Spatial Confidence Regions for Piecewise Continuous Processes

TL;DR

This work develops a novel convergence framework, convergence with restraint, for piecewise continuous processes to enable confidence regions for spatial excursion sets without requiring differentiability. It introduces suprema-preserving mappings and the concept of confinement to extend CRs to piecewise limits and operations such as intersections, unions, and symmetric differences. By deriving restrained central limit theorems and a SCoRE-style CR theory, the paper broadens spatial inference for neuroimaging, climatology, and cosmology settings where discontinuities and nonorientable piecewise behavior arise. The results provide practical, quantifiable uncertainty quantification for complex spatial predicates, with potential extensions to more general partitions and further connections to Skorokhod-type convergence.

Abstract

In scientific disciplines such as neuroimaging, climatology, and cosmology it is useful to study the uncertainty of excursion sets of imaging data. While the case of imaging data obtained from a single study condition has already been intensively studied, confidence statements about the intersection, or union, of the excursion sets derived from different subject conditions have only been introduced recently. Such methods aim to model the images from different study conditions as asymptotically Gaussian random processes with differentiable sample paths. In this work, we remove the restricting condition of differentiability and only require continuity of the sample paths. This allows for a wider range of applications including many settings which cannot be treated with the existing theory. To achieve this, we introduce a novel notion of convergence on piecewise continuous functions over finite partitions. This notion is of interest in its own right, as it implies convergence results for maxima of sequences of piecewise continuous functions over sequences of sets. Generalizing well-known results such as the extended continuous mapping theorem, this novel convergence notion also allows us to construct for the first time confidence regions for mathematically challenging examples such as symmetric differences of excursion sets.

Figures (13)

• Figure 1: (a) A spatially varying signal $\mu:\mathbb{R}^N\rightarrow\mathbb{R}$ (blue) thresholded at the levels $c=0$ and $c=1$ (purple). The sets $\mathcal{U}_c=\{s: \mu(s)>c\}$ and $\mathcal{L}_c=\{s: \mu(s)<c\}$, $c \in \{0,1\}$, are displayed below in dark red and dark blue, respectively. In practical settings, the signal $\mu$ and sets $\mathcal{U}_c$ and $\mathcal{L}_c$ are unknown and of empirical interest. (b) The sets $\mathcal{U}_1$ and $\mathcal{L}_1$, alongside potential CRs $\hat{\mathcal{U}}_1$ and $\hat{\mathcal{L}}_1$. In this case, since the level set $\{s: \mu(s)=1\}$ is a plateau, the boundaries of $\hat{\mathcal{U}}_1$ and $\hat{\mathcal{L}}_1$ do not resemble one another. (c) Potential CRs $\hat{\mathcal{U}}_0$ and $\hat{\mathcal{L}}_0$ overlayed on $\mathcal{U}_0$ and $\mathcal{L}_0$. As the boundaries of $\hat{\mathcal{U}}_0$ and $\hat{\mathcal{L}}_0$ are close to one another (both strongly resemble the circular level set on the right hand side of the image), we can infer that the point estimate for the level set, $\hat{\mu}_n^{-1}(0)$ (not depicted), which will lie within the 'circular band' $\mathcal{S}\setminus(\hat{\mathcal{U}}_0\cup\hat{\mathcal{L}}_0)$ by construction, is a reliable estimate.
• Figure 2: Left: the function $\gamma$ and an instance of $\hat{\gamma}_n$. Right: an instance of the noise process $\hat{G}_n$, alongside $-\hat{G}_n$ (dashed). In both plots vertical lines highlight the points at which $\gamma=0$.
• Figure 3: Left: An instance of the limiting noise process $G$, alongside $-G$. Right: The corresponding instance of the limiting process $H$. Whenever $\gamma$ changes sign this process switches from $-G$ to $+G$, from which it can be seen that $H$ is discontinuous with probability one.
• Figure 4: An illustration of $f$ and $\{f_n\}$ for $n=1,2,5,10$ and $100$ in Example \ref{['example:basic']}. Also shown is a shaded blue region of uniform width $\epsilon$ about $f$. If it were the case that $f_n\rightarrow f$ uniformly, $f_n$ would eventually have to lie within the shaded blue region. However, in this example, for any $n$, $f_n$ will always leave the shaded blue region near $s=0.5$.
• Figure 5: An illustration of $f$ and $\{f_n\}$ for $n=1,2$ and $5$ in Example \ref{['example:bad_converge']}. Also shown is a shaded blue region of uniform width $\epsilon$ about the graph of $f$. For any $n$, $f_n$ will always leave the shaded blue region near $s=0.5$ and therefore $f_n$ does not converge to $f$ uniformly.

Theorems & Definitions (63)

• Example 1
• Definition 1: Piecewise Continuous Function
• Example 2
• Example 3
• Example 4
• Example 5
• Theorem 1: Properties of Pointwise Convergence
• proof
• Definition 2: Restrained Bound
• Theorem 2: Sums of Restrained Bounds