Maximum Entropy Kernels for System Identification

A new nonparametric approach for system identification has been recently proposed where the impulse response is modeled as the realization of a zero-mean Gaussian process whose covariance (kernel) has to be estimated from data. In this scheme, quality of the estimates crucially depends on the parametrization of the covariance of the Gaussian process. A family of kernels that have been shown to be particularly effective in the system identification framework is the family of Diagonal/Correlated (DC) kernels. Maximum entropy properties of a related family of kernels, the Tuned/Correlated (TC) kernels, have been recently pointed out in the literature. In this paper we show that maximum entropy properties indeed extend to the whole family of DC kernels. The maximum entropy interpretation can be exploited in conjunction with results on matrix completion problems in the graphical models literature to shed light on the structure of the DC kernel. In particular, we prove that the DC kernel admits a closed-form factorization, inverse and determinant. These results can be exploited both to improve the numerical stability and to reduce the computational complexity associated with the computation of the DC estimator.Comment: Extends results of 2014 IEEE MSC Conference Proceedings (arXiv:1406.5706

Smooth parametric surfaces and n-sided patches

The theory of 'geometric continuity' within the subject of CAGD is reviewed. In particular, we are concerned with how parametric surface patches for CAGD can be pieced together to form a smooth Ck surface. The theory is applied to the problem of filling an n-sided hole occurring within a smooth rectangular patch complex. A number of solutions to this problem are surveyed

Numerical simulations of fuel droplet flows using a Lagrangian triangular mesh

The incompressible, Lagrangian, triangular grid code, SPLISH, was converted for the study of flows in and around fuel droplets. This involved developing, testing and incorporating algorithms for surface tension and viscosity. The major features of the Lagrangian method and the algorithms are described. Benchmarks of the algorithms are given. Several calculations are presented for kerosene droplets in air. Finally, extensions which make the code compressible and three dimensional are discussed

The dynamics of economics functions: modelling and forecasting the yield curve

The class of Functional Signal plus Noise (FSN) models is introduced that provides a new, general method for modelling and forecasting time series of economic functions. The underlying, continuous economic function (or "signal") is a natural cubic spline whose dynamic evolution is driven by a cointegrated vector autoregression for the ordinates (or "y-values") at the knots of the spline. The natural cubic spline provides flexible cross-sectional fit and results in a linear, state space model. This FSN model achieves dimension reduction, provides a coherent description of the observed yield curve and its dynamics as the cross-sectional dimension N becomes large, and can feasibly be estimated and used for forecasting when N is large. The integration and cointegration properties of the model are derived. The FSN models are then applied to forecasting 36-dimensional yield curves for US Treasury bonds at the one month ahead horizon. The method consistently outperforms the Diebold and Li (2006) and random walk forecasts on the basis of both mean square forecast error criteria and economically relevant loss functions derived from the realised profits of pairs trading algorithms. The analysis also highlights in a concrete setting the dangers of attempts to infer the relative economic value of model forecasts on the basis of their associated mean square forecast errors.Time-series analysis ; Forecasting ; Mathematical models ; Macroeconomics - Econometric models

The Falling Factorial Basis and Its Statistical Applications

We study a novel spline-like basis, which we name the "falling factorial basis", bearing many similarities to the classic truncated power basis. The advantage of the falling factorial basis is that it enables rapid, linear-time computations in basis matrix multiplication and basis matrix inversion. The falling factorial functions are not actually splines, but are close enough to splines that they provably retain some of the favorable properties of the latter functions. We examine their application in two problems: trend filtering over arbitrary input points, and a higher-order variant of the two-sample Kolmogorov-Smirnov test.Comment: Full version for the ICML paper with the same titl

Monotonicity preserving approximation of multivariate scattered data

This paper describes a new method of monotone interpolation and smoothing of multivariate scattered data. It is based on the assumption that the function to be approximated is Lipschitz continuous. The method provides the optimal approximation in the worst case scenario and tight error bounds. Smoothing of noisy data subject to monotonicity constraints is converted into a quadratic programming problem. Estimation of the unknown Lipschitz constant from the data by sample splitting and cross-validation is described. Extension of the method for locally Lipschitz functions is presented.<br /