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feat: add stats/covarmtk
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‎lib/node_modules/@stdlib/stats/covarmtk/README.md‎

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<!--
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@license Apache-2.0
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Copyright (c) 2026 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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# covarmtk
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> Compute the [covariance][covariance] of two 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 covarmtk = require( '@stdlib/stats/covarmtk' );
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```
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#### covarmtk( x, y, correction, meanx, meany\[, options] )
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Computes the [covariance][covariance] of two ndarrays 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 ndarray = require( '@stdlib/ndarray/ctor' );
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var scalar2ndarray = require( '@stdlib/ndarray/from-scalar' );
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var xbuf = new Float64Array( [ 1.0, -2.0, 2.0 ] );
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var ybuf = new Float64Array( [ 2.0, -2.0, 1.0 ] );
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var x = new ndarray( 'float64', xbuf, [ 3 ], [ 1 ], 0, 'row-major' );
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var y = new ndarray( 'float64', ybuf, [ 3 ], [ 1 ], 0, 'row-major' );
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var correction = scalar2ndarray( 1.0, {
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'dtype': 'float64'
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});
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var meanx = scalar2ndarray( 1.0/3.0, {
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'dtype': 'float64'
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});
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var meany = scalar2ndarray( 1.0/3.0, {
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'dtype': 'float64'
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});
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var out = covarmtk( x, y, correction, meanx, meany );
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// returns <ndarray>[ ~3.8333 ]
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```
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The function has the following parameters:
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- **x**: first input [ndarray][@stdlib/ndarray/ctor]. Must have a real-valued or "generic" [data type][@stdlib/ndarray/dtypes].
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- **y**: second input [ndarray][@stdlib/ndarray/ctor]. Must have a real-valued or "generic" [data type][@stdlib/ndarray/dtypes].
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- **correction**: zero-dimensional [ndarray][@stdlib/ndarray/ctor] specifying the 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 and `N` corresponds to the number of elements in each input [ndarray][@stdlib/ndarray/ctor]. 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**: zero-dimensional [ndarray][@stdlib/ndarray/ctor] specifying the mean of the first input [ndarray][@stdlib/ndarray/ctor].
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- **meany**: zero-dimensional [ndarray][@stdlib/ndarray/ctor] specifying the mean of the second input [ndarray][@stdlib/ndarray/ctor].
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- **options**: function options (_optional_).
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The function accepts the following options:
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- **dims**: list of dimensions over which to perform a reduction. If not provided, the function performs a reduction over all elements in a provided input [ndarray][@stdlib/ndarray/ctor].
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- **dtype**: output ndarray [data type][@stdlib/ndarray/dtypes]. Must be a real-valued floating-point or "generic" [data type][@stdlib/ndarray/dtypes].
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- **keepdims**: boolean indicating whether the reduced dimensions should be included in the returned [ndarray][@stdlib/ndarray/ctor] as singleton dimensions. Default: `false`.
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By default, the function performs a reduction over all elements in the provided input ndarrays. To perform a reduction over specific dimensions, provide a `dims` option.
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```javascript
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var Float64Array = require( '@stdlib/array/float64' );
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var ndarray = require( '@stdlib/ndarray/ctor' );
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var scalar2ndarray = require( '@stdlib/ndarray/from-scalar' );
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var xbuf = new Float64Array( [ 1.0, 2.0, 2.0, -7.0, -2.0, 3.0, 4.0, 2.0 ] );
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var ybuf = new Float64Array( [ 2.0, 1.0, 2.0, 1.0, -2.0, 2.0, 3.0, 4.0 ] );
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var x = new ndarray( 'float64', xbuf, [ 2, 2, 2 ], [ 4, 2, 1 ], 0, 'row-major' );
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var y = new ndarray( 'float64', ybuf, [ 2, 2, 2 ], [ 4, 2, 1 ], 0, 'row-major' );
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var correction = scalar2ndarray( 1.0, {
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'dtype': 'float64'
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});
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var meanx = scalar2ndarray( 1.25, {
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'dtype': 'float64'
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});
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var meany = scalar2ndarray( 1.25, {
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'dtype': 'float64'
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});
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var out = covarmtk( x, y, correction, meanx, meany, {
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'dims': [ 2 ]
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});
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// returns <ndarray>
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```
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#### covarmtk.assign( x, y, correction, meanx, meany, out\[, options] )
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Computes the [covariance][covariance] along one or more [ndarray][@stdlib/ndarray/ctor] dimensions and assigns results to a provided output [ndarray][@stdlib/ndarray/ctor].
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```javascript
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var Float64Array = require( '@stdlib/array/float64' );
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var ndarray = require( '@stdlib/ndarray/ctor' );
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var scalar2ndarray = require( '@stdlib/ndarray/from-scalar' );
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var zeros = require( '@stdlib/ndarray/zeros' );
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var xbuf = new Float64Array( [ 1.0, -2.0, 2.0 ] );
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var ybuf = new Float64Array( [ 2.0, -2.0, 1.0 ] );
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var x = new ndarray( 'float64', xbuf, [ 3 ], [ 1 ], 0, 'row-major' );
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var y = new ndarray( 'float64', ybuf, [ 3 ], [ 1 ], 0, 'row-major' );
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var correction = scalar2ndarray( 1.0, {
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'dtype': 'float64'
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});
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var meanx = scalar2ndarray( 1.0/3.0, {
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'dtype': 'float64'
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});
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var meany = scalar2ndarray( 1.0/3.0, {
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'dtype': 'float64'
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});
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var z = zeros( [], {
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'dtype': 'float64'
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});
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var out = covarmtk.assign( x, y, correction, meanx, meany, z );
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// returns <ndarray>[ ~3.8333 ]
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var bool = ( out === z );
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// returns true
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```
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The method has the following parameters:
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- **x**: first input [ndarray][@stdlib/ndarray/ctor]. Must have a real-valued or "generic" [data type][@stdlib/ndarray/dtypes].
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- **y**: second input [ndarray][@stdlib/ndarray/ctor]. Must have a real-valued or "generic" [data type][@stdlib/ndarray/dtypes].
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- **correction**: zero-dimensional [ndarray][@stdlib/ndarray/ctor] specifying the degrees of freedom adjustment.
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- **meanx**: zero-dimensional [ndarray][@stdlib/ndarray/ctor] specifying the mean of the first input [ndarray][@stdlib/ndarray/ctor].
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- **meany**: zero-dimensional [ndarray][@stdlib/ndarray/ctor] specifying the mean of the second input [ndarray][@stdlib/ndarray/ctor].
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- **out**: output [ndarray][@stdlib/ndarray/ctor].
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- **options**: function options (_optional_).
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The method accepts the following options:
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- **dims**: list of dimensions over which to perform a reduction. If not provided, the function performs a reduction over all elements in the provided input ndarrays.
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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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- Both input ndarrays must have the same shape.
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- Setting the `keepdims` option to `true` can be useful when wanting to ensure that the output [ndarray][@stdlib/ndarray/ctor] is [broadcast-compatible][@stdlib/ndarray/base/broadcast-shapes] with ndarrays having the same shape as the input ndarrays.
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- The output data type [policy][@stdlib/ndarray/output-dtype-policies] only applies to the main function and specifies that, by default, the function must return an [ndarray][@stdlib/ndarray/ctor] having a real-valued floating-point or "generic" [data type][@stdlib/ndarray/dtypes]. For the `assign` method, the output [ndarray][@stdlib/ndarray/ctor] is allowed to have any supported output [data type][@stdlib/ndarray/dtypes].
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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 uniform = require( '@stdlib/random/array/uniform' );
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var ndarray = require( '@stdlib/ndarray/ctor' );
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var scalar2ndarray = require( '@stdlib/ndarray/from-scalar' );
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var ndarray2array = require( '@stdlib/ndarray/to-array' );
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var covarmtk = require( '@stdlib/stats/covarmtk' );
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var opts = {
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'dtype': 'float64'
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};
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var xbuf = uniform( 40, -50.0, 50.0, opts );
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var x = new ndarray( opts.dtype, xbuf, [ 5, 2, 4 ], [ 8, 4, 1 ], 0, 'row-major' );
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var ybuf = uniform( 40, -50.0, 50.0, opts );
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var y = new ndarray( opts.dtype, ybuf, [ 5, 2, 4 ], [ 8, 4, 1 ], 0, 'row-major' );
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var correction = scalar2ndarray( 1.0, {
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'dtype': 'float64'
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});
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var meanx = scalar2ndarray( 0.0, {
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'dtype': 'float64'
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});
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var meany = scalar2ndarray( 0.0, {
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'dtype': 'float64'
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});
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var out = covarmtk( x, y, correction, meanx, meany, {
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'dims': [ 2 ]
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});
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console.log( ndarray2array( out ) );
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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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[covariance]: https://en.wikipedia.org/wiki/Covariance
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[@stdlib/ndarray/ctor]: https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/ndarray/ctor
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[@stdlib/ndarray/dtypes]: https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/ndarray/dtypes
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[@stdlib/ndarray/output-dtype-policies]: https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/ndarray/output-dtype-policies
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[@stdlib/ndarray/base/broadcast-shapes]: https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/ndarray/base/broadcast-shapes
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</section>
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<!-- /.links -->

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