Tolerance Bands for Functional Data

Abstract

Often the object of inference in biomedical applications is a range that brackets a given fraction of individual observations in a population. A classical estimate of this range for univariate measurements is a "tolerance interval." This article develops its natural extension for functional measurements, a "tolerance band," and proposes a methodology for constructing its pointwise and simultaneous versions that incorporates both sparse and dense functional data. Assuming that the measurements are observed with noise, the methodology uses functional principal component analysis in a mixed model framework to represent the measurements and employs bootstrapping to approximate the tolerance factors needed for the bands. The proposed bands also account for uncertainty in the principal components decomposition. Simulations show that the methodology has, generally, acceptable performance unless the data are quite sparse and unbalanced, in which case the bands may be somewhat liberal. The methodology is illustrated using two real datasets, a sparse dataset involving CD4 cell counts and a dense dataset involving core body temperatures.

A geometric approach for computing tolerance bounds for elastic functional data

“…Corresponding estimated confidence values using the methods in [24] and [18]. upper and lower bounds do not accurately capture the true underlying shape of the given data.…”

“…Again, comparing these bounds with those presented in Figure 5, we see some structural differences. Both the [24] 75.00 70.00 67.00 Lewis et al [18] 71.00 67.00 64.00 Table 1. Simulated confidence values of 90% coverage tolerance using the bootstrap-based approach (top).…”

“…Lastly, we compare the proposed tolerance bounds to those of Rathnayake and Choudhary [24] and Lewis et al [18] for different confidence levels at 99% coverage. These results are presented in Figure 10 for 90% (top), 95% (middle) and 99% (bottom) Figure 10.…”

“…These results are presented in Figure 10 for 90% (top), 95% (middle) and 99% (bottom) Figure 10. Comparison of the proposed method (Elastic) to [24] (Rathnayake) and [18] (Lewis). We use the simulated data to construct tolerance bounds for confidence levels 90% (top), 95% (middle), and 99% (bottom) at 99% coverage.…”

“…Recently, Storlie et al [30] developed a method to test the shape of a population of curves using a B-Spline basis, and a hierarchical Gaussian process approach to form confidence intervals. Rathnayake and Choudhary [24] developed tolerance bounds for functional data using functional Principal Component Analysis (fPCA). Further, Sun and Genton [31] developed a boxplot display for functional data, which provides a nice visualization technique for a sample of functions.…”