Complex Correlation Measure: a novel descriptor for Poincaré plot
- Chandan K Karmakar^{1}Email author,
- Ahsan H Khandoker^{1},
- Jayavardhana Gubbi^{1} and
- Marimuthu Palaniswami^{1}
https://doi.org/10.1186/1475-925X-8-17
© Karmakar et al; licensee BioMed Central Ltd. 2009
Received: 15 May 2009
Accepted: 13 August 2009
Published: 13 August 2009
Abstract
Background
Poincaré plot is one of the important techniques used for visually representing the heart rate variability. It is valuable due to its ability to display nonlinear aspects of the data sequence. However, the problem lies in capturing temporal information of the plot quantitatively. The standard descriptors used in quantifying the Poincaré plot (SD 1, SD 2) measure the gross variability of the time series data. Determination of advanced methods for capturing temporal properties pose a significant challenge. In this paper, we propose a novel descriptor "Complex Correlation Measure (CCM)" to quantify the temporal aspect of the Poincaré plot. In contrast to SD 1 and SD 2, the CCM incorporates point-to-point variation of the signal.
Methods
First, we have derived expressions for CCM. Then the sensitivity of descriptors has been shown by measuring all descriptors before and after surrogation of the signal. For each case study, lag-1 Poincaré plots were constructed for three groups of subjects (Arrhythmia, Congestive Heart Failure (CHF) and those with Normal Sinus Rhythm (NSR)), and the new measure CCM was computed along with SD 1 and SD 2. ANOVA analysis distribution was used to define the level of significance of mean and variance of SD 1, SD 2 and CCM for different groups of subjects.
Results
CCM is defined based on the autocorrelation at different lags of the time series, hence giving an in depth measurement of the correlation structure of the Poincaré plot. A surrogate analysis was performed, and the sensitivity of the proposed descriptor was found to be higher as compared to the standard descriptors. Two case studies were conducted for recognizing arrhythmia and congestive heart failure (CHF) subjects from those with NSR, using the Physionet database and demonstrated the usefulness of the proposed descriptors in biomedical applications. CCM was found to be a more significant (p = 6.28E-18) parameter than SD 1 and SD 2 in discriminating arrhythmia from NSR subjects. In case of assessing CHF subjects also against NSR, CCM was again found to be the most significant (p = 9.07E-14).
Conclusion
Hence, CCM can be used as an additional Poincaré plot descriptor to detect pathology.
Background
As mentioned earlier, standard descriptors SD 1 and SD 2 are linear statistics [2] and hence the measures do not directly quantify the nonlinear temporal variations in the time series contained in the Poincaré plot. Further, when applied to the data sets that form multiple clusters in a Poincaré plot due to complex dynamic behaviors, the SD 1/SD 2 statistics yields mixed results. This is because the technique relies on the existence of a single cluster or a defined pattern [13, 14]. Moreover, the limitations of the SD 1/SD 2 analysis are important to understand when attempting to investigate the physiological mechanisms in a time series, or when analyzing data where the occurrence of nonlinear behavior may be a distinguishing feature between health and disease. Such a study will also enable further studies in defining new descriptors for Poincaré plot which is currently not being addressed by researchers in this field. The necessity for such a study arises from the fact that the visual pattern is relied upon clinical scenarios and the application of the existing standard descriptors in various studies has resulted in limited success.
Therefore, we hypothesize that any descriptor that captures temporal information and is a function of multiple lag correlation, would provide more insight into the system rather than conventional measurements of variability of Poincaré plot (SD 1 and SD 2), which is a function of a lag-1 correlation. In this study, we propose a novel descriptor for Poincaré plot that can be applied to measure the multiple lag correlation of the signal. Unlike SD 1, SD 2 and SD 1/SD 2 terms, the proposed measure incorporates the temporal information of the time series. In this paper, we aim to evaluate all three descriptors (SD 1, SD 2 and CCM) of the Poincaré plot of RR intervals, and compare their performance in differentiating arrhythmia and CHF from normal subjects.
Standard Poincaré Plot Analysis
This section describes the standard descriptors of Poincaré plot and their limitations. In this paper, we have used RR interval time series signal to plot the Poincaré plot which is denoted by RR_{ n }. We assume that a finite number of RR intervals are available and a wide-stationarity of the RR interval as suggested in literature [2].
Standard Descriptors
where _{ γRR }(m) is the autocorrelation function for lag-m RR interval. This implies that the standard descriptors for any arbitrary m lag Poincaré plot is a function of autocorrelation of the signal at lag-0 and lag-m.
Limitations of Standard Descriptors
In a study [11], authors have shown that the measurement from multiple lag Poincaré plot provides more information than any measure from single lag Poincaré plot. Indeed, the Poincaré plot at any lag m is more of a generalized scenario, where other levels of temporal variation of any dynamic system are hidden.
Methods
The development of a new descriptor, Complex Correlation Measure (CCM) has been presented in this section. Firstly, the theoretical development has been given, followed by the analysis of the new measure with respect to the standard descriptors, SD1 and SD2. Finally, data and methodologies regarding the two case studies have been discussed which is followed by a brief description of statistical analysis used in this study.
Complex Correlation Measure
where m represents lag of Poincaré plot and C_{ n }is the normalizing constant which is defined as, C_{ n }= π * SD 1 * SD 2, represents the area of the fitted ellipse over Poincaré plot. The length of major and minor axis of the ellipse are 2*SD 1, 2*SD 2, where SD 1, SD 2 are the dispersion perpendicular to the line of identity (minor axis) and along the line of identity (major axis) respectively.
Analysis of Complex Correlation Measure(CCM)
In the previous section, we have given the mathematical definition of CCM and have clearly shown that CCM contains multiple lag correlation information of the signal. In this section, we explore the different properties of CCM with synthetic RR interval data. In our study, we used 4000 RR intervals of a synthetic RR interval (rr02) time series data from open access Physionet database [16] and the signal was divided into 20 windows with 200 RR intervals in each window.
Sensitivity to changes in temporal structure
Literally, the sensitivity is defined as the rate of change of the value due to the change in temporal structure of the signal. The change in temporal structure of the signal in a window is achieved by surrogating the signal (i.e, data points) in that window. In order to validate our assumption we calculated SD 1, SD 2 and CCM of a RR interval signal by randomly surrogating points of each window at a time.
where SD 1_{0} (= 0.36), SD 2_{0} (= 0.08) and CCM_{0} (= 0.16) were the parameters measured for the original data set without surrogation and j represents the window number whose data was surrogated. Moreover, SD 1_{ j }, SD 2_{ j }and CCM_{ j }represents the SD 1, SD 2 and CCM values respectively after surrogation of j^{ th }window. Since we divided the entire signal in 20 windows, it resulted in 20 values of SD 1, SD 2 and CCM.
Sensitivity to various lags of Poincaré plot
To verify the sensitivity of SD 1, SD 2 and CCM with various lags of Poincaré plot, values of all descriptors were calculated for different time delays or lags (m was varied from 1 to 100). At each step, lag-m Poincaré plot was constructed for the synthetic RR interval series and then SD 1, SD 2 and CCM values were calculated for the plot. As m was varied from 1 to 100, it resulted in 100 values of SD 1, SD 2 and CCM.
Case Studies
In order to validate the proposed measure – CCM, two case studies were conducted on RR interval data. The data from MIT-BIH Physionet database are [17] used in the experiments. Medical fraternity has utilized Poincaré plot, using both qualitative and quantitative approaches, for detecting and monitoring arrhythmia. Compared to arrhythmia, fewer attempts are made to utilize Poincaré plot to evaluate CHF. In this study, we have analyzed the performance of CCM and compared it with that of SD 1 and SD 2 for recognizing both arrhythmia and congestive heart failure using HRV signal.
HRV study of Arrhythmia and Normal Sinus Rhythm
In this study, we have used 54 long-term ECG recordings of subjects in normal sinus rhythm (30 men, aged 28.5 to 76, and 24 women, aged 58 to 73) from Physionet Normal Sinus Rhythm database [17].
Furthermore, we have also used NHLBI sponsored Cardiac Arrhythmia Suppression Trial (CAST) RR-Interval Sub-study database for the arrhythmia data set from Physionet. Subjects of CAST database had an acute myocardial infarction (MI) within the preceding 2 years and 6 or more ventricular premature complexes (PVCs) per hour during a pre-treatment (qualifying) long-term ECG (Holter) recording. Those subjects enrolled within 90 days of the index MI were required to have left ventricular ejection fractions less than or equal to 55%, while those enrolled after this 90 day window were required to have an ejection fraction less than or equal to 40%.
The database is divided into three different study groups among which we have used the Encainide (e) group data sets for our study. From that group we have chosen 272 subjects belong to subgroup baseline (no medication). The original long term ECG recordings were digitized at 128 Hz, and the beat annotations were obtained by automated analysis with manual review and correction [17]. lag-1 Poincaré plots were constructed for both normal and arrhythmia subjects and the new measure CCM was computed along with SD 1 and SD 2. The SD 1 and SD 2 were calculated to characterize the distribution of the plots whereas CCM were used for characterizing the temporal structure of the plots.
HRV study of Congestive Heart Failure(CHF) and Normal Sinus Rhythm
For this case study, we have used 29 long-term ECG recordings of subjects (aged 34 to 79) with CHF (NYHA classes I, II and III) from Physionet Congestive Heart Failure database along with 54 ECG recordings of subjects with normal sinus rhythm as discussed earlier [17]. Same ECG acquisition with beat annotations were used as discussed in previous case study. Similar to previous case study, lag-1 Poincaré plots were constructed for both normal and CHF subjects and the new descriptor CCM was computed as per traditional descriptors.
Statistical Analysis
In this study we have used ANOVA analysis assuming unknown and different variance for testing the hypothesis regarding mean i.e., the mean of NSR and Arrhythmia groups are equal. It suits our case studies as the sample size is small. The same test has been used to test the hypothesis for NSR and CHF group.
Results
Sensitivity to changes in temporal structure
Sensitivity to various lags of Poincaré plot
HRV studies of Arrhythmia and Normal Sinus Rhythm
Mean ± Standard deviation of all descriptors with p values for NSR and Arrhythmia subjects
SD1 | SD2 | CCM | |
---|---|---|---|
NSR | 0.03 ± 0.02 | 0.19 ± 0.04 | 0.05 ± 0.03 |
Arrhythmia | 1.92 ± 5.18 | 2.30 ± 5.86 | 0.26 ± 0.08 |
p value (ANOVA) | 7.60E-3 | 8.50E-3 | 6.28E-18 |
HRV studies of Congestive Heart Failure(CHF) and Normal Sinus Rhythm
Mean ± Standard deviation of all descriptors with p values for NSR and CHF subjects.
SD 1 | SD 2 | CCM | |
---|---|---|---|
NSR | 0.03 ± 0.02 | 0.19 ± 0.04 | 0.05 ± 0.03 |
CHF | 0.04 ± 0.02 | 0.11 ± 0.06 | 0.14 ± 0.06 |
p value (ANOVA) | 5.65E-4 | 5.04E-12 | 9.07E-14 |
Discussion
The main motivation for using Poincaré plot is to visualize the variability of any time series signal. In addition to this qualitative approach, we propose a novel quantitative measure, CCM, to extract underlying temporal dynamics in a Poincaré plot. Surrogate analysis showed that the standard quantitative descriptors SD 1 and SD 2 were not as significantly altered as did CCM, this is shown in figure 4. Both SD 1 and SD 2 are second order statistical measures [2], which are used to quantify the dispersion of the signal perpendicular and along the line of identity respectively. Moreover, SD 1 and SD 2 are functions of lag - m correlation of the signal for any m lag Poincaré plot. In contrast, CCM is a function of multiple lag (m - 2, m - 1, m, m + 1, m + 2) correlations and hence, this measure was found to be sensitive to changes in temporal structure of the signal as shown in figure 4.
From the theoretical definition of CCM it is obvious that the correlation information measured in SD 1 and SD 2 is already present in CCM. But this does not mean that, CCM is a derived measure from existing descriptors SD 1 and SD 2. Rather, CCM can be considered as an additional measure incorporating information obtained in SD 1 and SD 2 (as the lag m correlation is also included in CCM measure). In a Poincaré plot, it is expected that lag response is stronger at lower values of m and it attenuates with increasing values of m. This is due to the dependence of Poincaré descriptors on autocorrelation functions. The autocorrelation function monotonically decreases with increasing lags and in case of RR interval time series, usually the current beat influences only about six to eight successive beats [12]. In our study, we also found that all measured descriptors SD 1, SD 2 and CCM changed rapidly at lower lags and the values are stabilized with higher lag values (figure 5). Since, CCM is also a function of signals autocorrelations, it shows a similar lag response to that shown by SD 1 and SD 2. Therefore, CCM may be used to study the lag response behavior of any pathological condition in comparison with normal subjects, or controls.
HRV measure is considered to be a better marker for increased risk of arrhythmic events than any other noninvasive measure [18, 19]. An earlier study has shown that Poincaré plots exposed completely different 2D patterns in the case of arrhythmia subjects [20]. These abnormal medical conditions have complex patterns due to reduced autocorrelation of the RR intervals. Consequently due to the changes in autocorrelation, we have found that the variability measure using Poincaré (SD 1, SD 2) was higher than normal subjects (shown in table 1). Moreover, the fluctuations of these variability measures were also very high in the case of arrhythmias. This may be due to different types of arrhythmia along with subjective variation of HRV. In arrhythmia subjects, CCM was found to be higher compared to NSR subjects, but the deviation due to subjective variation is much smaller than SD 1 and SD 2. As a result, CCM linearly separates these two groups of subjects which means that the effect of different types of arrhythmia and subjective variation are reduced while using CCM than other variability measures. Therefore, we may conclude that CCM is a better marker for recognizing arrhythmia than the traditional variability measures of Poincaré plot.
In another case study, we have shown as to how Poincaré plot can be used to characterize CHF subjects from normal subjects using RR interval time series. Compared to SD 2, SD 1 and CCM values were found to be higher in CHF subjects. This findings might indicate that the short term variation in HRV is higher in CHF subjects, however, the long term variation is reduced. Since CCM captures the signal dynamics at short level (i.e, 3 points of the plot), it appears to be affected by short term variation of the signal in CHF subjects. In the case of recognition of CHF subjects, although SD 2 showed good result CCM was found to be more significant as shown in table 2.
Above discussion indicates that CCM is an additional descriptor of Poincaré plot with SD 1 and SD 2. This also implies that CCM is a more consistent descriptor compared to SD 1 and SD 2. Considering the presented case studies, it is clear that neither SD 1 nor SD 2 alone can independently distinguish between normal and pathology. However, in the same scenario, CCM has the ability to perform the classification task independently. This justifies the usefulness of the proposed descriptors as a feature in a pattern recognition scenario. Our primary motivation for detecting pathology with a novel descriptor like CCM rather than by observing visual pattern is achieved as shown by the case studies described. In this study, we have not looked at the physiological interpretation of CCM which remain to be studied in future. However, a few remarks on this would be appropriate. The Poincaré plot reflects the autocorrelation structure through the visual pattern of the plot. The standard descriptors SD 1 and SD 2 summarizes these correlation structure of RR interval data as shown by Brennan et. al. [2]. CCM is based on the autocorrelation at different lags of the time series hence giving an in-depth measurement of the correlation structure of the plot. Therefore, the value of CCM decreases with increased autocorrelation of the plot. In arrhythmia, the pattern of the Poincaré plots becomes more complex [20] and thus reducing the correlation of the signal (RR_{ i }, RR_{i+1}). In case of healthy subjects the value of CCM is lower than that of arrhythmic subjects. In future, it might be worth looking at the performance of CCM for other pathologies.
Conclusion
The proposed Complex Correlation Measure is based on the limitation of standard descriptors SD 1 and SD 2. The analysis carried out confirms the hypothesis that CCM measures the temporal variation of the Poincaré plot. In contrast to the standard descriptors, CCM evaluates point-to-point variation of the signal rather than gross variability of the signal. We have shown that CCM is more sensitive to changes in temporal variation of the signal. We have further demonstrated that CCM varies with different lags of Poincaré plot. Besides the mathematical definition of CCM and analyzing properties of the measure, we have also evaluated the performance of CCM using real world case studies. CCM was found to be effective in the assessment of both arrhythmia and CHF against normal sinus rhythm. In future, CCM may be used as an efficient feature for pathology detection.
Declarations
Acknowledgements
Dr. Mak Adam Daulatzai read the manuscript and made improvement.
Authors’ Affiliations
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