Prediction and Inference for Truncated Spatial Data

Abstract

Spatial data in mining, hydrology, and pollution monitoring commonly have a substantial proportion of zeros. One way to model such data is to suppose that some pointwise transformation of the observations follows the law of a truncated Gaussian random field. This article considers Monte Carlo methods for prediction and inference problems based on this model. In particular, a method for computing the conditional distribution of the random field at an unobserved location, given the data, is described. These results are compared to those obtained by simple kriging and indicator cokriging. Simple kriging is shown to give highly misleading results about conditional distributions; indicator cokriging does quite a bit better but still can give answers that are substantially different from the conditional distributions. A slight modification of this basic technique is developed for calculating the likelihood function for such models, which provides a method for computing maximum likelihood estimates of unknown parameters and Bayesian predictive distributions for values of the process at unobserved locations.     

A kriging procedure for processes indexed by graphs

We provide a new kriging procedure of processes on graphs. Based on the construction of Gaussian random processes indexed by graphs, we extend to this framework the usual linear prediction method for spatial random fields, known as kriging. We provide the expression of the estimator of such a random field at unobserved locations as well as a control for the prediction error.     

Some results on first passage times in one dimensional random walks

Analytic expressions are presented for the characteristic function of the first passage time distribution for biased random walk on a finite chain (and diffusion with drift on a finite line); of the first passage time distribution for a random walk on a chain, in which the events (jumps) are governed by an arbitrary renewal process; and of the distribution of the time of escape from a bounded set of points in the latter case. A fundamental relation between the first passage time distribution and the conditional probability for random walk (or diffusion) in one dimension is analyzed and generalized.     

Covariance tapering for prediction of large spatial data sets in transformed random fields

The best linear unbiased predictor (BLUP) is called a kriging predictor and has been widely used to interpolate a spatially correlated random process in scientific areas such as geostatistics. However, if an underlying random field is not Gaussian, the optimality of the BLUP in the mean squared error (MSE) sense is unclear because it is not always identical with the conditional expectation. Moreover, in many cases, data sets in spatial problems are often so large that a kriging predictor is impractically time-consuming. To reduce the computational complexity, covariance tapering has been developed for large spatial data sets. In this paper, we consider covariance tapering in a class of transformed Gaussian models for random fields and show that the BLUP using covariance tapering, the BLUP and the optimal predictor are asymptotically equivalent in the MSE sense if the underlying Gaussian random field has the Matérn covariance function.     

Representations of Gaussian Random Fields and Approximation of Elliptic PDEs with Lognormal Coefficients

Approximation of elliptic PDEs with random diffusion coefficients typically requires a representation of the diffusion field in terms of a sequence \(y=(y_j)_{j\ge 1}\) of scalar random variables. One may then apply high-dimensional approximation methods to the solution map \(y\mapsto u(y)\). Although Karhunen–Loève representations are commonly used, it was recently shown, in the relevant case of lognormal diffusion fields, that multilevel-type expansions may yield better approximation rates. Motivated by these results, we construct wavelet-type representations of stationary Gaussian random fields defined on arbitrary bounded domains. The size and localization properties of these wavelets are studied, and used to obtain polynomial approximation results for the related elliptic PDE which outperform those achievable when using Karhunen–Loève representations. Our construction is based on a periodic extension of the stationary random field, and the expansion on the domain is then obtained by simple restriction. This makes the approach easily applicable even when the computational domain of the PDE has a complicated geometry. In particular, we apply this construction to the class of Gaussian processes defined by the family of Matérn covariances. The proposed periodic continuation technique has other relevant applications such as fast simulation of trajectories. It can be regarded as a continuous analog of circulant embedding techniques introduced for Toeplitz matrices. One of its specific features is that the rate of decay of the eigenvalues of the covariance operator of the periodized process provably matches that of the Fourier transform of the covariance function of the original process.     

On the existence of minimum contrast estimates in binary response model

When random samples are drawn from a 3-parameter distribution with a shifted origin and the observations corresponding to each sample are binary, criteria for the existence of minimum contrast estimates are given. These criteria can be drived by a method, called the probability contents boundary analysis. The probabilities of the existence of maximum likelihood estimates and least squares estimates are evaluated, by simulation with 1000 replications, in the case where the underlying distribution is a 3-parameter lognormal distribution or a 3-parameter loglogistic distribution.     

A Sparse Grid Stochastic Collocation Discontinuous Galerkin Method for Constrained Optimal Control Problem Governed by Random Convection Dominated Diffusion Equations

A sparse grid stochastic collocation method combined with discontinuous Galerkin method is developed for solving convection dominated diffusion optimal control problem with random coefficients. By the optimal control theory, an optimality system is obtained for the problem, which consists of a state equation, a co-state equation and an inequality. Based on finite dimensional noise assumption of random field, the random coefficients are assumed to have finite term expansions depending on a finite number of mutually independent random variables in the probability space. An approximation scheme is established by using a discontinuous Galerkin method for the physical space and a sparse grid stochastic collocation method based on the Smolyak construction for the probability space, which leads to the solution of uncoupled deterministic problems. A priori error estimates are derived for the state, co-state and control variables. Numerical experiments are presented to illustrate the theoretical results.     

具有退化扩散系数It过程的Girsanov定理及其应用

对具有退化扩散系数的It过程,利用扩散系数矩阵的Moore-Penrose广义逆,给出Girsanov定理的一种便于应用的表述形式.应用此结果,给出具有有界随机漂移,退化而确定扩散的金融市场具有无套利机会的判据,此判据方便于应用.     

Moment equations and Hermite expansion for nonlinear stochastic differential equations with application to stock price models

Exact moment equations for nonlinear Itô processes are derived. Taylor expansion of the drift and diffusion coefficients around the first conditional moment gives a hierarchy of coupled moment equations which can be closed by truncation or a Gaussian assumption. The state transition density is expanded into a Hermite orthogonal series with leading Gaussian term and the Fourier coefficients are expressed in terms of the moments. The resulting approximate likelihood is maximized by using a quasi Newton algorithm with BFGS secant updates. A simulation study for the CEV stock price model compares the several approximate likelihood estimators with the Euler approximation and the exact ML estimator (Feller, in Ann Math 54: 173–182, 1951).     

Two-parameter diffusions random fields

This paper presents an alternative method for calculating the diffusion, drift, and mixed coefficients of an example of biparameter Gaussian diffusion defined as a solution of a linear hyperbolic stochastic partial differential equation (Nualart & Sanz , 1979). To derive the expression of these coefficients, we part from an integral stochastic repre , sentation given by these authors for this class of biparameter diffusion processes arising from biparameter Gaussian random fields verifying a particular Markov property     

Long time behavior of heat kernels of operators with unbounded drift terms

Given a second-order elliptic operator on Rd, with bounded diffusion coefficients and unbounded drift, which is the generator of a strongly continuous semigroup on L²(Rd) represented by an integral, we study the time behavior of the integral kernel and prove estimates on its decay at infinity. If the diffusion coefficients are symmetric, a local lower estimate is also proved.     

Bayesian adaptive Lasso

We propose the Bayesian adaptive Lasso (BaLasso) for variable selection and coefficient estimation in linear regression. The BaLasso is adaptive to the signal level by adopting different shrinkage for different coefficients. Furthermore, we provide a model selection machinery for the BaLasso by assessing the posterior conditional mode estimates, motivated by the hierarchical Bayesian interpretation of the Lasso. Our formulation also permits prediction using a model averaging strategy. We discuss other variants of this new approach and provide a unified framework for variable selection using flexible penalties. Empirical evidence of the attractiveness of the method is demonstrated via extensive simulation studies and data analysis.     

Response analysis for a vibroimpact Duffing system with bilateral barriers under external and parametric Gaussian white noises

In this paper, a vibroimpact Duffing oscillator with two barriers that are symmetrical with respect to the equilibrium point of the system is considered for the cases of external and parametric Gaussian white noise random excitations. According to the levels of the system energy, the motions of the unperturbed vibroimpact system are divided into two types: oscillations without impacts and oscillations with alternate impacts on both sides. Then, under the assumption that the vibroimpact Duffing system is quasi-conservative, the stochastic averaging method for energy envelope is applied to obtain the averaged drift and diffusion coefficients for the two types of motions, respectively. The Probability Density Functions (PDFs) of stationary responses are derived by solving the corresponding Fokker-Plank-Kolmogorov (FPK) equation. Lastly, results obtained from the proposed procedure are validated by directly numerical simulation. Meanwhile, effects of the position of bilateral barriers and the random excitations on the PDFs of the stationary responses are also discussed.     

How to estimate the Value at Risk under incomplete information

A key problem in financial and actuarial research, and particularly in the field of risk management, is the choice of models so as to avoid systematic biases in the measurement of risk. An alternative consists of relaxing the assumption that the probability distribution is completely known, leading to interval estimates instead of point estimates. In the present contribution, we show how this is possible for the Value at Risk, by fixing only a small number of parameters of the underlying probability distribution. We start by deriving bounds on tail probabilities, and we show how a conversion leads to bounds for the Value at Risk. It will turn out that with a maximum of three given parameters, the best estimates are always realized in the case of a unimodal random variable for which two moments and the mode are given. It will also be shown that a lognormal model results in estimates for the Value at Risk that are much closer to the upper bound than to the lower bound.     

On the Choice of Nugget in Kriging Modeling for Deterministic Computer Experiments

Kriging is commonly used for developing emulators as surrogates for computationally intensive simulations. One difficulty with kriging is the potential numerical instability in the computation of the inverse of the covariance matrix, which can lead to large variability and poor performance of the kriging predictor. First, we study some causes of ill-conditioning in kriging. We then study the use of nugget in kriging to overcome the numerical instability. Some asymptotic results on its interpolation bias and mean squared prediction errors are presented. Finally, we study the choice of the nugget parameter based on some algebraic lower bounds and use of a regularizing trace. A simulation study is performed to show the differences between kriging with and without nugget and to demonstrate the advantages of the former. This article has supplementary materials online.     

Maximum likelihood estimation for a branching diffusion process

This paper investigates the properties of the maximum likelihood estimators of the drift and diffusion coefficients under three sampling schemes for a branching diffusion process in which the branching process is a linear birth process and the diffusion is in accordance with the Brownian motion with drift.     

Estimates for the density of a nonlinear Landau process

The aim of this paper is to obtain estimates for the density of the law of a specific nonlinear diffusion process at any positive bounded time. This process is issued from kinetic theory and is called Landau process, by analogy with the associated deterministic Fokker-Planck-Landau equation. It is not Markovian, its coefficients are not bounded and the diffusion matrix is degenerate. Nevertheless, the specific form of the diffusion matrix and the nonlinearity imply the non-degeneracy of the Malliavin matrix and then the existence and smoothness of the density. In order to obtain a lower bound for the density, the known results do not apply. However, our approach follows the main idea consisting in discretizing the interval time and developing a recursive method. To this aim, we prove and use refined results on conditional Malliavin calculus. The lower bound implies the positivity of the solution of the Landau equation, and partially answers to an analytical conjecture. We also obtain an upper bound for the density, which again leads to an unusual estimate due to the bad behavior of the coefficients.     

Modelling and forecasting vehicle stocks using the trends of stochastic Gompertz diffusion models: The case of Spain

In the present study, we treat the stochastic homogeneous Gompertz diffusion process (SHGDP) by the approach of the Kolmogorov equation. Firstly, using a transformation in diffusion processes, we show that the probability transition density function of this process has a lognormal time‐dependent distribution, from which the trend and conditional trend functions and the stationary distribution are obtained. Second, the maximum likelihood approach is adapted to the problem of parameters estimation in the drift and the diffusion coefficient using discrete sampling of the process, then the approximated asymptotic confidence intervals of the parameter are obtained. Later, we obtain the corresponding inference of the stochastic homogeneous lognormal diffusion process as limit from the inference of SHGDP when the deceleration factor tends to zero. A statistical methodology, based on the above results, is proposed for trend analysis. Such a methodology is applied to modelling and forecasting vehicle stocks. Finally, an application is given to illustrate the methodology presented using real data, concretely the total vehicle stocks in Spain. Copyright © 2008 John Wiley & Sons, Ltd.     

Estimating Some Characteristics of the Conditional Distribution in Nonparametric Functional Models

This paper deals with a scalar response conditioned by a functional random variable. The main goal is to estimate nonparametrically some characteristics of this conditional distribution. Kernel type estimators for the conditional cumulative distribution function and the successive derivatives of the conditional density are introduced. Asymptotic properties are stated for each of these estimates, and they are applied to the estimations of the conditional mode and conditional quantiles. Our asymptotic results highlightes the importance of the concentration properties on small balls of the probability measure of the underlying functional variable. So, a special section is devoted to show how our results behave in several situations when the functional variable is a continuous time process, with special attention to diffusion processes and Gaussian processes. Even if the main purpose of our paper is theoretical, an application to some chemiometrical data set coming from food industry is presented in a short final section. This example illustrates the easy implementation of our method as well as its good behaviour for finite sample sizes.     

Mathematical foundation of nonequilibrium fluctuation–dissipation theorems for inhomogeneous diffusion processes with unbounded coefficients

Nonequilibrium fluctuation–dissipation theorems (FDTs) are one of the most important advances in stochastic thermodynamics over the past two decades. Here we provide rigorous mathematical proofs of two types of nonequilibrium FDTs for inhomogeneous diffusion processes with unbounded drift and diffusion coefficients by using the Schauder estimates for partial differential equations of parabolic type and the theory of weakly continuous semigroups. The FDTs proved in this paper apply to any forms of inhomogeneous and nonlinear external perturbations. Furthermore, we prove the uniqueness of the conjugate observables and clarify the precise mathematical conditions and ranges of applicability for the two types of FDTs. Examples are also given to illustrate the main results of this paper.