Non-parametric cumulants approach for outlier detection of multivariate financial data

Outlier detection has gained more relevance throughout the years, and, as of now, its fields of application range from medicine and engineering to finance. As for the latter, outliers can be the consequence of human error or fraudolent activities; similarly, financial crises can be viewed as anomalies since markets experience atypical behaviors in those periods. Because of this widespread practical relevance, many authors tackled this topic. Hence, the theory behind anomaly detection has unsurprisingly evolved, from the first studies which dealt with more simple instances, i.e., univariate Gaussian data, to more complex cases, such as multivariate data following nonparametric distributions. Especially when dealing with high dimensional multivariate data, many techniques aim to find outliers in univariate projections of such data to reduce the computational effort. For this reason, a number of studies have been devoted to determining the directions in which the data must be projected so as to exploit as much information as possible from the distribution. Therefore, for instance, [4] project the data onto the directions that maximize and minimize the kurtosis coefficient of the projection, while [3] chooses the direction that maximizes the fourth cumulant of the projection. Following this stream of literature, our work aims at detecting outliers, represented by financial crises, by projecting the data in the direction that maximizes the cumulant generating function (CGF). In our paper, we refine some theoretical results of [1] and [2]. More precisely, we prove that CGF is a convex function, and, then, we characterize the CGF maximization problem on the unit n-circle as a concave minimization problem. Then, we extend the PCA technique with the CGF maximization procedure for the outlier detection. Finally, we perform an extensive empirical analysis both on simulated and on historical data, and we compare our method with the aforementioned ones, along with a machine learning approach.