incrmcovariance

Compute a moving unbiased sample covariance incrementally.

For unknown population means, the unbiased sample covariance for a window n of size W is defined as

normal c normal o normal v Subscript normal n Baseline equals StartFraction 1 Over n minus 1 EndFraction sigma-summation Underscript i equals j Overscript j plus upper W minus 1 Endscripts left-parenthesis x Subscript i Baseline minus x overbar Subscript n Baseline right-parenthesis left-parenthesis y Subscript i Baseline minus y overbar Subscript n Baseline right-parenthesis

where j specifies the index of the value at which the window begins. For example, for a trailing (i.e., non-centered) window using zero-based indexing and j greater than or equal to W, j is the n-Wth value with n being the number of values thus analyzed.

For known population means, the unbiased sample covariance for a window n of size W is defined as

normal c normal o normal v Subscript normal n Baseline equals StartFraction 1 Over n EndFraction sigma-summation Underscript i equals j Overscript j plus upper W minus 1 Endscripts left-parenthesis x Subscript i Baseline minus mu Subscript x Baseline right-parenthesis left-parenthesis y Subscript i Baseline minus mu Subscript y Baseline right-parenthesis

Usage

var incrmcovariance = require( '@stdlib/stats/incr/mcovariance' );

incrmcovariance( window[, mx, my] )

Returns an accumulator function which incrementally computes a moving unbiased sample covariance. The window parameter defines the number of values over which to compute the moving unbiased sample covariance.

var accumulator = incrmcovariance( 3 );

If means are already known, provide mx and my arguments.

var accumulator = incrmcovariance( 3, 5.0, -3.14 );

accumulator( [x, y] )

If provided input values x and y, the accumulator function returns an updated unbiased sample covariance. If not provided input values x and y, the accumulator function returns the current unbiased sample covariance.

var accumulator = incrmcovariance( 3 );

var v = accumulator();
// returns null

// Fill the window...
v = accumulator( 2.0, 1.0 ); // [(2.0, 1.0)]
// returns 0.0

v = accumulator( -5.0, 3.14 ); // [(2.0, 1.0), (-5.0, 3.14)]
// returns ~-7.49

v = accumulator( 3.0, -1.0 ); // [(2.0, 1.0), (-5.0, 3.14), (3.0, -1.0)]
// returns -8.35

// Window begins sliding...
v = accumulator( 5.0, -9.5 ); // [(-5.0, 3.14), (3.0, -1.0), (5.0, -9.5)]
// returns -29.42

v = accumulator( -5.0, 1.5 ); // [(3.0, -1.0), (5.0, -9.5), (-5.0, 1.5)]
// returns -24.5

v = accumulator();
// returns -24.5

Notes

  • Input values are not type checked. If provided NaN or a value which, when used in computations, results in NaN, the accumulated value is NaN for at least W-1 future invocations. If non-numeric inputs are possible, you are advised to type check and handle accordingly before passing the value to the accumulator function.
  • As W (x,y) pairs are needed to fill the window buffer, the first W-1 returned values are calculated from smaller sample sizes. Until the window is full, each returned value is calculated from all provided values.

Examples

var randu = require( '@stdlib/random/base/randu' );
var incrmcovariance = require( '@stdlib/stats/incr/mcovariance' );

var accumulator;
var x;
var y;
var i;

// Initialize an accumulator:
accumulator = incrmcovariance( 5 );

// For each simulated datum, update the moving unbiased sample covariance...
for ( i = 0; i < 100; i++ ) {
    x = randu() * 100.0;
    y = randu() * 100.0;
    accumulator( x, y );
}
console.log( accumulator() );