Research

Hamiltonian Monte Carlo in State/Parameter Estimation

Chaos in dynamical systems has long been known to limit forecast predictability in state estimation. The work of Lorenz and the analysis of his 1963 and 1996 models have furthered scientific understanding of Data Assimilation methods in geophysical forecasting. In this short report, I investigated the effect of chaos on parameter estimation in the face of variable external forcing in the Lorenz 96 model. This effectively created a study of parameter fidelity in the face of ‘increasingly chaotic behavior’.

Full state estimation of a dynamical system $\mathbf{x}(t_i)$ for $i = 1, \dots, N$, or $\mathbf{x}_{1:N}$, given a series of observations $\mathbf{d}(t_m) = \mathbf{x}(t_m)$ for $m= 1, \dots, M$, or $\mathbf{d}(t_{1:M})$ is also known as the smoother estimate of the state. This estimate is a description of the posterior distribution $p(\mathbf{x}_{1:N} | \mathbf{d}_{1:M})$. There are many algorithms and sampling procedures for producing the smoother estimate (or approximation when exact calculation is unavailable). In particular, a type of Markov Chain Monte Carlo Method called Hamiltonian (or Hybrid) Monte Carlo, has been formulated for this problem by Alexander et al. (2005). In the paper above, this formulation has been adjusted and reapplied to perform state and parameter estimation on a chaotic Lorenz 96 SDE.

Bayesian Inference for Linear Inverse Models

Linear Inverse Models are popular in atmospheric science studies in order to estimate linear stochastic dynamical models describing the evolution of state variables. Often, the parameters of these models are estimated with Maximum Likelihood approximations made to the transition distribution. In an upcoming paper, I formulate a fully Bayesian scheme to estimate the parameters, and discuss the advantages over simple ML estimation.

Non-parametric Multi-Scale Bayesian Network Modeling

Inspired from “Multiresolution Optimal Interpolation and Statistical Analysis of TOPEXPOSEIDON Satellite Altimetry” by Fieguth et al and “Multiresolution Markov Models for Signal and Image Processing” by Willsky. Multi-Scale models using Bayesian networks is attractive because of the resulting tree structure is easy to perform inference on. These papers use linear functions between nodes, we consider non-parametric models that are fitted between scales.

Estimating Posterior Distribution of Chaotic Dynamics

The document Finite Element Solvers: A MATLAB and FEniCS Review provides a brief introduction and tutorial on solving 2 and 3 dimensional Poisson and Heat Equation problems using software packages for MATLAB and FEniCS (Python).

As an undergraduate, I assisted in research with Professor of Mathematics Ken Golden and Professor of Atmospheric Science Courtney Strong. As a part of his research group, we studied sea ice floes in a multi-scale framework. There, I focused on analyzing the Marginal Ice Zone from a macroscopic viewpoint.