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* @license Apache-2.0
*
* Copyright (c) 2020 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 isNonNegativeInteger = require( '@stdlib/assert/is-nonnegative-integer' ).isPrimitive;
var isndarrayLike = require( '@stdlib/assert/is-ndarray-like' );
var isCollection = require( '@stdlib/assert/is-collection' );
var flatten2dBy = require( '@stdlib/array/base/flatten2d-by' ).assign;
var Float64Array = require( '@stdlib/array/float64' );
var arrayShape = require( '@stdlib/array/shape' );
var ndarray = require( '@stdlib/ndarray/ctor' );
var numel = require( '@stdlib/ndarray/base/numel' );
var vind2bind = require( '@stdlib/ndarray/base/vind2bind' );
var ind2sub = require( '@stdlib/ndarray/base/ind2sub' );
var dsumpw = require( '@stdlib/blas/ext/base/dsumpw' );
var dscal = require( '@stdlib/blas/base/dscal' );
var dapx = require( '@stdlib/blas/ext/base/dapx' );
var dmin = require( '@stdlib/stats/strided/dmin' );
var abs = require( '@stdlib/math/base/special/abs' );
var min = require( '@stdlib/math/base/special/min' );
var format = require( '@stdlib/string/format' );
var chisqCDF = require( '@stdlib/stats/base/dists/chisquare/cdf' );
var defaults = require( './defaults.js' );
var validate = require( './validate.js' );
var Results = require( './results.js' );
// VARIABLES //
var MODE = 'throw';
// FUNCTIONS //
/**
* Callback invoked upon flattening an array-like object.
*
* @private
* @param {NonNegativeInteger} value - nested array element
* @param {NonNegativeIntegerArray} indices - element indices
* @param {Collection<Collection>} arr - input array
* @throws {TypeError} first argument must be a nonnegative integer
* @returns {NonNegativeInteger} nested array element
*/
function flattenClbk( value, indices ) {
if ( !isNonNegativeInteger( value ) ) {
throw new TypeError( format( 'invalid argument. First argument must contain nonnegative integers. Indices: (%s). Value: `%s`.', indices.join( ', ' ), String( value ) ) );
}
return value;
}
/**
* Copies ndarray data to a new data buffer.
*
* @private
* @param {ndarrayLike} x - input ndarray-like object
* @param {NonNegativeIntegerArray} shape - array shape
* @throws {TypeError} first argument must contain nonnegative integers
* @returns {Float64Array} data buffer
*/
function copy2buffer( x, shape ) {
var out;
var buf;
var idx;
var ord;
var sx;
var ox;
var N;
var v;
var i;
// Cache ndarray meta data:
buf = x.data;
ord = x.order;
sx = x.strides;
ox = x.offset;
// Compute the number of array elements:
N = numel( shape );
// Allocate a data buffer:
out = new Float64Array( N );
// Iterate over the ndarray and copy elements to a new buffer...
for ( i = 0; i < N; i++ ) {
idx = vind2bind( shape, sx, ox, ord, i, MODE );
v = buf[ idx ];
if ( !isNonNegativeInteger( v ) ) {
throw new TypeError( format( 'invalid argument. First argument must contain nonnegative integers. Indices: (%s). Value: `%s`.', ind2sub( shape, sx, ox, ord, i, MODE ), String( v ) ) );
}
out[ i ] = v;
}
return out;
}
/**
* Computes a sum along an axis.
*
* @private
* @param {Float64Array} x - strided array
* @param {NonNegativeInteger} nrows - number of rows
* @param {NonNegativeInteger} ncols - number of columns
* @param {NonNegativeInteger} axis - axis (0: sum across the rows; 1: sum down the columns)
* @returns {Float64Array} results
*
* @example
* var Float64Array = require( '@stdlib/array/float64' );
*
* // 2x3: [ [ 1.0, 2.0, 3.0 ], [ 4.0, 5.0, 6.0 ] ]
* var x = new Float64Array( [ 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 ] );
*
* var rsums = axisSum( x, 2, 3, 0 );
* // returns <Float64Array>[ 6.0, 15.0 ]
*
* var csums = axisSum( x, 2, 3, 1 );
* // returns <Float64Array>[ 5.0, 7.0, 9.0 ]
*/
function axisSum( x, nrows, ncols, axis ) {
var offset;
var out;
var dx0;
var dx1;
var S0;
var S1;
var i1;
// Resolve loop variables...
if ( axis === 0 ) {
// Sum across rows...
S0 = ncols;
S1 = nrows;
dx0 = 1;
dx1 = ncols;
} else { // axis === 1
// Sum down columns...
S0 = nrows;
S1 = ncols;
dx0 = ncols;
dx1 = 1;
}
out = new Float64Array( S1 );
offset = 0;
for ( i1 = 0; i1 < S1; i1++ ) {
out[ i1 ] = dsumpw.ndarray( S0, x, dx0, offset );
offset += dx1;
}
return out;
}
/**
* Computes the outer product.
*
* @private
* @param {Float64Array} x - first input array
* @param {Float64Array} y - second input array
* @returns {Float64Array} results
*/
function outer( x, y ) {
var out;
var io;
var M;
var N;
var v;
var i;
var j;
M = x.length;
N = y.length;
out = new Float64Array( M*N );
io = 0;
for ( i = 0; i < M; i++ ) {
v = x[ i ];
for ( j = 0; j < N; j++ ) {
out[ io ] = v * y[ j ];
io += 1;
}
}
return out;
}
/**
* Computes the element-wise absolute difference.
*
* @private
* @param {Float64Array} x - first input array
* @param {Float64Array} y - second input array
* @returns {Float64Array} results
*/
function absdiff( x, y ) {
var out;
var i;
out = new Float64Array( x.length );
for ( i = 0; i < x.length; i++ ) {
out[ i ] = abs( x[ i ] - y[ i ] );
}
return out;
}
/**
* Computes the test statistic.
*
* ## Notes
*
* - Mutates the first input array.
*
* @private
* @param {Float64Array} x - absolute differences
* @param {Float64Array} m - expected frequencies
* @returns {number} test statistic
*/
function testStatistic( x, m ) {
var i;
for ( i = 0; i < x.length; i++ ) {
x[ i ] = ( x[i]*x[i] ) / m[ i ];
}
return dsumpw( x.length, x, 1 );
}
// MAIN //
/**
* Performs a chi-square independence test.
*
* @param {(MatrixLike|Collection<Collection>)} x - two-way table of observed frequencies
* @param {Options} [options] - function options
* @param {number} [options.alpha=0.05] - significance level
* @param {boolean} [options.correct=true] - boolean indicating whether to use Yates' continuity correction when provided a 2x2 contingency table
* @throws {TypeError} first argument must be an array of arrays or a two-dimensional ndarray-like object
* @throws {TypeError} options argument must be an object
* @throws {TypeError} must provide valid options
* @throws {RangeError} significance level must be a number on the interval `[0,1]`
* @returns {Object} test results
*
* @example
* var x = [ [ 20, 30 ], [ 30, 20 ] ];
*
* var out = chi2test( x );
*
* var o = out.toJSON();
* // returns { 'rejected': false, 'alpha': 0.05, 'pValue': ~0.072, ... }
*/
function chi2test( x, options ) {
var correction;
var diffs;
var csums;
var rsums;
var means;
var opts;
var pval;
var stat;
var err;
var cnt;
var buf;
var dof;
var sh;
var M;
var N;
if ( isndarrayLike( x ) ) {
sh = x.shape;
if ( sh.length !== 2 ) {
throw new TypeError( format( 'invalid argument. First argument must be an array of arrays or a two-dimensional ndarray-like object. Number of input array dimensions: %u.', sh.length ) );
}
buf = copy2buffer( x, sh );
} else if ( isCollection( x ) ) {
sh = arrayShape( x );
if ( sh.length !== 2 ) {
throw new TypeError( format( 'invalid argument. First argument must be an array of arrays or a two-dimensional ndarray-like object. Number of input array dimensions: %u.', sh.length ) );
}
buf = new Float64Array( numel( sh ) );
flatten2dBy( x, sh, false, buf, 1, 0, flattenClbk );
} else {
throw new TypeError( format( 'invalid argument. First argument must be an array of arrays or a two-dimensional ndarray-like object. Value: `%s`.', x ) );
}
// NOTE: `buf` is now a single-segment contiguous `Float64Array` containing nonnegative integer values...
opts = defaults();
if ( arguments.length > 1 ) {
err = validate( opts, options );
if ( err ) {
throw err;
}
}
// Extract the array dimensions:
M = sh[ 0 ]; // number of rows
N = sh[ 1 ]; // number of columns
// Compute the total number of observations:
cnt = dsumpw( M*N, buf, 1 );
// Compute marginal sums:
rsums = axisSum( buf, M, N, 0 ); // sum across the rows (length: M)
csums = axisSum( buf, M, N, 1 ); // sum down the columns (length: N)
// Compute the outer product:
means = outer( rsums, csums ); // MxN
// Compute the expected frequencies:
means = dscal( means.length, 1.0/cnt, means, 1 );
// Compute the absolute differences between the observed and expected frequencies:
diffs = absdiff( buf, means );
// Apply Yates' continuity correction...
if ( M === 2 && N === 2 && opts.correct ) {
correction = min( 0.5, dmin( diffs.length, diffs, 1 ) );
dapx( diffs.length, -correction, diffs, 1 );
}
// Compute the test statistic:
stat = testStatistic( diffs, means );
// Compute the number of degrees of freedom:
dof = ( M-1 ) * ( N-1 );
// Compute the p-value:
pval = 1 - chisqCDF( stat, dof );
// Return test results:
means = new ndarray( 'float64', means, [ M, N ], [ N, 1 ], 0, 'row-major', {
'readonly': true
});
return new Results( pval, opts.alpha, stat, dof, means );
}
// EXPORTS //
module.exports = chi2test;
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