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feat: add stats/base/ndarray/scovarmtk
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‎lib/node_modules/@stdlib/stats/base/ndarray/scovarmtk/README.md

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<!--
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@license Apache-2.0
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Copyright (c) 2025 The Stdlib Authors.
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Licensed under the Apache License, Version 2.0 (the "License");
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you may not use this file except in compliance with the License.
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You may obtain a copy of the License at
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http://www.apache.org/licenses/LICENSE-2.0
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Unless required by applicable law or agreed to in writing, software
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distributed under the License is distributed on an "AS IS" BASIS,
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WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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See the License for the specific language governing permissions and
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limitations under the License.
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-->
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# scovarmtk
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> Calculate the [covariance][covariance] of two one-dimensional single-precision floating-point ndarrays provided known means and using a one-pass textbook algorithm.
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<section class="intro">
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The population [covariance][covariance] of two finite size populations of size `N` is given by
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<!-- <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."> -->
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```math
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\mathop{\mathrm{cov_N}} = \frac{1}{N} \sum_{i=0}^{N-1} (x_i - \mu_x)(y_i - \mu_y)
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```
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<!-- </equation> -->
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where the population means are given by
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<!-- <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."> -->
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```math
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\mu_x = \frac{1}{N} \sum_{i=0}^{N-1} x_i
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```
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<!-- </equation> -->
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and
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<!-- <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."> -->
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```math
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\mu_y = \frac{1}{N} \sum_{i=0}^{N-1} y_i
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```
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<!-- </equation> -->
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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`,
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<!-- <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."> -->
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```math
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\mathop{\mathrm{cov_n}} = \frac{1}{n-1} \sum_{i=0}^{n-1} (x_i - \bar{x}_n)(y_i - \bar{y}_n)
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```
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<!-- </equation> -->
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where sample means are given by
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<!-- <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."> -->
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```math
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\bar{x} = \frac{1}{n} \sum_{i=0}^{n-1} x_i
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```
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<!-- </equation> -->
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and
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<!-- <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."> -->
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```math
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\bar{y} = \frac{1}{n} \sum_{i=0}^{n-1} y_i
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```
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<!-- </equation> -->
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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.
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</section>
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<!-- /.intro -->
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<section class="usage">
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## Usage
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```javascript
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var scovarmtk = require( '@stdlib/stats/base/ndarray/scovarmtk' );
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```
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#### scovarmtk( arrays )
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Computes the covariance of two one-dimensional single-precision floating-point ndarrays provided known means and using a one-pass textbook algorithm.
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```javascript
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var Float32Array = require( '@stdlib/array/float32' );
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var scalar2ndarray = require( '@stdlib/ndarray/from-scalar' );
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var ndarray = require( '@stdlib/ndarray/base/ctor' );
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var opts = {
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'dtype': 'float32'
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};
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var xbuf = new Float32Array( [ 1.0, -2.0, 2.0 ] );
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var x = new ndarray( opts.dtype, xbuf, [ 3 ], [ 1 ], 0, 'row-major' );
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var ybuf = new Float32Array( [ 2.0, -2.0, 1.0 ] );
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var y = new ndarray( opts.dtype, ybuf, [ 3 ], [ 1 ], 0, 'row-major' );
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var meanx = scalar2ndarray( 1.0/3.0, opts );
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var meany = scalar2ndarray( 1.0/3.0, opts );
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var correction = scalar2ndarray( 1.0, opts );
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var v = scovarmtk( [ x, y, meanx, meany, correction ] );
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// returns ~3.8333
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```
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The function has the following parameters:
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- **arrays**: array-like object containing the following ndarrays in order:
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1. first one-dimensional input ndarray.
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2. second one-dimensional input ndarray.
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3. a zero-dimensional ndarray specifying mean of the first one-dimensional ndarray.
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4. a zero-dimensional ndarray specifying mean of the second one-dimensional ndarray.
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5. a zero-dimensional ndarray specifying 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).
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</section>
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<!-- /.usage -->
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<section class="notes">
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## Notes
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- If provided an empty one-dimensional ndarray, the function returns `NaN`.
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</section>
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<!-- /.notes -->
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<section class="examples">
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## Examples
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<!-- eslint no-undef: "error" -->
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```javascript
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var discreteUniform = require( '@stdlib/random/array/discrete-uniform' );
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var ndarray = require( '@stdlib/ndarray/base/ctor' );
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var ndarray2array = require( '@stdlib/ndarray/to-array' );
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var scalar2ndarray = require( '@stdlib/ndarray/from-scalar' );
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var scovarmtk = require( '@stdlib/stats/base/ndarray/scovarmtk' );
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// Define array options:
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var opts = {
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'dtype': 'float32'
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};
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// Create first one-dimensional ndarray containing pseudorandom integers drawn from a discrete uniform distribution:
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var xbuf = discreteUniform( 10, -50, 50, opts );
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var x = new ndarray( opts.dtype, xbuf, [ xbuf.length ], [ 1 ], 0, 'row-major' );
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console.log( ndarray2array( x ) );
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// Create second one-dimensional ndarray containing pseudorandom integers drawn from a discrete uniform distribution:
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var ybuf = discreteUniform( 10, -50, 50, opts );
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var y = new ndarray( opts.dtype, ybuf, [ ybuf.length ], [ 1 ], 0, 'row-major' );
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console.log( ndarray2array( y ) );
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// Specify the known means:
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var meanx = scalar2ndarray( 0.0, opts );
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var meany = scalar2ndarray( 0.0, opts );
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// Specify the degrees of freedom adjustment:
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var correction = scalar2ndarray( 1.0, opts );
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// Calculate the sample covariance:
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var v = scovarmtk( [ x, y, meanx, meany, correction ] );
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console.log( v );
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```
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</section>
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<!-- /.examples -->
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<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->
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<section class="related">
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</section>
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<!-- /.related -->
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<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
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<section class="links">
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</section>
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<!-- /.links -->
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[covariance]: https://en.wikipedia.org/wiki/Covariance
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</section>
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<!-- /.links -->

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

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/**
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* @license Apache-2.0
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*
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* Copyright (c) 2025 The Stdlib Authors.
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*
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* Licensed under the Apache License, Version 2.0 (the "License");
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* you may not use this file except in compliance with the License.
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* You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS,
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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*/
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'use strict';
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// MODULES //
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var bench = require( '@stdlib/bench' );
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var uniform = require( '@stdlib/random/array/uniform' );
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var isnan = require( '@stdlib/math/base/assert/is-nan' );
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var pow = require( '@stdlib/math/base/special/pow' );
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var ndarray = require( '@stdlib/ndarray/base/ctor' );
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var scalar2ndarray = require( '@stdlib/ndarray/base/from-scalar' );
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var pkg = require( './../package.json' ).name;
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var scovarmtk = require( './../lib' );
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// VARIABLES //
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var options = {
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'dtype': 'float32'
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};
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// FUNCTIONS //
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/**
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* Creates a benchmark function.
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*
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* @private
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* @param {PositiveInteger} len - array length
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* @returns {Function} benchmark function
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*/
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function createBenchmark( len ) {
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var correction;
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var meanx;
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var meany;
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var xbuf;
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var ybuf;
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var x;
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var y;
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xbuf = uniform( len, -10.0, 10.0, options );
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x = new ndarray( options.dtype, xbuf, [ len ], [ 1 ], 0, 'row-major' );
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ybuf = uniform( len, -10.0, 10.0, options );
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y = new ndarray( options.dtype, ybuf, [ len ], [ 1 ], 0, 'row-major' );
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meanx = scalar2ndarray( 0.0, options.dtype, 'row-major' );
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meany = scalar2ndarray( 0.0, options.dtype, 'row-major' );
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correction = scalar2ndarray( 1.0, options.dtype, 'row-major' );
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return benchmark;
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function benchmark( b ) {
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var v;
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var i;
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b.tic();
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for ( i = 0; i < b.iterations; i++ ) {
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v = scovarmtk( [ x, y, meanx, meany, correction ] );
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if ( isnan( v ) ) {
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b.fail( 'should not return NaN' );
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}
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}
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b.toc();
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if ( isnan( v ) ) {
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b.fail( 'should not return NaN' );
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}
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b.pass( 'benchmark finished' );
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b.end();
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}
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}
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// MAIN //
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/**
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* Main execution sequence.
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*
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* @private
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*/
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function main() {
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var len;
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var min;
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var max;
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var f;
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var i;
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min = 1; // 10^min
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max = 6; // 10^max
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for ( i = min; i <= max; i++ ) {
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len = pow( 10, i );
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f = createBenchmark( len );
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bench( pkg+':len='+len, f );
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}
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}
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main();

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

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{{alias}}( arrays )
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Computes the covariance of two one-dimensional single-precision
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floating-point ndarrays provided known means and using a one-pass textbook
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algorithm.
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If provided an empty ndarray, the function returns `NaN`.
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Parameters
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----------
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arrays: ArrayLikeObject<ndarray>
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The function expects the following ndarrays in order:
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- first one-dimensional input ndarray.
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- second one-dimensional input ndarray.
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- a zero-dimensional ndarray specifying mean of the first
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one-dimensional ndarray.
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- a zero-dimensional ndarray specifying mean of the second
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one-dimensional ndarray.
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- a zero-dimensional ndarray specifying degrees of freedom
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adjustment. Setting this parameter to a value other than `0` has the
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effect of adjusting the divisor during the calculation of the
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covariance according to `N-c` where `c` corresponds to the provided
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degrees of freedom adjustment. When computing the population
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covariance, setting this parameter to `0` is the standard choice (i.e.,
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the provided arrays contain data constituting entire populations). When
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computing the unbiased sample covariance, setting this parameter to `1`
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is the standard choice (i.e., the provided arrays contain data sampled
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from larger populations; this is commonly referred to as Bessel's
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correction).
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Returns
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-------
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out: number
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The covariance.
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Examples
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--------
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// Create the input ndarrays:
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> var xbuf = new {{alias:@stdlib/array/float32}}( [ 1.0, -2.0, 2.0 ] );
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> var ybuf = new {{alias:@stdlib/array/float32}}( [ 2.0, -2.0, 1.0 ] );
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> var dt = 'float32';
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> var sh = [ xbuf.length ];
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> var st = [ 1 ];
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> var oo = 0;
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> var ord = 'row-major';
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> var x = new {{alias:@stdlib/ndarray/ctor}}( dt, xbuf, sh, st, oo, ord );
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> var y = new {{alias:@stdlib/ndarray/ctor}}( dt, ybuf, sh, st, oo, ord );
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// Specify the known means:
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> var opts = { 'dtype': dt };
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> var meanx = new {{alias:@stdlib/ndarray/from-scalar}}( 1.0/3.0, opts );
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> var meany = new {{alias:@stdlib/ndarray/from-scalar}}( 1.0/3.0, opts );
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// Specify the degrees of freedom adjustment:
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> var correction = new {{alias:@stdlib/ndarray/from-scalar}}( 1.0, opts );
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// Calculate the sample covariance:
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> {{alias}}( [ x, y, meanx, meany, correction ] )
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~3.8333
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See Also
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--------
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