This vignette shows how to model multivariate distributions with copulas using `univariateML`

and the `copula`

package.

A **copula** is a function describing the dependency among a set of one-dimensional distributions. If both the marginal distributions and the copula is known, the entire multivariate distribution is known too.

Suppose we look at a multivariate distribution as the pair of a copula and its marginals. Then a natural model selection using is to

1.) select the marginal distributions using the AIC; 2.) select a copula using the marginal data transformed to the unit interval, again using the AIC.

This two-step procedure is commonly used due to its simplicity. The procedure must be carried out this order since the marginal data cannot be transformed to the unit interval unless we know the marginal distributions.

The `univariateML`

can be used for task 1, while the `copula`

package can be used to do task 2.

The `abalone`

data set is included in this package. It consists of \(9\) physical measurements of \(4177\) sea snails.

```
## sex length diameter height whole_weight shucked_weight viscera_weight
## 1 M 0.455 0.365 0.095 0.5140 0.2245 0.1010
## 2 M 0.350 0.265 0.090 0.2255 0.0995 0.0485
## 3 F 0.530 0.420 0.135 0.6770 0.2565 0.1415
## 4 M 0.440 0.365 0.125 0.5160 0.2155 0.1140
## 5 I 0.330 0.255 0.080 0.2050 0.0895 0.0395
## 6 I 0.425 0.300 0.095 0.3515 0.1410 0.0775
## shell_weight rings
## 1 0.150 15
## 2 0.070 7
## 3 0.210 9
## 4 0.155 10
## 5 0.055 7
## 6 0.120 8
```

Following Ko, Hjort, and Hobæk Haff (2019) we will take a look at four measurements of the abalones, namely `diameter`

, `height`

, `shell_weight`

and `age`

. The variable `age`

is not present in the `abalone`

data, but is defined as `age = rings + 1.5`

. Moreover, there are two outliers in the `height`

data at at \(1.13\) and \(0.52\). We will remove these outliers.

```
data = dplyr::filter(abalone, height < 0.5)
data$age = data$rings + 1.5
hist(data$height, main =" Abalone height", xlab = "Height in mm")
```

Let’s continue doing step 1. First we must decide on a set of models to try out.

```
models = c("mlgumbel", "mllaplace", "mllogis", "mlnorm", "mlexp", "mlgamma",
"mlinvgamma", "mlinvgauss", "mlinvweibull", "mlllogis", "mllnorm",
"mlrayleigh", "mlweibull", "mllgamma", "mlpareto", "mlbeta", "mlkumar",
"mllogitnorm")
length(models)
```

`## [1] 18`

The nest step is to define a function that calculates the AIC for all our candidate models over every margin.

```
#' Calculate AICs for a column
#' @param variable The variable to calculate AICs for.
#' @return A named vector of AICs.
AICs = function(variable) {
x = data[[variable]]
FUN = function(name, x = NULL) tryCatch(expr = eval(call(name, x)),
error = function(e) NULL)
fits = Filter(Negate(is.null), sapply(X = models, FUN = FUN, x = x))
sort(sapply(fits, AIC))
}
```

This function returns a vector of AICs for each model where `ml***`

didn’t fail. When it fails it is since the support of the data doesn’t not match the density, for instance when `-1`

is fed to the exponential density.

```
## mllnorm mlgumbel mlinvgauss mllgamma mlgamma
## 21033.17 21039.99 21055.60 21100.20 21104.82
## mlinvgamma mllogis mllaplace mlnorm mlweibull
## 21107.62 21295.85 21310.76 21627.91 21893.36
## mlinvweibull mlrayleigh mlexp mlpareto mlllogis
## 22050.05 23860.54 28697.63 31678.23 394429.50
```

But we are most interested in the models with smallest AICs.

```
## diameter height shell_weight age
## "mlkumar" "mlnorm" "mlweibull" "mllnorm"
```

Now we use the `fitCopula`

from the package `copula`

on the transformed margins of `abalone`

.

We will examine two elliptical copulas and three Archimedean copulas. The elliptical copulas are the Gaussian copula and the *t*-copula, while the Archimedean copulas are the Joe copula, the Clayton copula, and the Gumbel copula.

```
# Transform the marginals to the unit interval.
y = as.matrix(dplyr::transmute(data,
diameter = pml(diameter, mlkumar(diameter)),
height = pml(height, mlnorm(height)),
shell_weight = pml(shell_weight, mlweibull(shell_weight)),
age = pml(age, mllnorm(rings))))
# The copulas described above.
copulas = list(normal = copula::normalCopula(dim = 4, dispstr = "un"),
t = copula::tCopula(dim = 4, dispstr = "un"),
joe = copula::joeCopula(dim = 4),
clayton = copula::claytonCopula(dim = 4),
gumbel = copula::gumbelCopula(dim = 4))
fits = sapply(copulas,
function(x) copula::fitCopula(x, data = y, method = "mpl"))
sapply(fits, AIC)
```

```
## normal t joe clayton gumbel
## -20776.28 -21856.69 -10139.89 -14546.95 -13586.67
```

The *t*-copula is the clear winner of the AIC competition. The Archimedean copulas perform particularly poorly.

Hence our final model is the *t*-copula with Kumaraswamy (`mlkumar`

) marginal distribution for `diameter`

, normal marginal distribution for `height`

, Weibull marginal distribution for `shell_weight`

, and log normal marginal distribution for `age`

.

Ko, Vinnie, Nils Lid Hjort, and Ingrid Hobæk Haff. 2019. “Focused Information Criteria for Copulas.” *Scandinavian Journal of Statistics*.