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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 isSquareMatrix = require( '@stdlib/assert/is-square-matrix' );
var isVectorLike = require( '@stdlib/assert/is-vector-like' );
var Float64Array = require( '@stdlib/array/float64' );
var format = require( '@stdlib/string/format' );
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' );
}
/**
* 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 distance matrix.
*
* ## Method
*
* - For each sample Pearson product-moment correlation distance, 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.
*
* - The sample Pearson product-moment correlation distance is defined as
*
* ```tex
* d = 1 - r = 1 - \frac{\operatorname{cov}_n(x,y)}{\sigma_x \sigma_y}
* ```
*
* - The implementation thus computes each sample Pearson product-moment correlation coefficient \\(r\\) and subtracts each coefficient from 1.
*
* @param {(PositiveInteger|ndarray)} out - order of the correlation distance matrix or a square 2-dimensional output ndarray for storing the correlation distance 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 distance matrix dimensions
* @returns {Function} accumulator function
*
* @example
* var Float64Array = require( '@stdlib/array/float64' );
* var ndarray = require( '@stdlib/ndarray/ctor' );
*
* // Create an output correlation distance matrix:
* var buffer = new Float64Array( 4 );
* var shape = [ 2, 2 ];
* var strides = [ 2, 1 ];
* var offset = 0;
* var order = 'row-major';
*
* var dist = ndarray( 'float64', buffer, shape, strides, offset, order );
*
* // Create a correlation distance matrix accumulator:
* var accumulator = incrpcorrdistmat( dist );
*
* 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 === dist );
* // returns true
*
* vec.set( 0, -5.0 );
* vec.set( 1, 3.14 );
*
* out = accumulator( vec );
* // returns <ndarray>
*
* // Retrieve the correlation distance matrix:
* out = accumulator();
* // returns <ndarray>
*/
function incrpcorrdistmat( out, means ) {
var order;
var dist;
var M2;
var sd;
var mu;
var C;
var d;
var N;
N = 0;
if ( isPositiveInteger( out ) ) {
order = out;
dist = createMatrix( order, false );
} else if ( isSquareMatrix( out ) ) {
order = out.shape[ 0 ];
dist = out;
} else {
throw new TypeError( format( 'invalid argument. First argument must either specify the order of the correlation distance matrix or be a square two-dimensional ndarray for storing the correlation distance matrix. Value: `%s`.', out ) );
}
// 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( format( 'invalid argument. Second argument must be a one-dimensional ndarray. Value: `%s`.', means ) );
}
if ( numel( means.shape ) !== order ) {
throw new Error( format( 'invalid argument. The number of elements (means) in the second argument must match correlation distance matrix dimensions. Expected: `%u`. Actual: `%u`.', order, 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 distance matrix. If not provided a data vector, the accumulator function returns the current sample correlation distance matrix.
*
* @private
* @param {ndarray} [v] - data vector
* @throws {TypeError} must provide a 1-dimensional ndarray
* @throws {Error} vector length must match correlation distance matrix dimensions
* @returns {(ndarray|null)} sample correlation distance matrix or null
*/
function accumulator1( v ) {
var denom;
var rdx;
var cij;
var dij;
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 dist;
}
if ( !isVectorLike( v ) ) {
throw new TypeError( format( 'invalid argument. Must provide a one-dimensional ndarray. Value: `%s`.', v ) );
}
if ( v.shape[ 0 ] !== order ) {
throw new Error( format( 'invalid argument. Vector length must match correlation matrix dimensions. Expected: `%u`. Actual: `%u`.', order, 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 distance 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 );
// Update the co-moments and correlation distance matrix, recognizing that the matrices are symmetric...
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
dij = 1.0 - ( (cij/denom)/(sdi*sd[j]) );
dist.set( i, j, dij );
dist.set( j, i, dij ); // via symmetry
}
}
}
return dist;
}
/**
* If provided a data vector, the accumulator function returns an updated sample correlation distance matrix. If not provided a data vector, the accumulator function returns the current sample correlation distance matrix.
*
* @private
* @param {ndarray} [v] - data vector
* @throws {TypeError} must provide a 1-dimensional ndarray
* @throws {Error} vector length must match correlation distance matrix dimensions
* @returns {(ndarray|null)} sample correlation distance matrix or null
*/
function accumulator2( v ) {
var dij;
var cij;
var sdi;
var di;
var i;
var j;
if ( arguments.length === 0 ) {
if ( N === 0 ) {
return null;
}
return dist;
}
if ( !isVectorLike( v ) ) {
throw new TypeError( format( 'invalid argument. Must provide a one-dimensional ndarray. Value: `%s`.', v ) );
}
if ( v.shape[ 0 ] !== order ) {
throw new Error( format( 'invalid argument. Vector length must match correlation distance matrix dimensions. Expected: `%u`. Actual: `%u`.', order, v.shape[ 0 ] ) );
}
N += 1;
for ( i = 0; i < order; i++ ) {
// Compute the residual:
di = v.get( i ) - mu.get( i );
// Update standard deviation:
d[ i ] = di;
M2[ i ] += di * di;
sd[ i ] = sqrt( M2[i]/N );
// Update the co-moments and correlation distance 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
dij = 1.0 - ( (cij/N)/(sdi*sd[j]) );
dist.set( i, j, dij );
dist.set( j, i, dij ); // via symmetry
}
}
return dist;
}
}
// EXPORTS //
module.exports = incrpcorrdistmat;
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