114 lines
3.2 KiB
JavaScript
114 lines
3.2 KiB
JavaScript
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/**
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* @license Apache-2.0
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*
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* Copyright (c) 2020 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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// MAIN //
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/**
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* Computes the variance of a strided array using Welford's algorithm.
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*
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* ## Method
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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 {PositiveInteger} N - number of indexed elements
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* @param {number} correction - degrees of freedom adjustment
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* @param {NumericArray} x - input array
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* @param {integer} stride - stride length
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* @returns {number} variance
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*
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* @example
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* var x = [ 1.0, -2.0, 2.0 ];
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*
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* var v = variancewd( x.length, 1, x, 1 );
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* // returns ~4.3333
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*/
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function variancewd( N, correction, x, stride ) {
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var delta;
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var mu;
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var M2;
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var ix;
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var v;
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var n;
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var i;
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n = N - correction;
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if ( N <= 0 || n <= 0.0 ) {
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return NaN;
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}
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if ( N === 1 || stride === 0 ) {
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return 0.0;
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}
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if ( stride < 0 ) {
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ix = (1-N) * stride;
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} else {
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ix = 0;
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}
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M2 = 0.0;
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mu = 0.0;
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for ( i = 0; i < N; i++ ) {
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v = x[ ix ];
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delta = v - mu;
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mu += delta / (i+1);
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M2 += delta * ( v - mu );
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ix += stride;
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}
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return M2 / n;
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}
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// EXPORTS //
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module.exports = variancewd;
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