Terézia Hodásová 1,2; Jiří Holčík 1,3
Institute of Biostatistics and Analyses, Masaryk University, Brno, Czech Republic
1; Masaryk University, Brno, Czech Republic
Department of Mathematics and Statistics of the Faculty of Science
2; Institute of Measurement Science, Slovak Academy of Sciences, Bratislava, Slovakia
Lékař a technika - Clinician and Technology No. 2, 2012, 42, 15-18
Conference YBERC 2012
This paper deals with a processing of ECG signal parameters and modelling the parameter dynamics during the stresstest in horses. The standard Box-Jenkins methodology was applied for the linear model design. The order of the AR model (equal to 6) was estimated according to properties of autocorrelation and partial autocorrelation functions of the ECG parameter time series. Further, frequency responses of the model systems as well as distribution of their poles and zeros were determined for different stages of the stress test examination. The preliminary examination of the computed data and their dynamics indicates that they can carry useful information for evaluation the health and fitness conditions of the examined horses. All the computational procedures were implemented in MATLAB®.
Keywords: Stress-test examination in horses, ECG signal parameters, stationary process, transfer function, AR model
The aim of the work was to find proper mathematical
model of ECG parameter dynamics during the stresstest
in horses. The stress-test examination consists of
several stages. It starts with walk, lope and then gallop
stages follow. The gallop stages ordinarily take 1 or 2
minutes starting with speed of 7m/s. After that the
treadmill speed increases by 1 m/s usually up to 10 or
The parameters of ECG signal change depending on
the stress-test load. So the question is: is there any
functional dependency based on the ECG signal parameters,
which could describe the dynamics of the
ECG parameters and which could characterise either
the fitness level of the horse or state of its health?
Cardiovascular system condition can be characterised
by several ECG parameters as the lengths of RR
and/or QT intervals or duration of QRS complexes. In
spite of the fact that all of these parameters were available,
in this paper the lengths of RR intervals were
considered only. It is because this parameter provides
vets with a very fundamental information on the heart
rate, its variability and ways of cardiovascular control.
The problematic of modelling dynamic ECG parameters
is neglected part, therefore there were not
known related works.
The parameters were calculated from ECG signal recorded
throughout stress-test . The determined irregularly
sampled parameter time series were then
resampled by frequency of 10 Hz for next processing
(Fig.1). The origin samples were linear interpolated to
obtain the continuous signal and this signal was in next step sampled by frequency of 10 Hz to obtain equidistant
Stationarity and frequency spectrum
The basic well known approach for a description of a
time series dynamics is based on the Box-Jenkins
methodology. However, the approach prefers if the
data represent stationary processes. That is why it was
necessary to test how severe the non-stationarity of
stress test process is.
There are several types of tests for stationarity. Testing
stationarity in the mean proved to be sufficient for
our purpose because of data character. Stationarity in
the mean means that the value of the data mean does
not depend on time. Therefore the analysed time series
were divided to four intervals of the same lengths.
Values of the mean were calculated for each interval
and compared. Then the ANOVA test was used and the
hypotheses that the values are equal, was rejected with
p-value p=0 on significance level 0.05. The fact signified
non-stationarity of the process and that is why the
data had to be preprocessed to eliminate a data drift
To model the drift the frequency approach was used
based on knowledge of time series frequency spectrum
There are some significant frequency clusters in the
depicted spectrum. Each of them describes specific
process as breathing during the stress-test or noise
generated by some technical artificial sources
(electrode contact artefacts, movement of ECG leads,
From the Fig. 2 it is obvious that the most significant
cluster is that at very low frequencies, typical for the
slow drift in the data. Shape of the frequency spectrum
can be also used for specification of the cut-off frequency
of a low-pass filter that can be applied to remove
the drift. It was determined as a frequency, the
absolute value of which had the biggest difference
from the previous sample. Further, order of the moving
average filter (that was planned to use for removing of
the data drift) had to be chosen. Provided filter with
Hamming window  and sampling frequency of 10
Hz the degree of 700 samples proved to be of the best
performance based on the trial and error method. If the
degree was lower, the higher frequencies were much
more attenuated. The trend obtained by filtering signal
using the low-pass filter with Hamming window is
shown in Fig. 3.
It is possible to see corresponding changes in the frequency
spectrum in the Fig. 4. It is obvious from the
figure, that the higher frequencies were really not inconveniently
The stationary signal after drift elimination (by subtracting
estimated drift from the original RR interval
time series) is shown in the Fig. 3, too.
The test of stationarity in mean for the signal without
drift does not rejected the null hypothesis, the p-value
was p=0.1928 on significance level 0.05. That meant,
that the signal without drift could be supposed to represent
a stationary process.
Time series model
The proper linear model describing the RR interval
data was looked for generally in the ARMA system
As it was mentioned before, the dynamics of parameters
is connected to changes of the treadmill speed.
Therefore the signal was divided into parts belonging to each step of the stress-test before next processing.
These parts are obvious in the Fig. 3.
Each data interval was modelled individually, but the
structure of the model was the same for all the cases.
To determine such a structure it was necessary to decide
which type of the model is the most applicable for
the processed experimental data, then the order of
model had to be set.
Both the tasks were solved by means of properties of
an autocorrelation (ACF) and partial autocorrelation
(PACF) function (see Fig. 5), .
For every stress-test step the estimated ACF was
quickly decreasing curve shaped as attenuated sinusoid.
Such a type of ACF function is typical for an autoregressive
process, so the AR structure of the model
proved to be the best choice. The degree of 6 of the AR
model was estimated using the graph of PACF made
for each part. It is not so clear from the Fig. 5. PACFs
for all the stress-test steps were considered and rather
simpler model than that with more complicated structure
(higher order) was finally chosen.
On both of these graphs the 95% confidence intervals
(blue lines) are shown. This interval is useful especially
for estimation the degree of model. The values of
lag numbers represent the model coefficients. If their
values are outside of the confidence interval, then they
are statistically significant and therefore they had to be
included into the model.
Characteristics of AR(6)
The AR model of the order of 6 is defined by the difference
equation according to eq. (1), .
and its transfer function is
Character of the transfer function can be also described
by a map of its zeros and poles (see Figs. 6, 7,
8). In Fig. 6 there are obvious some clusters of poles
that belong to transfer functions for different stress-test
stages. These groups describe the movement of one
pole (sometimes the real pole splits into two complex
conjugate poles and vice versa). So the clusters can be
used to describe the trajectory of pole dynamics.
The question is, if this map can be useful for obtain
the fitness level in horses. Let us examine the maps for
two other horses, (Fig. 7 and 8). The pole map structure
is different in both the figures. The most concentrated
poles in one place are in Fig.7 (Line Honey, well
trained athlete with the best fitness level). That demonstrates
the low dynamics of heart rate (RR intervals) in
successive steps of the stress-test.
In the Fig. 8 (Jarys, horse with the worst level of fitness
in the group of examined horses) the concentration
of poles is rather poor, the poles are distributed
within the whole unit circle. That describes high level
dynamics of the parameter.
The concentration of poles for horse named Aragorn
(Fig.6) is somewhere in a middle. Aragorn runs the
races of the 3rd category. The previously mentioned
examples demonstrate that the movement of poles of the model describing time series of RR intervals could
provide us with useful information for assessing the
fitness level in horses.
Position of pole clusters in the complex Z plane (for
example defined by an angle between real axis and the
line connecting origin of the coordinates and centre of
the pole cluster) characterizes also frequencies of processes
that influence the heart rate. As it can be seen in
all the mentioned figures the examined horses could be
also assessed by means of such a parameter.
Experiments and calculations done with the experimental
data up to now demonstrate that AR models and
their characteristics could be considered as a useful
tool for assessing especially fitness level in horses.
To verify such an idea it will be necessary to verify
the hypothesis with larger set of experimental signals.
Then the algorithms for machine classification of fitness
level in horses will be developed.
The research was partly granted by the project of the Czech Science Foundation No.102/09/H083: „Information Technology in Biomedical Engineering“ and by the VEGA project No. 2/0210/10 “Methods and Systems for Multichannel Measurement and Evaluation of Bioelectric Signals of Heart and Brain”.
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