MULTIPLE TESTING OF LOCAL MAXIMA FOR DETECTION OF PEAKS IN 1D.

A topological multiple testing scheme for one-dimensional domains is proposed where, rather than testing every spatial or temporal location for the presence of a signal, tests are performed only at the local maxima of the smoothed observed sequence. Assuming unimodal true peaks with finite support and Gaussian stationary ergodic noise, it is shown that the algorithm with Bonferroni or Benjamini-Hochberg correction provides asymptotic strong control of the family wise error rate and false discovery rate, and is power consistent, as the search space and the signal strength get large, where the search space may grow exponentially faster than the signal strength. Simulations show that error levels are maintained for nonasymptotic conditions, and that power is maximized when the smoothing kernel is close in shape and bandwidth to the signal peaks, akin to the matched filter theorem in signal processing. The methods are illustrated in an analysis of electrical recordings of neuronal cell activity.

[1]  J. Potter,et al.  A data-analytic strategy for protein biomarker discovery: profiling of high-dimensional proteomic data for cancer detection. , 2003, Biostatistics.

[2]  Y. Benjamini,et al.  False Discovery Rates for Spatial Signals , 2007 .

[3]  K. Worsley,et al.  Detecting Sparse Signals in Random Fields, With an Application to Brain Mapping , 2007 .

[4]  Nava Rubin,et al.  Cluster-based analysis of FMRI data , 2006, NeuroImage.

[5]  Malcolm R Leadbetter,et al.  Stationary and Related Stochastic Processes: Sample Function Properties and Their Applications , 1967 .

[6]  R. Dougherty,et al.  FALSE DISCOVERY RATE ANALYSIS OF BRAIN DIFFUSION DIRECTION MAPS. , 2008, The annals of applied statistics.

[7]  S. Rice.  Mathematical analysis of random noise , 1944 .

[8]  Michael J. Berry,et al.  Recording spikes from a large fraction of the ganglion cells in a retinal patch , 2004, Nature Neuroscience.

[9]  Marvin K. Simon,et al.  Digital Communication Techniques: Signal Design and Detection , 2008 .

[10]  harald Cramer,et al.  Stationary And Related Stochastic Processes , 1967 .

[11]  Jeffrey S. Morris,et al.  Feature extraction and quantification for mass spectrometry in biomedical applications using the mean spectrum , 2005, Bioinform..

[12]  David C Christiani,et al.  Biomarker discovery for arsenic exposure using functional data. Analysis and feature learning of mass spectrometry proteomic data. , 2008, Journal of proteome research.

[13]  Y. Benjamini,et al.  Controlling the false discovery rate: a practical and powerful approach to multiple testing , 1995 .

[14]  Stephen M. Smith,et al.  Threshold-free cluster enhancement: Addressing problems of smoothing, threshold dependence and localisation in cluster inference , 2009, NeuroImage.

[15]  Hui Zhang,et al.  Cluster mass inference via random field theory , 2009, NeuroImage.

[16]  T. Speed,et al.  Deconvolution of Sparse Positive Spikes , 2004 .

[17]  Cun-Hui Zhang.  Nearly unbiased variable selection under minimax concave penalty , 2010, 1002.4734.

[18]  I. Verdinelli,et al.  False Discovery Control for Random Fields , 2004 .

[19]  R. Adler,et al.  Random Fields and Geometry , 2007 .

[20]  Karl J. Friston,et al.  Combining Spatial Extent and Peak Intensity to Test for Activations in Functional Imaging , 1997, NeuroImage.

[21]  Karl J. Friston,et al.  A unified statistical approach for determining significant signals in images of cerebral activation , 1996, Human brain mapping.

[22]  Alan C. Evans,et al.  Searching scale space for activation in PET images , 1996, Human brain mapping.

[23]  Anthony N. Sinclair,et al.  Recovery of a sparse spike time series by L1 norm deconvolution , 1994, IEEE Trans. Signal Process..

[24]  R. Tibshirani,et al.  Sparsity and smoothness via the fused lasso , 2005 .

[25]  Thomas E. Nichols,et al.  Controlling the familywise error rate in functional neuroimaging: a comparative review , 2003, Statistical methods in medical research.

[26]  Christopher R. Genovese,et al.  Scan clustering: A false discovery approach , 2007 .

[27]  K. Worsley,et al.  Unified univariate and multivariate random field theory , 2004, NeuroImage.

[28]  Thomas E. Nichols,et al.  Thresholding of Statistical Maps in Functional Neuroimaging Using the False Discovery Rate , 2002, NeuroImage.

[29]  Karl J. Friston,et al.  False discovery rate revisited: FDR and topological inference using Gaussian random fields , 2009, NeuroImage.

[30]  Terence Tao,et al.  The Dantzig selector: Statistical estimation when P is much larger than n , 2005, math/0506081.

[31]  M. Meister,et al.  Fast and Slow Contrast Adaptation in Retinal Circuitry , 2002, Neuron.

[32]  T. Speed,et al.  Parametric deconvolution of positive spike trains , 2000 .

[33]  Chi-Sang Poon,et al.  Analysis of First-Derivative Based QRS Detection Algorithms , 2008, IEEE Transactions on Biomedical Engineering.

[34]  Christopher J. Miller,et al.  Spike Hunting in Galaxy Spectra , 2005 .

[35]  Karl J. Friston,et al.  Topological FDR for neuroimaging , 2010, NeuroImage.