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sstdevpn

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sstdevpn

Calculate the standard deviation of a single-precision floating-point strided array using a two-pass algorithm.

The population standard deviation of a finite size population of size N is given by

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

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 standard deviation 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 standard deviation, the result is biased and yields an uncorrected sample standard deviation. To compute a corrected sample standard deviation for a sample of size n,

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

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 standard deviation and population standard deviation. 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 sstdevpn = require( '@stdlib/stats/strided/sstdevpn' );

sstdevpn( N, correction, x, strideX )

Computes the standard deviation of a single-precision floating-point strided array using a two-pass algorithm.

var Float32Array = require( '@stdlib/array/float32' );

var x = new Float32Array( [ 1.0, -2.0, 2.0 ] );

var v = sstdevpn( x.length, 1, x, 1 );
// returns ~2.0817

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 standard deviation according to N-c where c corresponds to the provided degrees of freedom adjustment. When computing the standard deviation 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 corrected sample standard deviation, 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 Float32Array.
  • 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 standard deviation of every other element in x,

var Float32Array = require( '@stdlib/array/float32' );

var x = new Float32Array( [ 1.0, 2.0, 2.0, -7.0, -2.0, 3.0, 4.0, 2.0 ] );

var v = sstdevpn( 4, 1, x, 2 );
// returns 2.5

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

var Float32Array = require( '@stdlib/array/float32' );

var x0 = new Float32Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] );
var x1 = new Float32Array( x0.buffer, x0.BYTES_PER_ELEMENT*1 ); // start at 2nd element

var v = sstdevpn( 4, 1, x1, 2 );
// returns 2.5

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

Computes the standard deviation of a single-precision floating-point strided array using a two-pass algorithm and alternative indexing semantics.

var Float32Array = require( '@stdlib/array/float32' );

var x = new Float32Array( [ 1.0, -2.0, 2.0 ] );

var v = sstdevpn.ndarray( x.length, 1, x, 1, 0 );
// returns ~2.0817

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 standard deviation for every other element in x starting from the second element

var Float32Array = require( '@stdlib/array/float32' );

var x = new Float32Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] );

var v = sstdevpn.ndarray( 4, 1, x, 2, 1 );
// returns 2.5

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 sstdevpn = require( '@stdlib/stats/strided/sstdevpn' );

var x = discreteUniform( 10, -50, 50, {
    'dtype': 'float32'
});
console.log( x );

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

C APIs

Usage

#include "stdlib/stats/strided/sstdevpn.h"

stdlib_strided_sstdevpn( N, correction, *X, strideX )

Computes the standard deviation of a single-precision floating-point strided array using a two-pass trial mean algorithm.

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

float v = stdlib_strided_sstdevpn( 3, 1.0f, x, 1 );
// returns ~2.0817f

The function accepts the following arguments:

  • N: [in] CBLAS_INT number of indexed elements.
  • correction: [in] float 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 standard deviation according to N-c where c corresponds to the provided degrees of freedom adjustment. When computing the standard deviation 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 corrected sample standard deviation, 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] float* input array.
  • strideX: [in] CBLAS_INT stride length for X.
float stdlib_strided_sstdevpn( const CBLAS_INT N, const float correction, const float *X, const CBLAS_INT strideX );

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

Computes the standard deviation of a single-precision floating-point strided array using a two-pass trial mean algorithm and alternative indexing semantics.

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

float v = stdlib_strided_sstdevpn_ndarray( 3, 1.0f, x, 1, 0 );
// returns ~2.0817f

The function accepts the following arguments:

  • N: [in] CBLAS_INT number of indexed elements.
  • correction: [in] float 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 standard deviation according to N-c where c corresponds to the provided degrees of freedom adjustment. When computing the standard deviation 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 corrected sample standard deviation, 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] float* input array.
  • strideX: [in] CBLAS_INT stride length for X.
  • offsetX: [in] CBLAS_INT starting index for X.
float stdlib_strided_sstdevpn_ndarray( const CBLAS_INT N, const float correction, const float *X, const CBLAS_INT strideX, const CBLAS_INT offsetX );

Examples

#include "stdlib/stats/strided/sstdevpn.h"
#include <stdio.h>

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

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

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

    // Compute the variance:
    float v = stdlib_strided_sstdevpn( N, 1.0f, x, strideX );

    // Print the result:
    printf( "sample standard deviation: %f\n", 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.
  • 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.

See Also