Fraud detection using copula regression with vine copulas

In the fraud detection setting, the response variable \(Y\) is dichotomous,
whereas the covariates \(\mathbf{X}\) typically consist of a mixture of continuous and discrete variables. Such combinations of variables are typically modelled with a regression type model, that specify the relationship between \(Y\) and the remaining variables, but do not account for complex dependencies between the covariates. One may of course include interaction effects, but that becomes complicated as the number of covariates increases. As the detected fraudulent cases typically are relatively few compared to the total number of cases, and moreover may have widely different characteristics, it may be decisive to capture the dependencies between the covariates.

Copulas (see for instance [1] for an introduction to copulas) have been proposed as alternatives to traditional regression models [2,3]. The idea is then to build a joint model for \((Y,\mathbf{X})\) and infer the conditional distribution of \(Y|\mathbf{X}\) from this model. Such a
joint model can be built with a copula.

Copulas are well suited for modelling the joint distribution of variables with dissimilar marginal behaviour, which is certainly the case for \((Y,\mathbf{X})\). The definition of a copula is a distribution of \(d\) (\(\geq 2\)\((0,1)\)) random variables, that marginally are uniformly distributed on the interval. According to [4], any \(d\)-variate cumulative distribution function \(F_{1\ldots d}\), with univariate margins \(F_{1},\ldots,F_{d}\), may be expressed as

\(F_{1\ldots d}(x_{1},\ldots,x_{d}) = C_{1\ldots d}\left(F_{1}(x_{1}),\ldots,F_{d}(x_{d})\right),\)

where \(C_{1\ldots d}\)is a copula. The idea is to build such a copula regression model, based on vine copulas, suitable for the fraud detection setting.

When the copula is Gaussian (a special case of vine copulas) and all the covariates
are continuous, the copula regression model is essentially a GLM with a probit link,
but with the Rosenblatt transformed variables

\(F_{1}(X_{1}),F_{2|1}(X_{2}|X_{1}),\ldots,F_{p|1\ldots p-1}(X_{p}|X_{1},\ldots,X_{p-1})\)

as covariates, i.e.

\(\mbox{P}(Y=1|X_{1}=x_{1},\ldots,X_{d}=x_{p}) = \Phi\left(\beta_{0}+\beta_{1}F_{1}(x_{1})+\ldots+\beta_{p}F_{p|1\ldots p-1}(x_{p}|x_{1},\ldots,x_{p-1})\right).\)

Here, the regression coefficients are functions of the correlation parameters of the Gaussian copula. The parameters can be estimated using the semi-parametric estimator proposed by [5]. The advantage of their method is that it does not require specifying the marginal distributions of the original covariates, but uses the empirical distributions instead. The disadvantage of their method is that it becomes very slow and computationally challenging in larger dimensions, as it requires optimisation over many parameters simultaneously.

One manner of speeding up the procedure is to perform it in two steps, taking advantage of the fact that the model is (approximately) a GLM, handling discrete covariates through jittering [6].  One might then estimate a Gaussian copula for the original covariates first. Subsequently, the Rosenblatt-transformed covariates can be computed from this copula. This was done in the thesis [7]. The idea in this thesis is to expand her work by letting the dependence between the original covariates be a more general vine copula, see [7], while the dependence between the response variable and the covariates is still modelled by Gaussian copulas. The resulting model is then still (approximately) a GLM with a probit link, with covariates that can be computed based on the vine copula.

Normally, one would also have to search for a good ordering of the variables in the vine copula for the covariates. However, in this problem, the ordering will, at least in the beginning, be assumed to be known (but with a possible expansion to unknown ordering). The added complexity compared to the thesis [7] lies among others fact that the copula between the covariates is no longer a Gaussian copula, but a vine copula, whose component copulas must be chosen, requiring model selection. The estimation procedure will also be more complicated.