In this article we are interested in modeling the relationship between a scalar, Y, and a functional predictor, X(t). We introduce a highly flexible approach called ”Functional Adaptive Model Estimation” (FAME) which extends generalized linear models (GLM), generalized additive models (GAM) and projection pursuit regression (PPR) to handle functional predictors. The FAME approach can model any of the standard exponential family of response distributions that are assumed for GLM or GAM while maintaining the flexibility of PPR. For example standard linear or logistic regression with functional predictors, as well as far more complicated models, can easily be applied using this approach. A functional principal components decomposition of the predictor functions is used to aid visualization of the relationship between X(t) and Y. We also show how the FAME procedure can be extended to deal with multiple functional and standard finite dimensional predictors, possibly with missing data. The FAME approach is illustrated on simulated data as well as on the prediction of arthritis based on bone shape. We end with a discussion of the relationships between standard regression approaches, their extensions to functional data and FAME.

We review and extend some statistical tools that have proved useful for analysing functional data. Functional data analysis primarily is designed for the analysis of random trajectories and infinite-dimensional data, and there exists a need for the development of adequate statistical estimation and inference techniques. While this field is in flux, some methods have proven useful. These include warping methods, functional principal component analysis, and conditioning under Gaussian assumptions for the case of sparse data. The latter is a recent development that may provide a bridge between functional and more classical longitudinal data analysis. Besides presenting a brief review of functional principal components and functional regression, we develop some concepts for estimating functional principal component scores in the sparse situation. An extension of the so-called generalized functional linear model to the case of sparse longitudinal predictors is proposed. This extension includes functional binary regression models for longitudinal data and is illustrated with data on primary biliary cirrhosis.

There has been substantial recent work on methods for estimating the slope function in linear regression for functional data analysis. However, as in the case of more conventional finite-dimensional regression, much of the practical interest in the slope centers on its application for the purpose of prediction, rather than on its significance in its own right. We show that the problems of slope-function estimation, and of prediction from an estimator of the slope function, have very different characteristics. While the former is intrinsically nonparametric, the latter can be either nonparametric or semiparametric. In particular, the optimal mean-square convergence rate of predictors is n −1, where n denotes sample size, if the predictand is a sufficiently smooth function. In other cases, convergence occurs at a polynomial rate that is strictly slower than n −1. At the boundary between these two regimes, the mean-square convergence rate is less than n −1 by only a logarithmic factor. More generally, the rate of convergence of the predicted value of the mean response in the regression model, given a particular value of the explanatory variable, is determined by a subtle interaction among the smoothness of the predictand, of the slope function in the model, and of the autocovariance function for the distribution of explanatory variables. 1. Introduction. In

Electronic commerce, and in particular online auctions, have received an extreme surge of popularity in recent years. While auction theory has been studied for a long time from a game-theory perspective, the electronic implementation of the auction mechanism poses new and challenging research questions. Although the body of empirical research on online auctions is growing, there is a lack of treatment of these data from a modern statistical point of view. In this work, we present a new source of rich auction data and introduce an innovative way of modelling and analyzing online bidding behavior. In particular, we use functional data analysis to investigate and scrutinize online auction dynamics. We describe the structure of such data and suggest suitable methods, including data smoothing and curve clustering, that allow one to profile online auctions and display different bidding behavior. We illustrate the methods on a set of eBay auction data and tie our results to the existing literature on online auctions. Key words and phrases: functional data analysis, smoothing, penalized splines, clustering, unsupervised

Abstract. Many real world data are sampled functions. As shown by Functional Data Analysis (FDA) methods, spectra, time series, images, gesture recognition data, etc. can be processed more efficiently if their functional nature is taken into account during the data analysis process. This is done by extending standard data analysis methods so that they can apply to functional inputs. A general way to achieve this goal is to compute projections of the functional data onto a finite dimensional sub-space of the functional space. The coordinates of the data on a basis of this sub-space provide standard vector representations of the functions. The obtained vectors can be processed by any standard method. In [43], this general approach has been used to define projection based Multilayer Perceptrons (MLPs) with functional inputs. We study in this paper important theoretical properties of the proposed model. We show in particular that MLPs with functional inputs are universal approximators: they can approximate to arbitrary accuracy any continuous mapping from a compact sub-space of a functional space to IR. Moreover, we provide a consistency result that shows that any mapping from a functional space to IR can be learned thanks to examples by a projection based MLP: the generalization mean square error of the MLP decreases to the smallest possible mean square error on the data when the number of examples goes to infinity.

The goal of this work is to derive models for forecasting the final price of ongoing online auctions. This forecasting task is important not only to the participants of an auction who compete against each other for the lowest price, but also to designers of bidder-side agents. Forecasting prices in online auctions is challenging from a statistical point-of-view because traditional forecasting models do not apply. The reasons for this are three typical features of online auction data: a) unequally spaced bids; b) the limited time horizon of an auction; c) the dynamics of bidding change drastically over time. We propose a dynamic forecasting model for the auction price that can overcome these challenges. We use modern functional data analysis methods that take into account the price velocity and the price acceleration as the basis for our forecasting model. We show that our model has high forecast accuracy and it outperforms traditional methods. Our results also allow for new statistical insight into auction forecasting. We find that the forecasting accuracy increases as we predict further into the future, that is, further towards the auction end, and we tie this finding together with existing auction theory. Key words and phrases: functional data analysis, smoothing, online auctions, bid sniping, price velocity, price acceleration, autoregressive models, exponential smoothing.

functional data for binary response

Abstract: Partial Least Squares (PLS) approach is employed for linear regression modeling when both the dependent variables and independent variables are functional data (curves). After the introduction of the constant-style mean, variance and the correlative coefficient of functional data, an approach for PLS regression modeling of functional data is proposed to overcome the multicollinearity existing in the independent variables set. An empirical study of the functional regression modeling shows that the proposed approach provides a tool for building regression model on functional data under the condition of multicollinearity. The empirical study conclusion, which is coincident with the wildly accepted economic theory, indicates that the Compensation of Employees is the most important variable that contributes to the

In this chapter we examine two different settings in which sparseness can be important in a functional data analysis (FDA). The first setting involves sparseness in the functions. The classical assumption of FDA is that each function has been measured at all time points. However, in practice it is often the case that the functions have only been observed at a relatively small number of points. Here we discuss different general approaches that can be applied in this setting, such as basis functions, mixed effects models and local smoothing, and examine the relative merits of each. Then we briefly outline several specific methodologies that have been developed for dealing with sparse functional data in the principal components, clustering, classification and regression paradigms. The second setting involves using sparsity ideas from high dimensional regression problems, where most of the regression coefficients are assumed to be zero, to perform a dimension reduction in the functional space. We discuss two specific approaches that have been suggested in the literature.

A significant problem with most functional data analyses is that of misaligned curves. Without adjustment, even an analysis as simple as estimation of the mean will fail. A common “synchronization” approach involves equating “landmarks ” such as peaks or troughs. The landmarks method can work well but will fail if marker events can not be identified or are missing from some curves. It may also involve a manual identification of marker events. We develop automated alignment methods based on equating the “moments ” of a given set of curves. These moments do not depend on the identification of markers. For example, the first moment is a measure of the average value of a curve in the x, or time, axis while the second moment measures its spread. We explore both linear and non-linear synchronization procedures. Finally, we discuss the advantages of utilizing, not only the “amplitude ” information, which measures the general shape of the curves, but also the “warping ” information, which measures the way the curves have been distorted in time. Illustrations are provided on functional analyses involving principal components, clustering, classification and regression.