SANGALLI, LAURA MARIA

SECCHI, PIERCESARE

VANTINI, SIMONE

Electronic Journal of Statistics

Archivio istituzionale della ricerca - Politecnico di Milano

Electronic Journal of StatisticsVol. 8 (2014) 1937–1939ISSN: 1935-7524DOI: 10.1214/14-EJS938REJRejoinder: Analysis of AneuRisk65 data∗Laura M. Sangalli, Piercesare Secchi and Simone VantiniReceived August 2013.We would like to thank the various groups who analyzed the AneuRisk65 data,providing a broad range of analyses with many interesting results. This datasetoffers the possibility of performing registration and analyses at various datalevels. Cheng et al. (2014) and Xie, Kurtek and Srivastava (2014) registeredthe three dimensional vessel centrelines, likewise in Sangalli et al. (2009, 2010).Staicu and Lu (2014) and Gervini (2014) opted instead for the correspondingone-dimensional curvature functions. The registration methods used are varied,some (see Cheng et al., 2014; Sangalli, Secchi and Vantini, 2014) only allowingfor shift or linear warpings of the asbcissa parameters, others (see Staicu and Lu,2014; Xie, Kurtek and Srivastava, 2014; Gervini, 2014) allowing for very flexiblewarpings. One difficulty with these data is that the portion of reconstructed ves-sel morphology has different length across subjects: for some subjects, only theterminal tract of the Inner Carotid Artery (ICA) is observed and reconstructed,for other patients a long portion of the arterial tract preceding the distal part isalso available. The observed vessel lengths are significantly different for the con-sidered groups of subjects – Upper and Lower-No groups – due to the differentlocation of the scanned volume. Some registration methods (see Cheng et al.,2014; Sangalli, Secchi and Vantini, 2014; Gervini, 2014) explicitly tackle thisissue, whilst others (see Staicu and Lu, 2014; Xie, Kurtek and Srivastava, 2014)map all curves to a common domain. On the other hand, we believe that themore flexible warping functions considered by these latter methods might havecounterbalanced the problem, at least to some extent, by re-stretching towardthe terminal part shorted arteries and by unwinding longer ones. After registra-tion, most groups (see Cheng et al., 2014; Staicu and Lu, 2014; Gervini, 2014)try to discriminate Upper and Lower-No groups analyzing the vessel shape orsome morphological feature such as vessel radius and curvature, similarly toSangalli et al. (2009). As expected, when a rich family of warping functions isused, the discrimination improves if the phase variability captured by the warp-ing functions is also explicitly considered in the discrimination method. Xie,Kurtek and Srivastava (2014) instead perform an unsupervised clustering of thevessels based on their three-dimensional shapes, analogously to Sangalli et al.(2010).Here are some comments specific to each data analysis.Cheng et al. (2014) analyze the data using techniques from shape anal-ysis. The vessel centrelines are registered in the three-dimensional space by∗Main article 10.1214/14-EJS938.19371938 L. M. Sangalli et al.considering rigid transformations (translations and rotations) of these three-dimensional curves and simple shifts along their curvilinear abscissas. The dif-ferent lengths of the vessel centerlines are appropriately considered by extrapo-lating shorter curves; the extrapolated tracts are then given nearly zero weightsin the weighted Procrustes method used for registration and in the subsequentanalysis. Differently from Sangalli, Secchi and Vantini (2014), since only shiftsin the abscissa are considered, the registration method does not try to matchcarotids with different sizes. We were very happy to see that the many in-sights the authors give on the data, concerning for instance vessel morpholog-ical features and differences between the Upper and Lower-No group’s center-lines, are in full agreement with our findings discovered during the AneuRiskproject.Staicu and Lu (2014) instead consider the one-dimensional vessel curvatureprofiles. These are registered by first mapping all curvature functions to a com-mon abscissa interval and then by applying the Fisher-Rao registration methodproposed in Srivastava et al. (2011). The optimal warping functions thus ob-tained are also used to register the vessel radius profiles. Hence, the amplitudevariability of the aligned radius profiles and that of the curvature profiles areexplored by means of functional principal component analysis, with a modifiedinner product that accounts for the phase variability captured by the warpingfunctions. Since the latter are computed after having mapped all vessels to acommon domain, these warpings implicitly include information about the vessellength, that is significantly different in the Upper and Lower-No subject groupsand is directly correlated with the location of the aneurism, or its absence, as re-called above. We believe that this feature may imply an artificial discriminationbetween the two groups. We were in any case positively impressed by seeing thatthe results fully support the findings in Cheng et al. (2014) and in Sangalli et al.(2009). This is particularly interesting to notice given the different data consid-ered in the registration (centerline curvature functions vs centrelines), and thedifferent registration method, which allows for very flexible warpings instead ofthe shift and linear warpings considered in Cheng et al. (2014) and in Sangalliet al. (2009) respectively.Xie, Kurtek and Srivastava (2014) use and extend the Fisher Rao approachto curves in three-dimensional space, warp the three-dimensional centerlinesand analyze their shape. Differently from Cheng et al. (2014) and similarly toStaicu and Lu (2014), the curves are first mapped to a common domain andvery flexible warpings are then allowed. The three-dimensional shapes of thevessel centrelines are hence studied by functional principal component analysisand clustering in the defined shape space.Finally, Gervini (2014) proposes a rather different approach. The author an-alyzes the curvature profiles and proposes a likelihood-based warping method,combined with a logistic regression model that aims at discriminating subjectsin the Upper and Lower-No groups. Short carotids are here treated as incom-plete data. Considering a rich class of warping functions, the author finds thatthe discrimination results improve significantly when the warping functions areincluded explicitly in the logistic model.Rejoinder 1939ReferencesCheng, W.,Dryden, I. L.,Hitchcock, D. B. and Le, H. (2014). Analysis ofAneuRisk65 data: Internal carotid artery shape analysis. Electron. J. Statist.8 1905–1913, Special Section on Statistics of Time Warpings and Phase Vari-ations.Gervini, D. (2014). Analysis of AneuRisk65 data: Warped logistic discrimina-tion. Electron. J. Statist. 8 1930–1936, Special Section on Statistics of TimeWarpings and Phase Variations.Sangalli, L. M., Secchi, P. and Vantini, S. (2014). Analysis of AneuRisk65data: k-mean alignment. Electron. J. Statist. 8 1891–1904, Special Section onStatistics of Time Warpings and Phase Variations.Sangalli, L. M., Secchi, P., Vantini, S. and Veneziani, A. (2009). A casestudy in exploratory functional data analysis: Geometrical features of theinternal carotid artery. J. Amer. Statist. Assoc. 104 37–48. MR2663032Sangalli, L. M., Secchi, P., Vantini, S. and Vitelli, V. (2010). K-meanalignment for curve clustering. Computational Statistics and Data Analysis54 1219–1233. MR2600827Srivastava, A., Wu, W., Kurtek, S., Klassen, E. and Mar-ron, J. S. (2011). Registration of functional data using Fisher-Rao metric.arXiv:1103.3817v2 [math.ST].Staicu, A. M. and Lu, X. (2014). Analysis of AneuRisk65 data: Classificationand curve registration. Electron. J. Statist. 8 1914–1919, Special Section onStatistics of Time Warpings and Phase Variations.Xie, Q., Kurtek, S. and Srivastava, A. (2014). Analysis of AneuRisk65data: Elastic shape registration of curves. Electron. J. Statist. 8 1920–1929,Special Section on Statistics of Time Warpings and Phase Variations.

Rejoinder: Analysis of AneuRisk65 data

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Rejoinder: Analysis of AneuRisk65 data

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