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Parametric multi-fidelity Monte Carlo estimation with applications to extremes

Minji Kim, Brendan Brown, Vladas Pipiras

Abstract

In a multi-fidelity setting, data are available from two sources, high- and low-fidelity. Low-fidelity data has larger size and can be leveraged to make more efficient inference about quantities of interest, e.g. the mean, for high-fidelity variables. In this work, such multi-fidelity setting is studied when the goal is to fit more efficiently a parametric model to high-fidelity data. Three multi-fidelity parameter estimation methods are considered, joint maximum likelihood, (multi-fidelity) moment estimation and (multi-fidelity) marginal maximum likelihood, and are illustrated on several parametric models, with the focus on parametric families used in extreme value analysis. An application is also provided concerning quantification of occurrences of extreme ship motions generated by two computer codes of varying fidelity.

Parametric multi-fidelity Monte Carlo estimation with applications to extremes

Abstract

In a multi-fidelity setting, data are available from two sources, high- and low-fidelity. Low-fidelity data has larger size and can be leveraged to make more efficient inference about quantities of interest, e.g. the mean, for high-fidelity variables. In this work, such multi-fidelity setting is studied when the goal is to fit more efficiently a parametric model to high-fidelity data. Three multi-fidelity parameter estimation methods are considered, joint maximum likelihood, (multi-fidelity) moment estimation and (multi-fidelity) marginal maximum likelihood, and are illustrated on several parametric models, with the focus on parametric families used in extreme value analysis. An application is also provided concerning quantification of occurrences of extreme ship motions generated by two computer codes of varying fidelity.

Paper Structure

This paper contains 26 sections, 127 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Multi-fidelity setting
  3. Goals and contributions of this work
  4. Related work
  5. Preliminaries
  6. Parametric MFMC estimation methods
  7. Baselines
  8. Joint maximum likelihood estimation
  9. Moment multi-fidelity estimation
  10. Marginal maximum likelihood multi-fidelity estimation
  11. Examples and numerical illustrations
  12. Bivariate Gaussian distribution
  13. Bivariate Gumbel distribution
  14. Bivariate distribution of binary outcomes
  15. Further discussion and results
  16. ...and 11 more sections

Figures (6)

  • Figure 1: Asymptotic variances of JML (red, circle), MML (green, triangle), MoM (blue, square), and baseline (dashed) estimators for $\mu_1$ and $\sigma_1$ across dependence parameter values $r$ for the bivariate Gumbel distribution.
  • Figure 2: Asymptotic variances of the MF (solid) and baseline (dashed) estimators for $p_1$ for Bernoulli random variables across different dependence levels and copula functions.
  • Figure 3: Left: Heave motion over time from LAMP and SC. Middle/Right: Normal quantile-quantile (QQ) plot for heave motion from LAMP/SC.
  • Figure 4: Left: Scatterplot of record maxima from LAMP and SC. The dashed line is the $45^\circ$ line. Middle: Histogram and estimated p.d.f. of SC heave record maxima based on 10,000 observations. Right: QQ plot for SC heave record maxima compared with the estimated Gumbel distribution.
  • Figure 5: Parametric baseline and MF methods for estimating location (top, $\mu_1$) and scale (bottom, $\sigma_1$) parameters of the Gumbel distribution fitted to heave motion data. The results show baseline and MF estimators along with their confidence intervals.
  • ...and 1 more figures

Theorems & Definitions (1)

  • proof