Functional Regression

Functional data analysis (FDA) involves the analysis of data whose ideal units of observation are functions defined on some continuous domain, and the observed data consist of a sample of functions taken from some population, sampled on a discrete grid. Ramsay and Silverman's 1997 textbook sparked the development of this field, which has accelerated in the past 10 years to become one of the fastest growing areas of statistics, fueled by the growing number of applications yielding this type of data. One unique characteristic of FDA is the need to combine information both across and within functions, which Ramsay and Silverman called replication and regularization, respectively. This article will focus on functional regression, the area of FDA that has received the most attention in applications and methodological development. First will be an introduction to basis functions, key building blocks for regularization in functional regression methods, followed by an overview of functional regression methods, split into three types: [1] functional predictor regression (scalar-on-function), [2] functional response regression (function-on-scalar) and [3] function-on-function regression. For each, the role of replication and regularization will be discussed and the methodological development described in a roughly chronological manner, at times deviating from the historical timeline to group together similar methods. The primary focus is on modeling and methodology, highlighting the modeling structures that have been developed and the various regularization approaches employed. At the end is a brief discussion describing potential areas of future development in this field

Integrated smoothed location model and data reduction approaches for multi variables classification

Smoothed Location Model is a classification rule that deals with mixture of continuous variables and binary variables simultaneously. This rule discriminates groups in a parametric form using conditional distribution of the continuous variables given each pattern of the binary variables. To conduct a practical classification analysis, the objects must first be sorted into the cells of a multinomial table generated from the binary variables. Then, the parameters in each cell will be estimated using the sorted objects. However, in many situations, the estimated parameters are poor if the number of binary is large relative to the size of sample. Large binary variables will create too many multinomial cells which are empty, leading to high sparsity problem and finally give exceedingly poor performance for the constructed rule. In the worst case scenario, the rule cannot be constructed. To overcome such shortcomings, this study proposes new strategies to extract adequate variables that contribute to optimum performance of the rule. Combinations of two extraction techniques are introduced, namely 2PCA and PCA+MCA with new cutpoints of eigenvalue and total variance explained, to determine adequate extracted variables which lead to minimum misclassification rate. The outcomes from these extraction techniques are used to construct the smoothed location models, which then produce two new approaches of classification called 2PCALM and 2DLM. Numerical evidence from simulation studies demonstrates that the computed misclassification rate indicates no significant difference between the extraction techniques in normal and non-normal data. Nevertheless, both proposed approaches are slightly affected for non-normal data and severely affected for highly overlapping groups. Investigations on some real data sets show that the two approaches are competitive with, and better than other existing classification methods. The overall findings reveal that both proposed approaches can be considered as improvement to the location model, and alternatives to other classification methods particularly in handling mixed variables with large binary size

ESTIMATING THE PROBABILITY OF MISCLASSIFICATIONS IN TWO-GROUPS DISCRIMINANT ANALYSIS

This paper is a survey study on estimation of the pro- bability of misclassifications in two-groups discriminant analysis using the linear discriminant function as the classification rule. Here we consider two groups of estimators, namely parametric esti- mators and empirical estimators. The results of some comparative studies on the performances of the considered estimators are also discussed

Pairwise local Fisher and naive Bayes: Improving two standard discriminants

Under embargo until: 2022-02-01The Fisher discriminant is probably the best known likelihood discriminant for continuous data. Another benchmark discriminant is the naive Bayes, which is based on marginals only. In this paper we extend both discriminants by modeling dependence between pairs of variables. In the continuous case this is done by local Gaussian versions of the Fisher discriminant. In the discrete case the naive Bayes is extended by taking geometric averages of pairwise joint probabilities. We also indicate how the two approaches can be combined for mixed continuous and discrete data. The new discriminants show promising results in a number of simulation experiments and real data illustrations.acceptedVersio

Supervised and Ensemble Classification of Multivariate Functional Data: Applications to Lupus Diagnosis

abstract: This dissertation investigates the classification of systemic lupus erythematosus (SLE) in the presence of non-SLE alternatives, while developing novel curve classification methodologies with wide ranging applications. Functional data representations of plasma thermogram measurements and the corresponding derivative curves provide predictors yet to be investigated for SLE identification. Functional nonparametric classifiers form a methodological basis, which is used herein to develop a) the family of ESFuNC segment-wise curve classification algorithms and b) per-pixel ensembles based on logistic regression and fused-LASSO. The proposed methods achieve test set accuracy rates as high as 94.3%, while returning information about regions of the temperature domain that are critical for population discrimination. The undertaken analyses suggest that derivate-based information contributes significantly in improved classification performance relative to recently published studies on SLE plasma thermograms.Dissertation/ThesisDoctoral Dissertation Applied Mathematics 201