Nan-Jung Hsu

Spatial Lasso With Applications to GIS Model Selection

Geographic information systems (GIS) organize spatial data in multiple two-dimensional arrays called layers. In many applications, a response of interest is observed on a set of sites in the landscape, and it is of interest to build a regression model from the GIS layers to predict the response at unsampled sites. Model selection in this context then consists not only of selecting appropriate layers, but also of choosing appropriate neighborhoods within those layers. We formalize this problem as a linear model and propose the use of Lasso to simultaneously select variables, choose neighborhoods, and estimate parameters. Spatially dependent errors are accounted for using generalized least squares and spatial smoothness in selected coefficients is incorporated through use of a priori spatial covariance structure. This leads to a modification of the Lasso procedure, called spatial Lasso. The spatial Lasso can be implemented by a fast algorithm and it performs well in numerical examples, including an application to prediction of soil moisture. The methodology is also extended to generalized linear models. Supplemental materials including R computer code and data analyzed in this article are available online.

A class of nearly long-memory time series models

Abstract We consider an autoregressive regime-switching model for the dynamic mean structure of a univariate time series. The model allows for a variety of stationary and nonstationary alternatives, and includes the possibility of approximate long-memory behavior. The proposed model includes as special cases white noise, first-order autoregression, and random walk models as well as regime-switching models and the random level-shift model proposed by Chen and Tiao, Journal of Business and Economic Statistics, 8 (1990) p. 83. We describe properties of the model, focusing on its resemblance to long-memory under a certain asymptotic parameterization. We develop a reversible-jump Markov chain Monte Carlo method for Bayesian inference on unknown model parameters and apply the methodology to the Nile River data.

Semiparametric Estimation and Selection for Nonstationary Spatial Covariance Functions

We propose a method for estimating nonstationary spatial covariance functions by representing a spatial process as a linear combination of some local basis functions with uncorrelated random coefficients and some stationary processes, based on spatial data sampled in space with repeated measurements. By incorporating a large collection of local basis functions with various scales at various locations and stationary processes with various degrees of smoothness, the model is flexible enough to represent a wide variety of nonstationary spatial features. The covariance estimation and model selection are formulated as a regression problem with the sample covariances as the response and the covariances corresponding to the local basis functions and the stationary processes as the predictors. A constrained least squares approach is applied to select appropriate basis functions and stationary processes as well as estimate parameters simultaneously. In addition, a constrained generalized least squares approach is proposed to further account for the dependencies among the response variables. A simulation experiment shows that our method performs well in both covariance function estimation and spatial prediction. The methodology is applied to a U.S. precipitation dataset for illustration. Supplemental materials relating to the application are available online.

Pile-up probabilities for the Laplace likelihood estimator of a non-invertible first order moving average

The flrst-order moving average model or MA(1) is given by Xt = Zt i µ0Zti1, with independent and identically distributed fZtg. This is ar- guably the simplest time series model that one can write down. The MA(1) with unit root (µ0 = 1) arises naturally in a variety of time series applications. For example, if an underlying time series consists of a linear trend plus white noise errors, then the difierenced series is an MA(1) with unit root. In such cases, testing for a unit root of the difierenced series is equivalent to testing the adequacy of the trend plus noise model. The unit root problem also arises naturally in a signal plus noise model in which the signal is modeled as a ran- dom walk. The difierenced series follows a MA(1) model and has a unit root if and only if the random walk signal is in fact a constant. The asymptotic theory of various estimators based on Gaussian likeli- hood has been developed for the unit root case and nearly unit root case (µ = 1+fl=n;fl • 0). Unlike standard 1= p n-asymptotics, these estimation pro- cedures have 1=n-asymptotics and a so-called pile-up efiect, in which P(^ = 1) converges to a positive value. One explanation for this pile-up phenomenon is the lack of identiflability of µ in the Gaussian case. That is, the Gaussian likelihood has the same value for the two sets of parameter values (µ;ae2) and (1=µ;µ 2 ae 2 ). It follows that µ = 1 is always a critical point of the likelihood function. In contrast, for non-Gaussian noise, µ is identiflable for all real values. Hence it is no longer clear whether or not the same pile-up phenomenon will persist in the non-Gaussian case. In this paper, we focus on limiting pile-up probabilities for estimates of µ0 based on a Laplace likelihood. In some cases, these estimates can be viewed as Least Absolute Deviation (LAD) estimates. Simulation results illustrate the limit theory.

Semiparametric Mixed Models for Increment-Averaged Data With Application to Carbon Sequestration in Agricultural Soils

Adoption of conservation tillage practice in agriculture offers the potential to mitigate greenhouse gas emissions. Studies comparing conservation tillage methods to traditional tillage pair fields under the two management systems and obtain soil core samples from each treatment. Cores are divided into multiple increments, and matching increments from one or more cores are aggregated and analyzed for carbon stock. These data represent not the actual value at a specific depth, but rather the total or average over a depth increment. A semiparametric mixed model is developed for such increment-averaged data. The model uses parametric fixed effects to represent covariate effects, random effects to capture correlation within studies, and an integrated smooth function to describe effects of depth. The depth function is specified as an additive model, estimated with penalized splines using standard mixed model software. Smoothing parameters are automatically selected using restricted maximum likelihood. The methodology is applied to the problem of estimating a change in carbon stock due to a change in tillage practice.

A diagnostic test for autocorrelation in increment-averaged data with application to soil sampling

Motivated by the problem of detecting spatial autocorrelation in increment- averaged data from soil core samples, we use the Cholesky decomposition of the inverse of an autocovariance matrix to derive a parametric linear regression model for autocovariances. In the absence of autocorrelation, the off-diagonal terms in the lower triangular matrix from the Cholesky decomposition should be identically zero, and so the regression coefficients should be identically zero. The standard F-test of this hypothesis and two bootstrapped versions of the test are evaluated as autocorrelation diagnostics via simulation. Size is assessed for a variety of heteroskedastic null hypotheses. Power is evaluated against autocorrelated alternatives, including increment-averaged Ornstein-Uhlenbeck and Matérn processes. The bootstrapped tests maintain approximately the correct size and have good power against moderately autocorrelated alternatives. The methods are applied to data from a study of carbon sequestration in agricultural soils.

LONG-RANGE DEPENDENT COMMON FACTOR MODELS: A BAYESIAN APPROACH

We propose a simulation-based Bayesian approach to analyze multivariate time series with possible common long-range dependent factors. A state-space approach is used to represent the likelihood function in a tractable manner. The approach taken here allows for extension to fit a non-Gaussian multivariate stochastic volatility (MVSV) model with common long-range dependent components. The method is illustrated for a set of stock returns for companies having similar annual sales.