401 lines
13 KiB
JavaScript
401 lines
13 KiB
JavaScript
/**
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* @license Apache-2.0
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*
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* Copyright (c) 2018 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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// MODULES //
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var isPositiveInteger = require( '@stdlib/assert/is-positive-integer' ).isPrimitive;
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var isNumber = require( '@stdlib/assert/is-number' ).isPrimitive;
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var isnan = require( '@stdlib/math/base/assert/is-nan' );
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var Float64Array = require( '@stdlib/array/float64' );
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// MAIN //
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/**
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* Returns an accumulator function which incrementally computes a moving unbiased sample covariance.
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*
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* ## Method
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*
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* - Let \\(W\\) be a window of \\(N\\) elements over which we want to compute an unbiased sample covariance.
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*
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* - We begin by defining the covariance \\( \operatorname{cov}_n(x,y) \\) for a window \\(n\\) as follows
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*
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* ```tex
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* \operatorname{cov}_n(x,y) &= \frac{C_n}{n}
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* ```
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*
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* where \\(C_n\\) is the co-moment, which is defined as
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*
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* ```tex
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* C_n = \sum_{i=1}^{N} ( x_i - \bar{x}_n ) ( y_i - \bar{y}_n )
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* ```
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*
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* and where \\(\bar{x}_n\\) and \\(\bar{y}_n\\) are the sample means for \\(x\\) and \\(y\\), respectively, and \\(i=1\\) specifies the first element in a window.
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*
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* - The sample mean is computed using the canonical formula
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*
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* ```tex
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* \bar{x}_n = \frac{1}{N} \sum_{i=1}^{N} x_i
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* ```
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*
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* which, taking into account a previous window, can be expressed
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*
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* ```tex
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* \begin{align*}
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* \bar{x}_n &= \frac{1}{N} \biggl( \sum_{i=0}^{N-1} x_i - x_0 + x_N \biggr) \\
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* &= \bar{x}_{n-1} + \frac{x_N - x_0}{N}
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* \end{align*}
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* ```
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*
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* where \\(x_0\\) is the first value in the previous window.
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*
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* - We can substitute into the co-moment equation
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*
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* ```tex
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* \begin{align*}
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* C_n &= \sum_{i=1}^{N} ( x_i - \bar{x}_n ) ( y_i - \bar{y}_n ) \\
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* &= \sum_{i=1}^{N} \biggl( x_i - \bar{x}_{n-1} - \frac{x_N - x_0}{N} \biggr) \biggl( y_i - \bar{y}_{n-1} - \frac{y_N - y_0}{N} \biggr) \\
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* &= \sum_{i=1}^{N} \biggl( \Delta x_{i,n-1} - \frac{x_N - x_0}{N} \biggr) \biggl( \Delta y_{i,n-1} - \frac{y_N - y_0}{N} \biggr)
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* \end{align*}
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* ```
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*
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* where
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*
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* ```tex
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* \Delta x_{i,k} = x_i - \bar{x}_{k}
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* ```
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*
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* - We can subsequently expand terms and apply a summation identity
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*
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* ```tex
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* \begin{align*}
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* C_n &= \sum_{i=1}^{N} \Delta x_{i,n-1} \Delta y_{i,n-1} - \sum_{i=1}^{N} \Delta x_{i,n-1} \biggl( \frac{y_N - y_0}{N} \biggr) - \sum_{i=1}^{N} \Delta y_{i,n-1} \biggl( \frac{x_N - x_0}{N} \biggr) + \sum_{i=1}^{N} \biggl( \frac{x_N - x_0}{N} \biggr) \biggl( \frac{y_N - y_0}{N} \biggr) \\
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* &= \sum_{i=1}^{N} \Delta x_{i,n-1} \Delta y_{i,n-1} - \biggl( \frac{y_N - y_0}{N} \biggr) \sum_{i=1}^{N} \Delta x_{i,n-1} - \biggl( \frac{x_N - x_0}{N} \biggr) \sum_{i=1}^{N} \Delta y_{i,n-1} + \frac{(x_N - x_0)(y_N - y_0)}{N}
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* \end{align*}
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* ```
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*
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* - Let us first consider the second term which we can reorganize as follows
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*
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* ```tex
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* \begin{align*}
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* \biggl( \frac{y_N - y_0}{N} \biggr) \sum_{i=1}^{N} \Delta x_{i,n-1} &= \biggl( \frac{y_N - y_0}{N} \biggr) \sum_{i=1}{N} ( x_i - \bar{x}_{n-1}) \\
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* &= \biggl( \frac{y_N - y_0}{N} \biggr) \sum_{i=1}^{N} x_i - \biggl( \frac{y_N - y_0}{N} \biggr) \sum_{i=1}^{N} \bar{x}_{n-1} \\
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* &= (y_N - y_0) \bar{x}_{n} - (y_N - y_0)\bar{x}_{n-1} \\
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* &= (y_N - y_0) (\bar{x}_{n} - \bar{x}_{n-1}) \\
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* &= \frac{(x_N - x_0)(y_N - y_0)}{N}
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* \end{align*}
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* ```
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*
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* - The third term can be reorganized in a manner similar to the second term such that
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*
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* ```tex
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* \biggl( \frac{x_N - x_0}{N} \biggr) \sum_{i=1}^{N} \Delta y_{i,n-1} = \frac{(x_N - x_0)(y_N - y_0)}{N}
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* ```
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*
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* - Substituting back into the equation for the co-moment
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*
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* ```tex
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* \begin{align*}
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* C_n &= \sum_{i=1}^{N} \Delta x_{i,n-1} \Delta y_{i,n-1} - \frac{(x_N - x_0)(y_N - y_0)}{N} - \frac{(x_N - x_0)(y_N - y_0)}{N} + \frac{(x_N - x_0)(y_N - y_0)}{N} \\
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* &= \sum_{i=1}^{N} \Delta x_{i,n-1} \Delta y_{i,n-1} - \frac{(x_N - x_0)(y_N - y_0)}{N}
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* \end{align*}
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* ```
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*
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* - Let us now consider the first term which we can modify as follows
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*
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* ```tex
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* \begin{align*}
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* \sum_{i=1}^{N} \Delta x_{i,n-1} \Delta y_{i,n-1} &= \sum_{i=1}^{N} (x_i - \bar{x}_{n-1})(y_i - \bar{y}_{n-1}) \\
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* &= \sum_{i=1}^{N-1} (x_i - \bar{x}_{n-1})(y_i - \bar{y}_{n-1}) + (x_N - \bar{x}_{n-1})(y_N - \bar{y}_{n-1}) \\
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* &= \sum_{i=1}^{N-1} (x_i - \bar{x}_{n-1})(y_i - \bar{y}_{n-1}) + (x_N - \bar{x}_{n-1})(y_N - \bar{y}_{n-1}) + (x_0 - \bar{x}_{n-1})(y_0 - \bar{y}_{n-1}) - (x_0 - \bar{x}_{n-1})(y_0 - \bar{y}_{n-1}) \\
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* &= \sum_{i=0}^{N-1} (x_i - \bar{x}_{n-1})(y_i - \bar{y}_{n-1}) + (x_N - \bar{x}_{n-1})(y_N - \bar{y}_{n-1}) - (x_0 - \bar{x}_{n-1})(y_0 - \bar{y}_{n-1})
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* \end{align*}
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* ```
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*
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* where we recognize that the first term equals the co-moment for the previous window
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*
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* ```tex
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* C_{n-1} = \sum_{i=0}^{N-1} (x_i - \bar{x}_{n-1})(y_i - \bar{y}_{n-1})
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* ```
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*
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* In which case,
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*
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* ```tex
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* \begin{align*}
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* \sum_{i=1}^{N} \Delta x_{i,n-1} \Delta y_{i,n-1} &= C_{n-1} + (x_N - \bar{x}_{n-1})(y_N - \bar{y}_{n-1}) - (x_0 - \bar{x}_{n-1})(y_0 - \bar{y}_{n-1}) \\
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* &= C_{n-1} + \Delta x_{N,n-1} \Delta y_{N,n-1} - \Delta x_{0,n-1} \Delta y_{0,n-1}
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* \end{align*}
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* ```
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*
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* - Substituting into the equation for the co-moment
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*
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* ```tex
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* C_n = C_{n-1} + \Delta x_{N,n-1} \Delta y_{N,n-1} - \Delta x_{0,n-1} \Delta y_{0,n-1} - \frac{(x_N - x_0)(y_N - y_0)}{N}
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* ```
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*
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* - We can make one further modification to the last term
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*
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* ```tex
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* \begin{align*}
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* \frac{(x_N - x_0)(y_N - y_0)}{N} &= \frac{(x_N - \bar{x}_{n-1} - x_0 + \bar{x}_{n-1})(y_N - \bar{y}_{n-1} - y_0 + \bar{y}_{n-1})}{N} \\
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* &= \frac{(\Delta x_{N,n-1} - \Delta x_{0,n-1})(\Delta y_{N,n-1} - \Delta y_{0,n-1})}{N}
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* \end{align*}
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* ```
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*
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* which, upon substitution into the equation for the co-moment, yields
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*
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* ```tex
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* C_n = C_{n-1} + \Delta x_{N,n-1} \Delta y_{N,n-1} - \Delta x_{0,n-1} \Delta y_{0,n-1} - \frac{(\Delta x_{N,n-1} - \Delta x_{0,n-1})(\Delta y_{N,n-1} - \Delta y_{0,n-1})}{N}
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* ```
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*
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*
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* @param {PositiveInteger} W - window size
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* @param {number} [meanx] - mean value
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* @param {number} [meany] - mean value
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* @throws {TypeError} first argument must be a positive integer
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* @throws {TypeError} second argument must be a number primitive
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* @throws {TypeError} third argument must be a number primitive
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* @returns {Function} accumulator function
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*
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* @example
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* var accumulator = incrmcovariance( 3 );
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*
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* var v = accumulator();
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* // returns null
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*
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* v = accumulator( 2.0, 1.0 );
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* // returns 0.0
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*
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* v = accumulator( -5.0, 3.14 );
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* // returns ~-7.49
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*
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* v = accumulator( 3.0, -1.0 );
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* // returns -8.35
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*
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* v = accumulator( 5.0, -9.5 );
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* // returns -29.42
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*
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* v = accumulator();
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* // returns -29.42
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*
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* @example
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* var accumulator = incrmcovariance( 3, -2.0, 10.0 );
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*/
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function incrmcovariance( W, meanx, meany ) {
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var buf;
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var dx0;
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var dxN;
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var dy0;
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var dyN;
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var mx;
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var my;
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var wi;
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var C;
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var N;
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var n;
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var i;
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if ( !isPositiveInteger( W ) ) {
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throw new TypeError( 'invalid argument. First argument must be a positive integer. Value: `' + W + '`.' );
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}
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buf = new Float64Array( 2*W ); // strided array
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n = W - 1;
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C = 0.0;
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wi = -1;
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N = 0;
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if ( arguments.length > 1 ) {
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if ( !isNumber( meanx ) ) {
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throw new TypeError( 'invalid argument. Second argument must be a number primitive. Value: `' + meanx + '`.' );
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}
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if ( !isNumber( meany ) ) {
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throw new TypeError( 'invalid argument. Third argument must be a number primitive. Value: `' + meany + '`.' );
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}
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mx = meanx;
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my = meany;
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return accumulator2;
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}
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mx = 0.0;
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my = 0.0;
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return accumulator1;
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/**
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* If provided a value, the accumulator function returns an updated unbiased sample covariance. If not provided a value, the accumulator function returns the current unbiased sample covariance.
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*
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* @private
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* @param {number} [x] - input value
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* @param {number} [y] - input value
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* @returns {(number|null)} unbiased sample covariance or null
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*/
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function accumulator1( x, y ) {
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var v1;
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var v2;
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var k;
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var j;
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if ( arguments.length === 0 ) {
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if ( N === 0 ) {
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return null;
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}
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if ( N === 1 ) {
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return 0.0;
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}
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if ( N < W ) {
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return C / (N-1);
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}
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return C / n;
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}
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// Update the window and strided array indices for managing the circular buffer:
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wi = (wi+1) % W;
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i = 2 * wi;
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// Case: an incoming value is NaN, the sliding co-moment is automatically NaN...
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if ( isnan( x ) || isnan( y ) ) {
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N = W; // explicitly set to avoid `N < W` branch
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C = NaN;
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}
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// Case: initial window...
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else if ( N < W ) {
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buf[ i ] = x; // update buffer
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buf[ i+1 ] = y;
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N += 1;
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dxN = x - mx;
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mx += dxN / N;
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my += ( y-my ) / N;
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C += dxN * ( y-my ); // Note: repeated `y-my` is intentional, as `my` is updated when used here
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if ( N === 1 ) {
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return 0.0;
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}
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return C / (N-1);
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}
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// Case: N = W = 1
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else if ( N === 1 ) {
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return 0.0;
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}
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// Case: an outgoing value is NaN, and, thus, we need to compute the accumulated values...
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else if ( isnan( buf[ i ] ) || isnan( buf[ i+1 ] ) ) {
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N = 1;
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mx = x;
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my = y;
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C = 0.0;
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for ( k = 0; k < W; k++ ) {
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j = 2 * k; // convert to a strided array index
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if ( j !== i ) {
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v1 = buf[ j ];
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v2 = buf[ j+1 ];
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if ( isnan( v1 ) || isnan( v2 ) ) {
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N = W; // explicitly set to avoid `N < W` branch
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C = NaN;
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break; // co-moment is automatically NaN, so no need to continue
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}
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N += 1;
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dxN = v1 - mx;
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mx += dxN / N;
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my += ( v2-my ) / N;
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C += dxN * ( v2-my ); // Note: repeated `y-my` is intentional, as `my` is updated when used here
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}
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}
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}
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// Case: neither the current co-moment nor the incoming values are NaN, so we need to update the accumulated values...
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else if ( isnan( C ) === false ) {
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dx0 = buf[ i ] - mx;
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dy0 = buf[ i+1 ] - my;
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dxN = x - mx;
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dyN = y - my;
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C += (dxN*dyN) - (dx0*dy0) - ( (dxN-dx0)*(dyN-dy0)/W );
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mx += ( dxN-dx0 ) / W;
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my += ( dyN-dy0 ) / W;
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}
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// Case: the current co-moment is NaN, so nothing to do until the buffer no longer contains NaN values...
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buf[ i ] = x;
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buf[ i+1 ] = y;
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return C / n;
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}
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/**
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* If provided a value, the accumulator function returns an updated unbiased sample covariance. If not provided a value, the accumulator function returns the current unbiased sample covariance.
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*
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* @private
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* @param {number} [x] - input value
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* @param {number} [y] - input value
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* @returns {(number|null)} unbiased sample covariance or null
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*/
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function accumulator2( x, y ) {
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var k;
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var j;
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if ( arguments.length === 0 ) {
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if ( N === 0 ) {
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return null;
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}
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if ( N < W ) {
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return C / N;
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}
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return C / W;
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}
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// Update the window and strided array indices for managing the circular buffer:
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wi = (wi+1) % W;
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i = 2 * wi;
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// Case: an incoming value is NaN, the sliding co-moment is automatically NaN...
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if ( isnan( x ) || isnan( y ) ) {
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N = W; // explicitly set to avoid `N < W` branch
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C = NaN;
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}
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// Case: initial window...
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else if ( N < W ) {
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buf[ i ] = x; // update buffer
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buf[ i+1 ] = y;
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N += 1;
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C += ( x-mx ) * ( y-my );
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return C / N;
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}
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// Case: an outgoing value is NaN, and, thus, we need to compute the accumulated values...
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else if ( isnan( buf[ i ] ) || isnan( buf[ i+1 ] ) ) {
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C = 0.0;
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for ( k = 0; k < W; k++ ) {
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j = 2 * k; // convert to a strided array index
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if ( j !== i ) {
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if ( isnan( buf[ j ] ) || isnan( buf[ j+1 ] ) ) {
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N = W; // explicitly set to avoid `N < W` branch
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C = NaN;
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break; // co-moment is automatically NaN, so no need to continue
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}
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C += ( buf[j]-mx ) * ( buf[j+1]-my );
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}
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}
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}
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// Case: neither the current co-moment nor the incoming values are NaN, so we need to update the accumulated values...
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else if ( isnan( C ) === false ) {
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// Use textbook formula along with simplification arising from difference of sums:
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C += ( (x-mx)*(y-my) ) - ( (buf[i]-mx)*(buf[i+1]-my) );
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}
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// Case: the current co-moment is NaN, so nothing to do until the buffer no longer contains NaN values...
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buf[ i ] = x;
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buf[ i+1 ] = y;
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return C / W;
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}
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}
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// EXPORTS //
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module.exports = incrmcovariance;
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