Ana M. Aguilera

Robust principal component analysis for functional data

A method for exploring the structure of populations of complex objects, such as images, is considered. The objects are summarized by feature vectors. The statistical backbone is Principal Component Analysis in the space of feature vectors. Visual insights come from representing the results in the original data space. In an ophthalmological example, endemic outliers motivate the development of a bounded influence approach to PCA.

Using principal components for estimating logistic regression with high-dimensional multicollinear data

The logistic regression model is used to predict a binary response variable in terms of a set of explicative ones. The estimation of the model parameters is not too accurate and their interpretation in terms of odds ratios may be erroneous, when there is multicollinearity (high dependence) among the predictors. Other important problem is the great number of explicative variables usually needed to explain the response. In order to improve the estimation of the logistic model parameters under multicollinearity and to reduce the dimension of the problem with continuous covariates, it is proposed to use as covariates of the logistic model a reduced set of optimum principal components of the original predictors. Finally, the performance of the proposed principal component logistic regression model is analyzed by developing a simulation study where different methods for selecting the optimum principal components are compared.

Principal component estimation of functional logistic regression: discussion of two different approaches

Over the last few years many methods have been developed for analyzing functional data with different objectives. The purpose of this paper is to predict a binary response variable in terms of a functional variable whose sample information is given by a set of curves measured without error. In order to solve this problem we formulate a functional logistic regression model and propose its estimation by approximating the sample paths in a finite dimensional space generated by a basis. Then, the problem is reduced to a multiple logistic regression model with highly correlated covariates. In order to reduce dimension and to avoid multicollinearity, two different approaches of functional principal component analysis of the sample paths are proposed. Finally, a simulation study for evaluating the estimating performance of the proposed principal component approaches is developed.

Functional PLS logit regression model

Functional logistic regression has been developed to forecast a binary response variable from a functional predictor. In order to fit this model, it is usual to assume that the functional observations and the parameter function of the model belong to a same finite space generated by a basis of functions. This consideration turns the functional model into a multiple logit model whose design matrix is the product of the matrix of sample paths basic coefficients and the matrix of the inner products between basic functions. The likelihood estimation of the parameter function of this model is very inaccurate due to the high dependence structure of the so obtained design matrix (multicollinearity). In order to solve this drawback several approaches have been proposed. These employ standard multivariate data analysis methods on the design matrix. This is the case of the functional principal component logistic regression model. As an alternative a functional partial least squares logit regression model is proposed, that has as covariates a set of partial least squares components of the design matrix of the multiple logit model associated to the functional one.

Approximation of estimators in the PCA of a stochastic process using B-splines

The objective of this paper is to estimate the principal factors of a continuous time real valued process when we have a collection of independent sample functions which are observed only at discrete time points. We propose to approximate the Principal Component Analysis (PCA) of the process, when the sample functions are regular, by means of the PCA of the natural cubic spline interpolation of the sample curves between the sampling time points. A physical application testing the accuracy of this approach by simulating sample functions of the harmonic oscillator stochastic process is also included. The approximated PCA of this well known process is compared with the exact one and with the classical PCA of the discrete time simulated data.

AN APPROXIMATED PRINCIPAL COMPONENT PREDICTION MODEL FOR CONTINUOUS-TIME STOCHASTIC PROCESSES

SUMMARY In this paper, a linear model for forecasting a continuous-time stochastic process in a future interval in terms of its evolution in a past interval is developed. This model is based on linear regression of the principal components in the future against the principal components in the past. In order to approximate the principal factors from discrete observations of a set of regular sample paths, cubic spline interpolation is used. An application for forecasting tourism evolution in Granada is also included. ( 1997 by John Wiley & Sons, Ltd.

Forecasting with unequally spaced data by a functional principal component approach

The Principal Component Regression model of multiple responses is extended to forccast a continuous-time stochastic process. Orthogonal projection on a subspace of trigonometric functions is applied in order to estimate the principal components using discrete-time observations from a sample of regular curves. The forecasts provided by this approach are compared with classical principal component regression on simulated data.

Modelling the mean of a doubly stochastic Poisson process by functional data analysis

A new procedure for estimating the mean process of a doubly stochastic Poisson process is introduced. The proposed estimation is based on monotone piecewise cubic interpolation of the sample paths of the mean. In order to estimate the continuous time structure of the mean process functional principal component analysis is applied to its trajectories previously adapted to their functional form. A validation of the estimation method is presented by means of some simulations.

Discussion of different logistic models with functional data. Application to Systemic Lupus Erythematosus

The relationship between time evolution of stress and flares in Systemic Lupus Erythematosus patients has recently been studied. Daily stress data can be considered as observations of a single variable for a subject, carried out repeatedly at different time points (functional data). In this study, we propose a functional logistic regression model with the aim of predicting the probability of lupus flare (binary response variable) from a functional predictor variable (stress level). This method differs from the classical approach, in which longitudinal data are considered as observations of different correlated variables. The estimation of this functional model may be inaccurate due to multicollinearity, and so a principal component based solution is proposed. In addition, a new interpretation is made of the parameter function of the model, which enables the relationship between the response and the predictor variables to be evaluated. Finally, the results provided by different logit approaches (functional and longitudinal) are compared, using a sample of Lupus patients.

Forecasting Pollen Concentration by a Two‐Step Functional Model

A functional regression model to forecast the cypress pollen concentration during a given time interval, considering the air temperature in a previous interval as the input, is derived by means of a two-step procedure. This estimation is carried out by functional principal component (FPC) analysis and the residual noise is also modeled by FPC regression, taking as the explicative process the pollen concentration during the earlier interval. The prediction performance is then tested on pollen data series recorded in Granada (Spain) over a period of 10 years.