varianceyc

Calculate the variance of a strided array using a one-pass algorithm proposed by Youngs and Cramer.

The population variance of a finite size population of size N is given by

sigma squared equals StartFraction 1 Over upper N EndFraction sigma-summation Underscript i equals 0 Overscript upper N minus 1 Endscripts left-parenthesis x Subscript i Baseline minus mu right-parenthesis squared

where the population mean is given by

mu equals StartFraction 1 Over upper N EndFraction sigma-summation Underscript i equals 0 Overscript upper N minus 1 Endscripts x Subscript i

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

s squared equals StartFraction 1 Over n minus 1 EndFraction sigma-summation Underscript i equals 0 Overscript n minus 1 Endscripts left-parenthesis x Subscript i Baseline minus x overbar right-parenthesis squared

where the sample mean is given by

x overbar equals StartFraction 1 Over n EndFraction sigma-summation Underscript i equals 0 Overscript n minus 1 Endscripts x Subscript i

The use of the term n-1 is commonly referred to as Bessel's correction. Note, however, that applying Bessel's correction can increase the mean squared error between the sample variance and population variance. Depending on the characteristics of the population distribution, other correction factors (e.g., n-1.5, n+1, etc) can yield better estimators.

Usage

var varianceyc = require( '@stdlib/stats/base/varianceyc' );

varianceyc( N, correction, x, stride )

Computes the variance of a strided array x using a one-pass algorithm proposed by Youngs and Cramer.

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

var v = varianceyc( x.length, 1, x, 1 );
// returns ~4.3333

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 variance according to N-c where c corresponds to the provided degrees of freedom adjustment. When computing the variance of a population, setting this parameter to 0 is the standard choice (i.e., the provided array contains data constituting an entire population). When computing the unbiased sample variance, setting this parameter to 1 is the standard choice (i.e., the provided array contains data sampled from a larger population; this is commonly referred to as Bessel's correction).
  • x: input Array or typed array.
  • stride: index increment for x.

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

var floor = require( '@stdlib/math/base/special/floor' );

var x = [ 1.0, 2.0, 2.0, -7.0, -2.0, 3.0, 4.0, 2.0 ];
var N = floor( x.length / 2 );

var v = varianceyc( N, 1, x, 2 );
// returns 6.25

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

var Float64Array = require( '@stdlib/array/float64' );
var floor = require( '@stdlib/math/base/special/floor' );

var x0 = new Float64Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] );
var x1 = new Float64Array( x0.buffer, x0.BYTES_PER_ELEMENT*1 ); // start at 2nd element

var N = floor( x0.length / 2 );

var v = varianceyc( N, 1, x1, 2 );
// returns 6.25

varianceyc.ndarray( N, correction, x, stride, offset )

Computes the variance of a strided array using a one-pass algorithm proposed by Youngs and Cramer and alternative indexing semantics.

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

var v = varianceyc.ndarray( x.length, 1, x, 1, 0 );
// returns ~4.33333

The function has the following additional parameters:

  • offset: starting index for x.

While typed array views mandate a view offset based on the underlying buffer, the offset parameter supports indexing semantics based on a starting index. For example, to calculate the variance for every other value in x starting from the second value

var floor = require( '@stdlib/math/base/special/floor' );

var x = [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ];
var N = floor( x.length / 2 );

var v = varianceyc.ndarray( N, 1, x, 2, 1 );
// returns 6.25

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.
  • Depending on the environment, the typed versions (dvarianceyc, svarianceyc, etc.) are likely to be significantly more performant.

Examples

var randu = require( '@stdlib/random/base/randu' );
var round = require( '@stdlib/math/base/special/round' );
var Float64Array = require( '@stdlib/array/float64' );
var varianceyc = require( '@stdlib/stats/base/varianceyc' );

var x;
var i;

x = new Float64Array( 10 );
for ( i = 0; i < x.length; i++ ) {
    x[ i ] = round( (randu()*100.0) - 50.0 );
}
console.log( x );

var v = varianceyc( x.length, 1, x, 1 );
console.log( v );

References

  • Youngs, Edward A., and Elliot M. Cramer. 1971. "Some Results Relevant to Choice of Sum and Sum-of-Product Algorithms." Technometrics 13 (3): 657–65. doi:10.1080/00401706.1971.10488826.