## Abstract

Myocardial ischaemia is hypothesized to stimulate the cardiac sympathetic excitatory afferents and, therefore, the spontaneous changes of heart period (approximated as the RR interval), and the QT interval in ischaemic dilated cardiomyopathy (IDC) patients might reflect this sympathetic activation. Symbolic analysis is a nonlinear and powerful tool for the extraction and classification of patterns in time-series analysis, which implies a transformation of the original series into symbols and the construction of patterns with the symbols. The aim of this work was to investigate whether symbolic transformations of RR and QT cardiac series can provide a better separation between IDC patients and healthy control (HC) subjects compared with traditional linear measures. The variability of these cardiac series was studied during daytime and night-time periods and also during the complete 24 h recording over windows of short data sequences of approximately 5 min. The IDC group was characterized by an increase in the occurrence rate of patterns without variations (0 V%) and a reduction in the occurrence rate of patterns with one variation (1 V%) and two variations (2 V%). Concerning the RR variability during the daytime, the highest number of patterns had 0 V%, whereas the rates of 1 V% and 2 V% were lower. During the night, 1 V% and 2 V% increased at the expense of diminishing 0 V%. Patterns with and without variations between consecutive symbols were able to increase the separation between the IDC and HC groups, allowing accuracies higher than 80%. With regard to entropy measures, an increase in RR regularity was associated with cardiac disease described by accuracy >70% in the RR series and by accuracy >60% in the QTc series. These results could be associated with an increase in the sympathetic tone in IDC patients.

## 1. Introduction

Hyperactivity of the sympathetic nervous system is triggered by both central and peripheral pathways that are associated with abnormal cardiovascular reflexes observed in a variety of disease states such as cardiac ischaemia, ventricular dysfunction, renal failure and obstructive sleep apnoea [1]. Patients with congestive heart failure (CHF) have evidence of increased activation of the sympathetic nervous system as reflected by the increase in plasma norepinephrine levels [2,3]. In addition, ischaemia can produce a rapid and massive increase in the concentration of endogenous catecholamines such as norepinephrine, epinephrine, endothelin and dopamine in the myocardial interstitial fluid, with a deleterious effect on the cardiac myocytes [4,5]. Ischaemic dilated cardiomyopathy (IDC) is a type of cardiomyopathy that is caused by ischaemic myocardial damage, that is, damage caused by insufficient coronary arterial flow, usually as a result of atherosclerotic coronary artery disease (CAD). In particular, the presence of CAD may accelerate the progression of CHF, explaining the higher mortality among ischaemic CHF patients than that among non-ischaemic CHF patients [4]. CAD is the initiating cause of approximately 70% of all cases of CHF [6]. Because of its relation with CHF, this suggests the hypothesis that IDC patients should exhibit sympathoexcitation, which may result from either an increase in excitatory influences or a decrease in inhibitory influences on brainstem cardiovascular centres [3]. Abnormalities in cardiac parasympathetic tone may be one factor that contributes to the increase in cardiac sympathetic activity seen in the setting of CHF [1,7].

Heart rate variability (HRV) has become a relevant indicator for cardiovascular risk in humans [8]. The effects of sympathetic and vagal modulation on normal HRV have been well characterized, as has the marked reduction in autonomically mediated HRV associated with CHF and myocardial infarction [8,9]. In general, healthy subjects show a more pronounced HRV than subjects with heart disease. Berger *et al*. [9] found that patients with dilated cardiomyopathy had a higher mean heart rate and lower HRV than control subjects, suggesting that this probably reflects withdrawal of parasympathetic tone accompanying CHF. Autonomic influences on ventricular repolarization have also been studied by means of QT interval analysis [10–12]. In the healthy heart, variability of the QT interval is intimately linked to that of heart rate, reflecting the rate adaptation of action potential duration of ventricular myocytes, which is influenced by the autonomic nervous system (ANS) and repolarization reserve [10]. Ventricular repolarization abnormalities may vary over time as a result of changes in the myocardial substrate, in response to changes in the ANS, changes due to circadian variations or changes due to the presence of ischaemia [11]. In ECG recordings from patients with CHF, the magnitude of QT interval variability can be accentuated owing to sympathetic hyperactivity and reduced cardiac vagal control [13]. An increased QT duration and increased QT interval variability reflecting variations in T-wave morphology were observed in patients with dilated cardiomyopathy [11].

A number of studies have been carried out aiming to characterize HRV and QT variability. One commonly used non-invasive mode of autonomic function testing is based on power spectral analysis of HRV and quantification of low-frequency (LF) and high-frequency (HF) power. It is generally accepted that HF power reflects respiratory sinus arrhythmia, which is mediated by the parasympathetic cholinergic system [14]. However, the origins and clinical significance of LF power have aroused persistent controversy [15]. Although it was suggested initially that LF power provides an index of cardiac sympathetic outflow, more recent studies support that LF power seems to provide a measure not of cardiac sympathetic tone but of baroreflex function, as indicated in [16]. Concerning ventricular repolarization, QT intervals are more under sympathetic control especially if the sympathetic drive is relevant [17].

Symbolic analysis is a nonlinear and powerful tool for pattern extraction and pattern classification in time-series analysis, including series from physiological systems. This technique implies a transformation of the original time series into symbols and the creation of patterns with the symbols. Different criteria can be applied to transform series into symbolic representations and create patterns. The analysis and identification of particular patterns can provide information about signal complexity without any *a priori* assumption of the signal behaviour and system process. These features guarantee the generality of symbolic analysis as a method for studying complex systems [8,18]. Although the symbolic analysis applied to physiological time series has included different approaches in order to obtain the symbols, two of them are widely used: one approach uses equidistant levels (EQs), which are obtained by dividing the span (range between the minimum and the maximum) of the time series in a fixed number of equal regions [19], and the other approach uses non-equidistant levels (NEQs), which are based on the amount of deviation from the average of the time series [20]. The complexity of the resulting symbolic patterns has been quantified using metrics such as the probability of occurrence, information entropies, forbidden patterns, classification according to the amount of variations in the patterns and others. These measures have provided complementary information for the spectral analysis of RR series [21–23].

The aim of this work was to investigate whether symbolic patterns derived from QT interval series and HRV improves separation between IDC patients and healthy subjects compared with traditional linear time-domain and frequency-domain analysis.

## 2. Methodology

### (a) Analysed database

The experimental dataset that we analysed belongs to the Intercity Digital ECG Alliance (IDEAL) database, which was organized by the University of Rochester Medical Center, Rochester, NY, USA [24]. Two groups of subjects have been selected for this study: 64 healthy control (HC) subjects and 44 patients with IDC. All subjects gave their written informed consent before study participation. Healthy subjects had no previous history of any heart disease or other chronic disorders. They were symptom free, off drugs, and had a normal physical examination and normal 12-lead ECG. If there was suspicion of any ECG changes, a normal electrocardiogram and normal exercise test were required. IDC patients were identified based on left ventricular dysfunction with a left ventricular ejection fraction less than or equal to 40% in the absence of ischaemic or valvular heart disease (angiographically confirmed). All IDC patients had to remain in sinus rhythm to be eligible for enrolment. The demographic characteristics of the subjects included in the study are shown in table 1, in which age, gender, aetiology, comorbidities and therapies are described.

A three-orthogonal-lead Holter ECG was recorded (Burkid-Spacelab, Milton, WI) during 24 h with a sampling frequency of 200 Hz. Heart period was approximated as the time interval between consecutive R-wave peaks on the electrocardiogram (RR), and QT intervals in each consecutive beat were obtained using a wavelet-based ECG detection algorithm [25]. The QT interval was defined as the time interval between the beginning of the QRS complex and the end of the T-wave. This system has proven to be quite robust against noise and morphological variations, even in the problematic T-wave delineation. In addition, algorithms based on a robust template-matching approach have been recommended for beat-to-beat variability measurement of the QT interval in body surface ECG [26].

The ECG lead with the best signal-to-noise ratio was selected for the analysis. All analysed subjects had less than 10% ectopic beats. Therefore, a possible alteration of the results produced by the filtering process can be discarded. In this way, the RR and QT series were filtered by replacing beat abnormalities if their deviation was more than a programmed tolerance of 15% and 10%, respectively, from the mean values of the previous five beats. In order to analyse QT values with a reduced effect of heart rate, we obtained the corrected QT (QTc) intervals. Calculation of QTc-values was according to Bazzet's formula [27].

Beat-to-beat RR, QT and QTc intervals were divided into windows of length *w*=300 samples, without overlapping. This was nearly the average data length in 5 min short-term RR interval series [14]. Besides that, awake (daytime) and sleep (night-time) periods were selected from the 24 h ECG recordings. Approximately, night-time corresponds to the interval from 00.00 to 06.00, and daytime from 09.00 to 21.00.

### (b) Linear time-domain and frequency-domain analysis

The standard temporal and spectral measures derived from RR and QT series were calculated according to the guidelines reported in [14]. The following linear time-domain measures were obtained: mean of the RR intervals (meanRR) and standard deviation of the NN intervals (SDRR); mean of the QT intervals (meanQT) and standard deviation of QT intervals (SDQT); mean of the QTc intervals (meanQTc) and standard deviation of QTc intervals (SDQTc). The power spectral measures of RR variability were: HFn, normalized power in high-frequency band (0.15–0.4 Hz); LFn, normalized power in low-frequency band (0.04–0.15 Hz); and VLF, power in very-low-frequency band (less than or equal to 0.04 Hz). Moreover, the ratio between LF/HF power was calculated.

### (c) Symbolic dynamics analysis

Symbolic dynamics is a nonlinear approach based on the transformations of a series into symbolic sequences with symbols from a given alphabet, for investigating the dynamic aspects of a time series. It provides simplified information of complex systems by means of a limited number of symbols [23]. Some detailed information is lost in this process, but the symbolic sequence obtained is useful to describe the dynamic behaviour of the original series [28]. In this study, two different approaches to transform cardiac series into a symbolic series are applied. These consist of dividing the cardiac series into EQ and NEQ levels.

#### (i) Transformation of the cardiac series into symbols: equidistant levels

In this approach, given a temporal series *x*={*x*(*i*),*i*=1,…,*N*}, the full range of dynamics of the series *x*(*i*) is divided into *ξ* quantization bins of size ℓ=(max(*x*(*i*))−min(*x*(*i*)))/*ξ* in order to obtain a symbolic cardiac series *S*_{max−min,i}, as is derived from the following algorithm (2.1):
2.1
Therefore, the transformation has the symbols *A*={1,…,*ξ*}. In this work, the number of quantization levels is set to *ξ*=4 and *ξ*=6 [21].

#### (ii) Transformation of the cardiac series into symbols: non-equidistant levels

This approach transforms the cardiac series *x*(*i*), *i*=1,…,*N*, into the symbol sequence *S*_{α,i} using quantization with NEQs [20], which are functions of the deviation from the average of *x*(*i*), as indicated by
2.2
This transformation uses four non-equidistant quantization levels where the set of symbols is *A*={1,2,3,4}, *μ* denotes the average of *x*(*i*), and (1+*α*)⋅*μ* and (1−*α*)⋅*μ* are thresholds, respectively, above and below the average according to the parameter *α*. In this study, *α* is set to 0.07. This means that the upper and lower thresholds are, respectively, 7% above and below the average. This choice has proven to yield reliable results if applied to cardiac series [29].

#### (iii) Patterns of symbolic sequences

Words of length *k*=3 consecutive symbols with an overlapping of two symbols are generated. Figure 1 shows an example of a QTc series quantized using *ξ*=6 EQs.

The words of length *k* are categorized in patterns according to their amount of variations between successive symbols as follows [21]:

— no variation (0 V) between successive symbols;

— one variation (1 V) between successive symbols; and

— two like (2 LV) or two unlike (2UV) variations between successive symbols (2 V). That is, two successive increases or decreases or one decrease followed by one increase, or vice versa.

This categorization was applied to the symbolic series *S*_{max−min,i} and *S*_{α,i} obtaining *ξ*^{k} different sequences of symbols. The categorization of the 6^{3}=216 different sequences results in: six with 0 V sequences, 0V={0Vd,0Vu} with 0Vd={111,222,333} and 0Vu={444,555,666}; 60 with 1 V sequences, 1 V={1Veu,1Vue,1Vde,1Ved}; 40 with 2 LV sequences and 110 with two UV sequences, 2 V={2Vdu,2Vud,2Vdd,2Vuu}. An example of how these sequences of *k*=3 symbols with *ξ*=6 vary is shown in figure 2. The categorization of the 4^{3}=64 different sequences results in: four with 0 V sequences, 0V={0Vd,0Vu} with 0Vd={111,222} and 0Vu={333,444}; 24 with 1V sequences, 1 V={1Veu,1Vue,1Vde,1Ved}; eight with 2LV sequences and 28 with 2UV sequences, 2 V={2Vdu,2Vud,2Vdd,2Vuu}.

#### (iv) Assessment of the regularity of the patterns

In this study, the following measures have been proposed to measure the regularity of the patterns:

— Occurrence rate of each pattern: 0 V%, {0Vd%, 0Vu%}; 1 V%, {1Veu%, 1Vue%, 1Vde%, 1Ved%}; 2V%, {2Vdu%, 2Vud%, 2Vdd%, 2Vuu%}.

— Number of patterns (PTHy) with probability higher than or equal to a threshold (THy), where THy={1%,3%,5%,10%,50%}.

— Number of forbidden words (fw01), defined as patterns with probability lower than 0.1%.

— Shannon entropy (SH) and Rényi entropy (Hq) 2.3 and 2.4 where the probabilities

*p*_{i}are the probabilities of the families 0, 1 and 2 V. Large probabilities dominantly influence Hq if*q*>1 and small probabilities mainly determine the value of this entropy if 0<*q*<1 [30]. When*q*tends to unity, Hq converges to SH. In this study, Hq was estimated for different values of the control parameter*q*={0.1,0.15,0.20,0.25,2,4,6}.— Conditional entropy (Hc) [19,31], calculated in a window of length

*w*using the definition 2.5 where*N*_{z}=*w*−(*k*−2),*N*_{j}=*w*−(*k*−1),*u*_{k}(*j*) represents a pattern of*k*consecutive samples,*p*(*u*_{k−1}(*z*)) denotes the joint probability of the pattern*u*_{k−1}(*z*),*p*(*u*_{k}(*j*)/*u*_{k−1}(*z*)) symbolizes the conditional probability of a pattern*u*_{k}(*j*) of*k*samples given a pattern*u*_{k−1}(*z*) of*k*−1 samples. The parameter*q*is a real number,*q*>0 and*q*≠1, that determines the manner in which the probabilities of the vectors*u*_{k}(*j*) are weighted. In this study, different values of parameter*q*={0.1,0.15,0.25,1,2,4} are taken into account, where*q*=1 indicates that the definition of SH was used, and the parameter*k*was fixed to*k*=3 [32]. The conditional entropy Hc will be indicated as Hc(*k*)_{q}, with*k*=3. Entropy Hc(3)_{q}was computed using the quantization based on dividing the amplitude of the cardiac series in EQs and in NEQs.

### (d) Statistical analysis

All measures from linear and nonlinear methodologies were expressed by the arithmetical mean±standard deviation. The separation ability between measures derived from HC and IDC populations inside the same period of analysis (24 h, daytime and night-time) was analysed using the Mann–Whitney *U*-test. The level of significance was set at *p*-value<0.05. In order to select the best measures able to separate IDC patients from the control group, a linear discriminant function was built for each individual measure with *p*-value<0.05, using the leave-one-out cross-validation procedure. In this statistical analysis, sensitivity (Sen), specificity (Spe) and accuracy (Acc) were taken into account. Correlations were assessed by Spearman's rank correlation coefficients (*ρ*) selected for those measures with the highest Acc values.

## 3. Results

### (a) Time-domain and frequency-domain analysis

Table 2 contains the results corresponding to the measures obtained from time-domain and frequency-domain analyses for the 24 h period, daytime and night-time, when the HC and IDC groups were compared. Table 2 contains only the measures with Acc higher than 60% in at least one of the periods of analysis (24 h, daytime or night-time). Although the measure meanRR was not significant, it was also included.

Regarding the results in table 2, the following can be observed. (i) The meanRR did not show a significant difference in any of the analysed periods. (ii) SDRR was statistically significant for all periods, with the 24 h period being the one with the highest Acc value (65.2%). The value of SDRR was higher in the HC group, suggesting that a significant decreased variability is present in the IDC group in all the analysed periods compared with the HC group. (iii) The meanQT and meanQTc measures showed Acc values higher than 70% in all analysed periods. The values of meanQT and meanQTc were higher in the IDC group, suggesting a significant lengthening of the repolarization intervals of the IDC patients. (iv) In relation to the results obtained from the frequency-domain analysis, LFn and LF/HF presented Acc values higher than 60% but only in the daytime. In this case, the values of LFn and LF/HF were significantly lower in the IDC group than in the HC group.

### (b) Symbolic dynamics

Many of the proposed measures derived from the application of the symbolic dynamics approach showed significant differences (*p*-value<0.05) with levels of sensitivity, specificity and accuracy higher than 60% (Sen>60%, Spe>60% and Acc>60%). The number of measures that met these conditions is indicated in table 3. The results were grouped by periods of time (24 h, daytime and night-time), by type of series (RR, QT and QTc), and by method of quantization (6EQ, six equidistant levels; 4EQ, four equidistant levels; and 4NEQ, four non-equidistant levels). The information contained in table 3 indicates that similar behaviour is observed in the RR series in the 24 h and daytime periods. For these RR series, a higher number of measures was obtained with the quantization method 4NEQ, whereas it was 4EQ for the night-time period. The number of measures was lower in the QTc series than in the RR series, with the quantization 6EQ being where more measures were obtained for the 24 h period and also for daytime and night-time. None of the proposed symbolic measures of the QT series showed *p*-value<0.05, Sen>60%, Spe>60% and Acc>60% simultaneously.

#### (i) Twenty-four hour period

Table 4 includes the symbolic measures derived from the RR and QTc series for the 24 h period. Only measures from 4NEQ levels for RR intervals and 6EQ levels for QTc intervals, with *p*-value<0.05, Sen>60% and Spe>60%, were included. In relation to the RR intervals, all the symbolic measures showed Acc values higher than 68%, all of which were superior to the Acc values achieved with the measures obtained from the time-domain and frequency-domain analyses. The measure 1Ved% had the highest Acc value (80.9%), which is an important improvement compared with the 65.2% obtained with the linear measure SDRR. In addition, three of the symbolic measures obtained in QTc intervals (1Vde%, 1Veu% and PTH1) showed higher Acc values than linear measures obtained in the time-domain analysis, with 80.6% being the highest Acc value. In general, the results in table 4 showed that (i) the occurrence of 0 V patterns was lower in the HC group than in the IDC group for the RR series; (ii) 0Vu% was higher in the HC group than in the IDC group, whereas 0Vd% showed the opposite; (iii) the rate of 1 V patterns (1V%, including 1Vde%, 1Ved%, 1Veu% and 1Vue%) was higher in the HC group than in the IDC group for the RR and QTc series; (iv) the number of patterns with probability higher than or equal to a threshold (PTH3, PTH5 and PTH10 for the RR series, and PTH1 and PTH3 for the QTc series) was higher in the HC group than in the IDC group; and (v) entropy measures (i.e. SH, Hq and Hc(3)) showed higher values in the HC group than in the IDC group.

#### (ii) Daytime and night-time periods

The results derived from symbolic analysis applied to the cardiac intervals during daytime and night-time are shown in figures 3 and 4. Only the measures from the 4NEQ and 4EQ levels are presented for the RR intervals during daytime and night-time periods, respectively, and from the 6EQ levels for QTc intervals during daytime as the most representative (as was seen in table 3). Figure 3 shows the occurrence rate of 0, 1, 2V patterns, and their decomposition into 0Vd%, 0Vu%, 1Veu%, 1Vue%, 1Vde%, 1Ved%, 2Vdu%, 2Vud%, 2Vdd%, 2Vuu%. Figure 4 shows the results of the patterns with probability higher than or equal to a threshold (PTH1, PTH3, PTH5, PTH10, PTH30 and PTH50) and the patterns with probability lower than 0.1% (fw01). The results corresponding to entropy measures (SH, Hq and Hc) are included in table 5.

Comparing Acc values between nonlinear measures (symbolic dynamics) and linear measures (time and frequency domains), it is possible to see (as was shown during the 24 h period) that nonlinear measures show better Acc values than linear measures computed in daytime and night-time. Indeed, the highest Acc value (81.8%) from RR intervals during the daytime was obtained with 1Ved% and 1Vue%, which was superior to the best Acc value (69.6%) obtained with linear measures during the same period of analysis. QTc intervals during the daytime reached the highest Acc value (78.0%) with 1V%, whereas Acc is 74.8% with linear measures. In relation to RR intervals from night-time, the highest Acc value with nonlinear measures was 80.8% (PTH1, fw01), whereas the best linear measure was SDRR—reaching Acc=60.6%. In the QTc series during night-time, the highest Acc values were obtained with both nonlinear and linear measures (1Veu% with 70.5% and meanQTc with 70.3%, respectively).

In general, the behaviour of the nonlinear measures during daytime and night-time was similar to the behaviour observed during the 24 h period, in both the RR and QTc intervals. Indeed, (i) the occurrence rate of 0 V patterns was lower in the HC group than in the IDC group; (ii) the occurrence rate of the 1V pattern (including 1Vde%, 1Ved%, 1Veu% and 1Vue%) was higher in the HC group than in the IDC group; (iii) measures PTH1, PTH3, PTH5 and PTH10 were higher in the HC group than in the IDC group; (iv) entropy measures (SH, Hq and Hc) showed higher values in the HC group than in the IDC group; (v) fw01 was lower in the HC group than in the IDC group. However, 0Vu% and 0Vd% showed a different behaviour during the daytime than during the night-time. While 0Vu% exhibited higher values and 0Vd% lower values in the HC group than in the IDC group during the daytime, 0Vu% and 0Vd% showed lower values in the HC group than in the IDC group during the night-time.

## 4. Discussion

The time-domain analysis showed that SDRR and the mean of the QT and QTc intervals were the only measures able to discriminate (*p*-value<0.05) and classify (Acc>60%) subjects in the HC and IDC groups. Our results indicate a reduction in the magnitude of HRV in IDC patients compared with the HC group. This behaviour was observed in all analysed periods (24 h, daytime and night-time). A similar response was reported in [32,33], where SDRR showed statistical differences during both 24 h and daytime, when low- and high-risk groups of IDC were compared. As to the mean of QT and QTc, the results indicate that IDC patients showed longer QT and QTc intervals than healthy subjects, suggesting that the repolarization of the myocardium is slower in patients with dilated cardiomyopathy. In agreement with our finding, other studies [10,11,34] have also shown an increased QTc interval evaluated in patients with dilated cardiomyopathy, where the lengthening of QTc intervals has been considered a marker of bad prognosis in different grades of patients. SDQT or SDQTc were not significantly different between groups despite a significantly different QTc mean. One limitation of this study is given by the relatively low sampling rate of 200 Hz, which produces an intrinsic error of 5 ms in the detection of cardiac intervals. Risk *et al*. [35] verified that the sampling rate affects the measurements of QT interval and duration of QRS complex; in both cases, these variables were overestimated at ECG sampling rates less than 300 Hz.

Considering that CAD can be the initiating cause of heart failure [6], and patients with CHF have an increased cardiac sympathetic outflow [3], it is expected that IDC patients will have similar responses. It has been reported that conditions such as CHF, hypertension and myocardial infarction are associated with baroreflex–cardiovagal failure and associated with a reduced LF power [15]. Notarius *et al*. [36] also reported, in patients with moderate and severe CHF, an absence of the spectral power in the LF range, which was very closely associated with the resting muscle sympathetic nerve activity. This is in agreement with our results where LFn and LF/HF were able to statistically discriminate subjects in the HC and IDC groups, with the lowest value being in the IDC group for both measures. The main statistical differences of LFn and LF/HF were during daytime, as was also reported in [32,33]. The measures extracted from time-domain and frequency-domain analysis of HRV have been shown to contain useful information [14,22]. However, these measures describe only linear features of the systems involved in heart rate control and may not capture the nonlinear features of such control.

The reported results showed in a clear way that nonlinear symbolic measures, according to the proposed methods, have provided a better separation of IDC patients from HC subjects than the linear measures obtained from the standard temporal and spectral RR measures. Many of the measures derived from symbolic dynamics, in RR or QTc intervals, were able to discriminate between the HC and IDC groups. These measures showed a better performance than the measures derived from the time domain and the frequency domain, reaching an accuracy higher than 80%. The occurrence rate of patterns without variations and patterns with variations between successive symbols was associated with sympathetic and parasympathetic modulations [18,22,37]. Indeed, Guzzetti *et al.* [22] reported that an increase in sympathetic modulation and a vagal withdrawal elicited an increased presence of patterns without variations and a decreased occurrence of patterns with variations between successive symbols, whereas parasympathetic dominance induced the opposite. For all analysed periods (24 h, daytime and night-time), our findings similarly showed an increase of 0 V% and a decrease of 1 V% in the IDC group, suggesting a reduction in the variability in RR and QTc intervals as a possible association with the increment in the sympathetic activity in this group. Although the 2 V% is also associated with a higher variability, these patterns had a lower performance than 1 V% (Acc<60%). All the patterns with one variation (1Veu%, 1Vue%, 1Vde%, 1Ved%) exhibited a similar tendency, with the number of these patterns being higher in the HC group than in the IDC group. This means that the result is almost the same, regardless of the type of variation (up or down) or if the variation occurs after or before a period without variation, and, therefore, all these measures are appropriate to indicate a reduction in the RR or QT variability in IDC patients.

As for the RR series during the daytime, the highest number of patterns (more than 80%) belongs to the 0 V family, whereas the 1V and 2 V families contain lower frequencies of patterns (less than 13%). During the night-time, the 1V and 2 V families increased their frequency values at the expense of diminishing 0 V%. Similar results were observed in Porta *et al*. [37], where a clear day–night variation of 0 V(%) and 2UV(%) with 0 V(%) decreasing and 2UV(%) increasing during the night-time was reported. This day–night modification leads to a higher regularity of the RR series during the daytime with the presence of 0 V patterns and a loss of regularity during the night-time. This behaviour was followed by both the HC and IDC groups, suggesting a preserved circadian pattern.

Similarly, the results from entropy measures, based on the occurrence probability of patterns, also showed a reduction in the variability in the IDC group compared with that in the HC group. Indeed, all the entropy measures (SH, Hq and Hc) reported in tables 4 and 5 had a higher value in the HC group, suggesting an increase in the regularity of the RR and QTc intervals in the IDC group. Porta *et al*. [19] pointed out that a reduced complexity through a reduction in the number of different patterns is associated with sympathetic activation. Therefore, the reduced entropy values obtained in IDC patients might be associated with sympathetic activation in those patients. Comparable results using entropy measures have been reported in [32,33], where an increase in heart rate regularity was associated with the evolution of cardiac disease severity. Conditional entropy Hc, which assesses the distribution of the pattern with length *k*=3 conditioned on the distribution of the pattern with length *k*=2, was one of the best entropies because it could describe the HC and IDC groups in all analysed periods with Acc>70% in the RR series and Acc>60% in the QTc series. While the SH and Hq entropies depend only on pattern distribution, Hc also considers the dynamic relations between patterns, providing more information about the regularity of the RR or QTc intervals. Entropy rates such as Hc can be affected by the short length of the series or the long length of the patterns, which can artificially reduce the value of conditional entropy, giving a false impression of determinism and requiring a correction factor [19]. In this study, we did not use it because we limited the evaluation of entropy to a very low embedding dimension [38]. Besides that, Valencia *et al*. [32] demonstrated that a non-uniform quantization of the dynamic range of the RR series, as applied in this work, can also improve the performance of Hc even with only a few quantization regions.

The correlation analysis between the measures with highest accuracy values obtained from nonlinear measures (0 V, 1 V, PTH1) and linear measures (meanQTc) has shown a weak correlation with the correlation coefficient values between 0.2 and 0.5 (*p*-value<0.05). Furthermore, no correlation was found between the nonlinear measures and measures from the frequency-domain analysis. Finally, consistent with Heitmann *et al*. [8], we have confirmed that indices from linear methods demonstrate only a weak differentiation between healthy subjects and patients with heart disease. By contrast, nonlinear indices have proven to have more significant discriminatory power, showing a statistical significance level of *p*-value<0.0005 in most of the measures. These significance levels also satisfied the Bonferroni criterion (*p*-value<0.0015). However, the observed differences between the HC and IDC groups may be influenced by a higher proportion of females in the HC group [17]. In relation to the QT and QTc intervals, the nonlinear measures of the individual QT have demonstrated that they could not statistically differentiate between the groups, but significant statistics were obtained for QTc (*p*-value<0.05 and Acc>60%). Considering the individual QT dependence on past RR intervals, the traditional approach for rate correction of the QT interval used was the Bazett correction. It is known that this correction and any other approach using a fixed formula will clearly over- or underestimate the true and individual QT/RR relationship and will lead to imprecise QTc values [39,40]. Furthermore, the QT/RR relationship is much more complicated when QT is analysed under different, non-standard conditions, as happens during the 24 h ECG. Some studies have directed efforts towards applying nonlinear dynamics methods to analyse the QT/RR, providing additional markers of repolarization instability. Indeed, Baranowski & Zebrowski [12] applied joint symbolic dynamics to the relation of QT on RR interval dynamics in healthy women and men. A similar nonlinear method was applied in [10], where the effect of ageing on the QT/RR relationship was assessed in healthy subjects. The results of the univariate symbolic dynamics applied to ECG cardiac intervals in the IDC and HC groups in this work indicate a requirement for further study based on joint symbolic dynamics to investigate the QT/RR relation of those patients.

## 5. Conclusion

This work investigated whether symbolic transformations applied to RR and QT variability can improve separation between IDC patients and a HC group, compared with traditional linear time and frequency-domain analyses. Two different approaches to transform the cardiac series into a symbolic series, consisting of dividing the cardiac series into EQs and NEQs, were applied to RR and QTc series belonging to HC and IDC subjects. Although measures extracted from the time-domain and frequency-domain analyses of the HRV were able to demonstrate some statistical differences between the IDC and HC groups, the measures derived from symbolic dynamics showed a better performance and were able to classify the subjects with more than 80% accuracy. Indeed, the rates of patterns without variation and with one variation indicated a reduction in the complexity in the RR and QTc series in IDC patients compared with healthy subjects. This reduction was confirmed by entropy measures, computed over the symbolic patterns. These results suggest that symbolic analysis is a useful method to obtain diagnostic information in IDC patients, and the symbolic analysis provides measures that might be related to the high sympathetic tone in IDC patients.

## Funding statement

This work was supported within the framework of the CICYT grant no. TEC2010-20886 from the Spanish Government. CIBER of Bioengineering, Biomaterials and Nanomedicine is an initiative of ISCIII.

## Appendix A

**(a) Dependency of the conditional entropy values on the pattern length**

Figure 5 shows the conditional entropy (Hc(*k*)_{q}) values as a function of the *k* consecutive samples in the defined patterns. This study was done on 30 random series of 10 000 samples with a Gaussian distribution with zero mean and unitary standard deviation. In figure 5, Hc was calculated using SH over windows of length *w*=300 samples and *w*=1000 samples. The windows were considered without overlapping. It can be observed that Hc(*k*)_{q} remained stable from *k*=1 to 3 even when windows of *w*=300 samples were taken into account.

**(b) Dependency of the conditional entropy values on the parameter q**

Figure 6 shows the conditional entropy (Hc(*k*)_{q}) values as function of the parameter *q*. The value of parameter *k* was set to 3 [32]. This study was done on the RR and QTc intervals of 64 HC subjects and 44 patients with IDC. The results of (Hc(3)_{q}) from the daytime and night-time periods are presented. It can be observed that during the daytime the best results of Hc(3) correspond to *q*={1,2} and during the night-time to *q*=1 for the RR series, but *q*≥2 for the QTc series.

## Footnotes

One contribution of 12 to a theme issue ‘Enhancing dynamical signatures of complex systems through symbolic computation’.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.