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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 incrmvmr = require( '@stdlib/stats/incr/mvmr' );
var isnan = require( '@stdlib/math/base/assert/is-nan' );
// MAIN //
/**
* Returns an accumulator function which incrementally computes a moving variance-to-mean ratio (VMR) while ignoring NaN values.
*
* ## Method
*
* - Let \\(W\\) be a window of \\(N\\) elements over which we want to compute a variance-to-mean ratio (VMR).
*
* - 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
*
* ```tex
* \Delta s^2 = s_{i+1}^2 - s_{i}^2
* ```
*
* - If we multiply both sides by \\(N-1\\),
*
* ```tex
* (N-1)(\Delta s^2) = (N-1)s_{i+1}^2 - (N-1)s_{i}^2
* ```
*
* - If we substitute the definition of the unbiased sample variance having the form
*
* ```tex
* \begin{align*}
* s^2 &= \frac{1}{N-1} \biggl( \sum_{i=1}^{N} (x_i - \bar{x})^2 \biggr) \\
* &= \frac{1}{N-1} \biggl( \sum_{i=1}^{N} (x_i^2 - 2\bar{x}x_i + \bar{x}^2) \biggr) \\
* &= \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) \\
* &= \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) \\
* &= \frac{1}{N-1} \biggl( \sum_{i=1}^{N} x_i^2 - 2N\bar{x}^2 + N\bar{x}^2 \biggr) \\
* &= \frac{1}{N-1} \biggl( \sum_{i=1}^{N} x_i^2 - N\bar{x}^2 \biggr)
* \end{align*}
* ```
*
* we return
*
* ```tex
* (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)
* ```
*
* - This can be further simplified by recognizing that subtracting the sums reduces to \\(x_N^2 - x_0^2\\); in which case,
*
* ```tex
* \begin{align*}
* (N-1)(\Delta s^2) &= x_N^2 - x_0^2 - N\bar{x}_{i+1}^2 + N\bar{x}_{i}^2 \\
* &= x_N^2 - x_0^2 - N(\bar{x}_{i+1}^2 - \bar{x}_{i}^2) \\
* &= x_N^2 - x_0^2 - N(\bar{x}_{i+1} - \bar{x}_{i})(\bar{x}_{i+1} + \bar{x}_{i})
* \end{align*}
* ```
*
* - Recognizing that the difference of means can be expressed
*
* ```tex
* \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}
* ```
*
* and substituting into the equation above
*
* ```tex
* (N-1)(\Delta s^2) = x_N^2 - x_0^2 - (x_N - x_0)(\bar{x}_{i+1} + \bar{x}_{i})
* ```
*
* - Rearranging terms gives us the update equation for the unbiased sample variance
*
* ```tex
* \begin{align*}
* (N-1)(\Delta s^2) &= (x_N - x_0)(x_N + x_0) - (x_N - x_0)(\bar{x}_{i+1} + \bar{x}_{i}) \\
* &= (x_N - x_0)(x_N + x_0 - \bar{x}_{i+1} - \bar{x}_{i}) \\
* &= (x_N - x_0)(x_N - \bar{x}_{i+1} + x_0 - \bar{x}_{i})
* \end{align*}
* ```
*
* @param {PositiveInteger} W - window size
* @param {number} [mean] - mean value
* @throws {TypeError} first argument must be a positive integer
* @throws {TypeError} second argument must be a number
* @returns {Function} accumulator function
*
* @example
* var accumulator = incrnanmvmr( 3 );
*
* var F = accumulator();
* // returns null
*
* F = accumulator( 2.0 );
* // returns 0.0
*
* F = accumulator( NaN );
* // returns 0.0
*
* F = accumulator( 1.0 );
* // returns ~0.33
*
* F = accumulator( 3.0 );
* // returns 0.5
*
* F = accumulator( NaN );
* // returns 0.5
*
* F = accumulator( 7.0 );
* // returns ~2.55
*
* F = accumulator();
* // returns ~2.55
*
* @example
* var accumulator = incrnanmvmr( 3, 2.0 );
*/
function incrnanmvmr( W, mean ) {
var acc;
// Initialize the wrapped accumulator:
if ( arguments.length > 1 ) {
acc = incrmvmr( W, mean );
} else {
acc = incrmvmr( W );
}
return accumulator;
/**
* 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.
* NaN input values are ignored.
*
* @private
* @param {number} [x] - input value
* @returns {(number|null)} accumulated value or null
*/
function accumulator( x ) {
if ( arguments.length === 0 ) {
return acc();
}
if ( isnan( x ) === false ) {
return acc( x );
}
return acc();
}
}
// EXPORTS //
module.exports = incrnanmvmr;
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