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

Fused kernel-spline smoothing for repeatedly measured outcomes in a generalized partially linear model with functional single index

We propose a generalized partially linear functional single index risk score model for repeatedly measured outcomes where the index itself is a function of time. We fuse the nonparametric kernel method and regression spline method, and modify the generalized estimating equation to facilitate estimation and inference. We use local smoothing kernel to estimate the unspecified coefficient functions of time, and use B-splines to estimate the unspecified function of the single index component. The covariance structure is taken into account via a working model, which provides valid estimation and inference procedure whether or not it captures the true covariance. The estimation method is applicable to both continuous and discrete outcomes. We derive large sample properties of the estimation procedure and show a different convergence rate for each component of the model. The asymptotic properties when the kernel and regression spline methods are combined in a nested fashion has not been studied prior to this work, even in the independent data case.Comment: Published at http://dx.doi.org/10.1214/15-AOS1330 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Regression modeling for digital test of ΣΔ modulators

The cost of Analogue and Mixed-Signal circuit testing is an important bottleneck in the industry, due to timeconsuming verification of specifications that require state-ofthe- art Automatic Test Equipment. In this paper, we apply the concept of Alternate Test to achieve digital testing of converters. By training an ensemble of regression models that maps simple digital defect-oriented signatures onto Signal to Noise and Distortion Ratio (SNDR), an average error of 1:7% is achieved. Beyond the inference of functional metrics, we show that the approach can provide interesting diagnosis information.Ministerio de Educación y Ciencia TEC2007-68072/MICJunta de Andalucía TIC 5386, CT 30

Non-parametric detection and estimation of structural change

SUMMARY: We propose a semi-non-parametric approach to the estimation and testing of structural change in time series regression models. Under the null of a given set of the coefficients being constant, we develop estimators of both the time-varying (non-parametric) and constant (parametric) components. Given the estimators under null and alternative, generalized F and Wald tests are developed. The asymptotic distributions of the estimators and test statistics are derived. A simulation study examines the finite-sample performance of the estimators and tests. The techniques are employed in the analysis of structural change in the US productivity and the Eurodollar term structure

GAMLSS for high-dimensional data – a flexible approach based on boosting

Generalized additive models for location, scale and shape (GAMLSS) are a popular semi-parametric modelling approach that, in contrast to conventional GAMs, regress not only the expected mean but every distribution parameter (e.g. location, scale and shape) to a set of covariates. Current fitting procedures for GAMLSS are infeasible for high-dimensional data setups and require variable selection based on (potentially problematic) information criteria. The present work describes a boosting algorithm for high-dimensional GAMLSS that was developed to overcome these limitations. Specifically, the new algorithm was designed to allow the simultaneous estimation of predictor effects and variable selection. The proposed algorithm was applied to data of the Munich Rental Guide, which is used by landlords and tenants as a reference for the average rent of a flat depending on its characteristics and spatial features. The net-rent predictions that resulted from the high-dimensional GAMLSS were found to be highly competitive while covariate-specific prediction intervals showed a major improvement over classical GAMs