Discussion on "Random-projection ensemble classification" by T. Cannings and R. Samworth

TL;DR

This discussion highlights the potential applications of the random-projection ensemble classification method in economics and suggests its extension to other classifiers, emphasizing the relevance of copula-based discriminant analysis.

Contribution

The paper proposes extending the random-projection ensemble classification approach to broader classifiers and discusses its applications in economic contexts like credit scoring.

Findings

Potential for applications in credit scoring and economics

Extension to more general classifiers possible

Highlights relevance of copula-based discriminant analysis

Abstract

Discussion on "Random-projection ensemble classification" by T. Cannings and R. Samworth. We believe that the proposed approach can find many applications in economics such as credit scoring (e.g. Altman (1968)) and can be extended to more general type of classifiers. In this discussion we would like to draw authors attention to the copula-based discriminant analysis (Han et al. (2013) and He et al. (2016)).

Equations

$$\Delta \left(s;{\pi }_0,{\Sigma }_0,{\Sigma }_1\right)=\log \left(\frac{{\pi }_1}{{\pi }_0}\right)-\frac{1}{2}\log \left(\frac{\det \left({\Sigma }_1\right)}{\det \left({\Sigma }_0\right)}\right)-\frac{1}{2}{s}^T\left[{\Sigma }_1^{-1}-{\Sigma }_0^{-1}\right]s.$$

$$S_{ji}$$

$${\hat{\Sigma }}_r^A$$

$$\displaystyle C_{n}^{\text{A-RQDA}}(s):=\left\{\begin{array}[]{ccc}1&&\Delta\left(s;\hat{\pi}_{0},\hat{\Sigma}^{A}_{0},\hat{\Sigma}^{A}_{1}\right)\geq 0,\\ 0&&\mathrm{otherwise}.\end{array}\right.$$

Full text

Discussion on “Random-projection ensemble classification” by T. Cannings and R. Samworth, written by Roberto Casarin†, Lorenzo Frattarolo*†* and Luca Rossini*†‡*111Corresponding author at: Free University of Bozen-Bolzano, piazza Università, 1, 39100, Bolzano-Bozen, Italy.

E-mail address: luca.rossini@unive.it (Luca Rossini)(†University Ca’ Foscari of Venice, Italy and ‡ Free University of Bozen-Bolzano, Italy).

The authors are to be congratulated on their excellent research, which has culminated in the development of a characterization of the approximation errors in random projection methods when applied to classification. We believe that the proposed approach can find many applications in economics such as credit scoring (e.g. Altman, (1968)) and can be extended to more general type of classifiers. In this discussion we would like to draw authors attention to the copula-based discriminant analysis (Han et al., (2013) and He et al., (2016)).

We consider $X|Y=r$ distributed as a $p$-dimensional meta Gaussian distribution and $\left.S\right|Y=r\sim\mathcal{N}_{p}\left(0,\Sigma_{r}\right)$, where $\Sigma_{r}$ is the linear correlation among variables. Given a $p\times d$ random projection $A$, $\left.AS\right|Y=r\sim\mathcal{N}_{d}\left(0,\Sigma^{A}_{r}\right)$, where $\Sigma^{A}_{r}=A\Sigma_{r}A^{T}$. If we assume that the information in the marginals is not relevant for the classification, the Bayes decision boundary depends only on the transformed variables $s_{i}=\Phi^{-1}\left(F(x_{i})\right)$ with $\Phi$ and $F$ the univariate normal and the marginal cdfs, respectively (Fang et al., (2002)), $s_{i}$ and $x_{i}$ the $i$-th element of $s$ and $x$, and the correlation of the two groups

[TABLE]

Analogously the classifier in the random projection ensamble will depend only on the random projection of the transformed variables and their covariances. We use the empirical distribution function to obtain the sample version of the transformed variables $S_{i}=(S_{1i},\dots,S_{pi})$, with

[TABLE]

The estimator of $\Sigma^{A}_{r}$ is obtained by maximizing the pseudo-likelihood:

[TABLE]

where the asymptotic normality is guaranteed by results in Genest et al., (1995) and recently in Segers et al., (2014). We propose the following robust QDA random-projection ensemble classifier:

[TABLE]

We are very pleased to be able to propose the vote of thanks to the authors for their work.