Optimal Classification for Functional Data

Abstract

A central topic in functional data analysis is how to design an optimal decision rule, based on training samples, to classify a data function. We exploit the optimal classification problem when data functions are Gaussian processes. Sharp convergence rates for minimax excess misclassification risk are derived in both settings that data functions are fully observed and discretely observed. We explore two easily implementable classifiers based on discriminant analysis and deep neural network, respectively, which are both proven to achieve optimality in Gaussian settings. Our deep neural network classifier is new in literature which demonstrates outstanding performance even when data functions are non-Gaussian. In case of discretely observed data, we discover a novel critical sampling frequency that governs the sharp convergence rates. The proposed classifiers perform favorably in finite-sample applications, as we demonstrate through comparisons with other functional classifiers in simulations and one real data application.

Deep neural network classifier for multidimensional functional data

“…On the other hand, under functional Gaussian data, researchers have proposed various functional classifiers, including functional quadratic discriminant analysis (FQDA) (Berrendero et al, 2018; Cai & Zhang, 2019a, 2019b; Dai et al, 2017; Delaigle et al, 2012; Delaigle & Hall, 2012, 2013; Galeano et al, 2015; Park et al, 2020; Shin, 2008). Gaussianity leads to a linear or quadratic polynomial $$$ {Q}^{\ast } $$$ which can be effectively estimated by FQDA, based on which Wang, Shang, et al (2021) showed that FQDA is minimax optimal. It is still unclear how to design optimal functional classifiers when data are non‐Gaussian, a gap that the present article attempts to close.…”

“…For instance, in either high‐ or infinite‐dimensional Gaussian data, it is well known that density ratio between two Gaussian population densities has an explicit expression in terms of mean difference and variance ratio, which impacts the sharp rate of MEMR. More precisely, in high‐dimensional Gaussian data classification, the rate depends on the number of nonzero components of mean difference vector (Cai & Zhang, 2019a, 2019b); in Gaussian functional data classification, the rate depends on the decay orders of both mean difference series and variance ratio series (Wang, Shang, et al, 2021). Nonetheless, in general non‐Gaussian case, likelihood ratio does not have an explicit expression, therefore, one cannot simply use mean or variance discrepancy to characterize the sharp rate of MEMR.…”

“…In this paper, we propose a new approach, called as functional deep neural network (FDNN), for multidimensional functional data classification. A mainstream technique in functional data classification is based on functional principal component analysis (FPCA) such as functional discriminant analysis (Adrover et al, 2004; Berrendero et al, 2018; Dai et al, 2017; Delaigle et al, 2012; Delaigle & Hall, 2012, 2013; Galeano et al, 2015; Park et al, 2020; Shin, 2008; Wang, Shang, et al, 2021). We start from FPCA to extract the functional principal components of the data functions, and then train a DNN‐based classifier on these FPCs as well as their corresponding class labels.…”

Minimax optimal high-dimensional classification using deep neural networks

“…Hence, t u can be viewed as an intrinsic dimension of g u . Structure (6) has been adopted by [21,20,25,14,17] in multivariate regression using deep learning to address the "curse of dimensionality." Examples of ( 6) include generalized additive model [9,16], tensor product space ANOVA model [15], among others.…”

Minimax optimal high‐dimensional classification using deep neural networks

“…Hence, can be viewed as an intrinsic dimension of . Structure (6) has been adopted by Schmidt‐Hieber (2020), Polson and Roˇcková (2018), Wang et al (2021), Li et al (2021) and Liu et al (2021) in multivariate regression using deep learning to address the “curse of dimensionality.” Examples of (6) include generalized additive models (Hastie & Tibshirani, 1990; Liu et al, 2022), tensor product space ANOVA model (Lin, 2000), among others. Specifically, the former corresponds to with being univariate smooth functions, being a set of indexes and so that , , , ; the latter corresponds to …”