Functional support vector machine

Abstract

Linear and generalized linear scalar-on-function modeling have been commonly used to understand the relationship between a scalar response variable (e.g. continuous, binary outcomes) and functional predictors. Such techniques are sensitive to model misspecification when the relationship between the response variable and the functional predictors is complex. On the other hand, support vector machines (SVMs) are among the most robust prediction models but do not take account of the high correlations between repeated measurements and cannot be used for irregular data. In this work, we propose a novel method to integrate functional principal component analysis with SVM techniques for classification and regression to account for the continuous nature of functional data and the nonlinear relationship between the scalar response variable and the functional predictors. We demonstrate the performance of our method through extensive simulation experiments and two real data applications: the classification of alcoholics using electroencephalography signals and the prediction of glucobrassicin concentration using near-infrared reflectance spectroscopy. Our methods especially have more advantages when the measurement errors in functional predictors are relatively large.

Keywords: EEG; functional data analysis; functional principal component analysis; scalar-on-function modeling; support vector machine.

Figure 1:

Simulation performance for binary outcomes of Scenarios 1–2. Left column: ${\eta }_{ij}\sim N(0,1)$; right column: ${\eta }_{ij}\sim N(0,10)$.

Figure 2:

Simulation performance for binary outcomes of Scenarios 3–5. Left column: ${\eta }_{ij}\sim N(0,1)$; right column: ${\eta }_{ij}\sim N(0,10)$.

Figure 3:

Simulation performance for continuous outcomes of Scenarios 1–4. Left column: ${\eta }_{ij}\sim N(0,1)$; Right column: ${\eta }_{ij}\sim N(0,10)$. $†$: the RMSEs of SVR linear were large and exceeded the figure when $N=50,J=40$.

Figure 4:

Waveform, classification accuracy, and AUC of EEG study. a) and b) Amplitude of the difference between matching condition and nonmatching condition of all subjects at P1 and P5 electrodes at 1 to 256 Hz, respectively. Solid lines: alcoholics; Dashed lines: controls. c) and d) Accuracy of classifying alcoholics and controls by P1 and P5 waveforms across 100 bootstrap samples. e) and f) AUC of classifying alcoholics and controls by P1 and P5 waveforms across 100 bootstrap samples.

Figure 5:

NIR spectra and root mean square error (RMSE) of NIR study. a) NIR spectra of all samples. Solid line: NIR spectra for each sample; Dashed line: average NIR spectra over samples. b) and c) RMSE of predicting dry and fresh weight GBS concentrations across 100 bootstrap samples.