time-to-botec

main.js (13688B)

---

 /**
 * @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 isSquareMatrix = require( '@stdlib/assert/is-square-matrix' );
 var isVectorLike = require( '@stdlib/assert/is-vector-like' );
 var Float64Array = require( '@stdlib/array/float64' );
 var sqrt = require( '@stdlib/math/base/special/sqrt' );
 var ctor = require( '@stdlib/ndarray/ctor' );
 var bctor = require( '@stdlib/ndarray/base/ctor' );
 var numel = require( '@stdlib/ndarray/base/numel' );
 
 
 // FUNCTIONS //
 
 /**
 * Returns a matrix.
 *
 * @private
 * @param {PositiveInteger} n - matrix order
 * @param {boolean} bool - boolean indicating whether to create a low-level ndarray
 * @returns {ndarray} matrix
 */
 function createMatrix( n, bool ) {
 	var strides;
 	var buffer;
 	var shape;
 	var f;
 
 	if ( bool ) {
 		f = bctor;
 	} else {
 		f = ctor;
 	}
 	buffer = new Float64Array( n*n );
 	shape = [ n, n ];
 	strides = [ n, 1 ];
 	return f( 'float64', buffer, shape, strides, 0, 'row-major' );
 }
 
 /**
 * Sets the values along the main diagonal of a square matrix.
 *
 * @private
 * @param {ndarray} matrix - input square matrix
 * @param {number} v - value
 * @returns {ndarray} input matrix
 */
 function diagonal( matrix, v ) {
 	var M = matrix.shape[ 0 ];
 	var i;
 	for ( i = 0; i < M; i++ ) {
 		matrix.set( i, i, v );
 	}
 	return matrix;
 }
 
 /**
 * Returns a vector.
 *
 * @private
 * @param {PositiveInteger} N - number of elements
 * @returns {ndarray} vector
 */
 function createVector( N ) {
 	var strides;
 	var buffer;
 	var shape;
 
 	buffer = new Float64Array( N );
 	shape = [ N ];
 	strides = [ 1 ];
 
 	return bctor( 'float64', buffer, shape, strides, 0, 'row-major' );
 }
 
 
 // MAIN //
 
 /**
 * Returns an accumulator function which incrementally computes a sample Pearson product-moment correlation matrix.
 *
 * ## Method
 *
 * -   For each sample Pearson product-moment correlation coefficient, we begin by defining the co-moment \\(C_{jn}\\)
 *
 *     ```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} + \frac{n-1}{n} (x_n - \bar{x}_{n-1}) (y_n - \bar{y}_{n-1})}{n} \\
 *     &= \frac{(n-1)\operatorname{cov}_{n-1}(x,y) + \frac{n-1}{n} (x_n - \bar{x}_{n-1}) (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) + \frac{n-1}{n} (x_n - \bar{x}_{n-1}) (y_n - \bar{y}_{n-1})}{n} \\
 *     &= \operatorname{cov}_{n-1}(x,y) + \frac{(x_n - \bar{x}_{n-1}) (y_n - \bar{y}_{n-1})}{n} \\
 *     &= \frac{C_{n-1}}{n-1} + \frac{(x_n - \bar{x}_{n-1}) (y_n - \bar{y}_{n-1})}{n}
 *     &= \frac{C_{n-1} + \frac{n-1}{n} (x_n - \bar{x}_{n-1}) (y_n - \bar{y}_{n-1})}{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 {(PositiveInteger|ndarray)} out - order of the correlation matrix or a square 2-dimensional output ndarray for storing the correlation matrix
 * @param {ndarray} [means] - mean values
 * @throws {TypeError} first argument must be either a positive integer or a 2-dimensional ndarray having equal dimensions
 * @throws {TypeError} second argument must be a 1-dimensional ndarray
 * @throws {Error} number of means must match correlation matrix dimensions
 * @returns {Function} accumulator function
 *
 * @example
 * var Float64Array = require( '@stdlib/array/float64' );
 * var ndarray = require( '@stdlib/ndarray/ctor' );
 *
 * // Create an output correlation matrix:
 * var buffer = new Float64Array( 4 );
 * var shape = [ 2, 2 ];
 * var strides = [ 2, 1 ];
 * var offset = 0;
 * var order = 'row-major';
 *
 * var corr = ndarray( 'float64', buffer, shape, strides, offset, order );
 *
 * // Create a correlation matrix accumulator:
 * var accumulator = incrpcorrmat( corr );
 *
 * var out = accumulator();
 * // returns null
 *
 * // Create a data vector:
 * buffer = new Float64Array( 2 );
 * shape = [ 2 ];
 * strides = [ 1 ];
 *
 * var vec = ndarray( 'float64', buffer, shape, strides, offset, order );
 *
 * // Provide data to the accumulator:
 * vec.set( 0, 2.0 );
 * vec.set( 1, 1.0 );
 *
 * out = accumulator( vec );
 * // returns <ndarray>
 *
 * var bool = ( out === corr );
 * // returns true
 *
 * vec.set( 0, -5.0 );
 * vec.set( 1, 3.14 );
 *
 * out = accumulator( vec );
 * // returns <ndarray>
 *
 * // Retrieve the correlation matrix:
 * out = accumulator();
 * // returns <ndarray>
 */
 function incrpcorrmat( out, means ) {
 	var order;
 	var corr;
 	var M2;
 	var sd;
 	var mu;
 	var C;
 	var d;
 	var N;
 
 	N = 0;
 	if ( isPositiveInteger( out ) ) {
 		order = out;
 		corr = createMatrix( order, false );
 	} else if ( isSquareMatrix( out ) ) {
 		order = out.shape[ 0 ];
 		corr = out;
 	} else {
 		throw new TypeError( 'invalid argument. First argument must either specify the order of the correlation matrix or be a square 2-dimensional ndarray for storing the correlation matrix. Value: `' + out + '`.' );
 	}
 	// Set the values along the correlation matrix diagonal to `1` (i.e., a random variable is always perfectly correlated with itself):
 	corr = diagonal( corr, 1.0 );
 
 	// Create a scratch array for storing residuals (i.e., `x_i - xbar_{i-1}`):
 	d = new Float64Array( order );
 
 	// Create a scratch array for storing second moments:
 	M2 = new Float64Array( order );
 
 	// Create a scratch array for storing standard deviations:
 	sd = new Float64Array( order );
 
 	// Create a low-level scratch matrix for storing co-moments:
 	C = createMatrix( order, true );
 
 	if ( arguments.length > 1 ) {
 		if ( !isVectorLike( means ) ) {
 			throw new TypeError( 'invalid argument. Second argument must be a 1-dimensional ndarray. Value: `' + means + '`.' );
 		}
 		if ( numel( means.shape ) !== order ) {
 			throw new Error( 'invalid argument. The number of elements (means) in the second argument must match correlation matrix dimensions. Expected: '+order+'. Actual: '+numel( means.shape )+'.' );
 		}
 		mu = means; // TODO: should we copy this? Otherwise, internal state could be "corrupted" due to mutation outside the accumulator
 		return accumulator2;
 	}
 	// Create an ndarray vector for storing sample means (note: an ndarray interface is not necessary, but it reduces implementation complexity by ensuring a consistent abstraction for accessing and updating sample means):
 	mu = createVector( order );
 
 	return accumulator1;
 
 	/**
 	* If provided a data vector, the accumulator function returns an updated sample correlation matrix. If not provided a data vector, the accumulator function returns the current sample correlation matrix.
 	*
 	* @private
 	* @param {ndarray} [v] - data vector
 	* @throws {TypeError} must provide a 1-dimensional ndarray
 	* @throws {Error} vector length must match correlation matrix dimensions
 	* @returns {(ndarray|null)} sample correlation matrix or null
 	*/
 	function accumulator1( v ) {
 		var denom;
 		var rdx;
 		var cij;
 		var rij;
 		var sdi;
 		var di;
 		var vi;
 		var m;
 		var n;
 		var r;
 		var i;
 		var j;
 		if ( arguments.length === 0 ) {
 			if ( N === 0 ) {
 				return null;
 			}
 			return corr;
 		}
 		if ( !isVectorLike( v ) ) {
 			throw new TypeError( 'invalid argument. Must provide a 1-dimensional ndarray. Value: `' + v + '`.' );
 		}
 		if ( v.shape[ 0 ] !== order ) {
 			throw new Error( 'invalid argument. Vector length must match correlation matrix dimensions. Expected: '+order+'. Actual: '+v.shape[ 0 ]+'.' );
 		}
 		n = N;
 		N += 1;
 		r = n / N;
 
 		denom = n || 1; // Bessel's correction (avoiding divide-by-zero below)
 
 		if ( N === 1 ) {
 			for ( i = 0; i < order; i++ ) {
 				vi = v.get( i );
 				m = mu.get( i );
 
 				// Compute the residual:
 				di = vi - m;
 
 				// Update the sample mean:
 				m += di / N;
 				mu.set( i, m );
 
 				// Update the sample standard deviation:
 				d[ i ] = di;
 				M2[ i ] += di * ( vi-m );
 				sd[ i ] = sqrt( M2[i]/denom );
 
 				// Update the co-moments and correlation matrix, recognizing that the matrices are symmetric...
 				rdx = r * d[i]; // if `n=0`, `r=0.0`
 				for ( j = 0; j < i; j++ ) {
 					cij = C.get( i, j ) + ( rdx*d[j] );
 					C.set( i, j, cij );
 					C.set( j, i, cij ); // via symmetry
 				}
 			}
 		} else {
 			for ( i = 0; i < order; i++ ) {
 				vi = v.get( i );
 				m = mu.get( i );
 
 				// Compute the residual:
 				di = vi - m;
 
 				// Update the sample mean:
 				m += di / N;
 				mu.set( i, m );
 
 				// Update the sample standard deviation:
 				d[ i ] = di;
 				M2[ i ] += di * ( vi-m );
 				sd[ i ] = sqrt( M2[i]/denom );
 
 				rdx = r * d[i];
 				sdi = sd[ i ];
 				for ( j = 0; j < i; j++ ) {
 					cij = C.get( i, j ) + ( rdx*d[j] );
 					C.set( i, j, cij );
 					C.set( j, i, cij ); // via symmetry
 
 					rij = ( cij/denom ) / ( sdi*sd[j] );
 					corr.set( i, j, rij );
 					corr.set( j, i, rij ); // via symmetry
 				}
 			}
 		}
 		return corr;
 	}
 
 	/**
 	* If provided a data vector, the accumulator function returns an updated sample correlation matrix. If not provided a data vector, the accumulator function returns the current sample correlation matrix.
 	*
 	* @private
 	* @param {ndarray} [v] - data vector
 	* @throws {TypeError} must provide a 1-dimensional ndarray
 	* @throws {Error} vector length must match correlation matrix dimensions
 	* @returns {(ndarray|null)} sample correlation matrix or null
 	*/
 	function accumulator2( v ) {
 		var rij;
 		var cij;
 		var sdi;
 		var di;
 		var i;
 		var j;
 		if ( arguments.length === 0 ) {
 			if ( N === 0 ) {
 				return null;
 			}
 			return corr;
 		}
 		if ( !isVectorLike( v ) ) {
 			throw new TypeError( 'invalid argument. Must provide a 1-dimensional ndarray. Value: `' + v + '`.' );
 		}
 		if ( v.shape[ 0 ] !== order ) {
 			throw new Error( 'invalid argument. Vector length must match correlation matrix dimensions. Expected: '+order+'. Actual: '+v.shape[ 0 ]+'.' );
 		}
 		N += 1;
 		for ( i = 0; i < order; i++ ) {
 			// Compute the residual:
 			di = v.get( i ) - mu.get( i );
 			d[ i ] = di;
 
 			// Update the standard deviation:
 			M2[ i ] += di * di;
 			sd[ i ] = sqrt( M2[i]/N );
 
 			// Update the co-moments and correlation matrix, recognizing that the matrices are symmetric...
 			sdi = sd[ i ];
 			for ( j = 0; j < i; j++ ) {
 				cij = C.get( i, j ) + ( di*d[j] );
 				C.set( i, j, cij );
 				C.set( j, i, cij ); // via symmetry
 
 				rij = ( cij/N ) / ( sdi*sd[j] );
 				corr.set( i, j, rij );
 				corr.set( j, i, rij ); // via symmetry
 			}
 		}
 		return corr;
 	}
 }
 
 
 // EXPORTS //
 
 module.exports = incrpcorrmat;