feat: add stats/base/ndarray/dcovarmtk

lib/node_modules/@stdlib/stats/base/ndarray/dcovarmtk/README.md

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+<!--
+
+@license Apache-2.0
+
+Copyright (c) 2025 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.
+
+-->
+
+# dcovarmtk
+
+> Calculate the [covariance][covariance] of two one-dimensional double-precision floating-point ndarrays provided known means and using a one-pass textbook algorithm.
+
+<section class="intro">
+
+The population [covariance][covariance] of two finite size populations of size `N` is given by
+
+<!-- <equation class="equation" label="eq:population_covariance" align="center" raw="\operatorname{\mathrm{cov_N}} = \frac{1}{N} \sum_{i=0}^{N-1} (x_i - \mu_x)(y_i - \mu_y)" alt="Equation for the population covariance."> -->
+
+```math
+\mathop{\mathrm{cov_N}} = \frac{1}{N} \sum_{i=0}^{N-1} (x_i - \mu_x)(y_i - \mu_y)
+```
+
+<!-- </equation> -->
+
+where the population means are given by
+
+<!-- <equation class="equation" label="eq:population_mean_for_x" align="center" raw="\mu_x = \frac{1}{N} \sum_{i=0}^{N-1} x_i" alt="Equation for the population mean for first array."> -->
+
+```math
+\mu_x = \frac{1}{N} \sum_{i=0}^{N-1} x_i
+```
+
+<!-- </equation> -->
+
+and
+
+<!-- <equation class="equation" label="eq:population_mean_for_y" align="center" raw="\mu_y = \frac{1}{N} \sum_{i=0}^{N-1} y_i" alt="Equation for the population mean for second array."> -->
+
+```math
+\mu_y = \frac{1}{N} \sum_{i=0}^{N-1} y_i
+```
+
+<!-- </equation> -->
+
+Often in the analysis of data, the true population [covariance][covariance] is not known _a priori_ and must be estimated from samples drawn from population distributions. If one attempts to use the formula for the population [covariance][covariance], the result is biased and yields a **biased sample covariance**. To compute an **unbiased sample covariance** for samples of size `n`,
+
+<!-- <equation class="equation" label="eq:unbiased_sample_covariance" align="center" raw="\operatorname{\mathrm{cov_n}} = \frac{1}{n-1} \sum_{i=0}^{n-1} (x_i - \bar{x}_n)(y_i - \bar{y}_n)" alt="Equation for computing an unbiased sample variance."> -->
+
+```math
+\mathop{\mathrm{cov_n}} = \frac{1}{n-1} \sum_{i=0}^{n-1} (x_i - \bar{x}_n)(y_i - \bar{y}_n)
+```
+
+<!-- </equation> -->
+
+where sample means are given by
+
+<!-- <equation class="equation" label="eq:sample_mean_for_x" align="center" raw="\bar{x} = \frac{1}{n} \sum_{i=0}^{n-1} x_i" alt="Equation for the sample mean for first array."> -->
+
+```math
+\bar{x} = \frac{1}{n} \sum_{i=0}^{n-1} x_i
+```
+
+<!-- </equation> -->
+
+and
+
+<!-- <equation class="equation" label="eq:sample_mean_for_y" align="center" raw="\bar{y} = \frac{1}{n} \sum_{i=0}^{n-1} y_i" alt="Equation for the sample mean for second array."> -->
+
+```math
+\bar{y} = \frac{1}{n} \sum_{i=0}^{n-1} y_i
+```
+
+<!-- </equation> -->
+
+The use of the term `n-1` is commonly referred to as Bessel's correction. Depending on the characteristics of the population distributions, other correction factors (e.g., `n-1.5`, `n+1`, etc) can yield better estimators.
+
+</section>
+
+<!-- /.intro -->
+
+<section class="usage">
+
+## Usage
+
+```javascript
+var dcovarmtk = require( '@stdlib/stats/base/ndarray/dcovarmtk' );
+```
+
+#### dcovarmtk( arrays )
+
+Computes the covariance of two one-dimensional double-precision floating-point ndarrays provided known means and using a one-pass textbook algorithm.
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+var scalar2ndarray = require( '@stdlib/ndarray/from-scalar' );
+var ndarray = require( '@stdlib/ndarray/base/ctor' );
+
+var opts = {
+    'dtype': 'float64'
+};
+
+var xbuf = new Float64Array( [ 1.0, -2.0, 2.0 ] );
+var x = new ndarray( opts.dtype, xbuf, [ 3 ], [ 1 ], 0, 'row-major' );
+
+var ybuf = new Float64Array( [ 2.0, -2.0, 1.0 ] );
+var y = new ndarray( opts.dtype, ybuf, [ 3 ], [ 1 ], 0, 'row-major' );
+
+var correction = scalar2ndarray( 1.0, opts );
+var meanx = scalar2ndarray( 1.0/3.0, opts );
+var meany = scalar2ndarray( 1.0/3.0, opts );
+
+var v = dcovarmtk( [ x, y, correction, meanx, meany ] );
+// returns ~3.8333
+```
+
+The function has the following parameters:
+
+-   **arrays**: array-like object containing the following ndarrays in order:
+
+    1.  first one-dimensional input ndarray.
+    2.  second one-dimensional input ndarray.
+    3.  a zero-dimensional ndarray specifying the degrees of freedom adjustment. Setting this parameter to a value other than `0` has the effect of adjusting the divisor during the calculation of the [covariance][covariance] according to `N-c` where `c` corresponds to the provided degrees of freedom adjustment and `N` corresponds to the number of elements in each input ndarray. When computing the population [covariance][covariance], setting this parameter to `0` is the standard choice (i.e., the provided arrays contain data constituting entire populations). When computing the unbiased sample [covariance][covariance], setting this parameter to `1` is the standard choice (i.e., the provided arrays contain data sampled from larger populations; this is commonly referred to as Bessel's correction).
+    4.  a zero-dimensional ndarray specifying the mean of the first one-dimensional ndarray.
+    5.  a zero-dimensional ndarray specifying the mean of the second one-dimensional ndarray.
+
+</section>
+
+<!-- /.usage -->
+
+<section class="notes">
+
+## Notes
+
+-   Both input ndarrays should have the same number of elements.
+-   If provided empty one-dimensional ndarrays, the function returns `NaN`.
+
+</section>
+
+<!-- /.notes -->
+
+<section class="examples">
+
+## Examples
+
+<!-- eslint no-undef: "error" -->
+
+```javascript
+var discreteUniform = require( '@stdlib/random/array/discrete-uniform' );
+var ndarray = require( '@stdlib/ndarray/base/ctor' );
+var ndarray2array = require( '@stdlib/ndarray/to-array' );
+var scalar2ndarray = require( '@stdlib/ndarray/from-scalar' );
+var dcovarmtk = require( '@stdlib/stats/base/ndarray/dcovarmtk' );
+
+// Define array options:
+var opts = {
+    'dtype': 'float64'
+};
+
+// Create one-dimensional ndarrays containing pseudorandom numbers:
+var xbuf = discreteUniform( 10, -50, 50, opts );
+var x = new ndarray( opts.dtype, xbuf, [ xbuf.length ], [ 1 ], 0, 'row-major' );
+console.log( ndarray2array( x ) );
+
+var ybuf = discreteUniform( 10, -50, 50, opts );
+var y = new ndarray( opts.dtype, ybuf, [ ybuf.length ], [ 1 ], 0, 'row-major' );
+console.log( ndarray2array( y ) );
+
+// Specify the degrees of freedom adjustment:
+var correction = scalar2ndarray( 1.0, opts );
+
+// Specify the known means:
+var meanx = scalar2ndarray( 0.0, opts );
+var meany = scalar2ndarray( 0.0, opts );
+
+// Calculate the sample covariance:
+var v = dcovarmtk( [ x, y, correction, meanx, meany ] );
+console.log( v );
+```
+
+</section>
+
+<!-- /.examples -->
+
+<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->
+
+<section class="related">
+
+</section>
+
+<!-- /.related -->
+
+<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
+
+<section class="links">
+
+</section>
+
+<!-- /.links -->
+
+[covariance]: https://en.wikipedia.org/wiki/Covariance
+
+</section>
+
+<!-- /.links -->

lib/node_modules/@stdlib/stats/base/ndarray/dcovarmtk/benchmark/benchmark.js

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+/**
+* @license Apache-2.0
+*
+* Copyright (c) 2025 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 bench = require( '@stdlib/bench' );
+var uniform = require( '@stdlib/random/array/uniform' );
+var isnan = require( '@stdlib/math/base/assert/is-nan' );
+var pow = require( '@stdlib/math/base/special/pow' );
+var ndarray = require( '@stdlib/ndarray/base/ctor' );
+var scalar2ndarray = require( '@stdlib/ndarray/base/from-scalar' );
+var pkg = require( './../package.json' ).name;
+var dcovarmtk = require( './../lib' );
+
+
+// VARIABLES //
+
+var options = {
+	'dtype': 'float64'
+};
+
+
+// FUNCTIONS //
+
+/**
+* Creates a benchmark function.
+*
+* @private
+* @param {PositiveInteger} len - array length
+* @returns {Function} benchmark function
+*/
+function createBenchmark( len ) {
+	var correction;
+	var meanx;
+	var meany;
+	var xbuf;
+	var ybuf;
+	var x;
+	var y;
+
+	xbuf = uniform( len, -10.0, 10.0, options );
+	x = new ndarray( options.dtype, xbuf, [ len ], [ 1 ], 0, 'row-major' );
+
+	ybuf = uniform( len, -10.0, 10.0, options );
+	y = new ndarray( options.dtype, ybuf, [ len ], [ 1 ], 0, 'row-major' );
+
+	correction = scalar2ndarray( 1.0, options.dtype, 'row-major' );
+	meanx = scalar2ndarray( 0.0, options.dtype, 'row-major' );
+	meany = scalar2ndarray( 0.0, options.dtype, 'row-major' );
+
+	return benchmark;
+
+	function benchmark( b ) {
+		var v;
+		var i;
+
+		b.tic();
+		for ( i = 0; i < b.iterations; i++ ) {
+			v = dcovarmtk( [ x, y, correction, meanx, meany ] );
+			if ( isnan( v ) ) {
+				b.fail( 'should not return NaN' );
+			}
+		}
+		b.toc();
+		if ( isnan( v ) ) {
+			b.fail( 'should not return NaN' );
+		}
+		b.pass( 'benchmark finished' );
+		b.end();
+	}
+}
+
+
+// MAIN //
+
+/**
+* Main execution sequence.
+*
+* @private
+*/
+function main() {
+	var len;
+	var min;
+	var max;
+	var f;
+	var i;
+
+	min = 1; // 10^min
+	max = 6; // 10^max
+
+	for ( i = min; i <= max; i++ ) {
+		len = pow( 10, i );
+		f = createBenchmark( len );
+		bench( pkg+':len='+len, f );
+	}
+}
+
+main();

lib/node_modules/@stdlib/stats/base/ndarray/dcovarmtk/docs/repl.txt

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+
+{{alias}}( arrays )
+    Computes the covariance of two one-dimensional double-precision floating-
+    point ndarrays provided known means and using a one-pass textbook algorithm.
+
+    Both input ndarrays should have the same number of elements.
+
+    If provided empty one-dimensional ndarrays, the function returns `NaN`.
+
+    Parameters
+    ----------
+    arrays: ArrayLikeObject<ndarray>
+        The function expects the following ndarrays in order:
+
+        -   first one-dimensional input ndarray.
+        -   second one-dimensional input ndarray.
+        -   a zero-dimensional ndarray specifying the degrees of freedom
+        adjustment. Setting this parameter to a value other than `0` has the
+        effect of adjusting the divisor during the calculation of the
+        covariance according to `N-c` where `c` corresponds to the provided
+        degrees of freedom adjustment and `N` corresponds to the number of
+        elements in each input ndarray. When computing the population
+        covariance, setting this parameter to `0` is the standard choice (i.e.,
+        the provided arrays contain data constituting entire populations). When
+        computing the unbiased sample covariance, setting this parameter to `1`
+        is the standard choice (i.e., the provided arrays contain data sampled
+        from larger populations; this is commonly referred to as Bessel's
+        correction).
+        -   a zero-dimensional ndarray specifying the mean of the first one-
+        dimensional ndarray.
+        -   a zero-dimensional ndarray specifying the mean of the second one-
+        dimensional ndarray.
+
+    Returns
+    -------
+    out: number
+        The covariance.
+
+    Examples
+    --------
+    // Create the input ndarrays:
+    > var xbuf = new {{alias:@stdlib/array/float64}}( [ 1.0, -2.0, 2.0 ] );
+    > var ybuf = new {{alias:@stdlib/array/float64}}( [ 2.0, -2.0, 1.0 ] );
+    > var dt = 'float64';
+    > var sh = [ xbuf.length ];
+    > var st = [ 1 ];
+    > var oo = 0;
+    > var ord = 'row-major';
+    > var x = new {{alias:@stdlib/ndarray/ctor}}( dt, xbuf, sh, st, oo, ord );
+    > var y = new {{alias:@stdlib/ndarray/ctor}}( dt, ybuf, sh, st, oo, ord );
+
+    // Specify the degrees of freedom adjustment:
+    > var opts = { 'dtype': dt };
+    > var correction = new {{alias:@stdlib/ndarray/from-scalar}}( 1.0, opts );
+
+    // Specify the known means:
+    > var meanx = new {{alias:@stdlib/ndarray/from-scalar}}( 1.0/3.0, opts );
+    > var meany = new {{alias:@stdlib/ndarray/from-scalar}}( 1.0/3.0, opts );
+
+    // Calculate the sample covariance:
+    > {{alias}}( [ x, y, correction, meanx, meany ] )
+    ~3.8333
+
+    See Also
+    --------
+