variancewd

Calculate the variance of a strided array using Welford's algorithm.

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

where the population mean is given by

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,

where the sample mean is given by

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.

Installation

npm install @stdlib/stats-base-variancewd

Usage

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

variancewd( N, correction, x, stride )

Computes the variance of a strided array x using Welford's algorithm.

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

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

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

Computes the variance of a strided array using Welford's algorithm and alternative indexing semantics.

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

var v = variancewd.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 = variancewd.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 (dvariancewd, svariancewd, 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 variancewd = require( '@stdlib/stats-base-variancewd' );

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

References

• Welford, B. P. 1962. "Note on a Method for Calculating Corrected Sums of Squares and Products." Technometrics 4 (3). Taylor & Francis: 419–20. doi:10.1080/00401706.1962.10490022.
• van Reeken, A. J. 1968. "Letters to the Editor: Dealing with Neely's Algorithms." Communications of the ACM 11 (3): 149–50. doi:10.1145/362929.362961.

See Also

• @stdlib/stats-base/dvariancewd: calculate the variance of a double-precision floating-point strided array using Welford's algorithm.
• @stdlib/stats-base/nanvariancewd: calculate the variance of a strided array ignoring NaN values and using Welford's algorithm.
• @stdlib/stats-base/stdevwd: calculate the standard deviation of a strided array using Welford's algorithm.
• @stdlib/stats-base/variance: calculate the variance of a strided array.

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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