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* @license Apache-2.0
*
* Copyright (c) 2025 The Stdlib Authors.
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
'use strict';
// MODULES //
var isCollection = require( '@stdlib/assert/is-collection' );
var isNumber = require( '@stdlib/assert/is-number' ).isPrimitive;
var dtypes = require( '@stdlib/array/dtypes' );
var dtype = require( '@stdlib/array/dtype' );
var contains = require( '@stdlib/array/base/assert/contains' );
var join = require( '@stdlib/array/base/join' );
var strided = require( '@stdlib/stats/base/variancewd' ).ndarray;
var format = require( '@stdlib/string/format' );
// VARIABLES //
var IDTYPES = dtypes( 'real_and_generic' );
var GENERIC_DTYPE = 'generic';
// MAIN //
/**
* Computes the standard deviation of an array using Welford's algorithm.
*
* ## Method
*
* - This implementation uses Welford's algorithm for efficient computation, which can be derived as follows. Let
*
* ```tex
* \begin{align*}
* S_n &= n \sigma_n^2 \\
* &= \sum_{i=1}^{n} (x_i - \mu_n)^2 \\
* &= \biggl(\sum_{i=1}^{n} x_i^2 \biggr) - n\mu_n^2
* \end{align*}
* ```
*
* Accordingly,
*
* ```tex
* \begin{align*}
* S_n - S_{n-1} &= \sum_{i=1}^{n} x_i^2 - n\mu_n^2 - \sum_{i=1}^{n-1} x_i^2 + (n-1)\mu_{n-1}^2 \\
* &= x_n^2 - n\mu_n^2 + (n-1)\mu_{n-1}^2 \\
* &= x_n^2 - \mu_{n-1}^2 + n(\mu_{n-1}^2 - \mu_n^2) \\
* &= x_n^2 - \mu_{n-1}^2 + n(\mu_{n-1} - \mu_n)(\mu_{n-1} + \mu_n) \\
* &= x_n^2 - \mu_{n-1}^2 + (\mu_{n-1} - x_n)(\mu_{n-1} + \mu_n) \\
* &= x_n^2 - \mu_{n-1}^2 + \mu_{n-1}^2 - x_n\mu_n - x_n\mu_{n-1} + \mu_n\mu_{n-1} \\
* &= x_n^2 - x_n\mu_n - x_n\mu_{n-1} + \mu_n\mu_{n-1} \\
* &= (x_n - \mu_{n-1})(x_n - \mu_n) \\
* &= S_{n-1} + (x_n - \mu_{n-1})(x_n - \mu_n)
* \end{align*}
* ```
*
* where we use the identity
*
* ```tex
* x_n - \mu_{n-1} = n (\mu_n - \mu_{n-1})
* ```
*
* ## 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](https://doi.org/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](https://doi.org/10.1145/362929.362961).
*
* @param {NumericArray} x - input array
* @param {number} [correction=1.0] - degrees of freedom adjustment
* @throws {TypeError} first argument must be an array-like object
* @throws {TypeError} first argument must have a supported data type
* @throws {TypeError} second argument must be a number
* @returns {number} variance
*
* @example
* var x = [ 1.0, -2.0, 2.0 ];
*
* var v = variancewd( x, 1.0 );
* // returns ~4.3333
*/
function variancewd( x ) {
var correction;
var dt;
if ( !isCollection( x ) ) {
throw new TypeError( format( 'invalid argument. First argument must be an array-like object. Value: `%s`.', x ) );
}
dt = dtype( x ) || GENERIC_DTYPE;
if ( !contains( IDTYPES, dt ) ) {
throw new TypeError( format( 'invalid argument. First argument must have one of the following data types: "%s". Data type: `%s`.', join( IDTYPES, '", "' ), dt ) );
}
if ( arguments.length > 1 ) {
correction = arguments[ 1 ];
if ( !isNumber( correction ) ) {
throw new TypeError( format( 'invalid argument. Second argument must be a number. Value: `%s`.', correction ) );
}
} else {
correction = 1.0;
}
return strided( x.length, correction, x, 1, 0 );
}
// EXPORTS //
module.exports = variancewd;
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