309 lines
8.4 KiB
JavaScript
309 lines
8.4 KiB
JavaScript
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/**
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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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// MAIN //
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/**
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* Returns an accumulator function which incrementally computes a moving variance-to-mean ratio (VMR).
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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 a variance-to-mean ratio (VMR).
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*
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* - The difference between the unbiased sample variance in a window \\(W_i\\) and the unbiased sample variance in a window \\(W_{i+1})\\) is given by
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*
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* ```tex
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* \Delta s^2 = s_{i+1}^2 - s_{i}^2
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* ```
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*
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* - If we multiply both sides by \\(N-1\\),
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*
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* ```tex
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* (N-1)(\Delta s^2) = (N-1)s_{i+1}^2 - (N-1)s_{i}^2
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* ```
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*
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* - If we substitute the definition of the unbiased sample variance having the form
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*
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* ```tex
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* \begin{align*}
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* s^2 &= \frac{1}{N-1} \biggl( \sum_{i=1}^{N} (x_i - \bar{x})^2 \biggr) \\
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* &= \frac{1}{N-1} \biggl( \sum_{i=1}^{N} (x_i^2 - 2\bar{x}x_i + \bar{x}^2) \biggr) \\
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* &= \frac{1}{N-1} \biggl( \sum_{i=1}^{N} x_i^2 - 2\bar{x} \sum_{i=1}^{N} x_i + \sum_{i=1}^{N} \bar{x}^2) \biggr) \\
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* &= \frac{1}{N-1} \biggl( \sum_{i=1}^{N} x_i^2 - \frac{2N\bar{x}\sum_{i=1}^{N} x_i}{N} + N\bar{x}^2 \biggr) \\
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* &= \frac{1}{N-1} \biggl( \sum_{i=1}^{N} x_i^2 - 2N\bar{x}^2 + N\bar{x}^2 \biggr) \\
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* &= \frac{1}{N-1} \biggl( \sum_{i=1}^{N} x_i^2 - N\bar{x}^2 \biggr)
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* \end{align*}
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* ```
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*
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* we return
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*
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* ```tex
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* (N-1)(\Delta s^2) = \biggl(\sum_{k=1}^N x_k^2 - N\bar{x}_{i+1}^2 \biggr) - \biggl(\sum_{k=0}^{N-1} x_k^2 - N\bar{x}_{i}^2 \biggr)
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* ```
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*
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* - This can be further simplified by recognizing that subtracting the sums reduces to \\(x_N^2 - x_0^2\\); in which case,
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*
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* ```tex
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* \begin{align*}
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* (N-1)(\Delta s^2) &= x_N^2 - x_0^2 - N\bar{x}_{i+1}^2 + N\bar{x}_{i}^2 \\
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* &= x_N^2 - x_0^2 - N(\bar{x}_{i+1}^2 - \bar{x}_{i}^2) \\
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* &= x_N^2 - x_0^2 - N(\bar{x}_{i+1} - \bar{x}_{i})(\bar{x}_{i+1} + \bar{x}_{i})
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* \end{align*}
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* ```
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*
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* - Recognizing that the difference of means can be expressed
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*
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* ```tex
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* \bar{x}_{i+1} - \bar{x}_i = \frac{1}{N} \biggl( \sum_{k=1}^N x_k - \sum_{k=0}^{N-1} x_k \biggr) = \frac{x_N - x_0}{N}
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* ```
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*
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* and substituting into the equation above
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*
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* ```tex
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* (N-1)(\Delta s^2) = x_N^2 - x_0^2 - (x_N - x_0)(\bar{x}_{i+1} + \bar{x}_{i})
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* ```
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*
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* - Rearranging terms gives us the update equation for the unbiased sample variance
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*
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* ```tex
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* \begin{align*}
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* (N-1)(\Delta s^2) &= (x_N - x_0)(x_N + x_0) - (x_N - x_0)(\bar{x}_{i+1} + \bar{x}_{i})
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* &= (x_N - x_0)(x_N + x_0 - \bar{x}_{i+1} - \bar{x}_{i}) \\
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* &= (x_N - x_0)(x_N - \bar{x}_{i+1} + x_0 - \bar{x}_{i})
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* \end{align*}
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* ```
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*
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* @param {PositiveInteger} W - window size
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* @param {number} [mean] - 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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* @returns {Function} accumulator function
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*
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* @example
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* var accumulator = incrmvmr( 3 );
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*
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* var F = accumulator();
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* // returns null
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*
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* F = accumulator( 2.0 );
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* // returns 0.0
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*
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* F = accumulator( 1.0 );
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* // returns ~0.33
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*
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* F = accumulator( 3.0 );
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* // returns 0.5
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*
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* F = accumulator( 7.0 );
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* // returns ~2.55
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*
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* F = accumulator();
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* // returns ~2.55
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*
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* @example
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* var accumulator = incrmvmr( 3, 2.0 );
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*/
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function incrmvmr( W, mean ) {
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var delta;
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var buf;
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var tmp;
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var M2;
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var mu;
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var d1;
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var d2;
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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. Must provide a positive integer. Value: `' + W + '`.' );
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}
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buf = new Array( W );
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n = W - 1;
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M2 = 0.0;
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i = -1;
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N = 0;
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if ( arguments.length > 1 ) {
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if ( !isNumber( mean ) ) {
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throw new TypeError( 'invalid argument. Must provide a number primitive. Value: `' + mean + '`.' );
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}
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mu = mean;
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return accumulator2;
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}
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mu = 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 accumulated value. If not provided a value, the accumulator function returns the current accumulated value.
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*
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* @private
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* @param {number} [x] - input value
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* @returns {(number|null)} accumulated value or null
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*/
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function accumulator1( x ) {
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var k;
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var v;
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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 ( isnan( M2 ) ) ? NaN : 0.0/mu;
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}
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if ( N < W ) {
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return ( M2/(N-1) ) / mu;
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}
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return ( M2/n ) / mu;
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}
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// Update the index for managing the circular buffer:
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i = (i+1) % W;
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// Case: incoming value is NaN, the sliding second moment is automatically NaN...
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if ( isnan( x ) ) {
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N = W; // explicitly set to avoid `N < W` branch
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mu = NaN;
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M2 = 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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N += 1;
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delta = x - mu;
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mu += delta / N;
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M2 += delta * (x - mu);
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if ( N === 1 ) {
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return 0.0 / mu;
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}
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return ( M2/(N-1) ) / mu;
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}
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// Case: N = W = 1
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else if ( N === 1 ) {
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mu = x;
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M2 = 0.0;
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return M2 / mu;
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}
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// Case: outgoing value is NaN, and, thus, we need to compute the accumulated values...
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else if ( isnan( buf[ i ] ) ) {
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N = 1;
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mu = x;
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M2 = 0.0;
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for ( k = 0; k < W; k++ ) {
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if ( k !== i ) {
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v = buf[ k ];
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if ( isnan( v ) ) {
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N = W; // explicitly set to avoid `N < W` branch
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mu = NaN;
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M2 = NaN;
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break; // second moment is automatically NaN, so no need to continue
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}
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N += 1;
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delta = v - mu;
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mu += delta / N;
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M2 += delta * (v - mu);
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}
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}
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}
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// Case: neither the current second moment nor the incoming value are NaN, so we need to update the accumulated values...
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else if ( isnan( M2 ) === false ) {
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tmp = buf[ i ];
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delta = x - tmp;
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d1 = tmp - mu;
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mu += delta / W;
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d2 = x - mu;
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M2 += delta * (d1 + d2);
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}
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// Case: the current second 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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return ( M2/n ) / mu;
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}
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/**
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* If provided a value, the accumulator function returns an updated accumulated value. If not provided a value, the accumulator function returns the current accumulated value.
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*
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* @private
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* @param {number} [x] - input value
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* @returns {(number|null)} accumulated value or null
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*/
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function accumulator2( x ) {
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var k;
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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 ( M2/N ) / mu;
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}
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return ( M2/W ) / mu;
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}
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// Update the index for managing the circular buffer:
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i = (i+1) % W;
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// Case: incoming value is NaN, the sliding second moment is automatically NaN...
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if ( isnan( x ) ) {
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N = W; // explicitly set to avoid `N < W` branch
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M2 = 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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N += 1;
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delta = x - mu;
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M2 += delta * delta;
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return ( M2/N ) / mu;
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}
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// Case: outgoing value is NaN, and, thus, we need to compute the accumulated values...
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else if ( isnan( buf[ i ] ) ) {
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M2 = 0.0;
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for ( k = 0; k < W; k++ ) {
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if ( k !== i ) {
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if ( isnan( buf[ k ] ) ) {
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N = W; // explicitly set to avoid `N < W` branch
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M2 = NaN;
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break; // second moment is automatically NaN, so no need to continue
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}
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delta = buf[ k ] - mu;
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M2 += delta * delta;
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}
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}
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}
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// Case: neither the current second moment nor the incoming value are NaN, so we need to update the accumulated values...
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else if ( isnan( M2 ) === false ) {
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tmp = buf[ i ];
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M2 += ( x-tmp ) * ( x-mu + tmp-mu );
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}
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// Case: the current second 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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return ( M2/W ) / mu;
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}
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}
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// EXPORTS //
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module.exports = incrmvmr;
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