Spatio-temporal functional regression on paleoecological data

There is much interest in predicting the impact of global warming on the genetic diversity of natural populations and the influence of climate on biodiversity is an important ecological question. Since Holocene, we face many climate perturbations and the geographical ranges of plant taxa have changed substantially. Actual genetic diversity of plant is a result of these processes and a first step to study the impact of future climate change is to understand the important features of reconstructed climate variables such as temperature or precipitation for the last 15,000 years on actual genetic diversity of forest. We model the relationship between genetic diversity in the European beech (Fagus sylvatica) forests and curves of temperature and precipitation reconstructed from pollen databases. Our model links the genetic measure to the climate curves. We adapt classical functional linear model to take into account interactions between climate variables as a bilinear form. Since the data are georeferenced, our extensions also account for the spatial dependence among the observations. The practical issues of these methodological extensions are discussed.     

Consistency of the recursive nonparametric regression estimation for dependent functional data

We consider the recursive estimation of a regression functional where the explanatory variables take values in some functional space. We prove the almost sure convergence of such estimates for dependent functional data. Also we derive the mean quadratic error of the considered class of estimators. Our results are established with rates and asymptotic appear bounds, under strong mixing condition. Finally, the feasibility of the proposed estimator is illustrated throughout an empirical study.     

Robust estimation for functional coefficient regression models with spatial data

A global smoothing procedure is developed using B-spline function approximation for estimating the unknown functions of a functional coefficient regression model with spatial data. A general formulation is used to treat mean regression, median regression, quantile regression and robust mean regression in one setting. The global convergence rates of the estimators of unknown coefficient functions are established. Various applications of the main results, including estimating conditional quantile coefficient functions and robustifying the mean regression coefficient functions are given. Finite sample properties of our procedures are studied through Monte Carlo simulations. A housing data example is used to illustrate the proposed methodology.     

Function-on-function regression for two-dimensional functional data

ABSTRACT

We present methods for modeling and estimation of a concurrent functional regression when the predictors and responses are two-dimensional functional datasets. The implementations use spline basis functions and model fitting is based on smoothing penalties and mixed model estimation. The proposed methods are implemented in available statistical software, allow the construction of confidence intervals for the bivariate model parameters, and can be applied to completely or sparsely sampled responses. Methods are tested to data in simulations and they show favorable results in practice. The usefulness of the methods is illustrated in an application to environmental data.     

Asymptotic normality of locally modelled regression estimator for functional data

We focus on the nonparametric regression of a scalar response on a functional explanatory variable. As an alternative to the well-known Nadaraya-Watson estimator for regression function in this framework, the locally modelled regression estimator performs very well [cf. [Barrientos-Marin, J., Ferraty, F., and Vieu, P. (2010), ‘Locally Modelled Regression and Functional Data’, Journal of Nonparametric Statistics, 22, 617–632]. In this paper, the asymptotic properties of locally modelled regression estimator for functional data are considered. The mean-squared convergence as well as asymptotic normality for the estimator are established. We also adapt the empirical likelihood method to construct the point-wise confidence intervals for the regression function and derive the Wilk's phenomenon for the empirical likelihood inference. Furthermore, a simulation study is presented to illustrate our theoretical results.     

Functional data analysis: estimation of the relative error in functional regression under random left-truncation model

In this paper, we investigate the relationship between a functional random covariable and a scalar response which is subject to left-truncation by another random variable. Precisely, we use the mean squared relative error as a loss function to construct a nonparametric estimator of the regression operator of these functional truncated data. Under some standard assumptions in functional data analysis, we establish the almost sure consistency, with rates, of the constructed estimator as well as its asymptotic normality. Then, a simulation study, on finite-sized samples, was carried out in order to show the efficiency of our estimation procedure and to highlight its superiority over the classical kernel estimation, for different levels of simulated truncated data.     

Automatic and location-adaptive estimation in functional single-index regression

This paper develops a new automatic and location-adaptive procedure for estimating regression in a Functional Single-Index Model (FSIM). This procedure is based on k-Nearest Neighbours (kNN) ideas. The asymptotic study includes results for automatically data-driven selected number of neighbours, making the procedure directly usable in practice. The local feature of the kNN approach insures higher predictive power compared with usual kernel estimates, as illustrated in some finite sample analysis. As by-product, we state as preliminary tools some new uniform asymptotic results for kernel estimates in the FSIM model.     

Optimal estimation in functional linear regression for sparse noise‐contaminated data

In this article, we propose a novel approach to fit a functional linear regression in which both the response and the predictor are functions. We consider the case where the response and the predictor processes are both sparsely sampled at random time points and are contaminated with random errors. In addition, the random times are allowed to be different for the measurements of the predictor and the response functions. The aforementioned situation often occurs in longitudinal data settings. To estimate the covariance and the cross‐covariance functions, we use a regularization method over a reproducing kernel Hilbert space. The estimate of the cross‐covariance function is used to obtain estimates of the regression coefficient function and of the functional singular components. We derive the convergence rates of the proposed cross‐covariance, the regression coefficient, and the singular component function estimators. Furthermore, we show that, under some regularity conditions, the estimator of the coefficient function has a minimax optimal rate. We conduct a simulation study and demonstrate merits of the proposed method by comparing it to some other existing methods in the literature. We illustrate the method by an example of an application to a real‐world air quality dataset. The Canadian Journal of Statistics 47: 524–559; 2019 © 2019 Statistical Society of Canada     

Cross-validation approximation in functional linear regression

Cross-validation has been widely used in the context of statistical linear models and multivariate data analysis. Recently, technological advancements give possibility of collecting new types of data that are in the form of curves. Statistical procedures for analysing these data, which are of infinite dimension, have been provided by functional data analysis. In functional linear regression, using statistical smoothing, estimation of slope and intercept parameters is generally based on functional principal components analysis (FPCA), that allows for finite-dimensional analysis of the problem. The estimators of the slope and intercept parameters in this context, proposed by Hall and Hosseini-Nasab [On properties of functional principal components analysis, J. R. Stat. Soc. Ser. B: Stat. Methodol. 68 (2006), pp. 109–126], are based on FPCA, and depend on a smoothing parameter that can be chosen by cross-validation. The cross-validation criterion, given there, is time-consuming and hard to compute. In this work, we approximate this cross-validation criterion by such another criterion so that we can turn to a multivariate data analysis tool in some sense. Then, we evaluate its performance numerically. We also treat a real dataset, consisting of two variables; temperature and the amount of precipitation, and estimate the regression coefficients for the former variable in a model predicting the latter one.     

On asymptotic distribution of prediction in functional linear regression

There has been substantial recent attention on problems involving a functional linear regression model with scalar response. Among them, there have been few works dealing with asymptotic distribution of prediction in functional linear regression models. In recent literature, the centeral limit theorem for prediction has been discussed, but the proof and conditions under which the random bias terms for a fixed predictor converge to zero have been ignored so that the impact of these terms on the convergence of the prediction has not been well understood. Clarifying the proof and conditions under which the bias terms converge to zero, we show that the asymptotic distribution of the prediction is normal. Furthermore, we have derived those results related to other terms that already obtained by others, under milder conditions. Finally, we conduct a simulation study to investigate performance of the asymptotic distribution under various parameter settings.     

Multivariate functional response low‐rank regression with an application to brain imaging data

We propose a multivariate functional response low‐rank regression model with possible high‐dimensional functional responses and scalar covariates. By expanding the slope functions on a set of sieve bases, we reconstruct the basis coefficients as a matrix. To estimate these coefficients, we propose an efficient procedure using nuclear norm regularization. We also derive error bounds for our estimates and evaluate our method using simulations. We further apply our method to the Human Connectome Project neuroimaging data to predict cortical surface motor task‐evoked functional magnetic resonance imaging signals using various clinical covariates to illustrate the usefulness of our results.     

Wavelet-based LASSO in functional linear quantile regression

ABSTRACT

In this paper, we develop an efficient wavelet-based regularized linear quantile regression framework for coefficient estimations, where the responses are scalars and the predictors include both scalars and function. The framework consists of two important parts: wavelet transformation and regularized linear quantile regression. Wavelet transform can be used to approximate functional data through representing it by finite wavelet coefficients and effectively capturing its local features. Quantile regression is robust for response outliers and heavy-tailed errors. In addition, comparing with other methods it provides a more complete picture of how responses change conditional on covariates. Meanwhile, regularization can remove small wavelet coefficients to achieve sparsity and efficiency. A novel algorithm, Alternating Direction Method of Multipliers (ADMM) is derived to solve the optimization problems. We conduct numerical studies to investigate the finite sample performance of our method and applied it on real data from ADHD studies.     

M-estimation for functional linear regression

This paper studies M-estimation in functional linear regression in which the dependent variable is scalar while the covariate is a function. An estimator for the slope function is obtained based on the functional principal component basis. The global convergence rate of the M-estimator of unknown slope function is established. The convergence rate of the mean-squared prediction error for the proposed estimators is also established. Monte Carlo simulation studies are conducted to examine the finite-sample performance of the proposed procedure. Finally, the proposed method is applied to analyze the Berkeley growth data.     

The kernel regression estimation for randomly censored functional stationary ergodic data

In this paper, we investigate the asymptotic properties of the kernel estimator for non parametric regression operator when the functional stationary ergodic data with randomly censorship are considered. More precisely, we introduce the kernel-type estimator of the non parametric regression operator with the responses randomly censored and obtain the almost surely convergence with rate as well as the asymptotic normality of the estimator. As an application, the asymptotic (1 ? ζ) confidence interval of the regression operator is also presented (0 < ζ < 1). Finally, the simulation study is carried out to show the finite-sample performances of the estimator.     

Smoothed functional canonical correlation analysis of humidity and temperature data

This paper focuses on smoothed functional canonical correlation analysis (SFCCA) to investigate the relationships and changes in large, seasonal and long-term data sets. The aim of this study is to introduce a guideline for SFCCA for functional data and to give some insights on the fine tuning of the methodology for long-term periodical data. The guidelines are applied on temperature and humidity data for 11 years between 2000 and 2010 and the results are interpreted. Seasonal changes or periodical shifts are visually studied by yearly comparisons. The effects of the ‘number of basis functions’ and the ‘selection of smoothing parameter’ on the general variability structure and on correlations between the curves are examined. It is concluded that the number of time points (knots), number of basis functions and the time span of evaluation (monthly, daily, etc.) should all be chosen harmoniously. It is found that changing the smoothing parameter does not have a significant effect on the structure of curves and correlations. The number of basis functions is found to be the main effector on both individual and correlation weight functions.     

On clustering shape data

ABSTRACT

Among the statistical methods to model stochastic behaviours of objects, clustering is a preliminary technique to recognize similar patterns within a group of observations in a data set. Various distances to measure differences among objects could be invoked to cluster data through numerous clustering methods. When variables in hand contain geometrical information of objects, such metrics should be adequately adapted. In fact, statistical methods for these typical data are endowed with a geometrical paradigm in a multivariate sense. In this paper, a procedure for clustering shape data is suggested employing appropriate metrics. Then, the best shape distance candidate as well as a suitable agglomerative method for clustering the simulated shape data are provided by considering cluster validation measures. The results are implemented in a real life application.     

Functional analysis of high-content high-throughput imaging data

High-content automated imaging platforms allow the multiplexing of several targets simultaneously to generate multi-parametric single-cell data sets over extended periods of time. Typically, standard simple measures such as mean value of all cells at every time point are calculated to summarize the temporal process, resulting in loss of time dynamics of the single cells. Multiple experiments are performed but observation time points are not necessarily identical, leading to difficulties when integrating summary measures from different experiments. We used functional data analysis to analyze continuous curve data, where the temporal process of a response variable for each single cell can be described using a smooth curve. This allows analyses to be performed on continuous functions, rather than on original discrete data points. Functional regression models were applied to determine common temporal characteristics of a set of single cell curves and random effects were employed in the models to explain variation between experiments. The aim of the multiplexing approach is to simultaneously analyze the effect of a large number of compounds in comparison to control to discriminate between their mode of action. Functional principal component analysis based on T-statistic curves for pairwise comparison to control was used to study time-dependent compound effects.     

Local functional data model for characterizing shape variation of similar curves

“Similar curves” in the present article refers to a family of curves whose major shape are similar, but who have variation coming from curve-specific sources. The goal here is to develop a general methodology to describe small changes among similar curves. Previous methods mainly focus on dimension reduction through FPCA, which are not appropriate for quantifying local variation. Here, we consider a local functional data model which divides data into segments adaptively and models each segment with a shape invariant model. Such model has great flexibility in characterizing local variation of curves, as illustrated by simulation and real data examples.     

Linear functional regression: The case of fixed design and functional response

The authors consider the problem of simple linear regression when the exogenous and endogenous variables are functional and the design is fixed. They propose an estimator for the underlying linear operator and prove its consistency under some conditions which ensure that the design is sufficiently informative. They consider the classical calibration (or inverse regression) problem and analyze a consistent estimator. They also give a simulation study. The proposed method is not hard to implement in practice.