main.js (4553B)
1 /** 2 * @license Apache-2.0 3 * 4 * Copyright (c) 2018 The Stdlib Authors. 5 * 6 * Licensed under the Apache License, Version 2.0 (the "License"); 7 * you may not use this file except in compliance with the License. 8 * You may obtain a copy of the License at 9 * 10 * http://www.apache.org/licenses/LICENSE-2.0 11 * 12 * Unless required by applicable law or agreed to in writing, software 13 * distributed under the License is distributed on an "AS IS" BASIS, 14 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 15 * See the License for the specific language governing permissions and 16 * limitations under the License. 17 */ 18 19 'use strict'; 20 21 // MODULES // 22 23 var isNumber = require( '@stdlib/assert/is-number' ).isPrimitive; 24 var isnan = require( '@stdlib/math/base/assert/is-nan' ); 25 26 27 // MAIN // 28 29 /** 30 * Returns an accumulator function which incrementally computes an unbiased sample variance. 31 * 32 * ## Method 33 * 34 * - This implementation uses Welford's algorithm for efficient computation, which can be derived as follows. Let 35 * 36 * ```tex 37 * \begin{align*} 38 * S_n &= n \sigma_n^2 \\ 39 * &= \sum_{i=1}^{n} (x_i - \mu_n)^2 \\ 40 * &= \biggl(\sum_{i=1}^{n} x_i^2 \biggr) - n\mu_n^2 41 * \end{align*} 42 * ``` 43 * 44 * Accordingly, 45 * 46 * ```tex 47 * \begin{align*} 48 * 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 \\ 49 * &= x_n^2 - n\mu_n^2 + (n-1)\mu_{n-1}^2 \\ 50 * &= x_n^2 - \mu_{n-1}^2 + n(\mu_{n-1}^2 - \mu_n^2) \\ 51 * &= x_n^2 - \mu_{n-1}^2 + n(\mu_{n-1} - \mu_n)(\mu_{n-1} + \mu_n) \\ 52 * &= x_n^2 - \mu_{n-1}^2 + (\mu_{n-1} - x_n)(\mu_{n-1} + \mu_n) \\ 53 * &= 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} \\ 54 * &= x_n^2 - x_n\mu_n - x_n\mu_{n-1} + \mu_n\mu_{n-1} \\ 55 * &= (x_n - \mu_{n-1})(x_n - \mu_n) \\ 56 * &= S_{n-1} + (x_n - \mu_{n-1})(x_n - \mu_n) 57 * \end{align*} 58 * ``` 59 * 60 * where we use the identity 61 * 62 * ```tex 63 * x_n - \mu_{n-1} = n (\mu_n - \mu_{n-1}) 64 * ``` 65 * 66 * ## References 67 * 68 * - 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). 69 * - 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). 70 * 71 * @param {number} [mean] - mean value 72 * @throws {TypeError} must provide a number primitive 73 * @returns {Function} accumulator function 74 * 75 * @example 76 * var accumulator = incrvariance(); 77 * 78 * var s2 = accumulator(); 79 * // returns null 80 * 81 * s2 = accumulator( 2.0 ); 82 * // returns 0.0 83 * 84 * s2 = accumulator( -5.0 ); 85 * // returns 24.5 86 * 87 * s2 = accumulator(); 88 * // returns 24.5 89 * 90 * @example 91 * var accumulator = incrvariance( 3.14 ); 92 */ 93 function incrvariance( mean ) { 94 var delta; 95 var mu; 96 var M2; 97 var N; 98 99 M2 = 0.0; 100 N = 0; 101 if ( arguments.length ) { 102 if ( !isNumber( mean ) ) { 103 throw new TypeError( 'invalid argument. Must provide a number primitive. Value: `' + mean + '`.' ); 104 } 105 mu = mean; 106 return accumulator2; 107 } 108 mu = 0.0; 109 return accumulator1; 110 111 /** 112 * If provided a value, the accumulator function returns an updated unbiased sample variance. If not provided a value, the accumulator function returns the current unbiased sample variance. 113 * 114 * @private 115 * @param {number} [x] - new value 116 * @returns {(number|null)} unbiased sample variance or null 117 */ 118 function accumulator1( x ) { 119 if ( arguments.length === 0 ) { 120 if ( N === 0 ) { 121 return null; 122 } 123 if ( N === 1 ) { 124 return ( isnan( M2 ) ) ? NaN : 0.0; 125 } 126 return M2 / (N-1); 127 } 128 N += 1; 129 delta = x - mu; 130 mu += delta / N; 131 M2 += delta * ( x - mu ); 132 if ( N < 2 ) { 133 return ( isnan( M2 ) ) ? NaN : 0.0; 134 } 135 return M2 / (N-1); 136 } 137 138 /** 139 * If provided a value, the accumulator function returns an updated unbiased sample variance. If not provided a value, the accumulator function returns the current unbiased sample variance. 140 * 141 * @private 142 * @param {number} [x] - new value 143 * @returns {(number|null)} unbiased sample variance or null 144 */ 145 function accumulator2( x ) { 146 if ( arguments.length === 0 ) { 147 if ( N === 0 ) { 148 return null; 149 } 150 return M2 / N; 151 } 152 N += 1; 153 delta = x - mu; 154 M2 += delta * delta; 155 return M2 / N; 156 } 157 } 158 159 160 // EXPORTS // 161 162 module.exports = incrvariance;