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Functional Additive Regression
, 2011
"... We suggest a new method, called “Functional Additive Regression”, or FAR, for efficiently performing high dimensional functional regression. FAR extends the usual linear regression model involving a functional predictor, X(t), and a scalar response, Y, in two key respects. First, FAR uses a penalize ..."
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We suggest a new method, called “Functional Additive Regression”, or FAR, for efficiently performing high dimensional functional regression. FAR extends the usual linear regression model involving a functional predictor, X(t), and a scalar response, Y, in two key respects. First, FAR uses a penalized least squares optimization approach to efficiently deal with high dimensional problems involving a large number of different functional predictors. Second, FAR extends beyond the standard linear regression setting to fit general nonlinear additive models. We demonstrate that FAR can be implemented with a wide range of penalty functions using a highly efficient coordinate descent algorithm. Theoretical results are developed which provide motivation for the FAR optimization criterion. Finally, we show through simulations and two real data sets that FAR can significantly outperform competing methods.
Methods for scalaronfunction regression
, 2015
"... Recent years have seen an explosion of activity in the field of functional data analysis (FDA), in which curves, spectra, images, etc. are considered as basic functional data units. A central problem in FDA is how to fit regression models with scalar responses and functional data points as predictor ..."
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Recent years have seen an explosion of activity in the field of functional data analysis (FDA), in which curves, spectra, images, etc. are considered as basic functional data units. A central problem in FDA is how to fit regression models with scalar responses and functional data points as predictors. We review some of the main approaches to this problem, categorizing the basic model types as linear, nonlinear and nonparametric. We discuss publicly available software packages, and illustrate some of the procedures by application to a functional magnetic resonance imaging dataset.
Functional Response Additive Model Estimation with Online Virtual Stock Markets
, 2012
"... While functional regression models have received increasing attention recently, most existing approaches assume both a linear relationship and a scalar response variable. We suggest a new method which extends the usual linear regression model to situations involving both functional responses, Xj(t), ..."
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While functional regression models have received increasing attention recently, most existing approaches assume both a linear relationship and a scalar response variable. We suggest a new method which extends the usual linear regression model to situations involving both functional responses, Xj(t), and functional predictors, Y (t). Our approach uses a penalized least squares optimization criterion to automatically perform variable selection in situations involving multiple functional predictors. In addition our method uses an efficient coordinate descent algorithm to fit general nonlinear additive relationships between the predictors and response. We apply our model to the context of forecasting product demand in the entertainment industry. In particular, we model the decay rate of demand for Hollywood movies using the predictive power of online virtual stock markets (VSMs). VSMs are online communities that, in a marketlike fashion, gather the crowds ’ opinion about a particular product. Our fully functional model captures the pattern of prerelease VSM trading values and provides superior predictive accuracy of a movie’s demand distribution in comparison to traditional methods. In addition, we propose graphical tools which give a glimpse into the causal relationship between market behavior and box office revenue patterns and hence provide valuable insight to movie decision makers. Key words and phrases: Functional data; nonlinear regression; penalty functions; forecasting; virtual markets; movies; Hollywood. 1
Index Models for Sparsely Sampled Functional Data
"... The regression problem involving functional predictors has many important applications and a number of functional regression methods have been developed. However, a common complication in functional data analysis is one of sparsely observed curves, that is predictors that are observed, with error, o ..."
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The regression problem involving functional predictors has many important applications and a number of functional regression methods have been developed. However, a common complication in functional data analysis is one of sparsely observed curves, that is predictors that are observed, with error, on a small subset of the possible time points. Such sparsely observed data induces an errorsinvariables model where one must account for measurement error in the functional predictors. Faced with sparsely observed data, most current functional regression methods simply estimate the unobserved predictors and treat them as fully observed; thus failing to account for the extra uncertainty from the measurement error. We propose a new functional errorsinvariables approach, “Sparse Index Model Functional Estimation ” (SIMFE), which uses a functional index model formulation to deal with sparsely observed predictors. SIMFE has several advantages over more traditional methods. First, the index model implements a nonlinear regression and uses an accurate supervised method to estimate the lower dimensional space into which the predictors should be projected. Second, SIMFE can be applied to both scalar and functional responses and multiple predictors. Finally, SIMFE uses a mixed effects model to effectively deal with very sparsely observed functional predictors and to correctly model the measurement error.
Index Models for Sparsely Sampled Functional Data
, 2014
"... The regression problem involving functional predictors has many important applications and a number of functional regression methods have been developed. However, a common complication in functional data analysis is one of sparsely observed curves, that is predictors that are observed, with error, ..."
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The regression problem involving functional predictors has many important applications and a number of functional regression methods have been developed. However, a common complication in functional data analysis is one of sparsely observed curves, that is predictors that are observed, with error, on a small subset of the possible time points. Such sparsely observed data induces an errorsinvariables model, where one must account for measurement error in the functional predictors. Faced with sparsely observed data, most current functional regression methods simply estimate the unobserved predictors and treat them as fully observed; thus failing to account for the extra uncertainty from the measurement error. We propose a new functional errorsinvariables approach, Sparse Index Model Functional Estimation (SIMFE), which uses a functional index model formulation to deal with sparsely observed predictors. SIMFE has several advantages over more traditional methods. First, the index model implements a nonlinear regression and uses an accurate supervised method to estimate the lower dimensional space into which the predictors should be projected. Second, SIMFE can be applied to both scalar and functional responses and multiple predictors. Finally, SIMFE uses a mixed effects model to effectively deal with very sparsely observed functional predictors and to correctly model the measurement error.