We give an alternative and unified derivation of the general framework developed in the last few years for analyzing nonstationary time series. A different approach for handling the resulting variational problem numerically is introduced. We further expand the framework by employing adaptive finite element algorithms and ideas from information theory to solve the problem of finding the most adequate model based on a maximum-entropy ansatz, thereby reducing the number of underlying probabilistic assumptions. In addition, we formulate and prove the result establishing the link between the optimal parametrizations of the direct and the inverse problems and compare the introduced algorithm to standard approaches like Gaussian mixture models, hidden Markov models, artificial neural networks and local kernel methods. Furthermore, based on the introduced general framework, we show how to create new data analysis methods for specific practical applications. We demonstrate the application of the framework to data samples from toy models as well as to real-world problems such as biomolecular dynamics, DNA sequence analysis and financial applications.

Dynamical systems with different characteristic behavior at multiple scales can be modeled with hybrid methods combining a discrete model (e.g., corresponding to the microscale) triggered by a continuous mechanism and vice versa. A data-driven black-box-type framework is proposed, where the discrete model is parametrized with adaptive regression techniques and the output of the continuous counterpart (e.g., output of partial differential equations) is coupled to the discrete system of interest in the form of a fixed exogenous time series of external factors. Data availability represents a significant issue for this type of coupled discrete-continuous model, and it is shown that missing information/observations can be incorporated in the model via a nonstationary and nonhomogeneous formulation. An unbiased estimator for the discrete model dynamics in presence of unobserved external impacts is derived and used to construct a data-based nonstationary and nonhomogeneous parameter estimator based on an appropriately regularized spatiotemporal clustering algorithm. One-step and long-term predictions are considered, and a new Bayesian approach to discrete data assimilation of hidden information is proposed. To illustrate our method, we apply it to synthetic data sets and compare it with standard techniques of the machine-learning community (such as maximumlikelihood estimation, artificial neural networks and support vector machines).

Abstract. This paper presents an application of the recently developed method for simultaneous dimension reduction and metastability analysis of high-dimensional time series in the context of computational finance. Further extensions are included to combine state-specific principal component analysis (PCA) and state-specific regressive trend models to handle the high-dimensional, nonstationary data. The identification of market phases allows one to control the involved phase-specific risk for futures portfolios. The numerical optimization strategy for futures portfolios based on Tikhonovtype regularization is presented. The application of proposed strategies to online detection of the market phases is exemplified first on the simulated data and then on historical futures prices for oil and wheat from [2005][2006][2007][2008] 1. Introduction. In today's financial industry, computational finance is used for a wide field of applications. Throughout this paper, we will concentrate on the portfolio optimization problem, where, for return maximization and risk minimization problems, the evolution of security returns is estimated with a variety of different methods, in order to find an allocation that optimizes a suitable performance criterion, while limiting the risk [27,31]. For practical reasons, computational methods have to address the following points:1. Time series analysis. Market dynamics are most likely not reflected by a stationary process, but involve (hidden) market phases, such as economic cycles, harvest cycles (for grain, fruits, and other kinds of plants), and yearly holidays, and many more aspects influence the markets. This poses fundamental problems for estimation: To obtain sensible estimates for mean returns even in stationary linear models, hundreds of years of data would be needed. In shorter intervals, estimation risk becomes significant.