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

Haymarket to the Heights: The Movement of Cleveland\u27s Orthodox Synagogues From Their Initial Meeting Places to the Heights

This document traces the movement, growth and demise of the small neighborhood synagogues, or shuls, established by newly-arrived Eastern European Jews in the Haymarket area as they migrated to the eastern suburbs.https://engagedscholarship.csuohio.edu/clevmembks/1022/thumbnail.jp

Beechwood, The Book

From the forward by Darrell A.Young: The city fathers have been called visionaries. The city has been studied by architects, planners, engineers and the like from all over the country. What is it about Beachwood that has attracted so much attention? To be certain, there is something magical that has taken place over the last 80 years in Beachwood and Jeffrey Morris has finally documented the historical blueprint from which we can study and learn. This book is the first opportunity to understand our heritage and to delve into the intellect that forged this wonderful community.https://engagedscholarship.csuohio.edu/clevmembks/1009/thumbnail.jp

Haymarket to the Heights: The Movement of Cleveland\u27s Orthodox Synagogues From Their Initial Meeting Places to the Heights

This document traces the movement, growth and demise of the small neighborhood synagogues, or shuls, established by newly-arrived Eastern European Jews in the Haymarket area as they migrated to the eastern suburbs.https://engagedscholarship.csuohio.edu/clevmembks/1022/thumbnail.jp

Automated analysis of quantitative image data using isomorphic functional mixed models, with application to proteomics data

Image data are increasingly encountered and are of growing importance in many areas of science. Much of these data are quantitative image data, which are characterized by intensities that represent some measurement of interest in the scanned images. The data typically consist of multiple images on the same domain and the goal of the research is to combine the quantitative information across images to make inference about populations or interventions. In this paper we present a unified analysis framework for the analysis of quantitative image data using a Bayesian functional mixed model approach. This framework is flexible enough to handle complex, irregular images with many local features, and can model the simultaneous effects of multiple factors on the image intensities and account for the correlation between images induced by the design. We introduce a general isomorphic modeling approach to fitting the functional mixed model, of which the wavelet-based functional mixed model is one special case. With suitable modeling choices, this approach leads to efficient calculations and can result in flexible modeling and adaptive smoothing of the salient features in the data. The proposed method has the following advantages: it can be run automatically, it produces inferential plots indicating which regions of the image are associated with each factor, it simultaneously considers the practical and statistical significance of findings, and it controls the false discovery rate.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS407 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Ordinal Probit Functional Regression Models with Application to Computer-Use Behavior in Rhesus Monkeys

Research in functional regression has made great strides in expanding to non-Gaussian functional outcomes, however the exploration of ordinal functional outcomes remains limited. Motivated by a study of computer-use behavior in rhesus macaques (\emph{Macaca mulatta}), we introduce the Ordinal Probit Functional Regression Model or OPFRM to perform ordinal function-on-scalar regression. The OPFRM is flexibly formulated to allow for the choice of different basis functions including penalized B-splines, wavelets, and O'Sullivan splines. We demonstrate the operating characteristics of the model in simulation using a variety of underlying covariance patterns showing the model performs reasonably well in estimation under multiple basis functions. We also present and compare two approaches for conducting posterior inference showing that joint credible intervals tend to out perform point-wise credible. Finally, in application, we determine demographic factors associated with the monkeys' computer use over the course of a year and provide a brief analysis of the findings

On Function-on-Scalar Quantile Regression

Existing work on functional response regression has focused predominantly on mean regression. However, in many applications, predictors may not strongly influence the conditional mean of functional responses, but other characteristics of their conditional distribution. In this paper, we study function-on-scalar quantile regression, or functional quantile regression (FQR), which can provide a comprehensive understanding of how scalar predictors influence the conditional distribution of functional responses. We introduce a scalable, distributed strategy to perform FQR that can account for intrafunctional dependence structures in the functional responses. This general distributed strategy first performs separate quantile regression to compute $M$-estimators at each sampling location, and then carries out estimation and inference for the entire coefficient functions by properly exploiting the uncertainty quantifications and dependence structures of $M$-estimators. We derive a uniform Bahadur representation and a strong Gaussian approximation result for the $M$-estimators on the discrete sampling grid, which are of independent interest and provide theoretical justification for this distributed strategy. Some large sample properties of the proposed coefficient function estimators are described. Interestingly, our rate calculations show a phase transition phenomenon that has been previously observed in functional mean regression. We conduct simulations to assess the finite sample performance of the proposed methods, and present an application to a mass spectrometry proteomics dataset, in which the use of FQR to delineate the relationship between functional responses and predictors is strongly warranted