dvariancewd

Calculate the variance of a double-precision floating-point strided array using Welford's algorithm.

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 dvariancewd = require( '@stdlib/stats/base/dvariancewd' );

dvariancewd( N, correction, x, strideX )

Computes the variance of a double-precision floating-point strided array x using Welford's algorithm.

var Float64Array = require( '@stdlib/array/float64' );

var x = new Float64Array( [ 1.0, -2.0, 2.0 ] );

var v = dvariancewd( 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 Float64Array.
  • strideX: stride length for x.

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

var Float64Array = require( '@stdlib/array/float64' );

var x = new Float64Array( [ 1.0, 2.0, 2.0, -7.0, -2.0, 3.0, 4.0, 2.0 ] );

var v = dvariancewd( 4, 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 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 v = dvariancewd( 4, 1, x1, 2 );
// returns 6.25

dvariancewd.ndarray( N, correction, x, strideX, offsetX )

Computes the variance of a double-precision floating-point strided array using Welford's algorithm and alternative indexing semantics.

var Float64Array = require( '@stdlib/array/float64' );

var x = new Float64Array( [ 1.0, -2.0, 2.0 ] );

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

The function has the following additional parameters:

  • offsetX: 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 element in x starting from the second element

var Float64Array = require( '@stdlib/array/float64' );

var x = new Float64Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] );

var v = dvariancewd.ndarray( 4, 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.

Examples

var discreteUniform = require( '@stdlib/random/array/discrete-uniform' );
var dvariancewd = require( '@stdlib/stats/base/dvariancewd' );

var x = discreteUniform( 10, -50, 50, {
    'dtype': 'float64'
});
console.log( x );

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

C APIs

Usage

#include "stdlib/stats/base/dvariancewd.h"

stdlib_strided_dvariancewd( N, correction, *X, strideX )

Computes the variance of a double-precision floating-point strided array using Welford's algorithm.

const double x[] = { 1.0, -2.0, 2.0 };

double v = stdlib_strided_dvariancewd( 3, 1.0, x, 1 );
// returns ~4.3333

The function accepts the following arguments:

  • N: [in] CBLAS_INT number of indexed elements.
  • correction: [in] double 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: [in] double* input array.
  • strideX: [in] CBLAS_INT stride length for X.
double stdlib_strided_dvariancewd( const CBLAS_INT N, const double correction, const double *X, const CBLAS_INT strideX );

stdlib_strided_dvariancewd_ndarray( N, correction, *X, strideX, offsetX )

Computes the variance of a double-precision floating-point strided array using Welford's algorithm and alternative indexing semantics.

const double x[] = { 1.0, -2.0, 2.0 };

double v = stdlib_strided_dvariancewd_ndarray( 3, 1.0, x, 1, 0 );
// returns ~4.3333

The function accepts the following arguments:

  • N: [in] CBLAS_INT number of indexed elements.
  • correction: [in] double 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: [in] double* input array.
  • strideX: [in] CBLAS_INT stride length for X.
  • offsetX: [in] CBLAS_INT starting index for X.
double stdlib_strided_dvariancewd_ndarray( const CBLAS_INT N, const double correction, const double *X, const CBLAS_INT strideX, const CBLAS_INT offsetX );

Examples

#include "stdlib/stats/base/dvariancewd.h"
#include <stdio.h>

int main( void ) {
    // Create a strided array:
    const double x[] = { 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0 };

    // Specify the number of elements:
    const int N = 4;

    // Specify the stride length:
    const int strideX = 2;

    // Compute the variance:
    double v = stdlib_strided_dvariancewd( N, 1, x, strideX );

    // Print the result:
    printf( "sample variance: %lf\n", 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

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