README.md

dnanstdevwd

Calculate the standard deviation of a one-dimensional double-precision floating-point ndarray, ignoring NaN values and using Welford's 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 dnanstdevwd = require( '@stdlib/stats/base/ndarray/dnanstdevwd' );

dnanstdevwd( arrays )

Computes the standard deviation of a one-dimensional double-precision floating-point ndarray, ignoring NaN values and using Welford's algorithm.

var Float64Array = require( '@stdlib/array/float64' );
var ndarray = require( '@stdlib/ndarray/base/ctor' );
var scalar2ndarray = require( '@stdlib/ndarray/from-scalar' );

var opts = {
    'dtype': 'float64'
};

var xbuf = new Float64Array( [ 1.0, -2.0, NaN, 2.0 ] );
var x = new ndarray( opts.dtype, xbuf, [ 4 ], [ 1 ], 0, 'row-major' );
var correction = scalar2ndarray( 1.0, opts );

var v = dnanstdevwd( [ x, correction ] );
// returns ~2.0817

The function has the following parameters:

• arrays: array-like object containing two elements: a one-dimensional input ndarray and a zero-dimensional ndarray specifying the degrees of freedom adjustment. Providing a non-zero degrees of freedom adjustment has the effect of adjusting the divisor during the calculation of the standard deviation according to N-c where N is the number of non-NaN elements in the input ndarray and 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).

Notes

• If provided an empty one-dimensional ndarray, the function returns NaN.
• If N - c is less than or equal to 0 (where N corresponds to the number of non-NaN elements in the input ndarray and c corresponds to the provided degrees of freedom adjustment), the function returns NaN.

Examples

var uniform = require( '@stdlib/random/base/uniform' );
var filledarrayBy = require( '@stdlib/array/filled-by' );
var bernoulli = require( '@stdlib/random/base/bernoulli' );
var ndarray = require( '@stdlib/ndarray/base/ctor' );
var scalar2ndarray = require( '@stdlib/ndarray/from-scalar' );
var ndarray2array = require( '@stdlib/ndarray/to-array' );
var dnanstdevwd = require( '@stdlib/stats/base/ndarray/dnanstdevwd' );

function rand() {
    if ( bernoulli( 0.8 ) < 1 ) {
        return NaN;
    }
    return uniform( -50.0, 50.0 );
}

var opts = {
    'dtype': 'float64'
};

var xbuf = filledarrayBy( 10, opts.dtype, rand );
var x = new ndarray( opts.dtype, xbuf, [ xbuf.length ], [ 1 ], 0, 'row-major' );
console.log( ndarray2array( x ) );

var correction = scalar2ndarray( 1.0, opts );
var v = dnanstdevwd( [ x, correction ] );
console.log( v );

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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.