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

Fast Covariance Estimation for High-dimensional Functional Data

For smoothing covariance functions, we propose two fast algorithms that scale linearly with the number of observations per function. Most available methods and software cannot smooth covariance matrices of dimension $J \times J$ with $J>500$; the recently introduced sandwich smoother is an exception, but it is not adapted to smooth covariance matrices of large dimensions such as $J \ge 10,000$. Covariance matrices of order $J=10,000$, and even $J=100,000$, are becoming increasingly common, e.g., in 2- and 3-dimensional medical imaging and high-density wearable sensor data. We introduce two new algorithms that can handle very large covariance matrices: 1) FACE: a fast implementation of the sandwich smoother and 2) SVDS: a two-step procedure that first applies singular value decomposition to the data matrix and then smoothes the eigenvectors. Compared to existing techniques, these new algorithms are at least an order of magnitude faster in high dimensions and drastically reduce memory requirements. The new algorithms provide instantaneous (few seconds) smoothing for matrices of dimension $J=10,000$ and very fast ($<$ 10 minutes) smoothing for $J=100,000$. Although SVDS is simpler than FACE, we provide ready to use, scalable R software for FACE. When incorporated into R package {\it refund}, FACE improves the speed of penalized functional regression by an order of magnitude, even for data of normal size ($J <500$). We recommend that FACE be used in practice for the analysis of noisy and high-dimensional functional data.Comment: 35 pages, 4 figure

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

Identifying Asset Poverty Thresholds New methods with an application to Pakistan and Ethiopia

Understanding how households escape poverty depends on understanding how they accumulate assets over time. Therefore, identifying the degree of linearity in household asset dynamics, and specifically any potential asset poverty thresholds, is of fundamental interest to the design of poverty reduction policies. If household asset holdings converged unconditionally to a single long run equilibrium, then all poor could be expected to escape poverty over time. In contrast, if there are critical asset thresholds that trap households below the poverty line, then households would need specific assistance to escape poverty. Similarly, the presence of asset poverty thresholds would mean that short term asset shocks could lead to long term destitution, thus highlighting the need for social safety nets. In addition to the direct policy relevance, identifying household asset dynamics and potential asset thresholds presents an interesting methodological challenge to researchers. Potential asset poverty thresholds can only be identified in a framework that allows multiple dynamic equilibria. Any unstable equilibrium points would indicate a potential poverty threshold, above which households are expected to accumulate further and below which households are on a trajectory that makes them poorer over time. The key empirical issue addressed in the paper is whether such threshold points exist in Pakistan and Ethiopia and, if so, where they are located. Methodologically, the paper explores what econometric technique is best suited for this type of analysis. The paper contributes to the small current literature on modeling nonlinear household welfare dynamics in three ways. First, it compares previously used techniques for identifying asset poverty traps by applying them to the same dataset, and examines whether, and how, the choice of estimation technique affects the result. Second, it explores whether other estimation techniques may be more suitable to locate poverty thresholds. Third, it adds the first study for a South Asian country and makes a comparison with Ethiopia. Household assets are combined into a single asset index using two techniques: factor analysis and regression. These indices are used to estimate asset dynamics and locate dynamic asset equilibria, first by nonparametric methods including LOWESS, kernel weighted local regression and spline smoothers, and then by global polynomial parametric techniques. To combine the advantages of nonparametric and parametric techniques - a flexible functional form and the ability to control for covariates, respectively - the paper adapts a mixed model representation of a penalized spline to estimate asset dynamics through a semiparametric partially linear model. This paper identifies a single dynamic asset equilibrium with a slightly concave dynamic asset accumulation path in each country. There is no evidence for multiple dynamic equilibria. This result is robust across econometric methods and across different ways of constructing the asset index. The concave accumulation path means that poorer households recover more slowly from asset shocks. Concavity also implies that greater initial equality of assets would lead to higher growth. Moreover, the dynamic asset equilibria are very low. In Pakistan it is below the average asset holdings of the poor households in the sample. In Ethiopia, the equilibrium is barely above the very low mean. This, together with the slow speed of asset accumulation for the poorest households, suggests that convergence towards the long run equilibrium may be slow and insufficient for rural households in Pakistan and Ethiopia to escape poverty.Poverty dynamics, Semiparametric Estimation, Penalized Splines, Pakistan, Ethiopia, Consumer/Household Economics, I32, C14, O12,