feat: add stats/strided/covarmtk

PR-URL: #7659
Co-authored-by: Athan Reines <kgryte@gmail.com>
Reviewed-by: Athan Reines <kgryte@gmail.com>

Author: gururaj1512

lib/node_modules/@stdlib/stats/strided/covarmtk/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.
+
+-->
+
+<!-- lint disable maximum-heading-length -->
+
+# covarmtk
+
+> Calculate the [covariance][covariance] of two strided arrays 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 covarmtk = require( '@stdlib/stats/strided/covarmtk' );
+```
+
+#### covarmtk( N, correction, meanx, x, strideX, meany, y, strideY )
+
+Computes the [covariance][covariance] of two strided arrays provided known means and using a one-pass textbook algorithm.
+
+```javascript
+var x = [ 1.0, -2.0, 2.0 ];
+var y = [ 2.0, -2.0, 1.0 ];
+
+var v = covarmtk( x.length, 1, 1.0/3.0, x, 1, 1.0/3.0, y, 1 );
+// returns ~3.8333
+```
+
+The function has the following parameters:
+
+-   **N**: number of indexed elements.
+-   **correction**: 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. 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).
+-   **meanx**: mean of `x`.
+-   **x**: first input [`Array`][mdn-array] or [`typed array`][mdn-typed-array].
+-   **strideX**: stride length for `x`.
+-   **meany**: mean of `y`.
+-   **y**: second input [`Array`][mdn-array] or [`typed array`][mdn-typed-array].
+-   **strideY**: stride length for `y`.
+
+The `N` and stride parameters determine which elements in the strided arrays are accessed at runtime. For example, to compute the [covariance][covariance] of every other element in `x` and `y`,
+
+```javascript
+var x = [ 1.0, 2.0, 2.0, -7.0, -2.0, 3.0, 4.0, 2.0 ];
+var y = [ 2.0, 1.0, 2.0, 1.0, -2.0, 2.0, 3.0, 4.0 ];
+
+var v = covarmtk( 4, 1, 1.25, x, 2, 1.25, y, 2 );
+// returns 5.25
+```
+
+Note that indexing is relative to the first index. To introduce an offset, use [`typed array`][mdn-typed-array] views.
+
+<!-- eslint-disable stdlib/capitalized-comments -->
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+
+var x0 = new Float64Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] );
+var y0 = new Float64Array( [ 2.0, -2.0, 2.0, 1.0, -2.0, 4.0, 3.0, 2.0 ] );
+
+var x1 = new Float64Array( x0.buffer, x0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
+var y1 = new Float64Array( y0.buffer, y0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
+
+var v = covarmtk( 4, 1, 1.25, x1, 2, 1.25, y1, 2 );
+// returns ~1.9167
+```
+
+#### covarmtk.ndarray( N, correction, meanx, x, strideX, offsetX, meany, y, strideY, offsetY )
+
+Computes the [covariance][covariance] of two strided arrays provided known means and using a one-pass textbook algorithm and alternative indexing semantics.
+
+```javascript
+var x = [ 1.0, -2.0, 2.0 ];
+var y = [ 2.0, -2.0, 1.0 ];
+
+var v = covarmtk.ndarray( x.length, 1, 1.0/3.0, x, 1, 0, 1.0/3.0, y, 1, 0 );
+// returns ~3.8333
+```
+
+The function has the following additional parameters:
+
+-   **offsetX**: starting index for `x`.
+-   **offsetY**: starting index for `y`.
+
+While [`typed array`][mdn-typed-array] views mandate a view offset based on the underlying buffer, the offset parameters support indexing semantics based on starting indices. For example, to calculate the [covariance][covariance] for every other element in `x` and `y` starting from the second element
+
+```javascript
+var x = [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ];
+var y = [ -7.0, 2.0, 2.0, 1.0, -2.0, 2.0, 3.0, 4.0 ];
+
+var v = covarmtk.ndarray( 4, 1, 1.25, x, 2, 1, 1.25, y, 2, 1 );
+// returns 6.0
+```
+
+</section>
+
+<!-- /.usage -->
+
+<section class="notes">
+
+## Notes
+
+-   If `N <= 0`, both functions return `NaN`.
+-   If `N - c` is less than or equal to `0` (where `c` corresponds to the provided degrees of freedom adjustment), both functions return `NaN`.
+-   Both functions support array-like objects having getter and setter accessors for array element access (e.g., [`@stdlib/array/base/accessor`][@stdlib/array/base/accessor]).
+-   Depending on the environment, the typed versions ([`dcovarmtk`][@stdlib/stats/strided/dcovarmtk], [`scovarmtk`][@stdlib/stats/strided/scovarmtk], etc.) are likely to be significantly more performant.
+
+</section>
+
+<!-- /.notes -->
+
+<section class="examples">
+
+## Examples
+
+<!-- eslint no-undef: "error" -->
+
+```javascript
+var discreteUniform = require( '@stdlib/random/array/discrete-uniform' );
+var covarmtk = require( '@stdlib/stats/strided/covarmtk' );
+
+var opts = {
+    'dtype': 'generic'
+};
+var x = discreteUniform( 10, -50, 50, opts );
+console.log( x );
+
+var y = discreteUniform( 10, -50, 50, opts );
+console.log( y );
+
+var v = covarmtk( x.length, 1, 0.0, x, 1, 0.0, y, 1 );
+console.log( v );
+```
+
+</section>
+
+<!-- /.examples -->
+
+* * *
+
+<section class="references">
+
+</section>
+
+<!-- /.references -->
+
+<!-- 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">
+
+[covariance]: https://en.wikipedia.org/wiki/Covariance
+
+[mdn-typed-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/TypedArray
+
+[mdn-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Array
+
+[@stdlib/array/base/accessor]: https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/array/base/accessor
+
+[@stdlib/stats/strided/dcovarmtk]: https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/stats/strided/dcovarmtk
+
+[@stdlib/stats/strided/scovarmtk]: https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/stats/strided/scovarmtk
+
+</section>
+
+<!-- /.links -->

lib/node_modules/@stdlib/stats/strided/covarmtk/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 pkg = require( './../package.json' ).name;
+var covarmtk = require( './../lib/main.js' );
+
+
+// VARIABLES //
+
+var options = {
+	'dtype': 'generic'
+};
+
+
+// FUNCTIONS //
+
+/**
+* Creates a benchmark function.
+*
+* @private
+* @param {PositiveInteger} len - array length
+* @returns {Function} benchmark function
+*/
+function createBenchmark( len ) {
+	var x = uniform( len, -10.0, 10.0, options );
+	return benchmark;
+
+	function benchmark( b ) {
+		var v;
+		var i;
+
+		b.tic();
+		for ( i = 0; i < b.iterations; i++ ) {
+			v = covarmtk( x.length, 1, 0.0, x, 1, 0.0, x, 1 );
+			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();