Robust domain selection for functional data via interval-wise testing and effect size mapping

Abstract

Among inferential problems in functional data analysis, domain selection is one of the practical interests aiming to identify sub-interval(s) of the domain where desired functional features are displayed. Motivated by applications in quantitative ultrasound signal analysis, we propose the robust domain selection method, particularly aiming to discover a subset of the domain presenting distinct behaviors on location parameters among different groups. By extending the interval testing approach, we propose to take into account multiple aspects of functional features simultaneously to detect the practically interpretable domain. To further handle potential outliers and missing segments on collected functional trajectories, we perform interval testing with a test statistic based on functional M-estimators for the inference. In addition, we introduce the effect size heatmap by calculating robustified effect sizes from the lowest to the largest scales over the domain to reflect dynamic functional behaviors among groups so that clinicians get a comprehensive understanding and select practically meaningful sub-interval(s). The performance of the proposed method is demonstrated through simulation studies and an application to motivating quantitative ultrasound measurements.

Figures (6)

• Figure 1: BSC versus frequency from EHS and LMTK tumors
• Figure 2: Examples of functional location parameters from two groups
• Figure 3: Illustrations of effect size heatmaps for $\ell=0$ (second row) and $\ell=0$ (third row) from various scenarios, calculated under \ref{['eqn:es']} from fine to coarse scales of $\Delta$. Effect size values within the triangle region are calculated based on the symmetric choice of lower and upper extremes, where the corresponding $\Delta$ in the y-axis exactly matches the length of the interval integrated to obtain $G^2_{fSNR}(t; \Delta)$. Values outside the triangle region are derived from asymmetric lower and upper extremes, implying the actual integrated interval length is shorter than $\Delta$.
• Figure 4: Illustration of fully observed trajectories from the scenarios of Gaussian process, $t_3$ process, curve outlier, and local outlier under the noise level $\sigma_e=3$ (top row). Boxplots of sensitivity from the proposed method and the comparison method, two variations of CTP-based method, under fully and partially observed trajectories and noise levels $\sigma_e = 1, 2, 3, 4$, when sample size $n = 50$.
• Figure 5: (Top row) Simulated partially observed trajectories under $t_3$ error processes and $n=100$, where $\sigma_e=1,\ldots,4$, respectively, where black and blue bold lines indicate true group location parameters featuring distinction over $t \in (0.34,1]$ with a dotted vertical line locating at $t=0.34$; (Second rows) Adjusted p-value functions $\tilde{p}_{D^0}(t)$ and $\tilde{p}_{D^1}(t)$ for testing on equality of two group parameters from trajectories of raw and the first-order derivatives. The gray regions illustrate the selected intervals under the proposed method; (Third and fourth rows) The robust effect size heatmaps under $\ell=0$ and 1, respectively. Effect size values within the triangle region are calculated based on the symmetric choice of lower and upper extremes (integrated interval length equals to $\Delta$), while values outside the triangle region are derived from asymmetric lower and upper extremes (integrated interval length shorter than $\Delta$).

Theorems & Definitions (3)

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