160 lines
4.1 KiB
JavaScript
160 lines
4.1 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 isNumber = require( '@stdlib/assert/is-number' ).isPrimitive;
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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 a variance-to-mean ratio (VMR).
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*
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* ## Method
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*
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* - This implementation uses [Welford's method][algorithms-variance] 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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* [algorithms-variance]: https://en.wikipedia.org/wiki/Algorithms_for_calculating_variance
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*
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* @param {number} [mean] - mean value
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* @throws {TypeError} must provide a number primitive
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* @returns {Function} accumulator function
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*
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* @example
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* var accumulator = incrvmr();
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*
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* var D = accumulator();
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* // returns null
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*
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* D = accumulator( 2.0 );
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* // returns 0.0
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*
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* D = accumulator( 1.0 );
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* // returns ~0.33
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*
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* D = accumulator();
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* // returns ~0.33
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*
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* @example
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* var accumulator = incrvmr( 3.14 );
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*/
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function incrvmr( mean ) {
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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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M2 = 0.0;
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N = 0;
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if ( arguments.length ) {
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if ( !isNumber( mean ) ) {
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throw new TypeError( 'invalid argument. Must provide a number primitive. Value: `' + mean + '`.' );
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}
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mu = mean;
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return accumulator2;
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}
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mu = 0.0;
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return accumulator1;
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/**
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* If provided a value, the accumulator function returns an updated accumulated value. If not provided a value, the accumulator function returns the current accumulated value.
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*
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* @private
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* @param {number} [x] - new value
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* @returns {(number|null)} accumulated value or null
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*/
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function accumulator1( 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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if ( N === 1 ) {
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return ( isnan( M2 ) ) ? NaN : 0.0/mu;
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}
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return ( M2/(N-1) ) / mu;
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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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if ( N < 2 ) {
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return ( isnan( M2 ) ) ? NaN : 0.0/mu;
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}
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return ( M2/(N-1) ) / mu;
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}
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/**
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* If provided a value, the accumulator function returns an updated accumulated value. If not provided a value, the accumulator function returns the current accumulated value.
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*
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* @private
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* @param {number} [x] - new value
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* @returns {(number|null)} accumulated value or null
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*/
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function accumulator2( 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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return ( M2/N ) / mu;
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}
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N += 1;
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delta = x - mu;
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M2 += delta * delta;
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return ( M2/N ) / mu;
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}
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}
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// EXPORTS //
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module.exports = incrvmr;
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