( '@stdlib/ndarray-base-ctor' );

var opts = {
    'dtype': 'generic'
};

var xbuf = [ 1.0, -2.0, 2.0 ];
var x = new ndarray( opts.dtype, xbuf, [ 3 ], [ 1 ], 0, 'row-major' );

var ybuf = [ 2.0, -2.0, 1.0 ];
var y = new ndarray( opts.dtype, ybuf, [ 3 ], [ 1 ], 0, 'row-major' );

var correction = scalar2ndarray( 1.0, opts );
var meanx = scalar2ndarray( 1.0/3.0, opts );
var meany = scalar2ndarray( 1.0/3.0, opts );

var v = covarmtk( [ x, y, correction, meanx, meany ] );
// returns ~3.8333

The function has the following parameters:

• arrays: array-like object containing the following ndarrays in order:

  1. first one-dimensional input ndarray.
  2. second one-dimensional input ndarray.
  3. a zero-dimensional ndarray specifying the 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 and N corresponds to the number of elements in each input ndarray. 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).
  4. a zero-dimensional ndarray specifying the mean of the first one-dimensional ndarray.
  5. a zero-dimensional ndarray specifying the mean of the second one-dimensional ndarray.

Notes

• Both input ndarrays should have the same number of elements.
• If provided empty one-dimensional ndarrays, the function returns NaN.

Examples

var discreteUniform = require( '@stdlib/random-array-discrete-uniform' );
var ndarray = require( '@stdlib/ndarray-base-ctor' );
var ndarray2array = require( '@stdlib/ndarray-to-array' );
var scalar2ndarray = require( '@stdlib/ndarray-from-scalar' );
var covarmtk = require( '@stdlib/stats-base-ndarray-covarmtk' );

// Define array options:
var opts = {
    'dtype': 'generic'
};

// Create one-dimensional ndarrays containing pseudorandom numbers:
var xbuf = discreteUniform( 10, -50, 50, opts );
var x = new ndarray( opts.dtype, xbuf, [ xbuf.length ], [ 1 ], 0, 'row-major' );
console.log( ndarray2array( x ) );

var ybuf = discreteUniform( 10, -50, 50, opts );
var y = new ndarray( opts.dtype, ybuf, [ xbuf.length ], [ 1 ], 0, 'row-major' );
console.log( ndarray2array( y ) );

// Specify the degrees of freedom adjustment:
var correction = scalar2ndarray( 1.0, opts );

// Specify the known means:
var meanx = scalar2ndarray( 0.0, opts );
var meany = scalar2ndarray( 0.0, opts );

// Calculate the sample covariance:
var v = covarmtk( [ x, y, correction, meanx, meany ] );
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

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