• Bernoulli
  • Volume 22, Number 4 (2016), 2113-2142.

The circular SiZer, inferred persistence of shape parameters and application to early stem cell differentiation

Stephan Huckemann, Kwang-Rae Kim, Axel Munk, Florian Rehfeldt, Max Sommerfeld, Joachim Weickert, and Carina Wollnik

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We generalize the SiZer of Chaudhuri and Marron (J. Amer. Statist. Assoc. 94 (1999) 807–823; Ann. Statist. 28 (2000) 408–428) for the detection of shape parameters of densities on the real line to the case of circular data. It turns out that only the wrapped Gaussian kernel gives a symmetric, strongly Lipschitz semi-group satisfying “circular” causality, that is, not introducing possibly artificial modes with increasing levels of smoothing. Some notable differences between Euclidean and circular scale space theory are highlighted. Based on this, we provide an asymptotic theory to make inference about the persistence of shape features. The resulting circular mode persistence diagram is applied to the analysis of early mechanically-induced differentiation in adult human stem cells from their actin-myosin filament structure. As a consequence, the circular SiZer based on the wrapped Gaussian kernel (WiZer) allows the verification at a controlled error level of the observation reported by Zemel et al. (Nat. Phys. 6 (2010) 468–473): Within early stem cell differentiation, polarizations of stem cells exhibit preferred directions in three different micro-environments.

Article information

Bernoulli Volume 22, Number 4 (2016), 2113-2142.

Received: April 2014
Revised: November 2014
First available in Project Euclid: 3 May 2016

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Zentralblatt MATH identifier

circular data circular scale spaces mode hunting multiscale process persistence inference stem cell differentiation variation diminishing wrapped Gaussian kernel estimator


Huckemann, Stephan; Kim, Kwang-Rae; Munk, Axel; Rehfeldt, Florian; Sommerfeld, Max; Weickert, Joachim; Wollnik, Carina. The circular SiZer, inferred persistence of shape parameters and application to early stem cell differentiation. Bernoulli 22 (2016), no. 4, 2113--2142. doi:10.3150/15-BEJ722.

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