Multiple testing of local maxima for detection of peaks in random fields

Abstract

A topological multiple testing scheme is presented for detecting peaks in images under stationary ergodic Gaussian noise, where tests are performed at local maxima of the smoothed observed signals. The procedure generalizes the one-dimensional scheme of [20] to Euclidean domains of arbitrary dimension. Two methods are developed according to two different ways of computing p-values: (i) using the exact distribution of the height of local maxima [6], available explicitly when the noise field is isotropic; (ii) using an approximation to the overshoot distribution of local maxima above a pre-threshold [6], applicable when the exact distribution is unknown, such as when the stationary noise field is non-isotropic. The algorithms, combined with the Benjamini-Hochberg procedure for thresholding p-values, provide asymptotic strong control of the False Discovery Rate (FDR) and power consistency, with specific rates, as the search space and signal strength get large. The optimal smoothing bandwidth and optimal pre-threshold are obtained to achieve maximum power. Simulations show that FDR levels are maintained in non-asymptotic conditions. The methods are illustrated in a nanoscopy image analysis problem of detecting fluorescent molecules against the image background.

Multiple Testing with the Structure-Adaptive Benjamini–Hochberg Algorithm

“…() and Cheng and Schwartzman () have developed a clusterwise FDR, which treats discoveries at the cluster level rather than counting individual points; in Chouldechova () this is done by modelling the detected cluster sizes for null versus non‐null clusters to adjust the usual pointwise FDR bounds and to obtain clusterwise FDR guarantees, whereas Schwartzman et al . () proceeded by testing only at the local maxima of the sequence (for a one‐dimensional spatial setting), and Cheng and Schwartzman () developed this idea further by testing extreme values of the derivative of the sequence, rather than the values of the sequence itself. The problem of detecting data‐determined clusters of non‐nulls was studied also by Siegmund et al .…”

Peak p-values and false discovery rate inference in neuroimaging

“…Suppose that the observed field isz(s) = σz(s) with mean 0 and variance σ 2 . Following definition (6), it is shown by Cheng and Schwartzman (2017) that Middle column: Histogram of peak heights over 10,000 simulated fields; superimposed is the theoretical peak height density with κ = 1. Right column: Empirical p-value distribution with confidence envelope.…”

“…As rightly pointed out by Friston (2009) andChumbley et al (2010), convolution with a kernel artificially extends the signal beyond its original domain. This additional extent is called transition region in Cheng and Schwartzman (2017). Different solutions to this problem have been offered in the literature.…”

“…The simulations in Cheng and Schwartzman (2017) were limited to two dimensions. To evaluate the methods in a situation closer to neuroimaging, we present here simulation Small support Large support Figure 9: Projection of the signal along the axis perpendicular to the page; in reality the three bumps belong to different slices.…”

Multiple testing of local maxima for detection of peaks on the (celestial) sphere

“…framework, in particular when noise is modeled as a stochastic process or a random field. In this setting, important progresses have been recently obtained combining ideas from two different streams of research, namely techniques from the multiple testing literature, such as False Discovery Rate (FDR) algorithms, and techniques to investigate excursion probabilities and local maxima for random fields; we refer for instance to [1,43,9,10] for further background and discussion. These works have covered applications in a univariate and multivariate Euclidean setting; analytic properties have been derived under a large sample asymptotic framework, i.e., under the assumption that the domain of observations is growing steadily, together with the signals to be detected.…”