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* @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' );
var Float64Array = require( '@stdlib/array/float64' );
var format = require( '@stdlib/string/format' );
// MAIN //
/**
* Returns an accumulator function which incrementally computes a moving unbiased sample covariance.
*
* ## Method
*
* - Let \\(W\\) be a window of \\(N\\) elements over which we want to compute an unbiased sample covariance.
*
* - We begin by defining the covariance \\( \operatorname{cov}_n(x,y) \\) for a window \\(n\\) as follows
*
* ```tex
* \operatorname{cov}_n(x,y) &= \frac{C_n}{n}
* ```
*
* where \\(C_n\\) is the co-moment, which is defined as
*
* ```tex
* C_n = \sum_{i=1}^{N} ( x_i - \bar{x}_n ) ( y_i - \bar{y}_n )
* ```
*
* 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.
*
* - The sample mean is computed using the canonical formula
*
* ```tex
* \bar{x}_n = \frac{1}{N} \sum_{i=1}^{N} x_i
* ```
*
* which, taking into account a previous window, can be expressed
*
* ```tex
* \begin{align*}
* \bar{x}_n &= \frac{1}{N} \biggl( \sum_{i=0}^{N-1} x_i - x_0 + x_N \biggr) \\
* &= \bar{x}_{n-1} + \frac{x_N - x_0}{N}
* \end{align*}
* ```
*
* where \\(x_0\\) is the first value in the previous window.
*
* - We can substitute into the co-moment equation
*
* ```tex
* \begin{align*}
* C_n &= \sum_{i=1}^{N} ( x_i - \bar{x}_n ) ( y_i - \bar{y}_n ) \\
* &= \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) \\
* &= \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)
* \end{align*}
* ```
*
* where
*
* ```tex
* \Delta x_{i,k} = x_i - \bar{x}_{k}
* ```
*
* - We can subsequently expand terms and apply a summation identity
*
* ```tex
* \begin{align*}
* 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) \\
* &= \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}
* \end{align*}
* ```
*
* - Let us first consider the second term which we can reorganize as follows
*
* ```tex
* \begin{align*}
* \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}) \\
* &= \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} \\
* &= (y_N - y_0) \bar{x}_{n} - (y_N - y_0)\bar{x}_{n-1} \\
* &= (y_N - y_0) (\bar{x}_{n} - \bar{x}_{n-1}) \\
* &= \frac{(x_N - x_0)(y_N - y_0)}{N}
* \end{align*}
* ```
*
* - The third term can be reorganized in a manner similar to the second term such that
*
* ```tex
* \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}
* ```
*
* - Substituting back into the equation for the co-moment
*
* ```tex
* \begin{align*}
* 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} \\
* &= \sum_{i=1}^{N} \Delta x_{i,n-1} \Delta y_{i,n-1} - \frac{(x_N - x_0)(y_N - y_0)}{N}
* \end{align*}
* ```
*
* - Let us now consider the first term which we can modify as follows
*
* ```tex
* \begin{align*}
* \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}) \\
* &= \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}) \\
* &= \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}) \\
* &= \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})
* \end{align*}
* ```
*
* where we recognize that the first term equals the co-moment for the previous window
*
* ```tex
* C_{n-1} = \sum_{i=0}^{N-1} (x_i - \bar{x}_{n-1})(y_i - \bar{y}_{n-1})
* ```
*
* In which case,
*
* ```tex
* \begin{align*}
* \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}) \\
* &= C_{n-1} + \Delta x_{N,n-1} \Delta y_{N,n-1} - \Delta x_{0,n-1} \Delta y_{0,n-1}
* \end{align*}
* ```
*
* - Substituting into the equation for the co-moment
*
* ```tex
* 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}
* ```
*
* - We can make one further modification to the last term
*
* ```tex
* \begin{align*}
* \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} \\
* &= \frac{(\Delta x_{N,n-1} - \Delta x_{0,n-1})(\Delta y_{N,n-1} - \Delta y_{0,n-1})}{N}
* \end{align*}
* ```
*
* which, upon substitution into the equation for the co-moment, yields
*
* ```tex
* 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}
* ```
*
* @param {PositiveInteger} W - window size
* @param {number} [meanx] - mean value
* @param {number} [meany] - mean value
* @throws {TypeError} first argument must be a positive integer
* @throws {TypeError} second argument must be a number
* @throws {TypeError} third argument must be a number
* @returns {Function} accumulator function
*
* @example
* var accumulator = incrmcovariance( 3 );
*
* var v = accumulator();
* // returns null
*
* v = accumulator( 2.0, 1.0 );
* // returns 0.0
*
* v = accumulator( -5.0, 3.14 );
* // returns ~-7.49
*
* v = accumulator( 3.0, -1.0 );
* // returns -8.35
*
* v = accumulator( 5.0, -9.5 );
* // returns -29.42
*
* v = accumulator();
* // returns -29.42
*
* @example
* var accumulator = incrmcovariance( 3, -2.0, 10.0 );
*/
function incrmcovariance( W, meanx, meany ) {
var buf;
var dx0;
var dxN;
var dy0;
var dyN;
var mx;
var my;
var wi;
var C;
var N;
var n;
var i;
if ( !isPositiveInteger( W ) ) {
throw new TypeError( format( 'invalid argument. First argument must be a positive integer. Value: `%s`.', W ) );
}
buf = new Float64Array( 2*W ); // strided array
n = W - 1;
C = 0.0;
wi = -1;
N = 0;
if ( arguments.length > 1 ) {
if ( !isNumber( meanx ) ) {
throw new TypeError( format( 'invalid argument. Second argument must be a number. Value: `%s`.', meanx ) );
}
if ( !isNumber( meany ) ) {
throw new TypeError( format( 'invalid argument. Third argument must be a number. Value: `%s`.', meany ) );
}
mx = meanx;
my = meany;
return accumulator2;
}
mx = 0.0;
my = 0.0;
return accumulator1;
/**
* 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.
*
* @private
* @param {number} [x] - input value
* @param {number} [y] - input value
* @returns {(number|null)} unbiased sample covariance or null
*/
function accumulator1( x, y ) {
var v1;
var v2;
var k;
var j;
if ( arguments.length === 0 ) {
if ( N === 0 ) {
return null;
}
if ( N === 1 ) {
return 0.0;
}
if ( N < W ) {
return C / (N-1);
}
return C / n;
}
// Update the window and strided array indices for managing the circular buffer:
wi = (wi+1) % W;
i = 2 * wi;
// Case: an incoming value is NaN, the sliding co-moment is automatically NaN...
if ( isnan( x ) || isnan( y ) ) {
N = W; // explicitly set to avoid `N < W` branch
C = NaN;
}
// Case: initial window...
else if ( N < W ) {
buf[ i ] = x; // update buffer
buf[ i+1 ] = y;
N += 1;
dxN = x - mx;
mx += dxN / N;
my += ( y-my ) / N;
C += dxN * ( y-my ); // Note: repeated `y-my` is intentional, as `my` is updated when used here
if ( N === 1 ) {
return 0.0;
}
return C / (N-1);
}
// Case: N = W = 1
else if ( N === 1 ) {
return 0.0;
}
// Case: an outgoing value is NaN, and, thus, we need to compute the accumulated values...
else if ( isnan( buf[ i ] ) || isnan( buf[ i+1 ] ) ) {
N = 1;
mx = x;
my = y;
C = 0.0;
for ( k = 0; k < W; k++ ) {
j = 2 * k; // convert to a strided array index
if ( j !== i ) {
v1 = buf[ j ];
v2 = buf[ j+1 ];
if ( isnan( v1 ) || isnan( v2 ) ) {
N = W; // explicitly set to avoid `N < W` branch
C = NaN;
break; // co-moment is automatically NaN, so no need to continue
}
N += 1;
dxN = v1 - mx;
mx += dxN / N;
my += ( v2-my ) / N;
C += dxN * ( v2-my ); // Note: repeated `y-my` is intentional, as `my` is updated when used here
}
}
}
// Case: neither the current co-moment nor the incoming values are NaN, so we need to update the accumulated values...
else if ( isnan( C ) === false ) {
dx0 = buf[ i ] - mx;
dy0 = buf[ i+1 ] - my;
dxN = x - mx;
dyN = y - my;
C += (dxN*dyN) - (dx0*dy0) - ( (dxN-dx0)*(dyN-dy0)/W );
mx += ( dxN-dx0 ) / W;
my += ( dyN-dy0 ) / W;
}
// Case: the current co-moment is NaN, so nothing to do until the buffer no longer contains NaN values...
buf[ i ] = x;
buf[ i+1 ] = y;
return C / n;
}
/**
* 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.
*
* @private
* @param {number} [x] - input value
* @param {number} [y] - input value
* @returns {(number|null)} unbiased sample covariance or null
*/
function accumulator2( x, y ) {
var k;
var j;
if ( arguments.length === 0 ) {
if ( N === 0 ) {
return null;
}
if ( N < W ) {
return C / N;
}
return C / W;
}
// Update the window and strided array indices for managing the circular buffer:
wi = (wi+1) % W;
i = 2 * wi;
// Case: an incoming value is NaN, the sliding co-moment is automatically NaN...
if ( isnan( x ) || isnan( y ) ) {
N = W; // explicitly set to avoid `N < W` branch
C = NaN;
}
// Case: initial window...
else if ( N < W ) {
buf[ i ] = x; // update buffer
buf[ i+1 ] = y;
N += 1;
C += ( x-mx ) * ( y-my );
return C / N;
}
// Case: an outgoing value is NaN, and, thus, we need to compute the accumulated values...
else if ( isnan( buf[ i ] ) || isnan( buf[ i+1 ] ) ) {
C = 0.0;
for ( k = 0; k < W; k++ ) {
j = 2 * k; // convert to a strided array index
if ( j !== i ) {
if ( isnan( buf[ j ] ) || isnan( buf[ j+1 ] ) ) {
N = W; // explicitly set to avoid `N < W` branch
C = NaN;
break; // co-moment is automatically NaN, so no need to continue
}
C += ( buf[j]-mx ) * ( buf[j+1]-my );
}
}
}
// Case: neither the current co-moment nor the incoming values are NaN, so we need to update the accumulated values...
else if ( isnan( C ) === false ) {
// Use textbook formula along with simplification arising from difference of sums:
C += ( (x-mx)*(y-my) ) - ( (buf[i]-mx)*(buf[i+1]-my) );
}
// Case: the current co-moment is NaN, so nothing to do until the buffer no longer contains NaN values...
buf[ i ] = x;
buf[ i+1 ] = y;
return C / W;
}
}
// EXPORTS //
module.exports = incrmcovariance;
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