AR versus ARX Modeling of Heart Rate Sequences Recorded during Stress-Tests ... Fig.1 Example of the RR interval sequence recorded during stress-test ...
AR versus ARX Modeling of Heart Rate Sequences Recorded during Stress-Tests J. Holcik1,2, T. Hodasova1,3, P. Jahn4, P. Melkova4, and J. Hanak4 1
Institute of Biostatistics and Analyses, Masaryk University, Brno, Czech Republic Institute of Measurement Science, Slovak Academy of Sciences, Bratislava, Slovakia 3 Department of Mathematics and Statistics, Faculty of Science, Brno, Czech Republic 4 Equine Clinic, University of Veterinary Sciences and Pharmacy, Brno, Czech Republic 2
Abstract—This paper presents some ideas about comparison of two fundamental linear approaches (AR and ARX models) for modeling RR interval series and its variability recorded in horses during stress-testing. Their theoretical background is discussed in brief and results of some computational experiments are given and analyzed, as well. In particular, problems of stationarity, determination of the response to changes of load, estimation of model order are examined here. Keywords—heart rate variability, AR model, ARX model, stress test.
I. INTRODUCTION Heart rate of an examined subject and in particular its dynamics is determined by a state of both the cardiovascular system and autonomic neural system that controls cardiovascular activity. Quality of a function of the mentioned physiological subsystems is crucial for determining level of fitness. That is why the global aim of our research is to attempt to rate fitness level of horse athletes based on heart rate sequences recorded during stress test. To facilitate the fitness level classification it seems appropriate to describe the heart rate sequences and their variability by means of some mathematical model, parameters of which could be used as features entering the classification algorithms. There are two significant approaches used for linear mathematical modeling heart rate sequences – Autoregressive (AR) models (e.g. [1] - [5]) and ARX models (Autoregressive, eXtra input) (e.g. [6]). Both the approaches have their advantages and disadvantages. The greatest disadvantage of the AR model is a requirement for stationarity of data that can be hardly fulfilled under conditions of stress test when the physical load increases. On the other hand, cardiovascular responses in horses usually change with an intensity of load. That is why it seems to be probably difficult to determine adequate optimum parameters of the time-invariant ARX model valid for the whole stress-test examination. Despite of the mentioned problems both the types of models were used for description of the cardiovascular responses in horses to physical load and the obtained results were compared.
II. DATA Stress-test in horses consists of several steps. It starts with an approx. 5 minute walk usually followed by lope (5min.) and then by gallop starting with treadmill speed of 7m/s (2 min) that increases after 1 minute by steps of 1m/s up to 10 or 11m/s depending on horse abilities. ECG signals were recorded from three bipolar chest leads which QRS complexes were detected in. After that the derived RR interval functions were interpolated by piecewise linear function and resampled by a frequency of 10 Hz to obtain equidistant time series. Altogether 15 data records were taken and processed, 6 of them recorded in a preliminary phase of examination with non-standard experimental arrangement of the load stages.
Fig.1 Example of the RR interval sequence recorded during stress-test III. AR MODELS As mentioned above the basic disadvantage of the AR model approach is the requirement for data stationarity. Unfortunately, according to an expectation based on practical experiences and also supported by numerous publications (e.g. [1] or [3]) the response to increasing load during stress-test is heavily time variant. Fig.1 depicts sequence of RR intervals determined from ECG signals recorded during the stress-test examination. It can be seen in the figure that the most significant non-stationarity is represented by responses to changes of the load. The moving average low-pass Hamming filter with an impulse response of 600
L.M. Roa Romero (ed.), XIII Mediterranean Conference on Medical and Biological Engineering and Computing 2013, IFMBE Proceedings 41, DOI: 10.1007/978-3-319-00846-2_247, © Springer International Publishing Switzerland 2014
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AR versus ARX Modeling of Heart Rate Sequences Recorded during Stress-Tests
Fig.2 Original RR interval sequence and its drift estimated by the narrow-band low-pass filter (upper part) and the difference sequence between the original sequence and its drift (lower part) samples was designed and used to remove the nonstationarity. The cut-off frequency was set according to frequency spectrum of the RR interval signal as the frequency separating band of lower frequencies with greater amplitudes from the components with higher frequencies (see Fig.2 - upper part). It means that it must be valid that
E(z) = RR(z) − H LP (z) ⋅ RR(z)
(1a)
RR(z) = H LP (z) ⋅ RR(z) + E(z)
(1b)
or
where RR(z) represents Z-transform of the original RR interval sequence, HLP(z) is a transfer function of the used Hamming low-pass filter and E(z) is a Z-transform of a difference sequence e(k) that should represents the stationary behavior of the examined horse during the whole stress-test. The sequence e(k) is modeled by a linear autoregressive system described by a transfer system function HAR(z) = 1/A(z). The function HAR(z) is proportional to power spectral density of sequence e(n) provided that input of the model system is a zero-mean white noise sequence n(k). In such a case it is
E( z ) =
1 ⋅ N(z), A( z )
(2)
where N(z) is the Z-transform of the white noise sequence n(k). Then the eq.(1b) can be rewrite as
RR (z) = H LP (z) ⋅ RR (z) +
1 ⋅ N( z) A( z)
(3)
Weak-sense stationarity (mean only) was verified per partes for partial sequences without drift corresponding to each step of the stress-test load. Because of lack of knowledge about the data statistical distribution non-parametric Kruskal – Wallis test was applied. In this way median of the subsequences proved to be sufficiently time invariant in the vast majority of the analyzed sequences. Small differences
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from stationarity came to pass in intervals just after the load changes (Fig.2). Even if the subtracting of the drift roughly ensures the stationarity of analyzed sequence (required for application of the AR model) it unfortunately removes substantial part of information on character of transients tied with the load changes. Although shortening the impulse response and related increasing of the cut-off frequency of the smoothing Hamming filter homogenizes data after subtracting the estimated drift neither the impulse response of 100 samples does not ensure the strict data stationarity. The principal task for identification of the AR model parameters is to determine its order that provides the best fit of the model to data being processed. Unfortunately, known algorithms for the AR model order estimation are not very reliable. The experimental results ([4], [7]) as well as theoretical studies (e.g. [8]) indicate that the practically used several statistical criteria do not usually yield definitive results, mostly tend to underestimation of the true order of the analyzed AR process. The model order for a partial time-series in every stage of the stress-test examination was searched for in the interval 6, 30. The value of 20 was then determined as the most frequent result computed for the given set of the experimental data sequences by the Akaike information criterion. Several different approaches are used for the final identification of the AR model parameters, each of them under specific conditions and with specific characteristics. We used two of them – the Yule – Walker method and the unconstrained least square method. The former, based on estimates of autocorrelation function of the analyzed sequence, can be assumed to be one of methods with maximum entropy. Therefore spectral characteristics of the determined model are relatively smooth in comparison with the latter and with relatively poor frequency resolution. On the other hand, parameters determined by the latter method make the resulting linear model much more frequency sensitive, however it does not have to be necessarily stable. Properties of the determined AR models can be illustrated either by their frequency responses or by distribution of transfer function poles in complex plane (see Fig.3 and Fig.4). Examples in the figures roughly confirm the above mentioned properties of both methods for identification of the AR model parameters. Frequency responses of the system with parameters obtained by means of the Yule – Walker method are smoother and not very sensitive to particular frequency components (baroreflex, breathing) incorporated in the analyzed signal if compared with results of application of the unconstrained least square method. We can carefully presume the existence of the mentioned harmonic components from the shape of phase characteristic only. However, the smoothness of the frequency responses results in less variability of positions of the transfer function poles.
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However, the load changes can be incorporated into the model structure as it is defined in a class of dynamical systems with external input which do not put any stacionarity demands on the analyzed data. There are three basic structures of these systems that differ in the random part of the definition formula – ARX models (AutoRegressive with eXternal input), ARMAX models (AutoRegressive Moving Average with eXternal input), and OE models (Output Erorr). The ARX systems are defined as
RR(z) =
Fig.3 Example of the AR(20) model frequency responses for the stages of the stress-test with parameters identified by means of Yule-Walker method (upper part) and distribution of its poles in complex Z-plane (lower part)
B(z) 1 ⋅ X(z) + ⋅ N( z ) A(z) A(z)
(4)
where X(z) is the Z-transform of a system input sequence that is determined by time-dependency of the load, A(z), and B(z), resp. are polynomials of the system transfer functions. As we can easily compare the formula is very similar to that in eq.(3). As well as in eq.(3) the first member at the right hand side of the formula represents response to the system input, now in more explicit form, and the second member represents response of the system to random interference. Even if there are quite different conditions under which both the described models could be used, from a theoretical viewpoint they differ in mathematical description of one member of the defining formula only (however, the meaning of both the expressions is essentially the same). Then we can write that
B(z). X(z) = H LP (z) ⋅ RR (z) A( z )
(5)
The only component in eq.(5) that is not determined on a base of some optimality criterion is the transfer function HLP(z) of the low-pass filter for filtering the experimental RR sequence. However, from the eq.(5) we can simply write
H LPopt (z) =
Fig.4 Example of the AR(20) model frequency responses in the stages of the stress-test with parameters identified by means of the unconstrained least square method (upper part) and distribution of its poles in complex Zplane (lower part)
IV. ARX MODELS As it was described above the requirements for stationarity can be hardly complied with completely due to the changes of experimental conditions during the stress-tests.
B(z) X(z) ⋅ , A(z) RR (z)
(6)
that should define both the optimum properties of the lowpass filter used for AR modeling of the RR interval time series and at the same time the model response to the change of the stress-test load that is the most important part of the model behavior. Then both the ways (using AR and ARX system) of the RR sequence analysis recorded during stress-test should be equivalent. Similarly as in the case of the AR model approach the fundamental task is to determine orders of the both the polynomials used in the eq.(4). Usually a parameter representing time shift between system input and output can be also determined. The polynomial orders were searched for again by means of the Akaike information criterion in interval of 6, 30 and the values of 20 – the order of the polynomial B(z), 20 – the order of the polynomial A(z), and 1 – the system time shift were chosen as the most frequent values for all the analyzed experimental records.
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AR versus ARX Modeling of Heart Rate Sequences Recorded during Stress-Tests
Coefficients of the polynomials A(z) and B(z) were computed by the unconstrained least square method. Example of frequency responses of the used model transfer function is given in Fig.5. Due to the unconstrained least square optimization models for some experimental data appeared unstable.
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To make interpretation of the model description as easy as possible the conversion of the ARMA structure (used in ARX model) or the low-pass MA filter (used in the AR model here) to AR representation can be considered [9]. If relatively simple model is required (necessary for adequate size of a feature space and complexity of the classification algorithms based on the model parameters – model order maximum up to 30) then our experimental results indicate that the spectral description of the RR interval sequence is still too smooth that means the model order is underestimated. A final decision about this fact can be done on the base of classification result only.
ACKNOWLEDGMENT This research was partially granted by the ESF project No. CZ.1.07/2.2.00/28.0043 “Interdisciplinary Development of the Study Programme in Mathematical Biology” and the project No. APVV-0513-10 „Measuring, Communication and Information Systems for Monitoring the Cardiovascular Risk”.
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Fig.5 Example of the ARX(20,20,1) model frequency responses – AR subsystem (upper part), ARMA subsystem (in the middle) and the distribution of its nulls and poles in complex Z-plane (lower part) V. CONCLUSIONS As it was shown above the mathematical structure of both the model types, the AR as well as the ARX, are basically the same. While the ARX models provide with well established procedure for determination of the model part that models the response to changes of the load, the approach based on the AR systems uses heuristic procedures for this purpose. However, the estimation of this part of the AR model can be done theoretically more precisely if the ARMA system computed for the ARX model is modified according to properties of the external input and the RR interval series (eq.(6)). On contrary, the AR systems look more suitable for description of variations in behavior of the examined subject in the particular stages of the whole stress-test.
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Jiri Holcik Institute of Biostatistics and Analyses, Masaryk University Kamenice 126/3 Brno Czech Republic holcik et iba.muni.cz
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