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# boxcoxinv

> Compute the inverse of a one-parameter [Box-Cox transformation][box-cox-transformation].

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<section class="intro">

To compute the inverse of a one-parameter [Box-Cox transformation][box-cox-transformation], one finds the `x` such that

<!-- <equation class="equation" label="eq:inverse_boxcox_transformation_one_parameter" align="center" raw="y = \begin{cases}\frac{x^{\lambda} - 1}{\lambda} & \textrm{if}\ \lambda \neq 0 \\ \ln(x) & \textrm{if}\ \lambda = 0 \end{cases}" alt="Inverse One-Parameter Box-Cox Transformation"> -->

<div class="equation" align="center" data-raw-text="y = \begin{cases}\frac{x^{\lambda} - 1}{\lambda} & \textrm{if}\ \lambda \neq 0 \\ \ln(x) & \textrm{if}\ \lambda = 0 \end{cases}" data-equation="eq:inverse_boxcox_transformation_one_parameter">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@1f2eca12bb7f86547d4b5b335364419bac5b97fb/lib/node_modules/@stdlib/math/base/special/boxcoxinv/docs/img/equation_inverse_boxcox_transformation_one_parameter.svg" alt="Inverse One-Parameter Box-Cox Transformation" />
    <br>
</div>

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</section>

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<!-- Package usage documentation. -->

<section class="usage">

## Usage

```javascript
var boxcoxinv = require( '@stdlib/math/base/special/boxcoxinv' );
```

#### boxcoxinv( y, lambda )

Computes the inverse of a one-parameter [Box-Cox transformation][box-cox-transformation].

```javascript
var v = boxcoxinv( 1.0, 2.5 );
// returns ~1.6505

v = boxcoxinv( 4.0, 2.5 );
// returns ~2.6095

v = boxcoxinv( 10.0, 2.5 );
// returns ~3.6812

v = boxcoxinv( 2.0, 0.0 );
// returns ~7.3891

v = boxcoxinv( -1.0, 2.5 );
// returns NaN

v = boxcoxinv( 0.0, -1.0 );
// returns 1.0

v = boxcoxinv( 1.0, NaN );
// returns NaN

v = boxcoxinv( NaN, 3.1 );
// returns NaN
```

</section>

<!-- /.usage -->

<!-- Package usage examples. -->

<section class="examples">

## Examples

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

```javascript
var incrspace = require( '@stdlib/array/base/incrspace' );
var boxcoxinv = require( '@stdlib/math/base/special/boxcoxinv' );

var y = incrspace( -1.0, 10.0, 1.0 );
var l = incrspace( -0.5, 5.0, 0.5 );

var b;
var i;
var j;
for ( i = 0; i < y.length; i++ ) {
    for ( j = 0; j < l.length; j++ ) {
        b = boxcoxinv( y[ i ], l[ j ] );
        console.log( 'boxcoxinv(%d, %d) = %d', y[ i ], l[ j ], b );
    }
}
```

</section>

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<!-- C interface documentation. -->

* * *

<section class="c">

## C APIs

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<section class="intro">

</section>

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<!-- C usage documentation. -->

<section class="usage">

### Usage

```c
#include "stdlib/math/base/special/boxcoxinv.h"
```

#### stdlib_base_boxcoxinv( y, lambda )

Computes the inverse of a one-parameter [Box-Cox transformation][box-cox-transformation].

```c
double out = stdlib_base_boxcoxinv( 1.0, 2.5 );
// returns ~1.6505

out = stdlib_base_boxcoxinv( 4.0, 2.5 );
// returns ~2.6095
```

The function accepts the following arguments:

-   **y**: `[in] double` input value.
-   **lambda**: `[in] double` power parameter.

```c
double stdlib_base_boxcoxinv ( const double y, const double lambda );
```

</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>

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<section class="examples">

### Examples

```c
#include "stdlib/math/base/special/boxcoxinv.h"
#include <stdio.h>

int main( void ) {
    const double x[] = { -1.0, 10.0, 1.0 };
    const double y[] = { -0.5, 5.0, 0.5 };

    double out;
    int i;
    int j;
    for ( i = 0; i < 3; i++ ) {
        for ( j = 0; j < 3; j++ ){
            out = stdlib_base_boxcoxinv( x[ i ], y[ j ] );
            printf ( "y: %lf, x: %lf, out: %lf\n", x[ i ], y[ j ], out );
        }
    }
}
```

</section>

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</section>

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<section class="references">

## References

-   Box, G. E. P., and D. R. Cox. 1964. "An Analysis of Transformations." _Journal of the Royal Statistical Society. Series B (Methodological)_ 26 (2). \[Royal Statistical Society, Wiley]: 211–52. <http://www.jstor.org/stable/2984418>.

</section>

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* * *

## See Also

-   <span class="package-name">[`@stdlib/math/base/special/boxcox`][@stdlib/math/base/special/boxcox]</span><span class="delimiter">: </span><span class="description">compute a one-parameter Box-Cox transformation.</span>
-   <span class="package-name">[`@stdlib/math/base/special/boxcox1p`][@stdlib/math/base/special/boxcox1p]</span><span class="delimiter">: </span><span class="description">compute a one-parameter Box-Cox transformation of 1+x.</span>
-   <span class="package-name">[`@stdlib/math/base/special/boxcox1pinv`][@stdlib/math/base/special/boxcox1pinv]</span><span class="delimiter">: </span><span class="description">compute the inverse of a one-parameter Box-Cox transformation for 1+x.</span>

</section>

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<section class="links">

[box-cox-transformation]: https://en.wikipedia.org/wiki/Power_transform#Box-Cox_transformation

<!-- <related-links> -->

[@stdlib/math/base/special/boxcox]: https://github.com/stdlib-js/math/tree/main/base/special/boxcox

[@stdlib/math/base/special/boxcox1p]: https://github.com/stdlib-js/math/tree/main/base/special/boxcox1p

[@stdlib/math/base/special/boxcox1pinv]: https://github.com/stdlib-js/math/tree/main/base/special/boxcox1pinv

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</section>

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