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Fast Empirical Scenarios

Michael Multerer, Paul Schneider, Rohan Sen

TL;DR

Two novel algorithms are proposed that pick important data points from states of the world that have already realized, and are consistent with higher-order sample moment information, and lend themselves to consistent scenario-based modeling and high-dimensional numerical integration.

Abstract

We seek to extract a small number of representative scenarios from large panel data that are consistent with sample moments. Among two novel algorithms, the first identifies scenarios that have not been observed before, and comes with a scenario-based representation of covariance matrices. The second proposal selects important data points from states of the world that have already realized, and are consistent with higher-order sample moment information. Both algorithms are efficient to compute and lend themselves to consistent scenario-based modeling and multi-dimensional numerical integration that can be used for interpretable decision-making under uncertainty. Extensive numerical benchmarking studies and an application in portfolio optimization favor the proposed algorithms.

Fast Empirical Scenarios

TL;DR

Two novel algorithms are proposed that pick important data points from states of the world that have already realized, and are consistent with higher-order sample moment information, and lend themselves to consistent scenario-based modeling and high-dimensional numerical integration.

Abstract

We seek to extract a small number of representative scenarios from large panel data that are consistent with sample moments. Among two novel algorithms, the first identifies scenarios that have not been observed before, and comes with a scenario-based representation of covariance matrices. The second proposal selects important data points from states of the world that have already realized, and are consistent with higher-order sample moment information. Both algorithms are efficient to compute and lend themselves to consistent scenario-based modeling and multi-dimensional numerical integration that can be used for interpretable decision-making under uncertainty. Extensive numerical benchmarking studies and an application in portfolio optimization favor the proposed algorithms.
Paper Structure (36 sections, 7 theorems, 59 equations, 8 figures, 4 algorithms)

This paper contains 36 sections, 7 theorems, 59 equations, 8 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Contributions
  4. Outline
  5. The Truncated Moment Problem
  6. Background
  7. Connection to multivariate quadrature
  8. Covariance scenarios
  9. The empirical moment problem
  10. Problem formulation
  11. Reformulation in RKHS
  12. Algorithms for scenario representation
  13. Step 1: Scenario extraction
  14. Step 2: Retrieval of probability weights
  15. Numerical experiments
  16. ...and 21 more sections

Key Result

Theorem 2.2

Let $\boldsymbol y \in \mathbb{R}^{m_{2q}}$ with $\mathop{\mathrm{rank}}\nolimits \boldsymbol M_{\boldsymbol y} = r$. Then $\boldsymbol y$ has a unique $r$-atomic representing measure on $\mathbb{R}^d$ iff the moment matrix $\boldsymbol M_{\boldsymbol y}$ is positive semi-definite and has a flat ext

Figures (8)

  • Figure 1: The OMP scenarios (red dots) from $10000$ random samples of a bivariate Gaussian mixture distribution with their contour lines. The PDF of the distribution is given in the first tile.
  • Figure 2: Relative error comparison for q=1. The data are generated from Gaussian mixture models according to Section \ref{['sec:gaussianmixture']}. The relative errors (left $y$-axis) are computed according to \ref{['eq:rel_error']}. The $x$-axis shows the dimension $d$ and the right $y$-axis shows the number of clusters entering the mixture distribution.
  • Figure 3: Relative error comparison for q=2. The data are generated from Gaussian mixture models according to Section \ref{['sec:gaussianmixture']}. The relative errors (left $y$-axis) are computed according to \ref{['eq:rel_error']}. The $x$-axis shows the dimension $d$ and the right $y$-axis shows the number of clusters entering the mixture distribution.
  • Figure 4: Number of scenarios comparison for q=1. The data are generated from Gaussian mixture models according to Section \ref{['sec:gaussianmixture']}. The left $y$-axis shows the number of scenarios. The $x$-axis shows the dimension $d$ and the right $y$-axis shows the number of clusters entering the mixture distribution.
  • Figure 5: Number of scenarios comparison for q=2. The data are generated from Gaussian mixture models according to Section \ref{['sec:gaussianmixture']}. The left $y$-axis shows the number of scenarios. The $x$-axis shows the dimension $d$ and the right $y$-axis shows the number of clusters entering the mixture distribution.
  • ...and 3 more figures

Theorems & Definitions (17)

  • Definition 2.1
  • Theorem 2.2: curtofialkow96
  • Theorem 2.3: bayerandteichmann06
  • Remark 2.4
  • Theorem 2.5
  • proof
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • ...and 7 more