A new solution approach to polynomial LPV system analysis and synthesis

Based on sum-of-squares (SOS) decomposition, we propose a new solution approach for polynomial LPV system analysis and control synthesis problems. Instead of solving matrix variables over a positive definite cone, the SOS approach tries to find a suitable decomposition to verify the positiveness of given polynomials. The complexity of the SOS-based numerical method is polynomial of the problem size. This approach also leads to more accurate solutions to LPV systems than most existing relaxation methods. Several examples have been used to demonstrate benefits of the SOS-based solution approach

Estimating Gram-Charlier Expansions with Positivity Constraints.

The Gram-Charlier expansion, where skewness and kurtosi directly appear as parameters, has become popular in Finance as a generalization of the normal density. We show how positivity constraints can be numerically implemented, thereby guaranteeing that the expansion defines a density. The constrained expansion can be referred to as a Gram-Charlier density. First, we apply our method to the estimation of risk neutral densities. Then, we assess the statistical properties of maximum-likelihood estimates of Gram-Charlier densities. Lastly, we apply the framework to the estimation of a GARCH model where the conditional density is a Gram-Charlier density.Hermite expansions ; Semi-nonparametric estimation ; Risk-neutral density ; GARCH model.

Non-parametric Estimation of Stochastic Differential Equations with Sparse Gaussian Processes

The application of Stochastic Differential Equations (SDEs) to the analysis of temporal data has attracted increasing attention, due to their ability to describe complex dynamics with physically interpretable equations. In this paper, we introduce a non-parametric method for estimating the drift and diffusion terms of SDEs from a densely observed discrete time series. The use of Gaussian processes as priors permits working directly in a function-space view and thus the inference takes place directly in this space. To cope with the computational complexity that requires the use of Gaussian processes, a sparse Gaussian process approximation is provided. This approximation permits the efficient computation of predictions for the drift and diffusion terms by using a distribution over a small subset of pseudo-samples. The proposed method has been validated using both simulated data and real data from economy and paleoclimatology. The application of the method to real data demonstrates its ability to capture the behaviour of complex systems

PRECONDITIONING STRATEGIES FOR 2D FINITE DIFFERENCE MATRIX SEQUENCES

In this paper we are concerned with the spectral analysis of the sequence of preconditioned matrices {P-1 n An(a, m1, m2, k)}n, where n = (n1, n2), N(n) = n1n2 and where An (a, m1, m2, k) 08 \u211dN(n) 7 N(n) is the symmetric two-level matrix coming from a high-order Finite Difference (FD) discretization of the problem (equation presented) with \u3bd denoting the unit outward normal direction and where m1 and m2 are parameters identifying the precision order of the used FD schemes. We assume that the coefficient a(x, y) is nonnegative and that the set of the possible zeros can be represented by a finite collection of curves. The proposed preconditioning matrix sequences correspond to two different choices: the Toeplitz sequence {An(1, m1, m2, k)}n and a Toeplitz based sequence that adds to the Toeplitz structure the informative content given by the suitable scaled diagonal part of An(a, m1, m2 k). The former case gives rise to optimal preconditioning sequences under the assumption of positivity and boundedness of a. With respect to the latter, the main result is the proof of the asymptotic clustering at unity of the eigenvalues of the preconditioned matrices, where the "strength" of the cluster depends on the order k, on the regularity features of a(x, y) and on the presence of zeros of a(x, y)

Estimating Semiparametric Panel Data Models by Marginal Integration

We propose a new methodology for estimating semiparametric panel data models, with a primary focus on the nonparametric component. We eliminate individual effects using first differencing transformation and estimate the unknown function by marginal integration. We extend our methodology to treat panel data models with both individual and time effects. And we characterize the asymptotic behavior of our estimators. Monte Carlo simulations show that our estimator behaves well in finite samples in both random effects and fixed effects settings.Semiparametric Panel Data Model, Partially Linear, First Differencing, Marginal Integration

Optimal Centers’ Allocation in Smoothing or Interpolating with Radial Basis Functions

This work was supported by FEDER/Junta de Andalucía-Consejería de Transformación Económica, Industria, Conocimiento y Universidades (Research Project A-FQM-76-UGR20, University of Granada) and by the Junta de Andalucía (Research Group FQM191).Function interpolation and approximation are classical problems of vital importance in many science/engineering areas and communities. In this paper, we propose a powerful methodology for the optimal placement of centers, when approximating or interpolating a curve or surface to a data set, using a base of functions of radial type. In fact, we chose a radial basis function under tension (RBFT), depending on a positive parameter, that also provides a convenient way to control the behavior of the corresponding interpolation or approximation method. We, therefore, propose a new technique, based on multi-objective genetic algorithms, to optimize both the number of centers of the base of radial functions and their optimal placement. To achieve this goal, we use a methodology based on an appropriate modification of a non-dominated genetic classification algorithm (of type NSGA-II). In our approach, the additional goal of maintaining the number of centers as small as possible was also taken into consideration. The good behavior and efficiency of the algorithm presented were tested using different experimental results, at least for functions of one independent variable.Junta de Andalucía-Consejería de Transformación Econímica, Industria, Conocimiento y Universidades A-FQM-76-UGR20Universidad de GranadaEuropean Regional Development FundJunta de Andalucía FQM19