npm package: @stdlib/stats-strided-covarmtk
Description: Calculate the covariance of two strided arrays provided known means and using a one-pass textbook algorithm.
Author: The Stdlib Authors
Latest version: 0.1.1

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# covarmtk

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> 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."> -->

<!-- </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."> -->

<!-- </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."> -->

<!-- </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."> -->

<!-- </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."> -->

<!-- </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."> -->

<!-- </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="installation">

## Installation

```bash
npm install @stdlib/stats-strided-covarmtk
```

</section>

<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>

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* * *

## Notice

This package is part of [stdlib][stdlib], a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop [stdlib][stdlib], see the main project [repository][stdlib].

#### Community

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---

## License

See [LICENSE][stdlib-license].


## Copyright

Copyright &copy; 2016-2026. The Stdlib [Authors][stdlib-authors].

</section>

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[deno-url]: https://github.com/stdlib-js/stats-strided-covarmtk/tree/deno
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[branches-url]: https://github.com/stdlib-js/stats-strided-covarmtk/blob/main/branches.md

[stdlib-license]: https://raw.githubusercontent.com/stdlib-js/stats-strided-covarmtk/main/LICENSE

[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://www.npmjs.com/package/@stdlib/array-base-accessor

[@stdlib/stats/strided/dcovarmtk]: https://www.npmjs.com/package/@stdlib/stats-strided-dcovarmtk

[@stdlib/stats/strided/scovarmtk]: https://www.npmjs.com/package/@stdlib/stats-strided-scovarmtk

</section>

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