time-to-botec

main.js (8629B)

---

 /**
 * @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 isPositiveInteger = require( '@stdlib/assert/is-positive-integer' ).isPrimitive;
 var isNumber = require( '@stdlib/assert/is-number' ).isPrimitive;
 var isnan = require( '@stdlib/math/base/assert/is-nan' );
 
 
 // MAIN //
 
 /**
 * Returns an accumulator function which incrementally computes a moving variance-to-mean ratio (VMR).
 *
 * ## 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 primitive
 * @returns {Function} accumulator function
 *
 * @example
 * var accumulator = incrmvmr( 3 );
 *
 * var F = accumulator();
 * // returns null
 *
 * F = accumulator( 2.0 );
 * // returns 0.0
 *
 * F = accumulator( 1.0 );
 * // returns ~0.33
 *
 * F = accumulator( 3.0 );
 * // returns 0.5
 *
 * F = accumulator( 7.0 );
 * // returns ~2.55
 *
 * F = accumulator();
 * // returns ~2.55
 *
 * @example
 * var accumulator = incrmvmr( 3, 2.0 );
 */
 function incrmvmr( W, mean ) {
 	var delta;
 	var buf;
 	var tmp;
 	var M2;
 	var mu;
 	var d1;
 	var d2;
 	var N;
 	var n;
 	var i;
 	if ( !isPositiveInteger( W ) ) {
 		throw new TypeError( 'invalid argument. Must provide a positive integer. Value: `' + W + '`.' );
 	}
 	buf = new Array( W );
 	n = W - 1;
 	M2 = 0.0;
 	i = -1;
 	N = 0;
 	if ( arguments.length > 1 ) {
 		if ( !isNumber( mean ) ) {
 			throw new TypeError( 'invalid argument. Must provide a number primitive. Value: `' + mean + '`.' );
 		}
 		mu = mean;
 		return accumulator2;
 	}
 	mu = 0.0;
 	return accumulator1;
 
 	/**
 	* 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.
 	*
 	* @private
 	* @param {number} [x] - input value
 	* @returns {(number|null)} accumulated value or null
 	*/
 	function accumulator1( x ) {
 		var k;
 		var v;
 		if ( arguments.length === 0 ) {
 			if ( N === 0 ) {
 				return null;
 			}
 			if ( N === 1 ) {
 				return ( isnan( M2 ) ) ? NaN : 0.0/mu;
 			}
 			if ( N < W ) {
 				return ( M2/(N-1) ) / mu;
 			}
 			return ( M2/n ) / mu;
 		}
 		// Update the index for managing the circular buffer:
 		i = (i+1) % W;
 
 		// Case: incoming value is NaN, the sliding second moment is automatically NaN...
 		if ( isnan( x ) ) {
 			N = W; // explicitly set to avoid `N < W` branch
 			mu = NaN;
 			M2 = NaN;
 		}
 		// Case: initial window...
 		else if ( N < W ) {
 			buf[ i ] = x; // update buffer
 			N += 1;
 			delta = x - mu;
 			mu += delta / N;
 			M2 += delta * (x - mu);
 			if ( N === 1 ) {
 				return 0.0 / mu;
 			}
 			return ( M2/(N-1) ) / mu;
 		}
 		// Case: N = W = 1
 		else if ( N === 1 ) {
 			mu = x;
 			M2 = 0.0;
 			return M2 / mu;
 		}
 		// Case: outgoing value is NaN, and, thus, we need to compute the accumulated values...
 		else if ( isnan( buf[ i ] ) ) {
 			N = 1;
 			mu = x;
 			M2 = 0.0;
 			for ( k = 0; k < W; k++ ) {
 				if ( k !== i ) {
 					v = buf[ k ];
 					if ( isnan( v ) ) {
 						N = W; // explicitly set to avoid `N < W` branch
 						mu = NaN;
 						M2 = NaN;
 						break; // second moment is automatically NaN, so no need to continue
 					}
 					N += 1;
 					delta = v - mu;
 					mu += delta / N;
 					M2 += delta * (v - mu);
 				}
 			}
 		}
 		// Case: neither the current second moment nor the incoming value are NaN, so we need to update the accumulated values...
 		else if ( isnan( M2 ) === false ) {
 			tmp = buf[ i ];
 			delta = x - tmp;
 			d1 = tmp - mu;
 			mu += delta / W;
 			d2 = x - mu;
 			M2 += delta * (d1 + d2);
 		}
 		// Case: the current second moment is NaN, so nothing to do until the buffer no longer contains NaN values...
 		buf[ i ] = x;
 
 		return ( M2/n ) / mu;
 	}
 
 	/**
 	* 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.
 	*
 	* @private
 	* @param {number} [x] - input value
 	* @returns {(number|null)} accumulated value or null
 	*/
 	function accumulator2( x ) {
 		var k;
 		if ( arguments.length === 0 ) {
 			if ( N === 0 ) {
 				return null;
 			}
 			if ( N < W ) {
 				return ( M2/N ) / mu;
 			}
 			return ( M2/W ) / mu;
 		}
 		// Update the index for managing the circular buffer:
 		i = (i+1) % W;
 
 		// Case: incoming value is NaN, the sliding second moment is automatically NaN...
 		if ( isnan( x ) ) {
 			N = W; // explicitly set to avoid `N < W` branch
 			M2 = NaN;
 		}
 		// Case: initial window...
 		else if ( N < W ) {
 			buf[ i ] = x; // update buffer
 			N += 1;
 			delta = x - mu;
 			M2 += delta * delta;
 			return ( M2/N ) / mu;
 		}
 		// Case: outgoing value is NaN, and, thus, we need to compute the accumulated values...
 		else if ( isnan( buf[ i ] ) ) {
 			M2 = 0.0;
 			for ( k = 0; k < W; k++ ) {
 				if ( k !== i ) {
 					if ( isnan( buf[ k ] ) ) {
 						N = W; // explicitly set to avoid `N < W` branch
 						M2 = NaN;
 						break; // second moment is automatically NaN, so no need to continue
 					}
 					delta = buf[ k ] - mu;
 					M2 += delta * delta;
 				}
 			}
 		}
 		// Case: neither the current second moment nor the incoming value are NaN, so we need to update the accumulated values...
 		else if ( isnan( M2 ) === false ) {
 			tmp = buf[ i ];
 			M2 += ( x-tmp ) * ( x-mu + tmp-mu );
 		}
 		// Case: the current second moment is NaN, so nothing to do until the buffer no longer contains NaN values...
 		buf[ i ] = x;
 
 		return ( M2/W ) / mu;
 	}
 }
 
 
 // EXPORTS //
 
 module.exports = incrmvmr;