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# Simultaneous confidence bands for the mean of functional data

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The mean function is a central object of inquiry in the analysis of functional data. Typical questions related to the mean function include quantifying estimation uncertainty, testing parametric models, and making comparisons between populations. To make probabilistic statements about the mean function over its entire domain, rather than at a single location, it is necessary to infer all of its values simultaneously. Pointwise inference is not appropriate for this task and indeed produces anticonservative results, i.e., the coverage level of confidence regions is too low and the significance level of hypothesis tests too high. In contrast, simultaneous confidence bands (SCB) provide a flexible framework for conducting simultaneous inference on the mean function and other functional parameters. They also offer powerful visualization tools for communicating analytic results to interdisciplinary audiences. The construction of SCB in the context of functional data requires specific theory and methods. In particular, it is not addressed by the nonparametric regression literature. Although software is available to perform individual steps of an SCB procedure, resources that provide end‐to‐end computations are scarce. Applications of SCB to one‐ and two‐sample inferences are illustrated here with the R package SCBmeanfd. WIREs Comput Stat 2017, 9:e1397. doi: 10.1002/wics.1397

Diffusion tensor imaging (DTI) data of 42 control subjects (left) and 334 multiple sclerosis patients (right). The x‐axis indicates location along a brain tract in the corpus callosum (arbitrary units). The y‐axis displays fractional anisotropy (FA), which measures water diffusivity and ranges in [0, 1]. Low FA values indicate possible neurological damage. The average profile (red line) is consistently lower in the patient group than in the control group. The same observation holds for the standard deviation function.
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Comparisons of mean tract profiles in diffusion tensor imaging fractional anisotropy data. Left: male control group versus female control group. Right: control group versus multiple sclerosis group (all genders).
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Significance trace of a specification test for the average aCGH profile μ of the cancer population. The null hypothesis H 0 : μ ≡ 0 (no gain/loss in DNA copy numbers) can be rejected at the 5% significance level for all bandwidths in both the normal and bootstrap tests. H 0 can be rejected at the 1% level for virtually all bandwidth values in the normal test and a relatively wide range of bandwidths in the bootstrap test.
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Simultaneous confidence bands of level 95% for the population‐level average aCGH profile of colorectal tumors. Chromosomal regions with significant gain/loss in DNA copy number are marked with black rectangles above the x‐axis. These regions may contain cancer causing genes.
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Estimated covariance (top) and correlation (bottom) functions for the brain tractography data of 42 control subjects (left) and 334 multiple sclerosis patients (right). After presmoothing the data by local linear smoothing (bandwidth h = 0.98 for the controls and h = 0.52 for the patients selected by leave‐one‐curve‐out cross validation), the population covariance (resp. correlation) function of each group was estimated by the corresponding sample covariance (resp. correlation) function. The estimated covariance functions display similar patterns in both groups although the scale (variance) is much smaller in the control group. The estimated correlation functions have similar global features in both groups but in the control group, it is rougher and closer to zero away from the diagonal.
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Comparison between pointwise confidence bands (PCBs) and simultaneous confidence bands (SCB) using artificial data. Here, X is a Gaussian process with mean function μ(t) = 10t 3 − 15t 4 + 6t 5 and covariance Γ(s, t) = (0.1)2exp(−2|st|); see the Supporting information for simulation details. The SCB covers μ whereas the PCB is too narrow and fails to cover it. For a nominal coverage level of 95% (in the pointwise sense for PCB, simultaneous sense for SCB), the observed simultaneous coverage level is 74.1 % for PCB and 92.8 % for SCB; the observed pointwise coverage level (averaged over $D$) is 94.2 % for PCB and 98.7 % for SCB.
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