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README.md

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.

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 );