Extending Functional kriging to a multivariate context

Environmental data usually have a spatio-temporal structure; pollutant concentrations, for example, are recorded along time and space. Generalized Additive Models (GAMs) represent a suitable tool to model spatial and/or temporal trends of this kind of data, that can be treated as functional, although they are collected as discrete observations. Frequently, the attention is focused on the prediction of a single pollutant at an unmonitored site and, at this aim, we extend kriging for functional data to a multivariate context by exploiting the correlation with the other pollutants. In particular, we propose two procedures: the first one (FKED) combines the regression of a variable (pollutant), of primary interest on the other variables, with functional kriging of the regression residuals; the second one (FCK) is based on linear unbiased prediction of spatially correlated multivariate random processes. The performance of the two proposed procedures is assessed by cross validation; data recorded during a year (2011) from the monitoring network of the state of California (USA) are considered

CLT in Functional Linear Regression Models

International audienceWe propose in this work to derive a CLT in the functional linear regression model to get confidence sets for prediction based on functional linear regression. The main difficulty is due to the fact that estimation of the functional parameter leads to a kind of ill-posed inverse problem. We consider estimators that belong to a large class of regularizing methods and we first show that, contrary to the multivariate case, it is not possible to state a CLT in the topology of the considered functional space. However, we show that we can get a CLT for the weak topology under mild hypotheses and in particular without assuming any strong assumptions on the decay of the eigenvalues of the covariance operator. Rates of convergence depend on the smoothness of the functional coefficient and on the point in which the prediction is made

Using basis expansions for estimating functional PLS regression. Applications with chemometric data

There are many chemometric applications, such as spectroscopy, where the objective is to explain a scalar response from a functional variable (the spectrum) whose observations are functions of wavelengths rather than vectors. In this paper, PLS regression is considered for estimating the linear model when the predictor is a functional random variable. Due to the infinite dimension of the space to which the predictor observations belong, they are usually approximated by curves/functions within a finite dimensional space spanned by a basis of functions. We show that PLS regression with a functional predictor is equivalent to finite multivariate PLS regression using expansion basis coefficients as the predictor, in the sense that, at each step of the PLS iteration, the same prediction is obtained. In addition, from the linear model estimated using the basis coefficients, we derive the expression of the PLS estimate of the regression coefficient function from the model with a functional predictor. The results provided by this functional PLS approach are compared with those given by functional PCR and discrete PLS and PCR using different sets of simulated and spectrometric data.Project P06-FQM-01470 from Consejería de Innovación, Ciencia y Empresa. Junta de Andalucía, SpainProject MTM2007-63793 from Dirección General de Investigación, Ministerio de Educación y Ciencia, Spai

Intraday forecasts of a volatility index: Functional time series methods with dynamic updating

As a forward-looking measure of future equity market volatility, the VIX index has gained immense popularity in recent years to become a key measure of risk for market analysts and academics. We consider discrete reported intraday VIX tick values as realisations of a collection of curves observed sequentially on equally spaced and dense grids over time and utilise functional data analysis techniques to produce one-day-ahead forecasts of these curves. The proposed method facilitates the investigation of dynamic changes in the index over very short time intervals as showcased using the 15-second high-frequency VIX index values. With the help of dynamic updating techniques, our point and interval forecasts are shown to enjoy improved accuracy over conventional time series models.Comment: 29 pages, 5 figures, To appear at the Annals of Operations Researc

Improving prediction performance of stellar parameters using functional models

This paper investigates the problem of prediction of stellar parameters, based on the star's electromagnetic spectrum. The knowledge of these parameters permits to infer on the evolutionary state of the star. From a statistical point of view, the spectra of different stars can be represented as functional data. Therefore, a two-step procedure decomposing the spectra in a functional basis combined with a regression method of prediction is proposed. We also use a bootstrap methodology to build prediction intervals for the stellar parameters. A practical application is also provided to illustrate the numerical performance of our approach

Mixture regression for observational data, with application to functional regression models

In a regression analysis, suppose we suspect that there are several heterogeneous groups in the population that a sample represents. Mixture regression models have been applied to address such problems. By modeling the conditional distribution of the response given the covariate as a mixture, the sample can be clustered into groups and the individual regression models for the groups can be estimated simultaneously. This approach treats the covariate as deterministic so that the covariate carries no information as to which group the subject is likely to belong to. Although this assumption may be reasonable in experiments where the covariate is completely determined by the experimenter, in observational data the covariate may behave differently across the groups. Thus the model should also incorporate the heterogeneity of the covariate, which allows us to estimate the membership of the subject from the covariate. In this paper, we consider a mixture regression model where the joint distribution of the response and the covariate is modeled as a mixture. Given a new observation of the covariate, this approach allows us to compute the posterior probabilities that the subject belongs to each group. Using these posterior probabilities, the prediction of the response can adaptively use the covariate. We introduce an inference procedure for this approach and show its properties concerning estimation and prediction. The model is explored for the functional covariate as well as the multivariate covariate. We present a real-data example where our approach outperforms the traditional approach, using the well-analyzed Berkeley growth study data

Functional data analysis methods for predicting disease status.

Introduction: Differential scanning calorimetry (DSC) is used to determine thermally-induced conformational changes of biomolecules within a blood plasma sample. Recent research has indicated that DSC curves (or thermograms) may have different characteristics based on disease status and, thus, may be useful as a monitoring and diagnostic tool for some diseases. Since thermograms are curves measured over a range of temperature values, they are often considered as functional data. In this dissertation we propose and apply functional data analysis (FDA) techniques to analyze DSC data from the Lupus Family Registry and Repository (LFRR). The aim is to develop FDA methods to create models for classifying lupus vs. control patients on the basis of the thermogram curves. Methods: In project 1 we examine how well standard functional regression is able to capture the differences in curves for cases and controls and compare this to a multivariate approach. In project 2 we develop a semiparametric model; the Generalized Functional Partially Linear Single-Index Model (GFPL). This model is useful when there exists some curvature or non-linearity in the logit, which cannot be modeled by the standard Functional Generalized Linear Model (FGLM). It also mitigates the curse of dimensionality, is a more flexible model, and yields interpretable results. In project 3, we propose a tree-based method: Local Basis Random Forests (LBRF) for Functional Data. This non-parametric method allows us to focus on significant parts of the functional covariates and reduce the noise level. Results: The standard functional logistic regression model with FPCA scores as the predictors gives an 81.25% correct classification rate on the test data, comparable to results from the multivariate approach. The proposed GFPL gives prediction accuracies and standard errors that are better than the standard FGLM when there is nonlinearity present. The LBRF for functional data yields high prediction accuracy (as high as 97% in simulations and 92% in the Lupus data), especially when the true signal is localized, and is able to capture where the true signal lies