The Annals of Applied Statistics

Fréchet means of curves for signal averaging and application to ECG data analysis

Jérémie Bigot

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Signal averaging is the process that consists in computing a mean shape from a set of noisy signals. In the presence of geometric variability in time in the data, the usual Euclidean mean of the raw data yields a mean pattern that does not reflect the typical shape of the observed signals. In this setting, it is necessary to use alignment techniques for a precise synchronization of the signals, and then to average the aligned data to obtain a consistent mean shape. In this paper, we study the numerical performances of Fréchet means of curves which are extensions of the usual Euclidean mean to spaces endowed with non-Euclidean metrics. This yields a new algorithm for signal averaging and for the estimation of the time variability of a set of signals. We apply this approach to the analysis of heartbeats from ECG records.

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Ann. Appl. Stat., Volume 7, Number 4 (2013), 2384-2401.

First available in Project Euclid: 23 December 2013

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Signal averaging mean shape Fréchet means curve registration geometric variability deformable models ECG data


Bigot, Jérémie. Fréchet means of curves for signal averaging and application to ECG data analysis. Ann. Appl. Stat. 7 (2013), no. 4, 2384--2401. doi:10.1214/13-AOAS676.

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  • Afsari, B. (2011). Riemannian $L^{p}$ center of mass: Existence, uniqueness, and convexity. Proc. Amer. Math. Soc. 139 655–673.
  • Allassonnière, S., Amit, Y. and Trouvé, A. (2007). Towards a coherent statistical framework for dense deformable template estimation. J. R. Stat. Soc. Ser. B Stat. Methodol. 69 3–29.
  • Antoniadis, A., Bigot, J. and Sapatinas, T. (2001). Wavelet estimators in nonparametric regression: A comparative simulation study. Journal of Statistical Software 6 1–83.
  • Beg, M. F., Miller, M. I., Trouvé, A. and Younes, L. (2005). Computing large deformation metric mappings via geodesic flows of diffeomorphisms. Int. J. Comput. Vis. 61 139–157.
  • Bigot, J. (2006). Landmark-based registration of curves via the continuous wavelet transform. J. Comput. Graph. Statist. 15 542–564.
  • Bigot, J. and Charlier, B. (2011). On the consistency of Fréchet means in deformable models for curve and image analysis. Electron. J. Stat. 5 1054–1089.
  • Bigot, J., Gadat, S. and Loubes, J.-M. (2009). Statistical M-estimation and consistency in large deformable models for image warping. J. Math. Imaging Vision 34 270–290.
  • Bigot, J. and Gadat, S. (2010). A deconvolution approach to estimation of a common shape in a shifted curves model. Ann. Statist. 38 2422–2464.
  • Bigot, J. and Gendre, X. (2013). Minimax properties of Fréchet means of discretely sampled curves. Ann. Statist. 41 923–956.
  • Birgé, L. and Massart, P. (2007). Minimal penalties for Gaussian model selection. Probab. Theory Related Fields 138 33–73.
  • Craven, P. and Wahba, G. (1978/79). Smoothing noisy data with spline functions. Estimating the correct degree of smoothing by the method of generalized cross-validation. Numer. Math. 31 377–403.
  • Fletcher, P. T., Lu, C., Pizer, S. M. and Joshi, S. (2004). Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Transactions on Medical Imaging 23 995–1005.
  • Fréchet, M. (1948). Les éléments aléatoires de nature quelconque dans un espace distancié. Ann. Inst. H. Poincaré 10 215–310.
  • Gamboa, F., Loubes, J.-M. and Maza, E. (2007). Semi-parametric estimation of shifts. Electron. J. Stat. 1 616–640.
  • Gasser, T. and Kneip, A. (1995). Searching for structure in curve samples. J. Amer. Statist. Assoc. 90 1179–1188.
  • Goldberger, A. L., Amaral, L. A. N., Glass, L., Hausdorff, J. M., Ivanov, P. C., Mark, R. G., Mietus, J. E., Moody, G. B., Peng, C. K. and Stanley, H. E. (2000). PhysioBank, PhysioToolkit, and PhysioNet: Components of a new research resource for complex physiologic signals. Circulation 101 e215–e220.
  • Goodall, C. (1991). Procrustes methods in the statistical analysis of shape. J. R. Stat. Soc. Ser. B Stat. Methodol. 53 285–339.
  • Guyton, A. C. and Hall, J. E. (2006). Textbook of Medical Physiology. Saunders Elsevier, Philadelphia, PA.
  • Huckemann, S. F. (2011). Intrinsic inference on the mean geodesic of planar shapes and tree discrimination by leaf growth. Ann. Statist. 39 1098–1124.
  • Klassen, E., Srivastava, A., Mio, M. and Joshi, S. H. (2004). Analysis of planar shapes using geodesic paths on shape spaces. IEEE Transactions on Pattern Analysis and Machine Intelligence 26 372–383.
  • Kneip, A. and Gasser, T. (1988). Convergence and consistency results for self-modeling nonlinear regression. Ann. Statist. 16 82–112.
  • Laciar, E., Jané, R. and Brooks, D. H. (2003). Improved alignment method for noisy high-resolution ECG and Holter records using multiscale cross-correlation. IEEE Trans. Biomed. Eng. 50 344–353.
  • Liu, X. and Müller, H.-G. (2004). Functional convex averaging and synchronization for time-warped random curves. J. Amer. Statist. Assoc. 99 687–699.
  • Ma, J., Miller, M. I., Trouvé, A. and Younes, L. (2008). Bayesian template estimation in computational anatomy. Neuroimage 42 252–261.
  • Mallows, C. L. (1973). Some comments on $C_{p}$. Technometrics 15 661–675.
  • Pan, J. and Tompkins, W. J. (1985). A real-time QRS detection algorithm. IEEE Trans. Biomed. Eng. 32 230–236.
  • Ramsay, J. O. and Li, X. (2001). Curve registration. J. R. Stat. Soc. Ser. B Stat. Methodol. 63 243–259.
  • Rompelman, O. and Ros, H. H. (1986a). Coherent averaging technique: A tutorial review. Part 1: Noise reduction and the equivalent filter. J. Biomed. Eng. 8 24–29.
  • Rompelman, O. and Ros, H. H. (1986b). Coherent averaging technique: A tutorial review. Part 2: Trigger jitter, overlapping responses and non-periodic stimulation. J. Biomed. Eng. 8 30–35.
  • Trigano, T., Isserles, U. and Ritov, Y. (2011). Semiparametric curve alignment and shift density estimation for biological data. IEEE Trans. Signal Process. 59 1970–1984.
  • Vimond, M. (2010). Efficient estimation for a subclass of shape invariant models. Ann. Statist. 38 1885–1912.
  • Wang, K. and Gasser, T. (1997). Alignment of curves by dynamic time warping. Ann. Statist. 25 1251–1276.
  • Wood, A. T. A. and Chan, G. (1994). Simulation of stationary Gaussian processes in $[0,1]^{d}$. J. Comput. Graph. Statist. 3 409–432.
  • Younes, L. (2010). Shapes and Diffeomorphisms. Applied Mathematical Sciences 171. Springer, Berlin.
  • Zhou, Y. and Sedransk, N. (2009). Functional data analytic approach of modeling ECG T-wave shape to measure cardiovascular behavior. Ann. Appl. Stat. 3 1382–1402.