Latent state estimation in a class of nonlinear systems

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.The problem of estimating latent or unobserved states of a dynamical system from observed data is studied in this thesis. Approximate filtering methods for discrete time series for a class of nonlinear systems are considered, which, in turn, require sampling from a partially specified discrete distribution. A new algorithm is proposed to sample from partially specified discrete distribution, where the specification is in terms of the first few moments of the distribution. This algorithm generates deterministic sigma points and corresponding probability weights, which match exactly a specified mean vector, a specified covariance matrix, the average of specified marginal skewness and the average of specified marginal kurtosis. Both the deterministic particles and the probability weights are given in closed form and no numerical optimization is required. This algorithm is then used in approximate Bayesian filtering for generation of particles and the associated probability weights which propagate higher order moment information about latent states. This method is extended to generate random sigma points (or particles) and corresponding probability weights that match the same moments. The algorithm is also shown to be useful in scenario generation for financial optimization. For a variety of important distributions, the proposed moment-matching algorithm for generating particles is shown to lead to approximation which is very close to maximum entropy approximation. In a separate, but related contribution to the field of nonlinear state estimation, a closed-form linear minimum variance filter is derived for the systems with stochastic parameter uncertainties. The expressions for eigenvalues of the perturbed filter are derived for comparison with eigenvalues of the unperturbed Kalman filter. Moment-matching approximation is proposed for the nonlinear systems with multiplicative stochastic noise

Analyzing and Quantifying the Intrinsic Distributional Robustness of CVaR Reformulation for Chance Constrained Stochastic Programs

Chance constrained program (CCP) is a popular stochastic optimization method in power system planning and operation problems. Conditional Value-at-Risk (CVaR) provides a convex approximation for chance constraints which are nonconvex. Although CCP assumes an exact empirical distribution, and the optimum of a stochastic programming model is thought to be sensitive in the designated probability distribution, this letter discloses that CVaR reformulation of chance constraint is intrinsically robust. A pair of indices are proposed to quantify the maximum tolerable perturbation of the probability distribution, and can be computed from a computationally-cheap dichotomy search. An example on the coordinated capacity optimization of energy storage and transmission line for a remote wind farm validates the main claims. The above results demonstrate that stochastic optimization methods are not necessarily vulnerable to distributional uncertainty, and justify the positive effect of the conservatism brought by the CVaR reformulation.©2020 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.fi=vertaisarvioitu|en=peerReviewed

Distributionally Robust Optimization: A Review

The concepts of risk-aversion, chance-constrained optimization, and robust optimization have developed significantly over the last decade. Statistical learning community has also witnessed a rapid theoretical and applied growth by relying on these concepts. A modeling framework, called distributionally robust optimization (DRO), has recently received significant attention in both the operations research and statistical learning communities. This paper surveys main concepts and contributions to DRO, and its relationships with robust optimization, risk-aversion, chance-constrained optimization, and function regularization

Bayesian inference for hedge funds with stable distribution of returns

Recently, a body of academic literature has focused on the area of stable distributions and their application potential for improving our understanding of the risk of hedge funds. At the same time, research has sprung up that applies standard Bayesian methods to hedge fund evaluation. Little or no academic attention has been paid to the combination of these two topics. In this paper, we consider Bayesian inference for alpha-stable distributions with particular regard to hedge fund performance and risk assessment. After constructing Bayesian estimators for alpha-stable distributions in the context of an ARMA-GARCH time series model with stable innovations, we compare our risk evaluation and prediction results to the predictions of several competing conditional and unconditional models that are estimated in both the frequentist and Bayesian setting. We find that the conditional Bayesian model with stable innovations has superior risk prediction capabilities compared with other approaches and, in particular, produced better risk forecasts of the abnormally large losses that some hedge funds sustained in the months of September and October 2008. --

Data-Driven Chance Constrained Programs over Wasserstein Balls

We provide an exact deterministic reformulation for data-driven chance constrained programs over Wasserstein balls. For individual chance constraints as well as joint chance constraints with right-hand side uncertainty, our reformulation amounts to a mixed-integer conic program. In the special case of a Wasserstein ball with the $1$-norm or the $\infty$-norm, the cone is the nonnegative orthant, and the chance constrained program can be reformulated as a mixed-integer linear program. Our reformulation compares favourably to several state-of-the-art data-driven optimization schemes in our numerical experiments.Comment: 25 pages, 9 figure

Evaluating dynamic covariance matrix forecasting and portfolio optimization

In this thesis we have evaluated the covariance forecasting ability of the simple moving average, the exponential moving average and the dynamic conditional correlation models. Overall we found that a dynamic portfolio can gain significant improvements by implementing a multivariate GARCH forecast. We further divided the global investment universe into sectors and regions in order to investigate the relative portfolio performance of several asset allocation strategies with both variance and conditional value at risk as a risk measure. We found that the choice of risk measure does not seem to heavily impact the asset allocation. As comparison to the dynamic portfolios we added regional/sector portfolios which where rebalanced after a 3% threshold rule. The regional portfolio was constructed to mimic the current strategy of the Norwegian Pension Fund Global. The max Sharpe portfolio for regions had the highest risk adjusted return, but suffered from a very high turnover. After being modified however, this strategy turned out to be superior even after transaction costs were imposed