Spatial Modeling, Applied Stochastics and Paleoclimatology

Speaker(s)

• Nathan Glatt-Holtz (Indiana University)
• Mickael Chekroun (University of California, Los Angeles)
• Yiannis Kamarianakis (IBM)
• Omar De la Cruz Cabrera (Case Western Reserve University)
• Zhu Wang (Virginia Tech)
• Chia Ying Lee (SAMSI)
• Murali Haran (Pennsylvania State University)
• Elizabeth Mannshardt (North Carolina State University)
• Peter Craigmile (Ohio State University)
• Bala Rajaratnum (Stanford University)
• Martin Tingley (Harvard University)
• Luis Barboza (Purdue University)

Description

Environmental science, climatology, and other emerging fields of great societal importance, have in common that they require sound mathematical and statistical tools, in order to properly draw inference for the physical process and accordingly quantify the uncertainty. In this session, a variety of new techniques currently being developed in spatial data modeling and stochastic analysis will be presented, and novel case studies in climatology will be illustrated. The presentations will include probabilists,statisticians, and applied mathematicians, who use the tools of their trade to address practical questions in atmospheric and climate science. Our session will foster discussions on how best to integrate these tools for specific topics including paleoclimate reconstruction, spatial downscaling, and sound stochastic modeling and inference in atmospheric science. 

Schedule

Sat, June 23 - Location: STEW 202

Time	Speaker	Title
8:30 - 9:55	Nathan Glatt-Holtz	Well-Posedness Results for the Stochastic Primitive Equations of the Ocean and Atmosphere
	Abstract: The primitive equations are widely regarded as a fundamental description of geophysical scale fluid flow and forms the core of the most advanced numerical general circulation models (GCMs). This system may be derived from the compressible Navier- Stokes equations with a combination of empirical observation and scale analysis. In view of the wide progress made in computation the need has appeared to better understand and model some of the uncertainties which are contained in these GCMs. In this context stochastic modeling has appeared as one of the major modes in the contemporary evolution of the field. While the mathematical theory for the deterministic primitive equations is now on a firm ground it seems that very little has been done so far on it stochastic counterpart. For this and other nonlinear SPDE's the issue of compactness remains a challenging problem especially for the case of nonlinear multiplicative noise. In this talk we discuss some recent work on the global existence and uniqueness of the primitive equations in both 2 and 3 spatial dimensions. This is joint work with A. Debussche, R. Temam, and M. Ziane.
	Mickael Chekroun	Nonlinear Stochastic Inverse Models with Memory, and Prediction of Climatic Phenomena
	Abstract: Any model-reduction strategy of climatic phenomena — theoretical or data-driven — faces the challenge of the so-called closure problem from (geophysical) turbulence theory. Closure requires an effects of the small scales on the large ones in order to resolve, in a suitably defined, averaged sense, the large-scale dynamics without having to resolve the evolution of all the variables. Recent theoretical works on stochastic Navier-Stokes equations and previous approaches coming from Statistical Mechanics such as the Mori-Zwanzig (MZ) formalism, have advocated the idea of representing the higher (unresolved) modes as functionals of the time history of the low (resolved) modes to deal with the closure problem. In the MZ formalism, these functional dependences arise typically in complicated integral terms obtained by repeated convolutions between decaying memory kernels and the resolved variables. In the case of a lack of scale separation, these memory kernels roughly decay at the same rate than the decorrelation rate of the solution itself; which renders challenging the numerical computations of these integral terms and thus the obtention of an efficient solution to the closure problem via this approach. By considering a specific class of memory kernels within the MZ formalism, it will be presented a numerically tractable data-driven approach to deal with this problem while allowing the cases where the separation of scales is not necessarily pronounced. The approach will be illustrated for the inverse modeling of two major tropical climatic phenomena: the El Nino-Southern Oscillation (ENSO) and the Madden-Julian Oscillation (MJO). Prediction capabilities of the resulting nonlinear stochastic inverse models will be discussed in each case. The "past noise forecasting method" recently developed (Chekroun et al., PNAS, 108 (29), 2011) will be then presented. It will be shown how this method allows to extend the prediction skill of our MJO and ENSO models by using one hand, information from the estimated path on which the inverse stochastic model lives; and on the other, dynamical features associated with the low frequency variability captured by these models. This talk is based on joint works with Dmitri Kondrashov (UCLA), Michael Ghil (UCLA) and Andrew Robertson (Columbia University).
	Yiannis Kamarianakis	Real-time road traffic forecasting using regime-switching space-time models and adaptive lasso
	Abstract: Smart transportation technologies require real-time traffic prediction to be both fast and scalable to full urban networks. We discuss a method that is able to meet this challenge while accounting for nonlinear traffic dynamics and space-time dependencies of traffic variables. Nonlinearity is taken into account by a union of non-overlapping linear regimes characterized by a sequence of temporal thresholds. In each regime, for each measurement location, a penalized estimation scheme, namely adaptive lasso, is implemented to perform both model selection and coefficient estimation. Both the robust to outliers least absolute deviation estimates and conventional lasso estimates are considered. The methodology is illustrated on five-minute average speed data from three road networks.
10:00-10:30AM	Break
10:30 - 11:55AM	Omar De la Cruz C.	Analysis of spatio-temporal variation in diversity using diffusion processes
	Abstract: We consider the use of several stochastic processes to model the spatial and temporal variation in the relative abundances of species, and the corresponding variation in measures of diversity. The processes are diffusions, in one or more dimensions, with values in either a finite- or infinite-dimensional space. Different correlation structures can be specified. Applications potentially include the study of diversity changes in extant ecosystems or those that have left fossil records (at the corresponding time scales). Another application is in the field of metagenomics, the study of communities of microorganisms using genomic methods.
	Zhu Wang	Numerical Methods for Stochastic Quasi-Geostrophic Equations
	Abstract: With the continuous increase in computational power, complex mathematical models are becoming more and more popular in the numerical simulation of oceanic and atmospheric flows. For some geophysical flows in which computational efficiency is of paramount importance, however, simplified mathematical models are central. For example, in climate modeling the Quasi-Geostrophic Equations (QGE), are commonly used in the numerical simulation of large scale wind-driven ocean circulations. Due to the requisite of long time integration in climate modeling, even for the simplified model, fast and accurate numerical algorithms are still desired. In this talk, we will pursue this direction and discuss efficient numerical approaches for QGE when uncertainty effects such as the random wind forcing are considered.
	Chia Ying Lee	A Malliavin calculus approach to random perturbations in SPDEs
	Abstract: We discuss a Malliavin calculus approach to modelling random perturbations in linear elliptic and parabolic SPDEs and the stochastic Navier-Stokes equation, based on Malliavin divergence operators and the Wick product. This approach interprets products as stochastic convolutions and is equivalent to the Ito formulation of SDEs, and furthermore has nice properties that make the random perturbation unbiased, even with the nonlinearity in the stochastic Navier-Stokes equation. Additionally, the Wiener chaos expansion and the availability of an adjoint operator, namely the Malliavin derivative, facilitate the analysis of solutions and numerical error estimates. We will see that the Wiener chaos expansion provides an efficient algorithm for numerical simulation, due to the lower triangular structure of the associated PDE system, and then compare numerical simulations of our stochastic model of the Navier-Stokes equations with the standard stochastic Navier-Stokes model.
12:00-1:30PM	Lunch
1:30 - 2:55	Murali Haran	Dimension Reduction and Alleviation of Spatial Confounding for Spatial Generalized Linear Mixed Models
	Abstract: Non-Gaussian spatial data are very common in many disciplines. For instance, count data are common in disease mapping and binary data are common in ecology problems. When fitting spatial regressions for such data one needs to account for dependence appropriately both for reliable inference regarding the regression coefficients as well as for accurate predictions. Spatial generalized linear mixed models (SGLMMs) are very popular and flexible approaches for modeling such data but they suffer from two major shortcomings: (i) the regression coefficient estimates obtained in SGLMMs are often uninterpretable due to confounding with the spatial random effects, (ii) the number of spatial random effects grows as the data set increases in size, making fully Bayesian inference for such models computationally infeasible in many cases. We propose a new sparse reparameterization of the SGLMM that simultaneously addresses both these issues by greatly reducing the dimension of the spatial random effects and providing more interpretable regression parameter estimates. We illustrate the application of our approach to both simulated and real data sets. This is joint work with John Hughes, University of Minnesota.
	Elizabeth Mannshardt	Statistical modeling of extreme value behavior in North American tree-ring density series
	Abstract: Many analyses of the paleoclimate record include conclusions about extremes, with a focus on the unprecedented nature of recent climate events. While the use of extreme value theory is becoming common in the analysis of the instrumental climate record,applications of this framework to the spatio-temporal analysis of paleoclimate records remain limited. This article develops a Bayesian hierarchical model to investigate spatially varying trends and dependencies in the parameters characterizing the extremes of a proxy data set, and applies it to the site-wise decadal maxima and minima of a gridded network of temperature sensitive tree ring density time series over northern North America. The statistical analysis reveals significant spatial associations in the temporal trends of the location parameters of the generalized extreme value distributions: maxima are increasing as a function of time, with stronger increases in the north and east of North America; minima are significantly increasing in the west, possibly decreasing in the east, and exhibit no changes in the center of the region. To the extent that the extremal behavior of the tree ring densities reflects extremal behavior in surface temperatures, results indicate that the distribution which describes temperature extremes varies as a function of both space and time, with temperature maxima becoming more extreme as a function of time and temperature minima having diverging spatial patterns. Results of this proxy-only analysis are a first step towards directly reconstructing extremal climate behavior, as opposed to mean climate behavior, by linking extremes in the proxy record to extremes in the instrumental record.
	Peter Craigmile	Regional climate model assessment using statistical upscaling and downscaling techniques
	Abstract: Climate models are mathematical models that describe the temporal evolution of climate, oceans, atmosphere, ice and land-use processes, across a spatial domain via systems of partial differential equations. Because these models cannot be solved analytically, the model output is generated numerically over grid boxes. Regional climate models (RCMs), or the dynamic downscaling of global climate models to regional scales, are often used for planning purposes, and it is important to assess carefully the uncertainty of such models. We evaluate the Swedish Meteorological and Hydrological Institute (SHMI) regional climate model by comparing its model output at the grid box level, with the predictions obtained from two observation-driven spatio-temporal statistical models. The ''downscaling model combines the spatially and temporally smoothed climate model output with temperature observations at synoptic stations in a spatio-temporal linear statistical model. The ''upscaling model describes the observational temperature alone at the daily scale, via a spatio-temporal model that includes a wavelet-based trend, spatially varying seasonality, along with volatility and long range dependence terms. Both statistical models have the ability to make predictions at a seasonal scale, both at point and grid box level. In the years 1962—2007 in South Central Sweden, we show that the climate model performs well in predicting the annual and seasonal average temperature at three reserved stations, but there are interesting differences among the model output and the statistical model-based predictions at the grid box level. This research is joint with Veronica Berrocal at The University of Michigan, and Peter Guttorp at the University of Washington and the Norwegian Computing Center.
3:00-3:30PM	Break
3:30 - 4:55	Bala Rajaratnum	A Markov Random Fields based approach to Multiproxy Paleoclimate Reconstructions
	Abstract: Various climate field reconstructions (CFR) methods have been proposed to infer past temperature from (paleoclimate) multiproxy networks, most notably the regularized EM algorithm (RegEM). We propose a new CFR method, called GraphEM, based on Gaussian graphical models (GGM)/ Gaussian Markov random fields. GGMs provide a flexible framework for modeling the inherent spatial heterogeneities of high-dimensional spatial fields and at the same time provide the parameter reduction necessary for obtaining precise and well-conditioned estimates of the covariance structure of the field, even when the sample size is much smaller than the number of variables. GraphEM discovers the graphical structure of the field via $\ell_1$-penalization methods and can subsequently be used to reconstruct past climate variations. The performance of GraphEM is then compared to published CFR methods using pseudoproxy experiments. Our results show that GraphEM can yield significant improvements over existing methods, with gains uniformly over space. The GraphEM algorithm is also useful for regional paleoclimate reconstructions, and can yield better uncertainty quantification. We demonstrate that the increase in performance is directly related to recovering the underlying sparsity in the covariance of the spatial field. We also provide compelling evidence that GraphEM performs well even at spatial locations with few proxies. This is joint work with D.Guillot and J. Emile-Geay.
	Martin Tingley	Arctic temperature extremes over the last 600 years
	Abstract: Evaluation of recent extremes in high northern latitude temperatures is better undertaken in the longer term context afforded by the paleoclimate record. Determining the probability that an event like the 2010 Russian heat wave is unprecedented requires a statistical treatment that permits for the imputation of temperatures in space, accounts for uncertainties in the instrumental and proxy observations, and permits for a probabilistic assessment of extreme values. In addition, the recent divergence of tree ring proxies from instrumental temperatures must be accounted for in any assessment of past extremes. While there is a rich tradition in the paleoclimate literature of describing recent observations as unprecedented, in this talk I present the first reconstruction, based on a Bayesian Hierarchical Model, that meets each of these necessary conditions for making inferences on extremes. I present a reconstruction of Arctic and sub-Arctic temperature anomalies over the last 600 years, and show that the summer of 2010 was with virtually certainty (p>0.99) the warmest in the last 600 years in Western Russia and very likely (p>0.9) the warmest in Western Greenland and the Canadian Arctic as well, while the modern rate of centennial-scale warming has only recently exceeded in magnitude the rate of cooling in the 1600s. The multi-proxy approach demonstrates that the recent tree ring divergence is unique to the last 600 years, and consistent (but less significant) results are obtained when tree ring records are excluded.
	Luis Barboza	Paleoclimate Reconstruction using Long Memory processes
	Abstract: We revisit the reconstruction of the Northern Hemisphere (NH) temperature anomalies over the past millennium by combining both the temperature proxies and external forcings while taking the possible long memory features of the involved stochastic processes into account. Our reconstruction is based on two linear models, one in which the unobserved temperature series is linearly related to the three main external forcings, and another in which the observed temperature proxy data (tree rings, ice cores, and others) is linearly related to the unobserved temperatures. Uncertainty about the linear relations is modeled using additive noise (errors). We find evidence for long memory in the linear regression errors, using standard data-based non-Bayesian non-parametric tests, we thus allow the error terms in both linear models to have long-range correlations in time (long memory fractional Gaussian noise). Our results give us reconstructions of the past unobserved land-air temperature anomalies and combined land-and-marine anomalies over the period 1000-1899. This contains precise reconstruction uncertainty that has the long-range error correlations taken into consideration, in the form of an empirical posterior distribution for the temperature series. This is a joint work with Prof Bo Li and Prof Frederi Viens from the Department of Statistics, Purdue University.