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* @license Apache-2.0
*
* Copyright (c) 2025 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 isnan = require( '@stdlib/math/base/assert/is-nan' );
var incrpcorr = require( '@stdlib/stats/incr/pcorr' );
// MAIN //
/**
* Returns an accumulator function which incrementally computes a sample Pearson product-moment correlation coefficient, ignoring `NaN` values.
*
* ## 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
* @throws {TypeError} second argument must be a number
* @returns {Function} accumulator function
*
* @example
* var accumulator = incrnanpcorr();
*
* var r = accumulator();
* // returns null
*
* r = accumulator( 2.0, 1.0 );
* // returns 0.0
*
* r = accumulator( -5.0, NaN );
* // returns 0.0
*
* r = accumulator( -5.0, 3.14 );
* // returns ~-1.0
*
* r = accumulator( NaN, 3.14 );
* // returns ~-1.0
*
* r = accumulator();
* // returns ~-1.0
*
* @example
* var accumulator = incrpcorr( 2.0, -3.0 );
*/
function incrnanpcorr( meanx, meany ) {
var pcorr;
var N;
if ( arguments.length ) {
pcorr = incrpcorr( meanx, meany );
} else {
pcorr = incrpcorr();
}
N = 0;
return accumulator;
/**
* Accumulator function that updates the sample correlation coefficient, ignoring `NaN` values.
*
* @private
* @param {number} [x] - new value
* @param {number} [y] - new value
* @returns {(number|null)} sample absolute correlation coefficient or null
*/
function accumulator( x, y ) {
if ( arguments.length === 0 ) {
if ( N === 0 ) {
return null;
}
return pcorr();
}
if ( isnan( x ) || isnan( y ) ) {
return pcorr();
}
N += 1;
return pcorr( x, y );
}
}
// EXPORTS //
module.exports = incrnanpcorr;
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