variancech

Calculate the variance of a strided array using a one-pass trial mean 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.

Usage

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

variancech( N, correction, x, stride )

Computes the variance of a strided array x using a one-pass trial mean algorithm.

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

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

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

Computes the variance of a strided array using a one-pass trial mean algorithm and alternative indexing semantics.

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

var v = variancech.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 = variancech.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.
The underlying algorithm is a specialized case of Neely's two-pass algorithm. As the variance is invariant with respect to changes in the location parameter, the underlying algorithm uses the first strided array element as a trial mean to shift subsequent data values and thus mitigate catastrophic cancellation. Accordingly, the algorithm's accuracy is best when data is unordered (i.e., the data is not sorted in either ascending or descending order such that the first value is an "extreme" value).
Depending on the environment, the typed versions (dvariancech, svariancech, 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 variancech = require( '@stdlib/stats/base/variancech' );

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

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.
Ling, Robert F. 1974. "Comparison of Several Algorithms for Computing Sample Means and Variances." Journal of the American Statistical Association 69 (348). American Statistical Association, Taylor & Francis, Ltd.: 859–66. doi:10.2307/2286154.
Chan, Tony F., Gene H. Golub, and Randall J. LeVeque. 1983. "Algorithms for Computing the Sample Variance: Analysis and Recommendations." The American Statistician 37 (3). American Statistical Association, Taylor & Francis, Ltd.: 242–47. doi:10.1080/00031305.1983.10483115.
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.