Bayesian Functional Data Modeling for Heterogeneous Volatility

This research implements two important main fields of the study which is functional data analysis and extreme value theory. The aims of this study are to convert the extreme rainfall data into functions and to propose method development for functional data analysis in extreme value theory. Functional data analysis is one of the techniques to convert or transform discrete observations into functions or curves. The daily rainfall data in Petaling Jaya from 1974 to 2003 will be tested on extreme value theory approach which is block maxima in order to practice the functional data analysis technique. The least squares and akaike criteria information method are used to smooth the fourier basis of the functional data analysis. A descriptive statistics which represents the behavior of a rainfall data for Petaling Jaya is analyzed and illustrated by a smooth curve from functional data analysis. This study indicates that there is an increase in rates of the rainfall in Petaling Jaya for the recent period compared to the last 30 years data both in the mean and standard deviation values.

BackgroundA key problem in systems biology is estimating dynamical models of gene regulatory networks. Traditionally, this has been done using regression or other ad-hoc methods when the model is linear. More detailed, realistic modeling studies usually employ nonlinear dynamical models, which lead to computationally difficult parameter estimation problems. Functional data analysis methods, however, offer a means to simplify fitting by transforming the problem from one of matching modeled and observed dynamics to one of matching modeled and observed time derivatives–a regression problem, albeit a nonlinear one.ResultsWe formulate a functional data analysis approach for estimating the parameters of nonlinear dynamical models and evaluate this approach on data from two real systems, the gap gene system of Drosophila melanogaster and the synthetic IRMA network, which was created expressly as a test case for genetic network inference. We also evaluate the approach on simulated data sets generated by the GeneNetWeaver program, the basis for the annual DREAM reverse engineering challenge. We assess the accuracy with which the correct regulatory relationships within the networks are extracted, and consider alternative methods of regularization for the purpose of overfitting avoidance. We also show that the computational efficiency of the functional data analysis approach, and the decomposability of the resulting regression problem, allow us to explicitly enumerate and evaluate all possible regulator combinations for every gene. This gives deeper insight into the the relevance of different regulators or regulator combinations, and lets one check for alternative regulatory explanations.ConclusionsFunctional data analysis is a powerful approach for estimating detailed nonlinear models of gene expression dynamics, allowing efficient and accurate estimation of regulatory architecture.

For detecting anomalies or interventions in the field of forest monitoring we propose an approach based on the spatial and temporal forecast of satellite time series data. For each pixel of the satellite image three different types of forecasts are provided, namely spatial, temporal and combined spatio-temporal forecast. Spatial forecast means that a clustering algorithm is used to group the time series data based on the features normalised difference vegetation index (NDVI) and the short-wave infrared band (SWIR). For estimation of the typical temporal trajectory of the NDVI and SWIR during the vegetation period of each spatial cluster, we apply several methods of functional data analysis including functional principal component analysis, and a novel form of random regression forests with online learning (streaming) capability. The temporal forecast is carried out by means of functional time series analysis and an autoregressive integrated moving average model. The combination of the temporal forecasts, which is based on the past of the considered pixel, and spatial forecasts, which is based on highly correlated pixels within one cluster and their past, is performed by functional data analysis, and a variant of random regression forests adapted to online learning capabilities. For evaluation of the methods, the approaches are applied to a study area in Germany for monitoring forest damages caused by wind-storm, and to a study area in Spain for monitoring forest fires.

Functional data, with basic observational units being functions (e.g., curves, surfaces) varying over a continuum, are frequently encountered in various applications. While many statistical tools have been developed for functional data analysis, the issue of smoothing all functional observations simultaneously is less studied. Existing methods often focus on smoothing each individual function separately, at the risk of removing important systematic patterns common across functions. We propose a nonparametric Bayesian approach to smooth all functional observations simultaneously and nonparametrically. In the proposed approach, we assume that the functional observations are independent Gaussian processes subject to a common level of measurement errors, enabling the borrowing of strength across all observations. Unlike most Gaussian process regression models that rely on pre-specified structures for the covariance kernel, we adopt a hierarchical framework by assuming a Gaussian process prior for the mean function and an Inverse-Wishart process prior for the covariance function. These prior assumptions induce an automatic mean-covariance estimation in the posterior inference in addition to the simultaneous smoothing of all observations. Such a hierarchical framework is flexible enough to incorporate functional data with different characteristics, including data measured on either common or uncommon grids, and data with either stationary or nonstationary covariance structures. Simulations and real data analysis demonstrate that, in comparison with alternative methods, the proposed Bayesian approach achieves better smoothing accuracy and comparable mean-covariance estimation results. Furthermore, it can successfully retain the systematic patterns in the functional observations that are usually neglected by the existing functional data analyses based on individual-curve smoothing.

Trends in ecologically meaningful water quality variates are important for establishing changes in water quality. Linear assessments of the trend at different monitoring sites are relatively easy to compare but may be misleading as they can poorly represent the nature of the trend. Non-linear trends, on the other hand, more faithfully capture the trend but have been much more difficult to compare between sites. In this article we investigate the ability of functional data analysis to explore between site differences in non-linear water quality trends. Functional data analysis is briefly reviewed, concentrating in particular on the techniques of functional principal component analysis, functional linear modelling and functional cluster analysis. These techniques are then applied to some nutrient and sediment trend data obtained from a number of monitoring sites in three dams in South East Queensland, Australia. Functional data analysis is found to be a valuable exploratory tool for providing insight into the between-site, depth and dam differences. Copyright © 2005 John Wiley & Sons, Ltd.

A nonparametric approach combining generative models and functional data analysis is presented in this paper for classifying functional data which arise naturally in a wide variety of signal processing applications, such as brain computer interfacing, speech recognition, or image classification. Based on a new and improved family of Bayesian classifiers, we extend hierarchical Bayesian classification methodology from vector to functional settings. We provide theoretical and practical motivations to our approach which relies on Dirichlet process mixtures and Gaussian processes. The performance is evaluated on phoneme recognition task, and compared to that of Functional Support Vector Machines (FSVMs).

We present a load carriage analysis for Malaysian soldier using functional data analysis (FDA) technique. The objectives of the study are to test the significance of curve registration in the analysis process flow and to investigate the effect of 10% load increment from 0% BW up to 40% BW on ground reaction force (GRF) for normal walking. Preliminary experiment on GRF was done using Vicon 1.4 motion analysis system, Kirstle and AMTI force plates and markers. Functional data analysis method was applied to analyze the data. B-spline smoothing, time normalization, landmark registration and permutation tests were performed. The result confirms that FDA is one of the best techniques in analyzing gait data. Curves registration can well distinguish the load increment and should be included in analysis sequences. The results show that 10 % BW load increment has significant effect on GRF for a Malaysian military.

Sign, Wilcoxon and Mann-Whitney tests are nonparametric methods in one or two-sample problems. The nonparametric methods are alternatives used for testing hypothesis when the standard methods based on the Gaussianity assumption are not suitable to be applied. Recently, the functional data analysis (FDA) has gained relevance in statistical modeling. In FDA, each observation is a curve or function which usually is a realization of a stochastic process. In the literature of FDA, several methods have been proposed for testing hypothesis with samples coming from Gaussian processes. However, when this assumption is not realistic, it is necessary to utilize other approaches. Clustering and regression methods, among others, for non-Gaussian functional data have been proposed recently. In this paper, we propose extensions of the sign, Wilcoxon and Mann-Whitney tests to the functional data context as methods for testing hypothesis when we have one or two samples of non-Gaussian functional data. We use random projections to transform the functional problem into a scalar one, and then we proceed as in the standard case. Based on a simulation study, we show that the proposed tests have a good performance. We illustrate the methodology by applying it to a real data set.

Left-censored linear regression models are quite popular models and have been deeply considered in the last three decades. In this paper, we consider a completely Bayesian approach for making a new Markov chain Monte Carlo (MCMC) algorithm with tractable full posteriors. Simulated consequences and real data analyses depict that the new Markov chain Monte Carlo algorithm has excellent mixing property and carry out very well than the present methods based on prediction accuracy.

Functional data analysis (FDA) is a relatively new branch of statistics devoted to describing and modelling data that are complete functions. Many relevant aspects of musical performance and perception can be understood and quantified as dynamic processes evolving as functions of time. In this paper, we show that FDA is a statistical methodology well suited for research into the field of quantitative musical performance analysis. To demonstrate this suitability, we consider tempo data for 28 performances of Schumann's Träumerei and analyse them by means of functional principal component analysis (one of the most powerful descriptive tools included in FDA). Specifically, we investigate the commonalities and differences between different performances regarding (expressive) timing, and we cluster similar performances together. We conclude that musical data considered as functional data reveal performance structures that might otherwise go unnoticed.

We consider modeling and analysis of functional data arising in method comparison studies. The observed data consist of repeated measurements of a continuous variable obtained using multiple methods of measurement on a sample of subjects. The data are treated as multivariate functional data that are observed with noise at a common set of discrete time points which may vary from subject to subject. The proposed methodology uses functional principal components analysis within the framework of a mixed-effects model to represent the observations in terms of a small number of method-specific principal components. Two approaches for estimating the unknowns in the model, both adaptations of general techniques developed for multivariate functional principal components analysis, are presented. Bootstrapping is employed to get estimates of bias and covariance matrix of model parameter estimates. These in turn are used to compute confidence intervals for parameters and functions thereof, such as the measures of similarity and agreement between the measurement methods, that are necessary for data analysis. The estimation approaches are evaluated using simulation. The methodology is illustrated by analyzing two datasets.

Application of functional data analysis to explore movements: walking, running and jumping - A systematic review

Functional data analysis (FDA) alternatives, based on the classical Mandel h and k statistics, are proposed to identify the laboratories that supply inconsistent results in interlaboratory studies (ILS). ILS is the procedure performed by a number of laboratories to test the precision of an analytical method, to measure the proficiency of laboratories in implementing an analytical procedure, to certify reference materials, and to evaluate a new experimental standard. The use of outlier tests, such as h and k Mandel statistics proposed by the ASTM E691, is crucial to assess these aims, estimating inter- and intra-laboratory data position and variability from a univariate point of view. Considering that experimental results obtained in analytical sciences are often functional, the use of FDA techniques can prevent the loss of important data information. The FDA approaches of h and k statistics are presented and point-wise obtained to deal with functional experimental data. Both functional statistics are estimated for each laboratory, their functional critical limits are obtained by bootstrap resampling, and new FDA versions of h and k graphics are presented. Real and synthetic thermogravimetric data are utilized to assess the good performance of the proposed FDA h and k statistics and their advantages with respect to the univariate approach.

Dynamic prediction of multivariate functional data based on Functional Kernel Partial Least Squares

The Functional Resonance Analysis Method (FRAM) is a novel research methodology that uses qualitative data collection methods to map and model complex healthcare processes by identifying and depicting the cumulative activities required to produce an outcome. The FRAM aims to identify the variability that can emerge in a process when healthcare activities are performed under dynamic conditions. With this knowledge, health care system design, safety, and quality improvement recommendations can be developed with a greater understanding of everyday process functionality. Researchers interested in using the FRAM require both an understanding of the methodology itself, as well as an understanding of how to effectively plan and conduct qualitative research. The purpose of this paper is to provide practical guidance to researchers on planning and operationalizing qualitative data collection and analysis methods to inform the building of a FRAM model. A combination of literature and practical experience will be used to examine and suggest appropriate ways for researchers to carry out this work.