163 lines
4.4 KiB
JavaScript
163 lines
4.4 KiB
JavaScript
/**
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* @license Apache-2.0
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*
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* Copyright (c) 2018 The Stdlib Authors.
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*
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* Licensed under the Apache License, Version 2.0 (the "License");
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* you may not use this file except in compliance with the License.
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* You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS,
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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*/
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'use strict';
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// MODULES //
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var isArrayLike = require( '@stdlib/assert/is-array-like-object' );
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var isnan = require( '@stdlib/math/base/assert/is-nan' );
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// MAIN //
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/**
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* Returns an accumulator function which incrementally computes an arithmetic mean and unbiased sample variance.
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*
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* ## Method
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*
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*
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* - This implementation uses Welford's algorithm for efficient computation, which can be derived as follows. Let
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*
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* ```tex
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* \begin{align*}
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* S_n &= n \sigma_n^2 \\
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* &= \sum_{i=1}^{n} (x_i - \mu_n)^2 \\
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* &= \biggl(\sum_{i=1}^{n} x_i^2 \biggr) - n\mu_n^2
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* \end{align*}
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* ```
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*
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* Accordingly,
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*
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* ```tex
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* \begin{align*}
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* 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 \\
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* &= x_n^2 - n\mu_n^2 + (n-1)\mu_{n-1}^2 \\
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* &= x_n^2 - \mu_{n-1}^2 + n(\mu_{n-1}^2 - \mu_n^2) \\
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* &= x_n^2 - \mu_{n-1}^2 + n(\mu_{n-1} - \mu_n)(\mu_{n-1} + \mu_n) \\
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* &= x_n^2 - \mu_{n-1}^2 + (\mu_{n-1} - x_n)(\mu_{n-1} + \mu_n) \\
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* &= 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} \\
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* &= x_n^2 - x_n\mu_n - x_n\mu_{n-1} + \mu_n\mu_{n-1} \\
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* &= (x_n - \mu_{n-1})(x_n - \mu_n) \\
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* &= S_{n-1} + (x_n - \mu_{n-1})(x_n - \mu_n)
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* \end{align*}
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* ```
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*
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* where we use the identity
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*
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* ```tex
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* x_n - \mu_{n-1} = n (\mu_n - \mu_{n-1})
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* ```
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*
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* ## References
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*
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* - 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).
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* - 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).
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*
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* @param {Collection} [out] - output array
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* @throws {TypeError} output argument must be array-like
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* @returns {Function} accumulator function
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*
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* @example
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* var accumulator = incrmeanvar();
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*
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* var mv = accumulator();
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* // returns null
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*
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* mv = accumulator( 2.0 );
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* // returns [ 2.0, 0.0 ]
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*
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* mv = accumulator( -5.0 );
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* // returns [ -1.5, 24.5 ]
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*
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* mv = accumulator( 3.0 );
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* // returns [ 0.0, 19.0 ]
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*
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* mv = accumulator( 5.0 );
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* // returns [ 1.25, ~18.92 ]
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*
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* mv = accumulator();
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* // returns [ 1.25, ~18.92 ]
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*/
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function incrmeanvar( out ) {
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var meanvar;
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var delta;
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var mu;
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var M2;
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var N;
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if ( arguments.length === 0 ) {
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meanvar = [ 0.0, 0.0 ];
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} else {
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if ( !isArrayLike( out ) ) {
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throw new TypeError( 'invalid argument. Output argument must be an array-like object. Value: `' + out + '`.' );
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}
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meanvar = out;
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}
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M2 = 0.0;
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mu = 0.0;
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N = 0;
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return accumulator;
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/**
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* If provided a value, the accumulator function returns updated results. If not provided a value, the accumulator function returns the current results.
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*
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* @private
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* @param {number} [x] - input value
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* @returns {(ArrayLikeObject|null)} output array or null
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*/
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function accumulator( x ) {
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if ( arguments.length === 0 ) {
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if ( N === 0 ) {
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return null;
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}
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meanvar[ 0 ] = mu; // Why? Because we cannot guarantee someone hasn't mutated the output array
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if ( N === 1 ) {
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if ( isnan( M2 ) ) {
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meanvar[ 1 ] = NaN;
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} else {
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meanvar[ 1 ] = 0.0;
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}
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return meanvar;
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}
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meanvar[ 1 ] = M2 / (N-1);
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return meanvar;
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}
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N += 1;
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delta = x - mu;
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mu += delta / N;
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M2 += delta * ( x - mu );
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meanvar[ 0 ] = mu;
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if ( N < 2 ) {
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if ( isnan( M2 ) ) {
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meanvar[ 1 ] = NaN;
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} else {
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meanvar[ 1 ] = 0.0;
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}
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return meanvar;
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}
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meanvar[ 1 ] = M2 / (N-1);
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return meanvar;
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}
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}
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// EXPORTS //
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module.exports = incrmeanvar;
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