Real event detection and the treatment of congestive heart failure: an efficient technique to help cardiologists to make crucial decisions Page: 4
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yielded the RR sequence, from which we generated the time series {T}.
The meaning of a given value Ti, as earlier stated, is the time distance between the i - th
and the (i + 1) - th pulse. The sequence {Ti} can be studied as a new time-series, with i
playing the role of "time". Moreover, the value Ti, expressed as a function of i with i > 1,
can be thought of as a function T(t), namely, as a function of a continuous time variable t.
The real curve T(t) looks erratic and disordered. However, our method of analysis
shows that it is quite different from a random process. For the reader to get an intuitive
understanding of this attractive but perplexing conclusion, let us describe first an ideal
model, with extended memory (EM), for the time evolution of T(t). First of all, we assume
that for a given time Tem, the curve T(t) keeps a given slope cz, then it abruptly gets a new
slope, ca', for an interval of time Tnm, after which a new abrupt transition to a new slope a"
takes place, for a time Tm, and so on. It is evident that the resulting T(t) has the form of a
zig-zag curve. We shall refer to the individual straight line intervals of this curve as laminar
regions. Any laminar region is associated with its own Tem. Then we introduce the extended
memory property. This is done by assuming for the waiting time distribution ((Tern) the
following inverse power law form
(Tem) = (V - 1) m,, (1)
[(Aem) (V - 1) + em]
with 2 < v < 3, where (Tem) is the average waiting time. This means that if the EM model
is directly observable, we can derive from T(t) the sequence {Te (j)}, where the discrete
index j denotes the time order of a given laminar region.
It is interesting to notice that this dynamic process is essentially equivalent to the strong
anomalous diffusion model recently proposed by the authors of Refs. [2,3] to explain the
effects of a ballistic mechanism in the Rayleigh-Bnard convection. This model, in turn, is
nothing but a generalization of the dynamic approach to Ldvy statistics proposed years ago
by the authors of Ref. [5]. In fact, the model of Ref. [5] is recovered from the model we
are adopting here, by assuming that the slope ac has only two values, of equal intensity and
opposite sign [6]. With this equivalence in mind, we adopt the specific walking prescription4
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Allegrini, Paolo; Balocchi, Rita; Chillemi, Santi; Grigolini, Paolo; Hamilton, P.; Maestri, Roberto et al. Real event detection and the treatment of congestive heart failure: an efficient technique to help cardiologists to make crucial decisions, paper, September 2002; (https://digital.library.unt.edu/ark:/67531/metadc174688/m1/4/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Arts and Sciences.