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

Abstract

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 without a reference template. We apply this approach to the estimation of a mean heart cycle from ECG records. Acknowledgements -The author acknowledges the support of the French Agence Nationale de la Recherche (ANR) under reference ANR-JCJC-SIMI1 DEMOS.Initialization : let θ 0 = 0, δ 0 = 1 ∇M (θ 0 ) , C(0) = M (θ 0 ), and set i = 0.Step 2 :

Time-Warping Invariants of Multidimensional Time Series

“…Example 1.2 Assume that there is a deterministic time series x ∈ R N which models some "prototype" evolution of a quantity, say the prototype heartbeat in a patient's ECG. This prototype is unknown, but one records a lot of samples of it run at different speeds and contaminated by noise (compare [6]). A model for these observations is then y ( ) n = x h ( ) (n) + w ( ) n , n= 1, .…”

“…The currently used method [6,31,33], consists in first trying to align the different samples, i.e., to estimate the time-changes h ( ) , and to average afterwards. This seems to work well in regimes where the noise w ( ) is small (large signal-to-noise ratio), but will break down if this is not the case.…”

Sets that maximize probability and a related variational problem

“…As we previously remarked in Section 1, this preference for manifold-valued random variables is because the assumption that the sample space is a vector or affine space is too restrictive for a wide class of real situations. Interested readers may refer to Bhattacharya & Dunson (2012) and Bigot (2013) (and references therein) to see additional realistic applications where the sample space of a random variable is modelled by a Riemannian manifold.…”

A functional supervised learning approach to the study of blood pressure data

“…Clearly, the best choice of registration parameters j , j for each subject j is the minimizer of M( , ) with respect to ( , ), ie, The solution to problem (12) produces the optimal values for registration parameters j , j for the sample, and by substituting these values into (10), we obtain the exact form of the discretized Fréchet mean. The optimization procedure relies on a gradient descent algorithm similar to the one proposed in Bigot et al, 23 in a different context. If needed, using a kernel approximation similar to the one described above, we obtain continuous or smooth approximations of the Fréchet mean estimator̂.…”