time-to-botec

main.js (7485B)

---

 /**
 * @license Apache-2.0
 *
 * Copyright (c) 2018 The Stdlib Authors.
 *
 * Licensed under the Apache License, Version 2.0 (the "License");
 * you may not use this file except in compliance with the License.
 * You may obtain a copy of the License at
 *
 *    http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */
 
 'use strict';
 
 // MODULES //
 
 var isNumber = require( '@stdlib/assert/is-number' ).isPrimitive;
 var isnan = require( '@stdlib/math/base/assert/is-nan' );
 var sqrt = require( '@stdlib/math/base/special/sqrt' );
 
 
 // MAIN //
 
 /**
 * Returns an accumulator function which incrementally computes a sample Pearson product-moment correlation coefficient.
 *
 * ## Method
 *
 * -   We begin by defining the co-moment \\(C_n\\)
 *
 *     ```tex
 *     C_n = \sum_{i=1}^{n} ( x_i - \bar{x}_n ) ( y_i - \bar{y}_n )
 *     ```
 *
 *     where \\(\bar{x}_n\\) and \\(\bar{y}_n\\) are the sample means for \\(x\\) and \\(y\\), respectively.
 *
 * -   Based on Welford's method, we know the update formulas for the sample means are given by
 *
 *     ```tex
 *     \bar{x}_n = \bar{x}_{n-1} + \frac{x_n - \bar{x}_{n-1}}{n}
 *     ```
 *
 *     and
 *
 *     ```tex
 *     \bar{y}_n = \bar{y}_{n-1} + \frac{y_n - \bar{y}_{n-1}}{n}
 *     ```
 *
 * -   Substituting into the equation for \\(C_n\\) and rearranging terms
 *
 *     ```tex
 *     C_n = C_{n-1} + (x_n - \bar{x}_n) (y_n - \bar{y}_{n-1})
 *     ```
 *
 *     where the apparent asymmetry arises from
 *
 *     ```tex
 *     x_n - \bar{x}_n = \frac{n-1}{n} (x_n - \bar{x}_{n-1})
 *     ```
 *
 *     and, hence, the update term can be equivalently expressed
 *
 *     ```tex
 *     \frac{n-1}{n} (x_n - \bar{x}_{n-1}) (y_n - \bar{y}_{n-1})
 *     ```
 *
 * -   The covariance can be defined
 *
 *     ```tex
 *     \begin{align*}
 *     \operatorname{cov}_n(x,y) &= \frac{C_n}{n} \\
 *     &= \frac{C_{n-1} + (x_n - \bar{x}_n) (y_n - \bar{y}_{n-1})}{n} \\
 *     &= \frac{(n-1)\operatorname{cov}_{n-1}(x,y) + (x_n - \bar{x}_n) (y_n - \bar{y}_{n-1})}{n}
 *     \end{align*}
 *     ```
 *
 * -   Applying Bessel's correction, we arrive at an update formula for calculating an unbiased sample covariance
 *
 *     ```tex
 *     \begin{align*}
 *     \operatorname{cov}_n(x,y) &= \frac{n}{n-1}\cdot\frac{(n-1)\operatorname{cov}_{n-1}(x,y) + (x_n - \bar{x}_n) (y_n - \bar{y}_{n-1})}{n} \\
 *     &= \operatorname{cov}_{n-1}(x,y) + \frac{(x_n - \bar{x}_n) (y_n - \bar{y}_{n-1})}{n-1} \\
 *     &= \frac{C_{n-1} + (x_n - \bar{x}_n) (y_n - \bar{y}_{n-1})}{n-1}
 *     &= \frac{C_{n-1} + (x_n - \bar{x}_{n-1}) (y_n - \bar{y}_n)}{n-1}
 *     \end{align*}
 *     ```
 *
 * -   To calculate the corrected sample standard deviation, we can use Welford's method, which can be derived as follows. We can express the variance as
 *
 *     ```tex
 *     \begin{align*}
 *     S_n &= n \sigma_n^2 \\
 *         &= \sum_{i=1}^{n} (x_i - \mu_n)^2 \\
 *         &= \biggl(\sum_{i=1}^{n} x_i^2 \biggr) - n\mu_n^2
 *     \end{align*}
 *     ```
 *
 *     Accordingly,
 *
 *     ```tex
 *     \begin{align*}
 *     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 \\
 *                   &= x_n^2 - n\mu_n^2 + (n-1)\mu_{n-1}^2 \\
 *                   &= x_n^2 - \mu_{n-1}^2 + n(\mu_{n-1}^2 - \mu_n^2) \\
 *                   &= x_n^2 - \mu_{n-1}^2 + n(\mu_{n-1} - \mu_n)(\mu_{n-1} + \mu_n) \\
 *                   &= x_n^2 - \mu_{n-1}^2 + (\mu_{n-1} - x_n)(\mu_{n-1} + \mu_n) \\
 *                   &= 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} \\
 *                   &= x_n^2 - x_n\mu_n - x_n\mu_{n-1} + \mu_n\mu_{n-1} \\
 *                   &= (x_n - \mu_{n-1})(x_n - \mu_n) \\
 *                   &= S_{n-1} + (x_n - \mu_{n-1})(x_n - \mu_n)
 *     \end{align*}
 *     ```
 *
 *     where we use the identity
 *
 *     ```tex
 *     x_n - \mu_{n-1} = n (\mu_n - \mu_{n-1})
 *     ```
 *
 * -   To compute the corrected sample standard deviation, we apply Bessel's correction and take the square root.
 *
 * -   The sample Pearson product-moment correlation coefficient can thus be calculated as
 *
 *     ```tex
 *     r = \frac{\operatorname{cov}_n(x,y)}{\sigma_x \sigma_y}
 *     ```
 *
 *     where \\(\sigma_x\\) and \\(\sigma_y\\) are the corrected sample standard deviations for \\(x\\) and \\(y\\), respectively.
 *
 * @param {number} [meanx] - mean value
 * @param {number} [meany] - mean value
 * @throws {TypeError} first argument must be a number primitive
 * @throws {TypeError} second argument must be a number primitive
 * @returns {Function} accumulator function
 *
 * @example
 * var accumulator = incrpcorr();
 *
 * var r = accumulator();
 * // returns null
 *
 * r = accumulator( 2.0, 1.0 );
 * // returns 0.0
 *
 * r = accumulator( -5.0, 3.14 );
 * // returns ~-1.0
 *
 * r = accumulator();
 * // returns ~-1.0
 *
 * @example
 * var accumulator = incrpcorr( 2.0, -3.0 );
 */
 function incrpcorr( meanx, meany ) {
 	var M2x;
 	var M2y;
 	var dy1;
 	var dy2;
 	var dy;
 	var dx;
 	var mx;
 	var my;
 	var sx;
 	var sy;
 	var C;
 	var N;
 
 	M2x = 0.0;
 	M2y = 0.0;
 	C = 0.0;
 	N = 0;
 	if ( arguments.length ) {
 		if ( !isNumber( meanx ) ) {
 			throw new TypeError( 'invalid argument. First argument must be a number primitive. Value: `' + meanx + '`.' );
 		}
 		if ( !isNumber( meany ) ) {
 			throw new TypeError( 'invalid argument. Second argument must be a number primitive. Value: `' + meany + '`.' );
 		}
 		mx = meanx;
 		my = meany;
 		return accumulator2;
 	}
 	mx = 0.0;
 	my = 0.0;
 	return accumulator1;
 
 	/**
 	* If provided input values, the accumulator function returns an updated sample correlation coefficient. If not provided input values, the accumulator function returns the current sample correlation coefficient.
 	*
 	* @private
 	* @param {number} [x] - new value
 	* @param {number} [y] - new value
 	* @returns {(number|null)} sample correlation coefficient or null
 	*/
 	function accumulator1( x, y ) {
 		var n;
 		if ( arguments.length === 0 ) {
 			if ( N === 0 ) {
 				return null;
 			}
 			if ( N === 1 ) {
 				return ( isnan( M2x ) || isnan( M2y ) ) ? NaN : 0.0;
 			}
 			return ( C/(N-1) ) / ( sx*sy );
 		}
 		N += 1;
 
 		dx = x - mx;
 		mx += dx / N;
 		M2x += dx * ( x-mx );
 
 		dy1 = y - my;
 		my += dy1 / N;
 		dy2 = y - my;
 		M2y += dy2 * dy1;
 
 		C += dx * dy2;
 		if ( N < 2 ) {
 			return ( isnan( M2x ) || isnan( M2y ) ) ? NaN : 0.0;
 		}
 		n = N - 1;
 		sx = sqrt( M2x/n );
 		sy = sqrt( M2y/n );
 		return ( C/n ) / ( sx*sy ); // Note: why all the dividing by `N`? To avoid overflow.
 	}
 
 	/**
 	* If provided input values, the accumulator function returns an updated sample correlation coefficient. If not provided input values, the accumulator function returns the current sample correlation coefficient.
 	*
 	* @private
 	* @param {number} [x] - new value
 	* @param {number} [y] - new value
 	* @returns {(number|null)} sample correlation coefficient or null
 	*/
 	function accumulator2( x, y ) {
 		if ( arguments.length === 0 ) {
 			if ( N === 0 ) {
 				return null;
 			}
 			return ( C/N ) / ( sx*sy );
 		}
 		N += 1;
 
 		dx = x - mx;
 		M2x += dx * dx;
 
 		dy = y - my;
 		M2y += dy * dy;
 
 		C += dx * dy;
 		sx = sqrt( M2x/N );
 		sy = sqrt( M2y/N );
 		return ( C/N ) / ( sx*sy ); // Note: why all the dividing by `N`? To avoid overflow.
 	}
 }
 
 
 // EXPORTS //
 
 module.exports = incrpcorr;