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feat: add stats/strided/dcovarmtk
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‎lib/node_modules/@stdlib/stats/strided/dcovarmtk/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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<!-- lint disable maximum-heading-length -->
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# dcovarmtk
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> Calculate the [covariance][covariance] of two double-precision floating-point strided arrays 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 a finite size population 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 variance** for a sample 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 dcovarmtk = require( '@stdlib/stats/strided/dcovarmtk' );
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```
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#### dcovarmtk( N, correction, meanx, x, strideX, meany, y, strideY )
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Computes the [covariance][covariance] of two double-precision floating-point strided arrays provided known means and using a one-pass textbook algorithm.
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```javascript
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var Float64Array = require( '@stdlib/array/float64' );
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var x = new Float64Array( [ 1.0, -2.0, 2.0 ] );
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var y = new Float64Array( [ 2.0, -2.0, 1.0 ] );
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var v = dcovarmtk( x.length, 1, 1.0/3.0, x, 1, 1.0/3.0, y, 1 );
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// returns ~7.6667
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```
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The function has the following parameters:
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- **N**: number of indexed elements.
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- **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).
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- **meanx**: mean of `x`.
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- **x**: first input [`Float64Array`][@stdlib/array/float64].
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- **strideX**: stride length for `x`.
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- **meany**: mean of `y`.
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- **y**: second input [`Float64Array`][@stdlib/array/float64].
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- **strideY**: stride length for `y`.
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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`,
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```javascript
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var Float64Array = require( '@stdlib/array/float64' );
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var x = new Float64Array( [ 1.0, 2.0, 2.0, -7.0, -2.0, 3.0, 4.0, 2.0 ] );
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var y = new Float64Array( [ -7.0, 2.0, 2.0, 1.0, -2.0, 2.0, 3.0, 4.0 ] );
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var v = dcovarmtk( 4, 1, 1.25, x, 2, 1.25, y, 2 );
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// returns 6.0
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```
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Note that indexing is relative to the first index. To introduce an offset, use [`typed array`][mdn-typed-array] views.
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<!-- eslint-disable stdlib/capitalized-comments -->
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```javascript
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var Float64Array = require( '@stdlib/array/float64' );
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var x0 = new Float64Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] );
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var y0 = new Float64Array( [ 2.0, -2.0, 2.0, 1.0, -2.0, 4.0, 3.0, 2.0 ] );
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var x1 = new Float64Array( x0.buffer, x0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
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var y1 = new Float64Array( y0.buffer, y0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
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var v = dcovarmtk( 4, 1, 1.25, x1, 2, 1.25, y1, 2 );
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// returns ~1.9167
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```
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#### dcovarmtk.ndarray( N, correction, meanx, x, strideX, offsetX, meany, y, strideY, offsetY )
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Computes the [covariance][covariance] of two double-precision floating-point strided arrays provided known means and using a one-pass textbook algorithm and alternative indexing semantics.
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```javascript
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var Float64Array = require( '@stdlib/array/float64' );
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var x = new Float64Array( [ 1.0, -2.0, 2.0 ] );
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var y = new Float64Array( [ 2.0, -2.0, 1.0 ] );
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var v = dcovarmtk.ndarray( x.length, 1, 1.0/3.0, x, 1, 0, 1.0/3.0, y, 1, 0 );
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// returns ~7.6667
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```
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The function has the following additional parameters:
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- **offsetX**: starting index for `x`.
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- **offsetY**: starting index for `y`.
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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
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```javascript
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var Float64Array = require( '@stdlib/array/float64' );
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var x = new Float64Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] );
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var y = new Float64Array( [ -7.0, 2.0, 2.0, 1.0, -2.0, 2.0, 3.0, 4.0 ] );
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var v = dcovarmtk.ndarray( 4, 1, 1.25, x, 2, 1, 1.25, y, 2, 1 );
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// returns 6.0
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```
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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 `N <= 0`, both functions return `NaN`.
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- If `N - c` is less than or equal to `0` (where `c` corresponds to the provided degrees of freedom adjustment), both functions return `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 dcovarmtk = require( '@stdlib/stats/strided/dcovarmtk' );
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var opts = {
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'dtype': 'float64'
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};
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var x = discreteUniform( 10, -50, 50, opts );
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console.log( x );
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var y = discreteUniform( 10, -50, 50, opts );
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console.log( y );
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var v = dcovarmtk( x.length, 1, 0.0, x, 1, 0.0, y, 1 );
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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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<!-- C interface documentation. -->
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* * *
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<section class="c">
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## C APIs
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<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->
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<section class="intro">
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</section>
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<!-- /.intro -->
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<!-- C usage documentation. -->
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<section class="usage">
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### Usage
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```c
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#include "stdlib/stats/strided/dcovarmtk.h"
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```
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#### stdlib_strided_dcovarmtk( N, correction, meanx, \*X, strideX, meany, \*Y, strideY )
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Computes the [covariance][covariance] of two double-precision floating-point strided arrays provided known means and using a one-pass textbook algorithm.
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```c
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const double x[] = { 1.0, -2.0, 2.0 };
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const double y[] = { 2.0, -2.0, 1.0 };
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double v = stdlib_strided_dcovarmtk( 3, 1.0, 1.0/3.0, x, 1, 1.0/3.0, y, 1 );
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// returns ~7.6667
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```
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The function accepts the following arguments:
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- **N**: `[in] CBLAS_INT` number of indexed elements.
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- **correction**: `[in] double` 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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- **meanx**: `[in] double` mean of `X`.
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- **X**: `[in] double*` first input array.
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- **strideX**: `[in] CBLAS_INT` stride length for `X`.
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- **meany**: `[in] double` mean of `Y`.
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- **Y**: `[in] double*` second input array.
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- **strideY**: `[in] CBLAS_INT` stride length for `Y`.
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```c
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double stdlib_strided_dcovarmtk( const CBLAS_INT N, const double correction, const double meanx, const double *X, const CBLAS_INT strideX, const double meanY, const double *Y, const CBLAS_INT strideY );
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```
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#### stdlib_strided_dcovarmtk_ndarray( N, correction, meanx, \*X, strideX, offsetX, meany, \*Y, strideY, offsetY )
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Computes the [covariance][covariance] of two double-precision floating-point strided arrays provided known means and using a one-pass textbook algorithm and alternative indexing semantics.
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```c
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const double x[] = { 1.0, -2.0, 2.0 };
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const double y[] = { 2.0, -2.0, 1.0 };
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double v = stdlib_strided_dcovarmtk_ndarray( 3, 1.0, 1.0/3.0, x, 1, 0, 1.0/3.0, y, 1, 0 );
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// returns ~7.6667
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```
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The function accepts the following arguments:
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- **N**: `[in] CBLAS_INT` number of indexed elements.
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- **correction**: `[in] double` 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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- **meanx**: `[in] double` mean of `X`.
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- **X**: `[in] double*` first input array.
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- **strideX**: `[in] CBLAS_INT` stride length for `X`.
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- **offsetX**: `[in] CBLAS_INT` starting index for `X`.
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- **meany**: `[in] double` mean of `Y`.
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- **Y**: `[in] double*` second input array.
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- **strideY**: `[in] CBLAS_INT` stride length for `Y`.
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- **offsetY**: `[in] CBLAS_INT` starting index for `Y`.
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```c
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double stdlib_strided_dcovarmtk_ndarray( const CBLAS_INT N, const double correction, const double meanx, const double *X, const CBLAS_INT strideX, const CBLAS_INT offsetX, const double meany, const double *Y, const CBLAS_INT strideY, const CBLAS_INT offsetY );
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```
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</section>
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<!-- /.usage -->
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<!-- C API usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
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<section class="notes">
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</section>
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<!-- /.notes -->
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<!-- C API usage examples. -->
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<section class="examples">
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### Examples
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```c
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#include "stdlib/stats/strided/dcovarmtk.h"
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#include <stdio.h>
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int main( void ) {
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// Create a strided array:
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const double x[] = { 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0 };
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// Specify the number of elements:
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const int N = 4;
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// Specify the stride length:
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const int strideX = 2;
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// Compute the covariance of `x` with itself:
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double v = stdlib_strided_dcovarmtk( N, 1, 4.5, x, strideX, 4.5, x, -strideX );
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// Print the result:
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printf( "covariance: %lf\n", v );
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}
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```
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</section>
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<!-- /.examples -->
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</section>
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<!-- /.c -->
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<section class="references">
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</section>
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<!-- /.references -->
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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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[covariance]: https://en.wikipedia.org/wiki/Covariance
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[@stdlib/array/float64]: https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/array/float64
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[mdn-typed-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/TypedArray
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</section>
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<!-- /.links -->

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