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# qad <a><img src='man/figures/hex.png' align="right" height="150" /></a>

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### Summary

The R-package qad (short for quantification of asymmetric dependence)
allows to estimate the (directed) dependence of two random variables *X*
and *Y*. The estimated population value *q*(*X*,*Y*) introduced in
\[1,3\] fulfills the following properties:

-   *q*(*X*,*Y*) = 1 if and only if *Y* is a function of *X* (knowing
    *X* means knowing *Y*)
-   *q*(*X*,*Y*) = 0 if and only if *X* and *Y* are independent (no
    information gain).

While the Pearson correlation coefficient assesses only linear and
Spearman rank correlation only monotonic relationships, qad is able to
detect any kind of association. For further information we refer to the
vignette or the related publications \[1,2,3\].

### Installation

The easiest way to get the package qad is:

``` r
install.packages("qad")
```

In order to install the development version of qad from GitHub:

``` r
# install devtools package
if (!requireNamespace("devtools", quietly = TRUE)) {
  install.packages("devtools")
}
# install package
devtools::install_github("griefl/qad", dependencies = TRUE)
```

### Usage

``` r
library(qad)

set.seed(314)
n <- 100
x <- rnorm(n)
y <- x^2 + rnorm(n, 0, 1)
plot(x,y, pch = 16)
```

<img src="man/figures/README-unnamed-chunk-4-1.png" width="70%" />

``` r
fit <- qad(x,y)
#> 
#> quantification of asymmetric dependence: 
#> 
#> Data: x1 := x
#>       x2 := y
#> 
#> Sample Size: 100
#> Number of unique ranks: x1: 100
#>                         x2: 100
#>                    (x1,x2): 100
#> Resolution: 10 x 10
#> 
#> Dependence measures:
#>                     q p.values
#>  q(x1,x2)       0.610    0.000
#>  q(x2,x1)       0.393    0.002
#>  max.dependence 0.610    0.000
#> 
#>                      a p.values
#>  asymmetry       0.217       NA
coef(fit)
#>         q(x1,x2)         q(x2,x1)   max.dependence        asymmetry 
#>            0.610            0.393            0.610            0.217 
#>       p.q(x1,x2)       p.q(x2,x1) p.max.dependence      p.asymmetry 
#>            0.000            0.002            0.000               NA

#Comparison with correlation
cor(x,y, method = "pearson")
#> [1] -0.04404337
cor(x,y, method = "spearman")
#> [1] 0.06546655
cor(x,y, method = "kendall")
#> [1] 0.05090909
```

### References

-   \[1\] R.R. Junker, F. Griessenberger, W. Trutschnig: Estimating
    scale-invariant directed dependence of bivariate distributions,
    *Computational Statistics and Data Analysis*, (2021), 153, 107058,
    <a href = "https://doi.org/10.1016/j.csda.2020.107058">https://doi.org/10.1016/j.csda.2020.107058</a>

-   \[2\] R.R. Junker, F. Griessenberger, W. Trutschnig: A copula-based
    measure for quantifying asymmetry in dependence and associations,
    <https://arxiv.org/abs/1902.00203>

-   \[3\] W. Trutschnig: On a strong metric on the space of copulas and
    its induced dependence measure, *Journal of Mathematical Analysis
    and Applications*, 2011, (384), 690-705.
    <a href="https://doi.org/10.1016/j.jmaa.2011.06.013">https://doi.org/10.1016/j.jmaa.2011.06.013</a>
