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dvarianceyc

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Calculate the variance of a double-precision floating-point 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

$$\sigma^2 = \frac{1}{N} \sum_{i=0}^{N-1} (x_i - \mu)^2$$

where the population mean is given by

$$\mu = \frac{1}{N} \sum_{i=0}^{N-1} x_i$$

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,

$$s^2 = \frac{1}{n-1} \sum_{i=0}^{n-1} (x_i - \bar{x})^2$$

where the sample mean is given by

$$\bar{x} = \frac{1}{n} \sum_{i=0}^{n-1} x_i$$

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

import dvarianceyc from 'https://cdn.jsdelivr.net/gh/stdlib-js/stats-base-dvarianceyc@esm/index.mjs';

dvarianceyc( N, correction, x, stride )

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

import Float64Array from 'https://cdn.jsdelivr.net/gh/stdlib-js/array-float64@esm/index.mjs';

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

var v = dvarianceyc( N, 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.
  • 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,

import Float64Array from 'https://cdn.jsdelivr.net/gh/stdlib-js/array-float64@esm/index.mjs';
import floor from 'https://cdn.jsdelivr.net/gh/stdlib-js/math-base-special-floor@esm/index.mjs';

var x = new Float64Array( [ 1.0, 2.0, 2.0, -7.0, -2.0, 3.0, 4.0, 2.0 ] );
var N = floor( x.length / 2 );

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

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

import Float64Array from 'https://cdn.jsdelivr.net/gh/stdlib-js/array-float64@esm/index.mjs';
import floor from 'https://cdn.jsdelivr.net/gh/stdlib-js/math-base-special-floor@esm/index.mjs';

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 = dvarianceyc( N, 1, x1, 2 );
// returns 6.25

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

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

import Float64Array from 'https://cdn.jsdelivr.net/gh/stdlib-js/array-float64@esm/index.mjs';

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

var v = dvarianceyc.ndarray( N, 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

import Float64Array from 'https://cdn.jsdelivr.net/gh/stdlib-js/array-float64@esm/index.mjs';
import floor from 'https://cdn.jsdelivr.net/gh/stdlib-js/math-base-special-floor@esm/index.mjs';

var x = new Float64Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] );
var N = floor( x.length / 2 );

var v = dvarianceyc.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.

Examples

<!DOCTYPE html>
<html lang="en">
<body>
<script type="module">

import randu from 'https://cdn.jsdelivr.net/gh/stdlib-js/random-base-randu@esm/index.mjs';
import round from 'https://cdn.jsdelivr.net/gh/stdlib-js/math-base-special-round@esm/index.mjs';
import Float64Array from 'https://cdn.jsdelivr.net/gh/stdlib-js/array-float64@esm/index.mjs';
import dvarianceyc from 'https://cdn.jsdelivr.net/gh/stdlib-js/stats-base-dvarianceyc@esm/index.mjs';

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

</script>
</body>
</html>

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.

See Also

  • @stdlib/stats-base/dnanvarianceyc: calculate the variance of a double-precision floating-point strided array ignoring NaN values and using a one-pass algorithm proposed by Youngs and Cramer.
  • @stdlib/stats-base/dstdevyc: calculate the standard deviation of a double-precision floating-point strided array using a one-pass algorithm proposed by Youngs and Cramer.
  • @stdlib/stats-base/dvariance: calculate the variance of a double-precision floating-point strided array.
  • @stdlib/stats-base/svarianceyc: calculate the variance of a single-precision floating-point strided array using a one-pass algorithm proposed by Youngs and Cramer.
  • @stdlib/stats-base/varianceyc: calculate the variance of a strided array using a one-pass algorithm proposed by Youngs and Cramer.

Notice

This package is part of stdlib, a standard library 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

Copyright © 2016-2024. The Stdlib Authors.