Identification of Pulmonary Hypertension Using Entropy Measure Analysis of Heart Sound Signal

This study introduced entropy measures to analyze the heart sound signals of people with and without pulmonary hypertension (PH). The lead II Electrocardiography (ECG) signal and heart sound signal were simultaneously collected from 104 subjects aged between 22 and 89. Fifty of them were PH patients and 54 were healthy. Eleven heart sound features were extracted and three entropy measures, namely sample entropy (SampEn), fuzzy entropy (FuzzyEn) and fuzzy measure entropy (FuzzyMEn) of the feature sequences were calculated. The Mann–Whitney U test was used to study the feature significance between the patient and health group. To reduce the age confounding factor, nine entropy measures were selected based on correlation analysis. Further, the probability density function (pdf) of a single selected entropy measure of both groups was constructed by kernel density estimation, as well as the joint pdf of any two and multiple selected entropy measures. Therefore, a patient or a healthy subject can be classified using his/her entropy measure probability based on Bayes’ decision rule. The results showed that the best identification performance by a single selected measure had sensitivity of 0.720 and specificity of 0.648. The identification performance was improved to 0.680, 0.796 by the joint pdf of two measures and 0.740, 0.870 by the joint pdf of multiple measures. This study showed that entropy measures could be a powerful tool for early screening of PH patients.


Introduction
Pulmonary hypertension (PH) is a hemodynamic and pathophysiological condition in which pulmonary artery pressure rises above a certain threshold. PH is a potentially fatal disease that can cause right heart failure [1]. If the PH is not diagnosed in a timely manner and no pretreatments are done actively, it will have serious consequences. In the early stage of PH, the symptoms are not apparent to be perceived by physicians but the mechanical activity of the heart has quietly changed and it can be reflected to some degree in the heart sound signals [2,3]. Therefore, the analysis of cardiac acoustic sound could play an important role in the initial diagnosis of PH.
Previous studies have shown that the time interval between the aortic component (A2) and the pulmonary component (P2) of the second heart sound (S2), as well as the dominant frequency of P2, increases in PH patients and they bring potential values to noninvasive diagnosis of PH [4,5]. The A2-P2 splitting interval (SI) was revealed to have links to pulmonary arterial pressure (PAP) [4][5][6].
Unfortunately, it was difficult to separate S2s, A2s, P2s from a heart sound recording reliably and precisely, as well as the splitting interval between A2 and P2 [7]. Some researchers turned their interests to extract features of the heart sounds that could provide relevant diagnostic information. For example, M Elgendi, P Bobhate and S Jain found useful features from time and frequency domains of a heart sound signal for distinguishing people with and without PH [8][9][10]. Another research team used machine learning algorithms to build models to classify people with and without PH based on features of the heart sounds [11]. Unlike the previous methods, this study introduced the entropy measures to analyze the heart sound signal of PH patients.
It is well-known that entropy is a quantitative measure of regularity of a sequence. It is reasonable to use entropy to investigate regularity differences of the heart sound features between people with and without PH. Eleven features of heart sounds were extracted and three entropy measures were used in this study to calculate their entropy values of each feature sequence. Typically, three entropy measures, i.e., sample entropy, fuzzy entropy and fuzzy measure entropy, were considered. Statistical analysis ranked the entropy measures according to the significance level. For identification purpose, the probability density function (pdf) of each entropy measure was built by kernel density estimation, as well as the joint pdf of two and multiple entropy measures. The results indicated that the entropy measures could be powerful to discriminate a PH patient from a healthy subject.

Data Collection
One-hundred and four subjects participated in this study. All subjects were given informed and written consent. The involved subjects were divided into two groups. One was the PH group (50 patients in the second attached hospital of Dalian Medical University) and the other was the healthy control group (54 subjects enrolled from Dalian University of Technology and Shandong University). The basic information of the PH patient group and the healthy control group are shown in Tables 1  and 2. A subject was asked to rest for 10 min prior to data collection. In the progress of data collection, each subject lied on his back quietly in a bed. The heart sound signal-Electrocardiography (ECG) lead II signal-were simultaneously recorded for at least five minutes at a sampling frequency of 2 kHz (PowerLab 8/35, ADInstruments, Sydney, Australia). A heart sound microphone sensor (MLT201, ADInstruments, Sydney, Australia) was placed at the left third intercostal space. The echocardiography was used to measure the pulmonary artery pressure for PH patients. The cardiologist in the hospital made the PH diagnose based on the measured data and symptoms. The PH patient group excludes those with atrial fibrillation or pacemaker. The subjects in the healthy control group were checked by ECG, echocardiography and a routine blood test to verify healthy condition. The R waves of the ECG signals, picked up by the Pan-Tomkins algorithm [12], were used to determine the beginning of a cardiac cycle. The first heart sound (S1) and the second heart sound (S2) were segmented by the Shannon energy envelope [13]. A manual interference was necessary to verify the segmentation if the envelope-based segmentation algorithm failed, especially for the PH patients' heart sound signals where heavy murmurs often occurred. Figures 1 and 2 were two examples of heart sound segmentation for a PH subject and a healthy subject.  The green circles showed the detected R waves; (b) the heart sound signal. The short and long red vertical lines indicated the detected S1s and S2s, respectively.

Figure 2.
A part of collected signals from a healthy subject who was a male aged 67 with 165 cm in height and 63 kg in weight. (a) The ECG signal. The green circles showed the detected R waves; (b) the heart sound signal. The short red vertical lines indicated the detected S1s and the long ones showed the S2s.

Feature Extraction
The time interval of a heart sound and cycle duration were considered as features in this study. The sound interval was the time duration from the start to the end of the heart sound, and the cycle duration was the time duration from the start of S1 to the start of the next S1, as illustrated in Figure 3. The authors defined "Int_s1" as the time interval of S1, "Int_s2" as the time interval of S2, and The green circles showed the detected R waves; (b) the heart sound signal. The short and long red vertical lines indicated the detected S1s and S2s, respectively. The green circles showed the detected R waves; (b) the heart sound signal. The short and long red vertical lines indicated the detected S1s and S2s, respectively.

Figure 2.
A part of collected signals from a healthy subject who was a male aged 67 with 165 cm in height and 63 kg in weight. (a) The ECG signal. The green circles showed the detected R waves; (b) the heart sound signal. The short red vertical lines indicated the detected S1s and the long ones showed the S2s.

Feature Extraction
The time interval of a heart sound and cycle duration were considered as features in this study. The sound interval was the time duration from the start to the end of the heart sound, and the cycle duration was the time duration from the start of S1 to the start of the next S1, as illustrated in Figure 3. The authors defined "Int_s1" as the time interval of S1, "Int_s2" as the time interval of S2, and

Feature Extraction
The time interval of a heart sound and cycle duration were considered as features in this study. The sound interval was the time duration from the start to the end of the heart sound, and the cycle duration was the time duration from the start of S1 to the start of the next S1, as illustrated in Figure 3. The authors defined "Int_s1" as the time interval of S1, "Int_s2" as the time interval of S2, and "Car_cycle" as the cardiac cycle duration. An "Int_s1", an "Int_s2" and a "Car_cycle" can be detected from one cardiac cycle signal of the heart sound recording. So, feature sequences of "Int_s1", "Int_s2" and "Car_cycle" were produced from a heart sound recording.
Entropy 2018, 20, x FOR PEER REVIEW 4 of 17 "Car_cycle" as the cardiac cycle duration. An "Int_s1", an "Int_s2" and a "Car_cycle" can be detected from one cardiac cycle signal of the heart sound recording. So, feature sequences of "Int_s1", "Int_s2" and "Car_cycle" were produced from a heart sound recording. Another four features were from the power spectral domain of S1 and S2. The Burg algorithm was used to calculate the power spectral density (PSD) because of its smoothing character where the order of the autoregressive model was empirically set as 4 [14]. The maximum magnitude of the power spectral density of S2, denoted as "Max_pow_s2" and the corresponding frequency value, denoted as "Max_f_s2" were extracted as two features, as illustrated in Figure 4. The "Max_pow_s1"and "Max_f_s1" were extracted in a similar way. Another four features considered in energy domain were the average energy of S1 (denoted as "Ener_s1"), the average energy of S2 (denoted as "Ener_s2"), the average Shannon energy of S1 (denoted as "ShanEner_s1") and the average Shannon energy of S2 (denoted as "ShanEner_s2"), as given in the following     Another four features were from the power spectral domain of S1 and S2. The Burg algorithm was used to calculate the power spectral density (PSD) because of its smoothing character where the order of the autoregressive model was empirically set as 4 [14]. The maximum magnitude of the power spectral density of S2, denoted as "Max_pow_s2" and the corresponding frequency value, denoted as "Max_f_s2" were extracted as two features, as illustrated in Figure 4. The "Max_pow_s1"and "Max_f_s1" were extracted in a similar way. "Car_cycle" as the cardiac cycle duration. An "Int_s1", an "Int_s2" and a "Car_cycle" can be detected from one cardiac cycle signal of the heart sound recording. So, feature sequences of "Int_s1", "Int_s2" and "Car_cycle" were produced from a heart sound recording. Another four features were from the power spectral domain of S1 and S2. The Burg algorithm was used to calculate the power spectral density (PSD) because of its smoothing character where the order of the autoregressive model was empirically set as 4 [14]. The maximum magnitude of the power spectral density of S2, denoted as "Max_pow_s2" and the corresponding frequency value, denoted as "Max_f_s2" were extracted as two features, as illustrated in Figure 4. The "Max_pow_s1"and "Max_f_s1" were extracted in a similar way. Another four features considered in energy domain were the average energy of S1 (denoted as "Ener_s1"), the average energy of S2 (denoted as "Ener_s2"), the average Shannon energy of S1 (denoted as "ShanEner_s1") and the average Shannon energy of S2 (denoted as "ShanEner_s2"), as given in the following Another four features considered in energy domain were the average energy of S1 (denoted as "Ener_s1"), the average energy of S2 (denoted as "Ener_s2"), the average Shannon energy of S1 (denoted as "ShanEner_s1") and the average Shannon energy of S2 (denoted as "ShanEner_s2"), as given in the following Entropy 2018, 20, 389 where s 1 (i) and s 2 (i) were the S1 and S2 detected by the envelope-based algorithm. L s1 and L s2 were the number of sampling points of the heart sounds. The 11 features used in this study were summarized in Table 3. An example to illustrate some of the feature sequences extracted from a PH subject and a healthy subject were given in Figure 5. were the number of sampling points of the heart sounds. The 11 features used in this study were summarized in Table 3. An example to illustrate some of the feature sequences extracted from a PH subject and a healthy subject were given in Figure 5.  Max_pow_s1 Maximum magnitude of the power spectral density of S1 5 Max_f_s1 The frequency value corresponding to "Max_pow_s1" 6 Max_pow_s2 Maximum magnitude of the power spectral density of S2 7 Max_f_s2 The frequency value corresponding to "Max_pow_s2" 8 Ener_s1 Average energy of S1 9 Ener_s2 Average energy of S2 10 ShanEner_s1 Average Shannon energy of S1 11 ShanEner_s2 Average Shannon energy of S2  Maximum magnitude of the power spectral density of S1 5 Max_f_s1 The frequency value corresponding to "Max_pow_s1" 6 Max_pow_s2 Maximum magnitude of the power spectral density of S2 7 Max_f_s2 The frequency value corresponding to "Max_pow_s2" 8 Ener_s1 Average energy of S1 9 Ener_s2 Average energy of S2 10 ShanEner_s1 Average Shannon energy of S1 11 ShanEner_s2 Average Shannon energy of S2

Sample Entropy
As a measure of the complexity of a digital sequence, the entropy measures have been widely used in physiological signals such as electrocardiogram signals, heart rate variability signals and heart sound signals [15][16][17]. To quantify the regularity of the short and noisy physiological time series, approximate entropy (ApEn) was firstly proposed by Pincus [18]. The calculation of ApEn was very easy and it was quickly applied to the various clinical cardiovascular studies [19,20]. However, ApEn lacked consistency and the results depended on data length. In the meanwhile, it could cause a biased estimation for the complexity of physiological signals because of self-matching. In order to overcome the defects above, Richman and Moorman proposed the sample entropy (SampEn) [21], which was relatively consistent and dependent on data length less. Here SampEn was used to calculate the entropy value of a feature sequence as an entropy measure. The algorithm to calculate SampEn is as the follows.
For a feature sequence x(i) (1 ≤ i ≤ N), given the embedding dimension m and threshold parameter r to form the vector X m where N is the number of feature samples. The vector X m i represents m consecutive samples of where d(.) is the function to calculate the maximum absolute difference. Then B m i (r) can be calculated as where A(·) is the Heaviside function Then increase m by 1 and repeat the steps above to get B m+1 (r). Finally, SampEn is defined by In this study, the embedding dimension m = 2 and threshold parameter r is set to be 0.2 times of the sequence' standard deviation [20].

Fuzzy Entropy
SampEn is based on the Heaviside function of the classical sets which is a two-state classifier that judges two vectors as either "similar" or "dissimilar" with no intermediate states and which influences the statistical stability of results. To enhance the statistical stability, another entropy named fuzzy entropy (FuzzyEn) was proposed and the Heaviside function was replaced by the Zadeh fuzzy sets that provided a smooth similarity classifier [22,23]. The FuzzyEn and the SampEn algorithm are basically the same, except that (7) and (8) are replaced by Finally, FuzzyEn is defined by In this study, the embedding dimension m = 2 and threshold parameter r is set to be 0.2 times of the sequence' standard deviation [20].

Fuzzy Measure Entropy
Fuzzy entropy focuses only on the local characteristics of the sequence and the global fluctuation may affect the results. In order to combine both the local and global similarity of the time series, the fuzzy measure entropy (FuzzyMEn) method was proposed [24], which is described as follows.
For a feature sequence x(i) (1 ≤ i ≤ N), given the embedding dimension m to form the local vector XL m i and the global vector The vector XL m i represents m consecutive samples in x(i) starting from the index i and it removes the local baseline x(i), which is defined as The vector XG m i indicates m consecutive samples in x(i) from the index i and it removes the global mean value x of x(i), which is defined as: Then the distance of the local sequence between XL m i and XL m j is defined as dL m i,j and the distance of the global sequence between XG m i and XG m j is defined as dG m i,j . The dL m i,j and dG m i,j are calculated as Then compute the local similarity degree DL m i,j (n L , r L ) between XL m i and XL m j by the fuzzy function µL(dL m i,j , n L , r L ), and calculate the global similarity degree DG m i,j (n G , r G ) between XG m i and XG m j by the fuzzy function µG(dG m i,j , n G , r G ). where the r L and r G are the thresholds, which are both set as 0.2 times of the sequences' standard deviation [20]. The n L and n G are weights of sequences' similarity, which are set to be 3 and 2, respectively [24]. Define the function φL m (n L , r L ) and φG m (n G , r G ) as Increase m by 1 and repeat the steps above to get φL m+1 (n L , r L ) and φG m+1 (n G , r G ). Then the fuzzy local measure entropy (FuzzyLMEn) and the fuzzy global measure entropy (FuzzyGMEn) are defined as FuzzyLMEn(m, n L , r L , N) = − ln(φL m+1 (n L , r L )/φL m (n L , r L )) FuzzyGMEn(m, n G , r G , N) = − ln(φG m+1 (n G , r G )/φG m (n G , r G )) .
Finally, FuzzyMEn is defined as This study used the three entropies to measure the regularity difference of a feature sequence between PH patients and healthy subjects. Eleven feature sequences were extracted from a heart sound recording and sample entropy, fuzzy entropy and fuzzy measure entropy were calculated for each feature sequence. In total, three entropies combined with 11 feature sequences yield 33 entropy measures for a heart sound recording. For example, the sample entropy measure of a Max_pow_s2 sequence is abbreviated as SampEn_Max_pow_s2. Similarly, the fuzzy entropy measure of a Max_pow_s2 sequence is abbreviated as FuzzyEn_Max_pow_s2.

Statistical Tests
It is known from the mechanism of heart sound generation that the proposed heart sound features reflect physiological and pathological condition of heart hemodynamics. So, the complexity embedded in a heart sound feature sequence, measured by the entropy, will vary by body condition. The entropy value is hopefully different in different body condition even for the same subject. On the other hand, the estimated entropy value also changes in noisy environments. The authors in this study took the entropy values as random numbers and made an assumption that the entropy values had different distribution for PH subject and healthy subject. A question arises as to how to evaluate the significance of an entropy measure between the PH group and the healthy group. In this study, the Mann-Whitney U test was used to achieve this purpose. The Mann-Whitney U test is a nonparametric test applicable to non-Gaussian distribution data. Based on this test, one can draw a conclusion that any two entropy measures come from the same group as long as they have an equal median [25]. The significance level of each entropy measure calculated by the test was used. In a typical case, a threshold of 0.05 is set for the significance level. However, this study considered multiple tests of statistical significance on the same data. As such, a Bonferroni correction was further used to improve the significance test. If the significance level was less than the threshold, it is unlikely that the two entropy measures are from the same group, i.e., they are from the PH group and health group, respectively.

Probability Density Function of an Entropy Measure Fitted by Kernel Density Estimation
To characterize the random numbers, it is necessary to build the probability density function (pdf). For a single entropy measure, its pdf can be constructed by the nonparametric kernel density estimation (KDE) based on a Gaussian kernel function [26]. Suppose e(i) (1 ≤ i ≤ n) is an entropy Entropy 2018, 20, 389 9 of 17 measure of a feature sequence and n is the number of a subject group. The pdf of an entropy measure can be estimated byf where h is a smoothing parameter called bandwidth and K 1 (·) is the single-variable kernel function The Silverman's rule of thumb for the bandwidth is used to get the best one [27] h = 1.06σn −1/5 .
A joint pdf of multiple entropy measure can be generalized from (20). Let e = [e 1 e 2 · · · e d ] be a d-dimensional entropy measure vector. The joint pdf f d (e) of the entropy measures can be obtained by where e i = (e i1 , e i2 , . . . , e id ) T , i = 1, 2, . . . n. K 2 (·) is the multiple-variable kernel function which is a symmetric multivariate density. The standard multivariate normal kernel was used here.
where µ is a d-dimensional vector. H is the d × d bandwidth (or smoothing) matrix which is symmetric and positive definite. Similarly, the Silverman's rule of thumb gives the best bandwidth matrix H [27][28][29]: where σ i is the standard deviation of the i-th variable and H ij = 0, i = j.

Identification of a PH Patient from a Healthy Subject Using the pdf Based on the Bayes' Decision Rule
The pdf of PH group and the health control group were estimated by Section 2.5. If the Mann-Whitney U test shows that an entropy measure is significant between the two groups, the pdf of the entropy measure for PH group must be somewhat different from that of health group. The authors proposed algorithms to identify an unknown subject based on Bayes' decision rule, as seen in Tables 4 and 5. Table 4. Identification algorithm using a single entropy measure.

Algorithm 1: identification using a single entropy measure
Letf p (e) be the pdf of an entropy measure of the PH patient group andf h (e) be the pdf of the entropy measure of the health control group. The entropy measure of an unknown subject is e u . Iff p (e u ) >f h (e u ) then the unknown subject is judged as a PH patient else the unknown subject is judged as a healthy subject Table 5. Identification algorithm using joint entropy measures.

Algorithm 2: identification using joint entropy measures
Letf d p (e) be the pdf of joint entropy measures of the PH patient group andf d h (e) be the joint pdf of the entropy measures of the health control group. The entropy measure vector of an unknown subject is e u . Iff d p (e u ) >f d h (e u ) then the unknown subject is judged as a PH patient else the unknown subject is judged as a healthy subject An unknown subject could be classified correctly or incorrectly. So, four cases may occur in the results. A case that a PH subject is correctly identified as a PH patient is called true positive (TP). A case that a healthy subject wrongly is wrongly identified as a PH patient is called false positive (FP). A case that a healthy subject is correctly identified as a healthy subject is called true negative (TN). A case that a PH subject is wrongly identified as a healthy subject is called false negative (FN). Using the above identification algorithms, an unknown subject was identified. The number of TP, FN, TN and FP are defined as num_TP, num_FN, num_TN, and num_FP, respectively. The sensitivity and specificity is therefore calculated as [30] where Sen and Spe represent the values of sensitivity and specificity. Then the overall evaluation index was defined by the accuracy to measure the identification performance

Significance of the Features and Reduction of the Age Confounding Factor
To investigate the significance of the 33 entropy measures, the Mann-Whitney U test was performed, as shown in Table 6. The least p value, 1.20 × 10 −9 , was of SampEn_max_pow_s2 (No. 13). The measures related to max_pow_s2 had very low p values around 10 −9 E-09 (No. [13][14][15]). The p values of the measures from No. 1 to No. 30 were small. The maximum p value, 4.22 × 10 −1 , was of SampEn_Int_s1 (No. 31). The p values of entropy measures related to Int_s1 (No. 31-33) were much greater than others, which meant that these measures were not significant between PH and healthy group.
In this study, the average age of PH group and health group was 69.4 and 32.6, respectively, as seen in Tables 1 and 2. There was significant difference. There is doubt whether the significance of the proposed measures is likely attributed to age difference or not. As such, age is a confounding factor. To investigate the effect of the age confounding factor, the authors performed Pearson correlation analysis between the measures and the age values. The correlation coefficients (CC.) were given in Table 6. The measure, SampEn_max_f_s2 (No. 16), had maximum correlation coefficient of 0.43. The minimum correlation coefficient came from FuzzyEn_max_f_s1 (No. 8), 0.14. A further check showed that the measures from No. 12 to No. 29 had somewhat high correlation coefficients. The correlation analysis showed that the age did contribute measure difference between the two groups to some degree. To reduce the age confounding factor, it is reasonable to discard those measures which have high correlation coefficients with age. The authors set a threshold, 0.30, for the coefficients. The measures with absolute coefficients less than 0.30 are believed to be weakly correlated with age. Therefore, nine measures indicated by bold text (No. 1, 2, 4, 5, 6, 7, 8, 9 and 11) were selected as candidate measures for identification purpose.
A threshold for significance level is usually set as 0.05. However, it is known that a Bonferroni correction is a safeguard against multiple tests of statistical significance on the same data [31]. So, the significance level in this study was set as 0.05/9, i.e., 5.6 × 10 −3 . That is to say, an entropy measure will be safely significant if its significance level is less than 5.6 × 10 −3 . It was found that the selected nine measures (except No. 9 which was close to the threshold) were highly significant between the two groups. It revealed that the selected nine measures were unlikely from the same group. They were reasonable to be used for identification.

Identification Performance of a Single Entropy Measure
For a selected entropy measure, both pdfs of the PH group and the health control group were fitted respectively by KDE as given in Section 2.5. The leave-one-out cross-validation was used to evaluate the identification performance. That is, one subject was taken out and the other was involved in pdf estimation. The curves of estimated pdf pairs were shown in Figure 6. The red and black curves were for health group and PH group, respectively. A subject can be identified by the Algorithm 1 proposed in Section 2.6. The corresponding Receiver Operating Characteristic (ROC) curves were given in Figure 7. The summary of identification performance was showed in Table 7 in term of sensitivity, specificity, accuracy and area under curve (AUC).
It was seen in Table 7 that the entropy measures related to Ener_s1 had the best identification performance whose AUC was higher than 0.70. For example, the sensitivity, specificity and accuracy of SampEn_Ener_s1 was 0.720, 0.648 and 0.683. The performance of entropy measures related to Max_f_s1 was the worst. The identification performance of the entropy measures can also be highly reflected by the overlapping of the pdf pairs. A careful check to the pdf pairs shown in Figure 6 may indicate that the pdf pairs related to Ener_s1 overlapped least as seen in Figure 6a1,a2. However, the pdf pairs related to Max_f_s1 had the maximum overlapping as in Figure 6c1-c3. A close look at Tables 6 and 7 showed that the entropy measures were ranked in almost the same order. This evidence showed that the significance level drawn from the Mann-Whitney U test and the pdf overlapping were consistent and compatible. This proved that the Mann-Whitney U test was a useful tool to find effective entropy measures for identification purposes.
overlapping were consistent and compatible. This proved that the Mann-Whitney U test was a useful tool to find effective entropy measures for identification purposes.    overlapping were consistent and compatible. This proved that the Mann-Whitney U test was a useful tool to find effective entropy measures for identification purposes.

Identification Performance of Two Joint-Entropy Measures
To improve identification performance, it is natural to use joint pdf of two selected entropy measures. A selection of any two measures out of the nine measures yields 36 combinations. The authors investigated the identification performance of the combinations one by one based on leave-one-out validation. The joint pdfs of the best six and the worst three combinations were shown in Figure 8. So, a subject can be classified as a PH patient or a healthy subject by Algorithm 2 based on the joint pdf proposed in Section 2.6 where the entropy measure vector is a two-dimensional vector. Visual observation to Figure 8a-f indicated the joint pdf of PH patients and healthy subjects had less overlapping where the peaks can be seen separately and clearly. However, the joint pdfs in Figure 8g-i overlapped to a high degree and the peaks were much closer than those in Figure 8a-f. So, a conclusion could be drawn from the observation that the identification performance of the joint measures of Figure 8a-f was much better than those in Figure 8g-i. The quantitative performance indicators of the nine combinations were shown in Table 8. The nine joint measures in Table 8 corresponded to the joint pdfs in Figure 8a-i. The best performance was achieved by the joint of SampEn_Ener_s1 and SampEn_ShanEner_s2 which resulted in that the sensitivity, specificity, accuracy and AUC were 0.680, 0.796, 0.740, and 0.770. The quantitative performance in Table 8 confirmed the observation in Figure 8. Comparison between Tables 7 and 8 revealed that the identification performance got improvement by joint pdf of two entropy measures.

Identification Performance of Two Joint-Entropy Measures
To improve identification performance, it is natural to use joint pdf of two selected entropy measures. A selection of any two measures out of the nine measures yields 36 combinations. The authors investigated the identification performance of the combinations one by one based on leave-one-out validation. The joint pdfs of the best six and the worst three combinations were shown in Figure 8. So, a subject can be classified as a PH patient or a healthy subject by Algorithm 2 based on the joint pdf proposed in Section 2.6 where the entropy measure vector is a two-dimensional vector. Visual observation to Figure 8a-f indicated the joint pdf of PH patients and healthy subjects had less overlapping where the peaks can be seen separately and clearly. However, the joint pdfs in Figure 8g-i overlapped to a high degree and the peaks were much closer than those in Figure 8a-f. So, a conclusion could be drawn from the observation that the identification performance of the joint measures of Figure 8a-f was much better than those in Figure 8g-i. The quantitative performance indicators of the nine combinations were shown in Table 8. The nine joint measures in Table 8 corresponded to the joint pdfs in Figure 8a-i. The best performance was achieved by the joint of SampEn_Ener_s1 and SampEn_ShanEner_s2 which resulted in that the sensitivity, specificity, accuracy and AUC were 0.680, 0.796, 0.740, and 0.770. The quantitative performance in Table 8 confirmed the observation in Figure 8. Comparison between Tables 7 and 8 revealed that the identification performance got improvement by joint pdf of two entropy measures.       Following the reasoning of the joint pdf of two entropy measures, the authors tried to use joint multiple measures to obtain better identification performance. Similarly, the joint pdf of multiple measures of PH group and health group can be built by the KDE estimation based on leave-one-out cross-validation. Then, the proposed Algorithm 2 was used to identify a subject as a patient or a healthy one where the entropy measure vector was a multidimensional vector. Unfortunately, the multidimensional pdfs cannot be visualized normally. The authors investigated the identification performance of all possible combinations of the entropy measures. The top ten results were shown in Table 9. These results were better than those in Table 8 (joint pdf of two entropy measures) and Table 7 (pdf of a single entropy measure). For example, in the first line in Table 9, the joint five entropy measures yielded the best identification performance with sensitivity of 0.740, specificity of 0.870, accuracy of 0.808, and AUC of 0.829. Similar excellent performance can also be obtained by other combinations, such as line 2 to line 10 in Table 9. A careful check to Tables 8 and 9 revealed that the identification performance can be improved by joint multiple measures, as the authors expected. Table 9. Top ten identification performance of joint pdf of multiple entropy measures based on leave-one-out cross-validation.

Summary and Discussions
Pulmonary hypertension (PH) is often diagnosed late because early identification is very difficult even after the onset of symptoms. Therefore, early diagnosing the clinical PH is needed. Heart sound, acoustic vibration generated by the interaction between the heart hemodynamics and valves, chambers and great vessels, is an important physiological signal needed exploration for PH detection further. In the previous studies, four papers have studied the similar topic [8][9][10][11]. The four papers analyzed multiple heart sound features for normal subjects and patients. But, three studies [8][9][10] were for children and only one study [11] was for adults. The classification performance for children showed a sensitivity of 0.93 and a specificity of 0.92. The identification accuracy from the fourth paper for adults was 0.77. The authors verified for adults that the regularity of heart sound features of a PH patient varied comparing to that of a healthy subject. Sample entropy, fuzzy entropy and fuzzy measure entropy were calculated for the proposed 11 feature sequences. The identification was achieved by the difference of entropy measure probability occurred in PH group and health group. The results showed that the difference in entropy measure probability between the PH group and health group did exist. A subject can be classified by the estimated pdf based on the Bayes' decision rule and the classification performance could be improved by joint entropy measures.