dvariancewd.c (3277B)
1 /** 2 * @license Apache-2.0 3 * 4 * Copyright (c) 2020 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 #include "stdlib/stats/base/dvariancewd.h" 20 #include <stdint.h> 21 22 /** 23 * Computes the variance of a double-precision floating-point strided array using Welford's algorithm. 24 * 25 * ## Method 26 * 27 * - This implementation uses Welford's algorithm for efficient computation, which can be derived as follows. Let 28 * 29 * ```tex 30 * \begin{align*} 31 * S_n &= n \sigma_n^2 \\ 32 * &= \sum_{i=1}^{n} (x_i - \mu_n)^2 \\ 33 * &= \biggl(\sum_{i=1}^{n} x_i^2 \biggr) - n\mu_n^2 34 * \end{align*} 35 * ``` 36 * 37 * Accordingly, 38 * 39 * ```tex 40 * \begin{align*} 41 * 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 \\ 42 * &= x_n^2 - n\mu_n^2 + (n-1)\mu_{n-1}^2 \\ 43 * &= x_n^2 - \mu_{n-1}^2 + n(\mu_{n-1}^2 - \mu_n^2) \\ 44 * &= x_n^2 - \mu_{n-1}^2 + n(\mu_{n-1} - \mu_n)(\mu_{n-1} + \mu_n) \\ 45 * &= x_n^2 - \mu_{n-1}^2 + (\mu_{n-1} - x_n)(\mu_{n-1} + \mu_n) \\ 46 * &= 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} \\ 47 * &= x_n^2 - x_n\mu_n - x_n\mu_{n-1} + \mu_n\mu_{n-1} \\ 48 * &= (x_n - \mu_{n-1})(x_n - \mu_n) \\ 49 * &= S_{n-1} + (x_n - \mu_{n-1})(x_n - \mu_n) 50 * \end{align*} 51 * ``` 52 * 53 * where we use the identity 54 * 55 * ```tex 56 * x_n - \mu_{n-1} = n (\mu_n - \mu_{n-1}) 57 * ``` 58 * 59 * ## References 60 * 61 * - 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). 62 * - 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). 63 * 64 * @param N number of indexed elements 65 * @param correction degrees of freedom adjustment 66 * @param X input array 67 * @param stride stride length 68 * @return output value 69 */ 70 double stdlib_strided_dvariancewd( const int64_t N, const double correction, const double *X, const int64_t stride ) { 71 double delta; 72 int64_t ix; 73 int64_t i; 74 double mu; 75 double M2; 76 double n; 77 double v; 78 79 n = (double)N - correction; 80 if ( N <= 0 || n <= 0.0 ) { 81 return 0.0 / 0.0; // NaN 82 } 83 if ( N == 1 || stride == 0 ) { 84 return 0.0; 85 } 86 if ( stride < 0 ) { 87 ix = (1-N) * stride; 88 } else { 89 ix = 0; 90 } 91 M2 = 0.0; 92 mu = 0.0; 93 for ( i = 0; i < N; i++ ) { 94 v = X[ ix ]; 95 delta = v - mu; 96 mu += delta / (double)(i+1); 97 M2 += delta * ( v - mu ); 98 ix += stride; 99 } 100 return M2 / n; 101 }