Modeling and Analysis of Some Heavy Tailed Time Series

The thesis has covered various aspects of modeling and analysis of finite mean time series with symmetric stable distributed innovations. Time series analysis based on Box and Jenkins methods are the most popular approaches where the models are linear and errors are Gaussian. We highlighted the limitations of classical time series analysis tools and explored some generalized tools and organized the approach parallel to the classical set up. In the present thesis we mainly studied the estimation and prediction of signal plus noise model. Here we assumed the signal and noise follow some models with symmetric stable innovations.We start the thesis with some motivating examples and application areas of alpha stable time series models. Classical time series analysis and corresponding theories based on finite variance models are extensively discussed in second chapter. We also surveyed the existing theories and methods correspond to infinite variance models in the same chapter. We present a linear filtering method for computing the filter weights assigned to the observation for estimating unobserved signal under general noisy environment in third chapter. Here we consider both the signal and the noise as stationary processes with infinite variance innovations. We derived semi infinite, double infinite and asymmetric signal extraction filters based on minimum dispersion criteria. Finite length filters based on Kalman-Levy filters are developed and identified the pattern of the filter weights. Simulation studies show that the proposed methods are competent enough in signal extraction for processes with infinite variance.Parameter estimation of autoregressive signals observed in a symmetric stable noise environment is discussed in fourth chapter. Here we used higher order Yule-Walker type estimation using auto-covariation function and exemplify the methods by simulation and application to Sea surface temperature data. We increased the number of Yule-Walker equations and proposed a ordinary least square estimate to the autoregressive parameters. Singularity problem of the auto-covariation matrix is addressed and derived a modified version of the Generalized Yule-Walker method using singular value decomposition.In fifth chapter of the thesis we introduced partial covariation function as a tool for stable time series analysis where covariance or partial covariance is ill defined. Asymptotic results of the partial auto-covariation is studied and its application in model identification of stable auto-regressive models are discussed. We generalize the Durbin-Levinson algorithm to include infinite variance models in terms of partial auto-covariation function and introduce a new information criteria for consistent order estimation of stable autoregressive model.In chapter six we explore the application of the techniques discussed in the previous chapter in signal processing. Frequency estimation of sinusoidal signal observed in symmetric stable noisy environment is discussed in this context. Here we introduced a parametric spectrum analysis and frequency estimate using power transfer function. Estimate of the power transfer function is obtained using the modified generalized Yule-Walker approach. Another important problem in statistical signal processing is to identify the number of sinusoidal components in an observed signal. We used a modified version of the proposed information criteria for this purpose.

ARMA modeling of time series based on rational approximation of spectral density function

This study is concerned with Autoregressive Moving Average (ARMA) models of time series. ARMA models form a subclass of the class of general linear models which represents stationary time series, a phenomenon encountered most often in practice by engineers, scientists and economists. It is always desirable to employ models which use parameters parsimoniously. Parsimony will be achieved by ARMA models because it has only finite number of parameters. Even though the discussion is primarily concerned with stationary time series, later we will take up the case of homogeneous non stationary time series which can be transformed to stationary time series. Time series models, obtained with the help of the present and past data is used for forecasting future values. Physical science as well as social science take benefits of forecasting models. The role of forecasting cuts across all fields of management-—finance, marketing, production, business economics, as also in signal process, communication engineering, chemical processes, electronics etc. This high applicability of time series is the motivation to this study.

On Time Series Analysis of Public Health and Biomedical Data

A time series is a sequence of observations made over time. Examples in public health include daily ozone concentrations, weekly admissions to an emergency department or annual expenditures on health care in the United States. Time series models are used to describe the dependence of the response at each time on predictor variables including covariates and possibly previous values in the series. Time series methods are necessary to account for the correlation among repeated responses over time. This paper gives an overview of time series ideas and methods used in public health research.

The Time-Series Properties on Housing Prices: A Case Study of the Southern California Market

We examine the time-series relationship between housing prices in eight Southern California metropolitan statistical areas (MSAs). First, we perform cointegration tests of the housing price indexes for the MSAs, finding seven cointegrating vectors. Thus, the evidence suggests that one common trend links the housing prices in these eight MSAs, a purchasing power parity finding for the housing prices in Southern California. Second, we perform temporal Granger causality tests revealing intertwined temporal relationships. The Santa Anna MSA leads the pack in temporally causing housing prices in six of the other seven MSAs, excluding only the San Luis Obispo MSA. The Oxnard MSA experienced the largest number of temporal effects from other MSAs, six of the seven, excluding only Los Angeles. The Santa Barbara MSA proved the most isolated in that it temporally caused housing prices in only two other MSAs (Los Angels and Oxnard) and housing prices in the Santa Anna MSA temporally caused prices in Santa Barbara. Third, we calculate out-of-sample forecasts in each MSA, using various vector autoregressive (VAR) and vector error-correction (VEC) models, as well as Bayesian, spatial, and causality versions of these models with various priors. Different specifications provide superior forecasts in the different MSAs. Finally, we consider the ability of theses time-series models to provide accurate out-of-sample predictions of turning points in housing prices that occurred in 2006:Q4. Recursive forecasts, where the sample is updated each quarter, provide reasonably good forecasts of turning points.

The effect of introducing radiographer reporting on the availability of reports for Accident and Emergency and General Practitioner examinations: a time-series analysis

This study explores whether the introduction of selectively trained radiographers reporting Accident and Emergency (A&E) X-ray examinations or the appendicular skeleton affected the availability of reports for A&E and General Practitioner (GP) examinations at it typical district general hospital. This was achieved by analysing monthly data on A&E and GP examinations for 1993 1997 using structural time-series models. Parameters to capture stochastic seasonal effects and stochastic time trends were included ill the models. The main outcome measures were changes in the number, proportion and timeliness of A&E and GP examinations reported. Radiographer reporting X-ray examinations requested by A&E was associated with it 12% (p = 0.050) increase in the number of A&E examinations reported and it 37% (p

Structural change in macroeconomic time series: A complex systems perspective

We demonstrate that the process of generating smooth transitions Call be viewed as a natural result of the filtering operations implied in the generation of discrete-time series observations from the sampling of data from an underlying continuous time process that has undergone a process of structural change. In order to focus discussion, we utilize the problem of estimating the location of abrupt shifts in some simple time series models. This approach will permit its to address salient issues relating to distortions induced by the inherent aggregation associated with discrete-time sampling of continuous time processes experiencing structural change, We also address the issue of how time irreversible structures may be generated within the smooth transition processes. (c) 2005 Elsevier Inc. All rights reserved.

Structural Time Series Modeling: an Application to Environmental Variables

A structural time series model is one which is set up in terms of components which have a direct interpretation. In this paper, the discussion focuses on the dynamic modeling procedure based on the state space approach (associated to the Kalman filter), in the context of surface water quality monitoring, in order to analyze and evaluate the temporal evolution of the environmental variables, and thus identify trends or possible changes in water quality (change point detection). The approach is applied to environmental time series: time series of surface water quality variables in a river basin. The statistical modeling procedure is applied to monthly values of physico- chemical variables measured in a network of 8 water monitoring sites over a 15-year period (1999-2014) in the River Ave hydrological basin located in the Northwest region of Portugal.

Groundwater visualisation system (GVS) : a software framework for integrated display and interrogation of conceptual hydrogeological models, data and time-series animation

Management of groundwater systems requires realistic conceptual hydrogeological models as a framework for numerical simulation modelling, but also for system understanding and communicating this to stakeholders and the broader community. To help overcome these challenges we developed GVS (Groundwater Visualisation System), a stand-alone desktop software package that uses interactive 3D visualisation and animation techniques. The goal was a user-friendly groundwater management tool that could support a range of existing real-world and pre-processed data, both surface and subsurface, including geology and various types of temporal hydrological information. GVS allows these data to be integrated into a single conceptual hydrogeological model. In addition, 3D geological models produced externally using other software packages, can readily be imported into GVS models, as can outputs of simulations (e.g. piezometric surfaces) produced by software such as MODFLOW or FEFLOW. Boreholes can be integrated, showing any down-hole data and properties, including screen information, intersected geology, water level data and water chemistry. Animation is used to display spatial and temporal changes, with time-series data such as rainfall, standing water levels and electrical conductivity, displaying dynamic processes. Time and space variations can be presented using a range of contouring and colour mapping techniques, in addition to interactive plots of time-series parameters. Other types of data, for example, demographics and cultural information, can also be readily incorporated. The GVS software can execute on a standard Windows or Linux-based PC with a minimum of 2 GB RAM, and the model output is easy and inexpensive to distribute, by download or via USB/DVD/CD. Example models are described here for three groundwater systems in Queensland, northeastern Australia: two unconfined alluvial groundwater systems with intensive irrigation, the Lockyer Valley and the upper Condamine Valley, and the Surat Basin, a large sedimentary basin of confined artesian aquifers. This latter example required more detail in the hydrostratigraphy, correlation of formations with drillholes and visualisation of simulation piezometric surfaces. Both alluvial system GVS models were developed during drought conditions to support government strategies to implement groundwater management. The Surat Basin model was industry sponsored research, for coal seam gas groundwater management and community information and consultation. The “virtual” groundwater systems in these 3D GVS models can be interactively interrogated by standard functions, plus production of 2D cross-sections, data selection from the 3D scene, rear end database and plot displays. A unique feature is that GVS allows investigation of time-series data across different display modes, both 2D and 3D. GVS has been used successfully as a tool to enhance community/stakeholder understanding and knowledge of groundwater systems and is of value for training and educational purposes. Projects completed confirm that GVS provides a powerful support to management and decision making, and as a tool for interpretation of groundwater system hydrological processes. A highly effective visualisation output is the production of short videos (e.g. 2–5 min) based on sequences of camera ‘fly-throughs’ and screen images. Further work involves developing support for multi-screen displays and touch-screen technologies, distributed rendering, gestural interaction systems. To highlight the visualisation and animation capability of the GVS software, links to related multimedia hosted online sites are included in the references.

Removing observational noise from fisheries-independent time series data using ARIMA models

Abundance indices derived from fishery-independent surveys typically exhibit much higher interannual variability than is consistent with the within-survey variance or the life history of a species. This extra variability is essentially observation noise (i.e. measurement error); it probably reflects environmentally driven factors that affect catchability over time. Unfortunately, high observation noise reduces the ability to detect important changes in the underlying population abundance. In our study, a noise-reduction technique for uncorrelated observation noise that is based on autoregressive integrated moving average (ARIMA) time series modeling is investigated. The approach is applied to 18 time series of finfish abundance, which were derived from trawl survey data from the U.S. northeast continental shelf. Although the a priori assumption of a random-walk-plus-uncorrelated-noise model generally yielded a smoothed result that is pleasing to the eye, we recommend that the most appropriate ARIMA model be identified for the observed time series if the smoothed time series will be used for further analysis of the population dynamics of a species.

A general framework for integrating environmental time series into stock assessment models: model description, simulation testing, and example

We present a method to integrate environmental time series into stock assessment models and to test the significance of correlations between population processes and the environmental time series. Parameters that relate the environmental time series to population processes are included in the stock assessment model, and likelihood ratio tests are used to determine if the parameters improve the fit to the data significantly. Two approaches are considered to integrate the environmental relationship. In the environmental model, the population dynamics process (e.g. recruitment) is proportional to the environmental variable, whereas in the environmental model with process error it is proportional to the environmental variable, but the model allows an additional temporal variation (process error) constrained by a log-normal distribution. The methods are tested by using simulation analysis and compared to the traditional method of correlating model estimates with environmental variables outside the estimation procedure. In the traditional method, the estimates of recruitment were provided by a model that allowed the recruitment only to have a temporal variation constrained by a log-normal distribution. We illustrate the methods by applying them to test the statistical significance of the correlation between sea-surface temperature (SST) and recruitment to the snapper (Pagrus auratus) stock in the Hauraki Gulf–Bay of Plenty, New Zealand. Simulation analyses indicated that the integrated approach with additional process error is superior to the traditional method of correlating model estimates with environmental variables outside the estimation procedure. The results suggest that, for the snapper stock, recruitment is positively correlated with SST at the time of spawning.

Dynamic Covariance Models for Multivariate Financial Time Series

The accurate prediction of time-changing covariances is an important problem in the modeling of multivariate financial data. However, some of the most popular models suffer from a) overfitting problems and multiple local optima, b) failure to capture shifts in market conditions and c) large computational costs. To address these problems we introduce a novel dynamic model for time-changing covariances. Over-fitting and local optima are avoided by following a Bayesian approach instead of computing point estimates. Changes in market conditions are captured by assuming a diffusion process in parameter values, and finally computationally efficient and scalable inference is performed using particle filters. Experiments with financial data show excellent performance of the proposed method with respect to current standard models.

Simultaneous Learning of Nonlinear Manifold and Dynamical Models for High-dimensional Time Series

The goal of this work is to learn a parsimonious and informative representation for high-dimensional time series. Conceptually, this comprises two distinct yet tightly coupled tasks: learning a low-dimensional manifold and modeling the dynamical process. These two tasks have a complementary relationship as the temporal constraints provide valuable neighborhood information for dimensionality reduction and conversely, the low-dimensional space allows dynamics to be learnt efficiently. Solving these two tasks simultaneously allows important information to be exchanged mutually. If nonlinear models are required to capture the rich complexity of time series, then the learning problem becomes harder as the nonlinearities in both tasks are coupled. The proposed solution approximates the nonlinear manifold and dynamics using piecewise linear models. The interactions among the linear models are captured in a graphical model. By exploiting the model structure, efficient inference and learning algorithms are obtained without oversimplifying the model of the underlying dynamical process. Evaluation of the proposed framework with competing approaches is conducted in three sets of experiments: dimensionality reduction and reconstruction using synthetic time series, video synthesis using a dynamic texture database, and human motion synthesis, classification and tracking on a benchmark data set. In all experiments, the proposed approach provides superior performance.