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Nonlinear additive autoregressive model-based analysis of short-term heart rate variability

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Abstract

In this contribution we test the hypothesis that nonlinear additive autoregressive model-based data analysis improves the diagnostic ability based on short-term heart rate variability. For this purpose, a nonlinear regression approach, namely, the maximal correlation method is applied to the data of 37 patients with dilated cardiomyopathy as well as of 37 age- and sex-matched healthy subjects. We find that this approach is a powerful tool in discriminating both groups and promising for further model-based analyses.

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Acknowledgments

We would like to thank Henning Voss for providing the Matlab-code and very helpful discussions as well as the Deutsche Forschungsgemeinshaft (DFG BA 1581/4-1, BR 1303/8-1, KU 837/20-1) for supporting the study.

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Correspondence to Niels Wessel.

Appendix: the ACE algorithm

Appendix: the ACE algorithm

This appendix provides a short description of the ACE algorithm of Breiman and Friedman [6], the computer programs used can be obtained from the authors (http://tocsy.agnld.uni-potsdam.de/). In the following section, the same notations as introduced in Sect. II are used.

Generally, the estimation of functions that are optimal for correlation is equivalent to the estimation of functions that are optimal for regression. Therefore, the problem

$$\Psi(S,T_{1},\ldots,T_{k}) = \max_{\theta,\phi_i}\left|\rho(\theta(S), \sum\limits_{i=1}^k\phi_i(T_i))\right| $$
(8)

may also be expressed as the regression problem

$$E \left[\left(\theta(S) - \sum\limits_{i=1}^k \phi_i(T_i)\right)^2\right] \stackrel{!}{=} \min . $$
(9)

Here, the functions θ and ϕ j (j=1,...,k) are varied in the space of Borel measurable functions, and the constraints onto these functions are that they have vanishing expectation and finite variances to exclude trivial solutions.

For the one-dimensional case (k=1), the ACE algorithm works as follows: When denoting the conditional expectation of ϕ1 (T 1) with respect to S by E1 (T 1)|S], then the function \(\bar{\phi}_{0} (S) = E[\phi_{1} (T_{1})|S]\) minimizes (9) with respect to θ(S) for a given ϕ1(T 1). Similarly, \(\bar{\phi}_{1} (T_{1}) = E[\theta(S)|T_{1}]/||E[\theta(S)|T_1]||,\) where the norm is defined by \(||Z||=\sqrt{\hbox{var} [Z]},\) minimizes (9) with respect to ϕ1 (T 1) for a given θ (S), keeping E 21 (T 1)]=1. Now, the ACE algorithm consists of the following iterative procedure: Starting with the initial function

$$\phi_{1}^{(1)} (T_{1}) = E[S|T_{1}], $$
(10)

from i=2 it is calculated recursively

$$\theta^{(i)}(S) = E[\phi_{1}^{(i-1)}(T_{1})|S] $$
(11)

and

$$\phi_{1}^{(i)} (T_{1}) = E[\theta^{(i)}(S)|T_1]/||E[\theta^{(i)}(S)|T_1]||, $$
(12)

until E[(ϕ (i)1 (T 1) − θ(i)(S))2] fails to decrease. The limit values are then estimates for the optimal transformations θ and ϕ1. For the minimization of the right hand side of Eq. 9 a double-loop algorithm is used. In the additional inner loop, the functions

$$ \phi_{j}^{(i)}(T_{j}) = E \left[\theta^{(i)}(S) - \sum\limits_{p\ne j}\phi_p^{(i,i-1)}(T_p)\,|\,T_{j}\right] $$

are calculated. In the sum, the superscript “.(i)” is used for p < j and “.(i-1)” for p > j. For k > 1, the ACE algorithm works similarly.

There are several possibilities of estimating conditional expectations from finite data-sets. In our examples, local smoothing of the data is used. This smoothing can be achieved with different kernel estimators. We use a simple boxcar window, i.e. the conditional expectation value E[y|x] is estimated at each site i via

$$ \hat{E} [y|x_{i}] = \frac{1}{2N+1} \sum\limits_{j=-N}^{N} y_{i+j} $$

for a fixed half window size N. In all examples of this paper, n=30 is used to account for a reliable estimate of the mean value.

Furthermore, to allow for a better estimation in the case of inhomogeneous distributions, the data are transformed to have rank-ordered distributions prior to the application of the ACE algorithm [i.e. the data-set X is sorted in ascending order, resulting in the vector Y and all further calculations are performed with the corresponding index vector I, where Y=X(I)]. This allows for a more precise estimation of expectation values independently of the form of the data distribution and simplifies the algorithm considerably. It is allowed, since the rank transformation is invertible and the maximal correlation is invariant under invertible transformations by definition. Proofs of convergence and consistency of the function estimates are given in Ref. [6].

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Wessel, N., Malberg, H., Bauernschmitt, R. et al. Nonlinear additive autoregressive model-based analysis of short-term heart rate variability. Med Bio Eng Comput 44, 321–330 (2006). https://doi.org/10.1007/s11517-006-0038-0

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