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dmeanvarpn

Calculate the mean and variance of a double-precision floating-point strided array using a two-pass algorithm.

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.

Installation

npm install @stdlib/stats-base-dmeanvarpn

Alternatively,

• To load the package in a website via a script tag without installation and bundlers, use the ES Module available on the esm branch (see README).
• If you are using Deno, visit the deno branch (see README for usage intructions).
• For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

Usage

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

dmeanvarpn( N, correction, x, strideX, out, strideOut )

Computes the mean and variance of a double-precision floating-point strided array x using a two-pass algorithm.

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

var x = new Float64Array( [ 1.0, -2.0, 2.0 ] );
var out = new Float64Array( 2 );

var v = dmeanvarpn( x.length, 1, x, 1, out, 1 );
// returns <Float64Array>[ ~0.3333, ~4.3333 ]

var bool = ( v === out );
// returns true

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: index increment for x.
• out: output Float64Array for storing results.
• strideOut: index increment for out.

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

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

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

var v = dmeanvarpn( N, 1, x, 2, out, 1 );
// returns <Float64Array>[ 1.25, 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 out0 = new Float64Array( 4 );
var out1 = new Float64Array( out0.buffer, out0.BYTES_PER_ELEMENT*2 ); // start at 3rd element

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

var v = dmeanvarpn( N, 1, x1, 2, out1, 1 );
// returns <Float64Array>[ 1.25, 6.25 ]

dmeanvarpn.ndarray( N, correction, x, strideX, offsetX, out, strideOut, offsetOut )

Computes the mean and variance of a double-precision floating-point strided array using a two-pass algorithm and alternative indexing semantics.

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

var x = new Float64Array( [ 1.0, -2.0, 2.0 ] );
var out = new Float64Array( 2 );

var v = dmeanvarpn.ndarray( x.length, 1, x, 1, 0, out, 1, 0 );
// returns <Float64Array>[ ~0.3333, ~4.3333 ]

The function has the following additional parameters:

• offsetX: starting index for x.
• offsetOut: starting index for out.

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

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

var x = new Float64Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] );
var out = new Float64Array( 4 );
var N = floor( x.length / 2 );

var v = dmeanvarpn.ndarray( N, 1, x, 2, 1, out, 2, 1 );
// returns <Float64Array>[ 0.0, 1.25, 0.0, 6.25 ]

Notes

• If N <= 0, both functions return a mean and variance equal to NaN.
• If N - c is less than or equal to 0 (where c corresponds to the provided degrees of freedom adjustment), both functions return a variance equal to NaN.

Examples

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

var out;
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 );

out = new Float64Array( 2 );
dmeanvarpn( x.length, 1, x, 1, out, 1 );
console.log( out );

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References

• Neely, Peter M. 1966. "Comparison of Several Algorithms for Computation of Means, Standard Deviations and Correlation Coefficients." Communications of the ACM 9 (7). Association for Computing Machinery: 496–99. doi:10.1145/365719.365958.
• Schubert, Erich, and Michael Gertz. 2018. "Numerically Stable Parallel Computation of (Co-)Variance." In Proceedings of the 30th International Conference on Scientific and Statistical Database Management. New York, NY, USA: Association for Computing Machinery. doi:10.1145/3221269.3223036.

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See Also

• @stdlib/stats-base/dmeanpn: calculate the arithmetic mean of a double-precision floating-point strided array using a two-pass error correction algorithm.
• @stdlib/stats-base/dmeanstdevpn: calculate the mean and standard deviation of a double-precision floating-point strided array using a two-pass algorithm.
• @stdlib/stats-base/dmeanvar: calculate the mean and variance of a double-precision floating-point strided array.
• @stdlib/stats-base/dvariancepn: calculate the variance of a double-precision floating-point strided array using a two-pass algorithm.

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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.

Community

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License

See LICENSE.

Copyright

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