npm package: @stdlib/stats-strided-dcovarmtk
Description: Calculate the covariance of two double-precision floating-point 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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# 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.

<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-dcovarmtk
```

</section>

<section class="usage">

## Usage

```javascript
var dcovarmtk = require( '@stdlib/stats-strided-dcovarmtk' );
```

#### dcovarmtk( N, correction, meanx, x, strideX, meany, y, strideY )

Computes the [covariance][covariance] of two double-precision floating-point strided arrays provided known means and using a one-pass textbook algorithm.

```javascript
var Float64Array = require( '@stdlib/array-float64' );

var x = new Float64Array( [ 1.0, -2.0, 2.0 ] );
var y = new Float64Array( [ 2.0, -2.0, 1.0 ] );

var v = dcovarmtk( 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 [`Float64Array`][@stdlib/array/float64].
-   **strideX**: stride length for `x`.
-   **meany**: mean of `y`.
-   **y**: second input [`Float64Array`][@stdlib/array/float64].
-   **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 Float64Array = require( '@stdlib/array-float64' );

var x = new Float64Array( [ 1.0, 2.0, 2.0, -7.0, -2.0, 3.0, 4.0, 2.0 ] );
var y = new Float64Array( [ 2.0, 1.0, 2.0, 1.0, -2.0, 2.0, 3.0, 4.0 ] );

var v = dcovarmtk( 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 = dcovarmtk( 4, 1, 1.25, x1, 2, 1.25, y1, 2 );
// returns ~1.9167
```

#### dcovarmtk.ndarray( N, correction, meanx, x, strideX, offsetX, meany, y, strideY, offsetY )

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.

```javascript
var Float64Array = require( '@stdlib/array-float64' );

var x = new Float64Array( [ 1.0, -2.0, 2.0 ] );
var y = new Float64Array( [ 2.0, -2.0, 1.0 ] );

var v = dcovarmtk.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 Float64Array = require( '@stdlib/array-float64' );

var x = new Float64Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] );
var y = new Float64Array( [ -7.0, 2.0, 2.0, 1.0, -2.0, 2.0, 3.0, 4.0 ] );

var v = dcovarmtk.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`.

</section>

<!-- /.notes -->

<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

```javascript
var discreteUniform = require( '@stdlib/random-array-discrete-uniform' );
var dcovarmtk = require( '@stdlib/stats-strided-dcovarmtk' );

var opts = {
    'dtype': 'float64'
};
var x = discreteUniform( 10, -50, 50, opts );
console.log( x );

var y = discreteUniform( 10, -50, 50, opts );
console.log( y );

var v = dcovarmtk( x.length, 1, 0.0, x, 1, 0.0, y, 1 );
console.log( v );
```

</section>

<!-- /.examples -->

<!-- C interface documentation. -->

* * *

<section class="c">

## C APIs

<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->

<section class="intro">

</section>

<!-- /.intro -->

<!-- C usage documentation. -->

<section class="usage">

### Usage

```c
#include "stdlib/stats/strided/dcovarmtk.h"
```

#### stdlib_strided_dcovarmtk( N, correction, meanx, \*X, strideX, meany, \*Y, strideY )

Computes the [covariance][covariance] of two double-precision floating-point strided arrays provided known means and using a one-pass textbook algorithm.

```c
const double x[] = { 1.0, -2.0, 2.0 };
const double y[] = { 2.0, -2.0, 1.0 };

double v = stdlib_strided_dcovarmtk( 3, 1.0, 1.0/3.0, x, 1, 1.0/3.0, y, 1 );
// returns ~3.8333
```

The function accepts the following arguments:

-   **N**: `[in] CBLAS_INT` number of indexed elements.
-   **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).
-   **meanx**: `[in] double` mean of `X`.
-   **X**: `[in] double*` first input array.
-   **strideX**: `[in] CBLAS_INT` stride length for `X`.
-   **meany**: `[in] double` mean of `Y`.
-   **Y**: `[in] double*` second input array.
-   **strideY**: `[in] CBLAS_INT` stride length for `Y`.

```c
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 );
```

#### stdlib_strided_dcovarmtk_ndarray( N, correction, meanx, \*X, strideX, offsetX, meany, \*Y, strideY, offsetY )

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.

```c
const double x[] = { 1.0, -2.0, 2.0 };
const double y[] = { 2.0, -2.0, 1.0 };

double v = stdlib_strided_dcovarmtk_ndarray( 3, 1.0, 1.0/3.0, x, 1, 0, 1.0/3.0, y, 1, 0 );
// returns ~3.8333
```

The function accepts the following arguments:

-   **N**: `[in] CBLAS_INT` number of indexed elements.
-   **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).
-   **meanx**: `[in] double` mean of `X`.
-   **X**: `[in] double*` first input array.
-   **strideX**: `[in] CBLAS_INT` stride length for `X`.
-   **offsetX**: `[in] CBLAS_INT` starting index for `X`.
-   **meany**: `[in] double` mean of `Y`.
-   **Y**: `[in] double*` second input array.
-   **strideY**: `[in] CBLAS_INT` stride length for `Y`.
-   **offsetY**: `[in] CBLAS_INT` starting index for `Y`.

```c
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 );
```

</section>

<!-- /.usage -->

<!-- C API usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="notes">

</section>

<!-- /.notes -->

<!-- C API usage examples. -->

<section class="examples">

### Examples

```c
#include "stdlib/stats/strided/dcovarmtk.h"
#include <stdio.h>

int main( void ) {
    // Create a strided array:
    const double x[] = { 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0 };

    // Specify the number of elements:
    const int N = 4;

    // Specify the stride length:
    const int strideX = 2;

    // Compute the covariance of `x` with itself:
    double v = stdlib_strided_dcovarmtk( N, 1.0, 4.5, x, strideX, 4.5, x, -strideX );

    // Print the result:
    printf( "covariance: %lf\n", v );
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

<section class="references">

</section>

<!-- /.references -->

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<section class="related">

</section>

<!-- /.related -->

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* * *

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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].

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---

## License

See [LICENSE][stdlib-license].


## Copyright

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

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