[Paper Review] Peak Detection as Multiple Testing

Armin Schwartzman, Yulia Gavrilov+1 more|arXiv (Cornell University)|Aug 11, 2010

Statistical Methods and Inference25 references6 citations

TL;DR

This paper proposes a multiple testing framework for detecting equal-shaped, non-overlapping unimodal peaks in noisy data, using kernel smoothing followed by p-value computation at local maxima. It achieves strong control of family-wise error rate and false discovery rate asymptotically, with optimal detection power when smoothing bandwidth matches peak width.

ABSTRACT

This paper considers the problem of detecting equal-shaped non-overlapping unimodal peaks in the presence of Gaussian ergodic stationary noise, where the number, location and heights of the peaks are unknown. A multiple testing approach is proposed in which, after kernel smoothing, the presence of a peak is tested at each observed local maximum. The procedure provides strong control of the family wise error rate and the false discovery rate asymptotically as both the signal-to-noise ratio (SNR) and the search space get large, where the search space may grow exponentially as a function of SNR. Simulations assuming a Gaussian peak shape and a Gaussian autocorrelation function show that desired error levels are achieved for relatively low SNR and are robust to partial peak overlap. Simulations also show that detection power is maximized when the smoothing bandwidth is close to the bandwidth of the signal peaks, akin to the well-known matched filter theorem in signal processing. The procedure is illustrated in an analysis of electrical recordings of neuronal cell activity.

Motivation & Objective

• Develop a general, statistically rigorous peak detection method that controls global error rates in the presence of unknown peak numbers and locations.
• Address the limitation of ad-hoc thresholds in existing algorithms by formalizing peak detection as a multiple testing problem.
• Ensure strong error control (FWER and FDR) under Gaussian ergodic stationary noise with asymptotic validity as signal-to-noise ratio and search space grow.
• Provide a computationally efficient and scalable procedure suitable for large-scale data, such as genomic or neuronal recordings.
• Establish theoretical guarantees for error control and detection power under realistic assumptions of equal-shaped, non-overlapping unimodal peaks.

Proposed method

• Apply kernel smoothing to the observed data to enhance signal-to-noise ratio and reduce dimensionality.
• Identify candidate peaks as local maxima in the smoothed sequence to reduce the number of tests from all data points to only potential peak locations.
• Compute p-values at each local maximum using the distribution of peak heights under the null hypothesis of pure noise, derived via Palm calculus for Gaussian processes.
• Apply multiple testing corrections—Bonferroni for family-wise error rate control and Benjamini-Hochberg for false discovery rate control—using the computed p-values.
• Use asymptotic approximations for the thresholding procedures to ensure theoretical error control as the signal-to-noise ratio and search space grow.
• Derive theoretical conditions under which the error rates are controlled and detection power converges to one.

Experimental results

Research questions

• RQ1 Can peak detection be formalized as a multiple testing problem to ensure rigorous error control?
• RQ2 Does the proposed method achieve strong control of both family-wise error rate and false discovery rate under realistic noise assumptions?
• RQ3 How does the choice of smoothing bandwidth affect detection power, and is there an optimal bandwidth matching the signal peak width?
• RQ4 What is the asymptotic behavior of the error rates and detection power as the signal-to-noise ratio and search space increase?
• RQ5 How robust is the method to partial peak overlap and low signal-to-noise ratios in practical settings?

Key findings

• The proposed method achieves strong asymptotic control of both family-wise error rate and false discovery rate under Gaussian ergodic stationary noise.
• Desired error levels are achieved even at relatively low signal-to-noise ratios, as demonstrated by simulations with Gaussian peak shapes and autocorrelated noise.
• Detection power is maximized when the smoothing bandwidth is close to the true bandwidth of the signal peaks, consistent with the matched filter theorem.
• The method is robust to partial peak overlap, maintaining accurate error control and high detection power in overlapping peak scenarios.
• Theoretical analysis confirms that detection power converges to one in probability when using deterministic thresholds, and this result extends to random thresholds due to their asymptotic equivalence.
• Simulations show that the Benjamini-Hochberg procedure maintains FDR control asymptotically, with the threshold converging to a fixed limit under a fixed proportion of true peaks.