Annals of Statistics

Multiple testing of local maxima for detection of peaks in 1D

Armin Schwartzman, Yulia Gavrilov, and Robert J. Adler

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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.

Article information

Ann. Statist., Volume 39, Number 6 (2011), 3290-3319.

First available in Project Euclid: 5 March 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H15: Hypothesis testing
Secondary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

False discovery rate Gaussian process kernel smoothing matched filter topological inference


Schwartzman, Armin; Gavrilov, Yulia; Adler, Robert J. Multiple testing of local maxima for detection of peaks in 1D. Ann. Statist. 39 (2011), no. 6, 3290--3319. doi:10.1214/11-AOS943.

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  • Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York.
  • Adler, R. J., Taylor, J. E. and Worsley, K. J. (2010). Applications of random fields and geometry: Foundations and case studies. Available at
  • Arzeno, N. M., Deng, Z.-D. and Poon, C.-S. (2008). Analysis of first-derivative based QRS detection algorithms. IEEE Trans. Biomed. Eng. 55 478–484.
  • Baccus, S. A. and Meister, M. (2002). Fast and slow contrast adaptation in retinal circuitry. Neuron 36 909–919.
  • Benjamini, Y. and Heller, R. (2007). False discovery rates for spatial signals. J. Amer. Statist. Assoc. 102 1272–1281.
  • Benjamini, Y. and Hochberg, Y. (1995). Controlling the false discovery rate: A practical and powerful approach to multiple testing. J. Roy. Statist. Soc. Ser. B 57 289–300.
  • Brutti, P., Genovese, C. R., Miller, C. J., Nichol, R. C. and Wasserman, L. (2005). Spike hunting in galaxy spectra. Technical report, Libera Univ. Internazionale degli Studi Sociali Guido Carli di Roma. Available at
  • Candes, E. and Tao, T. (2007). The Dantzig selector: Statistical estimation when p is much larger than n. Ann. Statist. 35 2313–2351.
  • Chumbley, J. R. and Friston, K. J. (2009). False discovery rate revisited: FDR and topological inference using Gaussian random fields. Neuroimage 44 62–70.
  • Chumbley, J. R., Worsley, K., Flandin, G. and Friston, K. J. (2010). Topological fdr for neuroimaging. Neuroimage 49 3057–3064.
  • Cramér, H. and Leadbetter, M. R. (1967). Stationary and Related Stochastic Processes. Sample Function Properties and Their Applications. Wiley, New York.
  • Genovese, C. R., Lazar, N. A. and Nichols, T. E. (2002). Thresholding of statistical maps in functional neuroimaging using the false discovery rate. Neuroimage 15 870–878.
  • Harezlak, J., Wu, M. C., Wang, M., Schwartzman, A., Christiani, D. C. and Lin, X. (2008). Biomarker discovery for arsenic exposure using functional data. Analysis and feature learning of mass spectrometry proteomic data. J. Proteome Res. 7 217–224.
  • Heller, R., Stanley, D., Yekutieli, D., Rubin, N. and Benjamini, Y. (2006). Cluster-based analysis of FMRI data. Neuroimage 33 599–608.
  • Li, L. and Speed, T. P. (2000). Parametric deconvolution of positive spike trains. Ann. Statist. 28 1279–1301.
  • Li, L. M. and Speed, T. P. (2004). Deconvolution of sparse positive spikes. J. Comput. Graph. Statist. 13 853–870.
  • Morris, J. S., Coombes, K. R., Koomen, J., Baggerly, K. A. and Kobayashi, R. (2006). Feature extraction and quantification for mass spectrometry in biomedical applications using the mean spectrum. Bioinformatics 21 1764–1775.
  • Nichols, T. and Hayasaka, S. (2003). Controlling the familywise error rate in functional neuroimaging: A comparative review. Stat. Methods Med. Res. 12 419–446.
  • O’Brien, M. S., Sinclair, A. N. and Kramer, S. M. (1994). Recovery of a sparse spike train time series by l1 norm deconvolution. IEEE Trans. Signal Process. 42 3353–3365.
  • Perone Pacifico, M., Genovese, C., Verdinelli, I. and Wasserman, L. (2004). False discovery control for random fields. J. Amer. Statist. Assoc. 99 1002–1014.
  • Perone Pacifico, M., Genovese, C., Verdinelli, I. and Wasserman, L. (2007). Scan clustering: A false discovery approach. J. Multivariate Anal. 98 1441–1469.
  • Poline, J. B., Worsley, K. J., Evans, A. C. and Friston, K. J. (1997). Combining spatial extent and peak intensity to test for activations in functional imaging. Neuroimage 5 83–96.
  • Pratt, W. K. (1991). Digital Image Processing. Wiley, New York.
  • Rice, S. O. (1945). Mathematical analysis of random noise. Bell System Tech. J. 24 46–156.
  • Schwartzman, A., Dougherty, R. F. and Taylor, J. E. (2008). False discovery rate analysis of brain diffusion direction maps. Ann. Appl. Stat. 2 153–175.
  • Segev, R., Goodhouse, J., Puchalla, J. and Berry, M. J. II (2004). Recording spikes from a large fraction of the ganglion cells in a retinal patch. Nature Neuroscience 7 1155–1162.
  • Simon, M. (1995). Digital Communication Techniques: Signal Design and Detection. Prentice Hall, Englewood Cliffs, NJ.
  • Smith, S. M. and Nichols, T. E. (2009). Threshold-free cluster enhancement: Addressing problems of smoothing, threshold dependence and localisation in cluster inference. Neuroimage 44 83–98.
  • Taylor, J. E. and Worsley, K. J. (2007). Detecting sparse signals in random fields, with an application to brain mapping. J. Amer. Statist. Assoc. 102 913–928.
  • Tibshirani, R., Saunders, M., Rosset, S., Zhu, J. and Knight, K. (2005). Sparsity and smoothness via the fused lasso. J. R. Stat. Soc. Ser. B Stat. Methodol. 67 91–108.
  • Worsley, K. J., Marrett, S., Neelin, P. and Evans, A. C. (1996a). Searching scale space for activation in PET images. Human Brain Mapping 4 74–90.
  • Worsley, K. J., Marrett, S., Neelin, P., Vandal, A. C., Friston, K. J. and Evans, A. C. (1996b). A unified statistical approach for determining significant signals in images of cerebral activation. Human Brain Mapping 4 58–73.
  • Worsley, K. J., Taylor, J. E., Tomaiuolo, F. and Lerch, J. (2004). Unified univariate and multivariate random field theory. Neuroimage 23 S189–195.
  • Yasui, Y., Pepe, M., Thompson, M. L., Bao-Ling, A., Wright, J. G. L., Yinsheng, Q., Potter, J. D., Winget, M., Thornquist, M. and Ziding, F. (2003). A data-analytic strategy for protein biomarker discovery: Profiling of high-dimensional proteomic data for cancer detection. Biostatistics 4 449–463.
  • Zhang, C.-H. (2010). Nearly unbiased variable selection under minimax concave penalty. Ann. Statist. 38 894–942.
  • Zhang, H., Nichols, T. E. and Johnson, T. D. (2009). Cluster mass inference via random field theory. Neuroimage 44 51–61.