Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Distributionally Robust Optimization as a Scalable Framework to Characterize Extreme Value Distributions

Patrick Kuiper, Ali Hasan, Wenhao Yang, Yuting Ng, Hoda Bidkhori, Jose Blanchet, Vahid Tarokh

Abstract

The goal of this paper is to develop distributionally robust optimization (DRO) estimators, specifically for multidimensional Extreme Value Theory (EVT) statistics. EVT supports using semi-parametric models called max-stable distributions built from spatial Poisson point processes. While powerful, these models are only asymptotically valid for large samples. However, since extreme data is by definition scarce, the potential for model misspecification error is inherent to these applications, thus DRO estimators are natural. In order to mitigate over-conservative estimates while enhancing out-of-sample performance, we study DRO estimators informed by semi-parametric max-stable constraints in the space of point processes. We study both tractable convex formulations for some problems of interest (e.g. CVaR) and more general neural network based estimators. Both approaches are validated using synthetically generated data, recovering prescribed characteristics, and verifying the efficacy of the proposed techniques. Additionally, the proposed method is applied to a real data set of financial returns for comparison to a previous analysis. We established the proposed model as a novel formulation in the multivariate EVT domain, and innovative with respect to performance when compared to relevant alternate proposals.

Distributionally Robust Optimization as a Scalable Framework to Characterize Extreme Value Distributions

Abstract

The goal of this paper is to develop distributionally robust optimization (DRO) estimators, specifically for multidimensional Extreme Value Theory (EVT) statistics. EVT supports using semi-parametric models called max-stable distributions built from spatial Poisson point processes. While powerful, these models are only asymptotically valid for large samples. However, since extreme data is by definition scarce, the potential for model misspecification error is inherent to these applications, thus DRO estimators are natural. In order to mitigate over-conservative estimates while enhancing out-of-sample performance, we study DRO estimators informed by semi-parametric max-stable constraints in the space of point processes. We study both tractable convex formulations for some problems of interest (e.g. CVaR) and more general neural network based estimators. Both approaches are validated using synthetically generated data, recovering prescribed characteristics, and verifying the efficacy of the proposed techniques. Additionally, the proposed method is applied to a real data set of financial returns for comparison to a previous analysis. We established the proposed model as a novel formulation in the multivariate EVT domain, and innovative with respect to performance when compared to relevant alternate proposals.
Paper Structure (37 sections, 9 theorems, 44 equations, 7 figures, 1 table, 1 algorithm)

This paper contains 37 sections, 9 theorems, 44 equations, 7 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and Problem Formulation
  3. Extreme Value Theory Background
  4. Spectral Representation of Component-wise Maxima and Asymptotic Characterization
  5. Decomposing Max-Stable Random Variables
  6. Wasserstein DRO Formulations for EVDs
  7. Robustification of the Poisson Point Process
  8. Optimizations for Specific Loss Functions
  9. Robustifying the CDF
  10. Probability an Event in a Rare Set Occurs
  11. Conditional Value at Risk
  12. Experiments and Results
  13. Computational Implementations
  14. Experiments with Synthetic EVT Data
  15. Synthetic Datasets
  16. ...and 22 more sections

Key Result

Lemma 2.1

For ${X^{(k)}}$ with standard Frechét marginal distributions, we have: where $\Delta_{d-1}$ represents the unit $d-$dimensional simplex and $H(\cdot)$ satisfies: $\int w_{k} H(dw)=d^{-1}$ for all $k \in \{1, \ldots, d\}$.

Figures (7)

  • Figure 1: Illustration of the error in expected loss with two models under different constraints. Here uncertainty size describes a confidence in the data used for extrapolating EVT distributions, usually quantified by the amount of data available for this process.
  • Figure 2: Two dimensional space for robustificaiton.
  • Figure 3: Visualization of Synthetic datasets.
  • Figure 4: DRO Estimators for MEV Distributions.
  • Figure 5: Comparison of adversarial formulations using $A^{(n)}$ and $V^{(n)}$ parameterization of fixed "true value" CDF. As $\delta$ is increased there is greater model uncertainty.
  • ...and 2 more figures

Theorems & Definitions (21)

  • Lemma 2.1: A corollary of Theorem 1 in dehaan
  • Theorem 3.1: Primal and Dual Form for Point Processes
  • proof
  • Corollary 3.2
  • Theorem 4.1: Formulation for CDF
  • proof
  • Corollary 4.2: Minimizer of CDF
  • proof
  • Theorem 4.3: Formulation for Rare Set
  • proof
  • ...and 11 more