Parameter estimation of partial differential equation models

Partial differential equation (PDE) models are commonly used to model complex dynamic systems in applied sciences such as biology and finance. The forms of these PDE models are usually proposed by experts based on their prior knowledge and understanding of the dynamic system. Parameters in PDE models often have interesting scientific interpretations, but their values are often unknown and need to be estimated from the measurements of the dynamic system in the presence of measurement errors. Most PDEs used in practice have no analytic solutions, and can only be solved with numerical methods. Currently, methods for estimating PDE parameters require repeatedly solving PDEs numerically under thousands of candidate parameter values, and thus the computational load is high. In this article, we propose two methods to estimate parameters in PDE models: a parameter cascading method and a Bayesian approach. In both methods, the underlying dynamic process modeled with the PDE model is represented via basis function expansion. For the parameter cascading method, we develop two nested levels of optimization to estimate the PDE parameters. For the Bayesian method, we develop a joint model for data and the PDE and develop a novel hierarchical model allowing us to employ Markov chain Monte Carlo (MCMC) techniques to make posterior inference. Simulation studies show that the Bayesian method and parameter cascading method are comparable, and both outperform other available methods in terms of estimation accuracy. The two methods are demonstrated by estimating parameters in a PDE model from long-range infrared light detection and ranging data. Supplementary materials for this article are available online

Where is the Economics in Spatial Econometrics?

Spatial econometrics has been criticized by some economists because some model specifications have been driven by data-analytic considerations rather than having a firm foundation in economic theory. In particular this applies to the so-called W matrix, which is integral to the structure of endogenous and exogenous spatial lags, and to spatial error processes, and which are almost the sine qua non of spatial econometrics. Moreover it has been suggested that the significance of a spatially lagged dependent variable involving W may be misleading, since it may be simply picking up the e¤ects of omitted spatially dependent variables, incorrectly suggesting the existence of a spillover mechanism. In this paper we review the theoretical and empirical rationale for network dependence and spatial externalities as embodied in spatially lagged variables, arguing that failing to acknowledge their presence at least leads to biased inference, can be a cause of inconsistent estimation, and leads to an incorrect understanding of true causal processes.Spatial econometrics, endogenous spatial lag, exogenous spatial lag, spatially dependent errors, network dependence, externalities, the W matrix, panel data with spatial effects, multilevel models with spatial effects.

Reassessing the Paradigms of Statistical Model-Building

Statistical model-building is the science of constructing models from data and from information about the data-generation process, with the aim of analysing those data and drawing inference from that analysis. Many statistical tasks are undertaken during this analysis; they include classification, forecasting, prediction and testing. Model-building has assumed substantial importance, as new technologies enable data on highly complex phenomena to be gathered in very large quantities. This creates a demand for more complex models, and requires the model-building process itself to be adaptive. The word “paradigm” refers to philosophies, frameworks and methodologies for developing and interpreting statistical models, in the context of data, and applying them for inference. In order to solve contemporary statistical problems it is often necessary to combine techniques from previously separate paradigms. The workshop addressed model-building paradigms that are at the frontiers of modern statistical research. It tried to create synergies, by delineating the connections and collisions among different paradigms. It also endeavoured to shape the future evolution of paradigms

Individual Claims Reserving: Using Machine Learning Methods

To date, most methods for loss reserving are still used on aggregate data arranged in a triangular form such as the Chain-Ladder (CL) method and the over-dispersed Poisson (ODP) method. With the booming of machine learning methods and the significant increment of computing power, the loss of information resulting from the aggregation of the individual claims data into accident and development year buckets is no longer justifiable. Machine learning methods like Neural Networks (NN) and Random Forest (RF) are then applied and the results are compared with the traditional methods on both simulated data and real data (aggregate at company level)

ESTIMATING THE CONDITIONAL HAZARD FUNCTION OF JOINT LATENT CLASS MIXED MODELS USING HAZARD REGRESSION WITH APPLICATIONS TO PSYCHOLOGY AND NEUROSCIENCE

Research within the fields psychology and neuroscience often have interest in estimating the change of a latent variable over repeated measurements. While the inference of interest lies within this latent variable, observed variables that express this latent variable are instead measured. If the latent variable is observed by a change in a specific biomarker as well as the time-to-event, then a joint model may be more suitable for estimating the change of the latent variable than a typical single-model approach. Given a heterogeneous sample comprising a mixture of classes of this latent variable, a joint latent class mixed model may prove effective.Within this joint model, the Cox proportional hazards model is a popular choice for modeling the time-to-event due to its robustness and minimal assumptions. However, the Cox model still requires a proportionality assumption. Hazard regression (HARE) was developed with relaxing this assumption in mind. HARE uses an adaptive B-spline method to estimate the conditional log-hazard function of the survival model not requiring that the hazard function follow this proportionality assumption. The B-splines in HARE can take the form of covariates of interest, time, or a tensor product of the two. An adaptive regression method estimates the conditional log-hazard function via a partial likelihood method.The purpose of this proposal is to introduce the HARE methodology to estimating the class-specific conditional log-hazard function of a joint latent class mixed model with applications to psychology and neuroscience research. This method is then tested on a study of simulated data, on a subset of data from the Paquid longitudinal cohort study, data from an experiment assessing visual information from infants as they engage in a social arousal task, and longitudinal data from the Infant Brain Imaging Study. The novelty and utility of HARE within the joint latent class mixed model are explored and discussed.Doctor of Public Healt

Statistical Inference in Inverse Problems

Inverse problems have gained popularity in statistical research recently. This dissertation consists of two statistical inverse problems: a Bayesian approach to detection of small low emission sources on a large random background, and parameter estimation methods for partial differential equation (PDE) models. Source detection problem arises, for instance, in some homeland security applications. We address the problem of detecting presence and location of a small low emission source inside an object, when the background noise dominates. The goal is to reach the signal-to-noise ratio levels on the order of 10^-3. We develop a Bayesian approach to this problem in two-dimension. The method allows inference not only about the existence of the source, but also about its location. We derive Bayes factors for model selection and estimation of location based on Markov chain Monte Carlo simulation. A simulation study shows that with sufficiently high total emission level, our method can effectively locate the source. Differential equation (DE) models are widely used to model dynamic processes in many fields. The forward problem of solving equations for given parameters that define the DEs has been extensively studied in the past. However, the inverse problem of estimating parameters based on observed state variables is relatively sparse in the statistical literature, and this is especially the case for PDE models. We propose two joint modeling schemes to solve for constant parameters in PDEs: a parameter cascading method and a Bayesian treatment. In both methods, the unknown functions are expressed via basis function expansion. For the parameter cascading method, we develop the algorithm to estimate the parameters and derive a sandwich estimator of the covariance matrix. For the Bayesian method, we develop the joint model for data and the PDE, and describe how the Markov chain Monte Carlo technique is employed to make posterior inference. A straightforward two-stage method is to first fit the data and then to estimate parameters by the least square principle. The three approaches are illustrated using simulated examples and compared via simulation studies. Simulation results show that the proposed methods outperform the two-stage method

Climate change threatens the future of rain forest ringtail possums by 2050

Aim: The increasing frequency and intensity of extreme weather escalate the pressure of global warming on biodiversity. Globally, synergistic effects of multiple components of climate change have driven local extinctions and community collapses, raising concern about the irreversible deterioration of ecosystems. Here, we disentangle the pressure of increasing warming and frequency of extreme heatwaves on the population dynamics of tropical ringtail possums (family: Pseudocheiridae). Location: The Australian Wet Tropics World Heritage Area. Method: Ringtail possums' population dynamics were estimated between 1992 and 2021 using a hierarchical population model that explicitly described the state process and accounted for imperfect detection. Under our model, we propagated the estimated mechanisms governing the system by forecasting ringtails' population dynamics between 2022 and 2050. Derived from this process, we calculated the probability of absolute and quasi-extinction using different population viability thresholds. Results: We find a strong negative effect of climate change on population dynamics, particularly extreme heatwaves, resulting in a rapid and severe decline in ringtails' population size in the last three decades. Main Conclusions: Forecasted increases in temperature and heatwaves threaten the collapse of rain forest ringtail possums by 2050, with populations falling below viability thresholds within three decades