Modelling Aggregation on the Large Scale and Regularity on the Small Scale in Spatial Point Pattern Datasets

We consider a dependent thinning of a regular point process with the aim of obtaining aggregation on the large scale and regularity on the small scale in the resulting target point process of retained points. Various parametric models for the underlying processes are suggested and the properties of the target point process are studied. Simulation and inference procedures are discussed when a realization of the target point process is observed, depending on whether the thinned points are observed or not. The paper extends previous work by Dietrich Stoyan on interrupted point processes.     

Intriguing yet simple skewness: kurtosis relation in economic and demographic data distributions,pointing to preferential attachment processes

In this paper, we propose that relations between high-order moments of data distributions, for example, between the skewness (S) and kurtosis (K), allow to point to theoretical models with understandable structural parameters. The illustrative data concern two cases: (i) the distribution of income taxes and (ii) that of inhabitants, after aggregation over each city in each province of Italy in 2011. Moreover, from the rank-size relationship, for either S or K, in both cases, it is shown that one obtains the parameters of the underlying (hypothetical) modeling distribution: in the present cases, the 2-parameter Beta function, itself related to the Yule–Simon distribution function, whence suggesting a growth model based on the preferential attachment process.     

A Case Study for Modelling Cancer Incidence Using Bayesian Spatio‐Temporal Models

Researchers familiar with spatial models are aware of the challenge of choosing the level of spatial aggregation. Few studies have been published on the investigation of temporal aggregation and its impact on inferences regarding disease outcome in space–time analyses. We perform a case study for modelling individual disease outcomes using several Bayesian hierarchical spatio‐temporal models, while taking into account the possible impact of spatial and temporal aggregation. Using longitudinal breast cancer data from South East Queensland, Australia, we consider both parametric and non‐parametric formulations for temporal effects at various levels of aggregation. Two temporal smoothness priors are considered separately; each is modelled with fixed effects for the covariates and an intrinsic conditional autoregressive prior for the spatial random effects. Our case study reveals that different model formulations produce considerably different model performances. For this particular dataset, a classical parametric formulation that assumes a linear time trend produces the best fit among the five models considered. Different aggregation levels of temporal random effects were found to have little impact on model goodness‐of‐fit and estimation of fixed effects.     

Identification of composite (∑+II) arma models by relatively simpler models

In the real world situations, many time series are aggregates of two or more time series. An aggregation may take place due to an addition or the product or both of two or more time series. We are often interested in the study of the properties of aggregates which are, in turn, dependent on the properties of the constituent series. Motivated by this problem, the authors study in this paper the properties of models generated by the operator (Σ+II) on autoregressive-moving-average (ARMA) processes of orders (p_i,q_i), i = l→n . A few practical examples where such models have been used are given in the introduction and an illustrative numerical example is discussed at the end of the paper.     

The effect of temporal aggregation on the estimation accuracy of ARMA models

ABSTRACT

Autoregressive Moving Average (ARMA) time series model fitting is a procedure often based on aggregate data, where parameter estimation plays a key role. Therefore, we analyze the effect of temporal aggregation on the accuracy of parameter estimation of mixed ARMA and MA models. We derive the expressions required to compute the parameter values of the aggregate models as functions of the basic model parameters in order to compare their estimation accuracy. To this end, a simulation experiment shows that aggregation causes a severe accuracy loss that increases with the order of aggregation, leading to poor accuracy.     

A self-normalizing approach to the specification test of mixed-frequency models

In econometrics and finance, variables are collected at different frequencies. One straightforward regression model is to aggregate the higher frequency variable to match the lower frequency with a fixed weight function. However, aggregation with fixed weight functions may overlook useful information in the higher frequency variable. On the other hand, keeping all higher frequencies may result in overly complicated models. In literature, mixed data sampling (MIDAS) regression models have been proposed to balance between the two. In this article, a new model specification test is proposed that can help decide between the simple aggregation and the MIDAS model.     

Non‐stationary Cross‐Covariance Models for Multivariate Processes on a Globe

Abstract. In geophysical and environmental problems, it is common to have multiple variables of interest measured at the same location and time. These multiple variables typically have dependence over space (and/or time). As a consequence, there is a growing interest in developing models for multivariate spatial processes, in particular, the cross‐covariance models. On the other hand, many data sets these days cover a large portion of the Earth such as satellite data, which require valid covariance models on a globe. We present a class of parametric covariance models for multivariate processes on a globe. The covariance models are flexible in capturing non‐stationarity in the data yet computationally feasible and require moderate numbers of parameters. We apply our covariance model to surface temperature and precipitation data from an NCAR climate model output. We compare our model to the multivariate version of the Matérn cross‐covariance function and models based on coregionalization and demonstrate the superior performance of our model in terms of AIC (and/or maximum loglikelihood values) and predictive skill. We also present some challenges in modelling the cross‐covariance structure of the temperature and precipitation data. Based on the fitted results using full data, we give the estimated cross‐correlation structure between the two variables.     

Estimating the duration of dynamic effects with temporally aggregated observations

This paper discusses how an Ordinary Least Squares (OLS) estimator can be used to obtain reasonably accurate estimates of the duration of dynamic effects in a Koyck model framework without knowledge of the true level of temporal aggregation of the data. With proper changes in the analytic derivation, the approach can be extended to other dynamic models.     

The effect of aggregation on nonlinearity

This paper investigates the interaction between aggregation and nonlinearity through a monte carlo study. Various tests for neglected nonlinearity are used to compare the power of the tests for different nonlinear models to different levels of aggregation. Three types of aggregation, namely, cross-sectional aggregation, temporal aggregation and systematic sampling are considered. Aggregation is inclined to simplify nonlinearity. The degree to which nonlinearity is reduced depends on the importance of common factor and extent of the aggregation. The effect is larger when the size of common factor is smaller and when the extent of the aggregation is larger.     

The effect of aggregation on nonlinearity

This paper investigates the interaction between aggregation and nonlinearity through a monte carlo study. Various tests for neglected nonlinearity are used to compare the power of the tests for different nonlinear models to different levels of aggregation. Three types of aggregation, namely, cross-sectional aggregation, temporal aggregation and systematic sampling are considered. Aggregation is inclined to simplify nonlinearity. The degree to which nonlinearity is reduced depends on the importance of common factor and extent of the aggregation. The effect is larger when the size of common factor is smaller and when the extent of the aggregation is larger.     

Some aspects of modern population mathematics

The purpose of this paper is to survey a number of the technical tools and models that have found use in the study of human and other populations, and to indicate some problems of current interest. These tools and models are varied: integral equations, nonlinear oscillations, differential geometry, dynamical systems, nonlinear operations, bifurcation theory, semigroup theory, martingale theory, Markov processes, diffusion processes, branching processes, ergodic theory, prediction theory and state-space models. A fairly extensive bibliography is provided. Also an Appendix has been added describing the analysis of a classical entomological data set.     

Bivariate degradation modelling with marginal heterogeneous stochastic processes

In this paper, we consider that the degradation of two performance characteristics of a product can be modelled by stochastic processes and jointly by copula functions, but different stochastic processes govern the behaviour of each performance characteristic (PC) degradation. Different heterogeneous and homogeneous models are presented considering copula functions and different combinations of the most used stochastic processes in degradation analysis as marginal distributions. This is an important aspect to consider because the behaviour of the degradation of each PC may be different in its nature. As the joint distributions of the proposed models result in complex distributions, the estimation of the parameters of interest is performed via MCMC. A simulation study is performed to compare heterogeneous and homogeneous models. In addition, the proposed models are implemented to crack propagation data of two terminals of an electronic device, and some insights are provided about the product reliability under heterogeneous models.     

Hierarchical Bayesian models for predicting spatially correlated curves

Functional data analysis has emerged as a new area of statistical research with a wide range of applications. In this paper, we propose novel models based on wavelets for spatially correlated functional data. These models enable one to regularize curves observed over space and predict curves at unobserved sites. We compare the performance of these Bayesian models with several priors on the wavelet coefficients using the posterior predictive criterion. The proposed models are illustrated in the analysis of porosity data.     

Estimation for binary models generated by Gaussian autoregressive processes

Regression-type and partial likelihood models are presented for binary data obtained by clipping a Gaussian autoregressive process. Five methods for estimating parameters of the model are proposed and compared via a simulation study. A real data analysis is also presented.     

Flowgraph Models for Generalized Phase Type Distributions Having Non-Exponential Waiting Times

ABSTRACT. Aalen (1995) introduced phase type distributions based on Markov processes for modelling disease progression in survival analysis. For tractability and to maintain the Markov property, these use exponential waiting times for transitions between states. This article extends the work of Aalen (1995) by generalizing these models to semi-Markov processes with non-exponential waiting times. The generalization allows more realistic modelling of the stages of a disease where the Markov property and exponential waiting times may not hold. Flowgraph models are introduced to provide a closed form for the distributions in situations involving non-exponential waiting times. Flowgraph models work where traditional methods of stochastic processes are intractable. Saddlepoint approximations are used in the analysis. Together, generalized phase type distributions, flowgraphs, and saddlepoint approximations create exciting and innovative prospects for the analysis of survival data.     

Accelerated Degradation Models for Failure Based on Geometric Brownian Motion and Gamma Processes

Based on a generalized cumulative damage approach with a stochastic process describing degradation, new accelerated life test models are presented in which both observed failures and degradation measures can be considered for parametric inference of system lifetime. Incorporating an accelerated test variable, we provide several new accelerated degradation models for failure based on the geometric Brownian motion or gamma process. It is shown that in most cases, our models for failure can be approximated closely by accelerated test versions of Birnbaum–Saunders and inverse Gaussian distributions. Estimation of model parameters and a model selection procedure are discussed, and two illustrative examples using real data for carbon-film resistors and fatigue crack size are presented.     

Spatiotemporal prediction for log-Gaussian Cox processes

Space–time point pattern data have become more widely available as a result of technological developments in areas such as geographic information systems. We describe a flexible class of space–time point processes. Our models are Cox processes whose stochastic intensity is a space–time Ornstein–Uhlenbeck process. We develop moment-based methods of parameter estimation, show how to predict the underlying intensity by using a Markov chain Monte Carlo approach and illustrate the performance of our methods on a synthetic data set.     

Data cloning estimation of GARCH and COGARCH models

GARCH models include most of the stylized facts of financial time series and they have been largely used to analyse discrete financial time series. In the last years, continuous-time models based on discrete GARCH models have been also proposed to deal with non-equally spaced observations, as COGARCH model based on Lévy processes. In this paper, we propose to use the data cloning methodology in order to obtain estimators of GARCH and COGARCH model parameters. Data cloning methodology uses a Bayesian approach to obtain approximate maximum likelihood estimators avoiding numerically maximization of the pseudo-likelihood function. After a simulation study for both GARCH and COGARCH models using data cloning, we apply this technique to model the behaviour of some NASDAQ time series.     

Statistical models for disaggregation and reaggregation of natural gas consumption data

In this paper, we present a unified framework for natural gas consumption modeling and forecasting. This consists of models of GAM class and their nonlinear extension, tailored for easy estimation, aggregation and treatment of the delayed relationship between temperature and consumption. Since the consumption data for households and small commercial customers are routinely available in many countries only as long-term sum meter readings, their disaggregation and possibly reaggregation to different time intervals is necessary for a variety of purposes. We show some examples of specific models based on the presented framework and then we demonstrate their use in practice, especially for the disaggregation and reaggregation tasks.     

Numerical evaluation of observed sojourn time distributions for a single ion channel incorporating time interval omission

The dynamical aspects of single ion channel gating can be modelled by a semi-Markov process. There is aggregation of states, corresponding to the receptor channel being open or closed, and there is time interval omission, brief sojourns in either the open or closed classes of states not being detected. This paper is concerned with the computation of the probability density functions of observed open (closed) sojourn-times incorporating time interval omission. A system of Volterra integral equations is derived, whose solution governs the required density function. Numerical procedures, using iterative and multistep methods, are described for solving these equations. Examples are given, and in the special case of Markov models results are compared with those obtained by alternative methods. Probabilistic interpretations are given for the iterative methods, which also give lower bounds for the solutions.