Chapter 3. Data Analysis

Transcription

1 Chapter 3 Data Analysis

2 CHAPTER 3 DATA ANALYSIS The Heart Rate Variability (HRV) data profiles from a number of healthy, cardiac diseased (AF and CHF) subjects, and non-cardiac diseased subjects such as Hypothyroid and Depression subjects have been collected for the characterisation of diseased subjects. The various procedures of data analysis and the algorithms developed for estimating the nonlinear parameters such as Entropies, Power Spectral Density and Correlation Dimension are discussed in this chapter. 3.1 Data Acquisition This study covered five different populations. The first group consisted subjects of congestive Heart Failure (CHF), second group consisted of Atrial Fibrillations (AF). The third group consisted of Hypothyroid with a known history of Hypothyroidism. The fourth group consisted of Depression and fifth group healthy subjects. The healthy, Hypothyroid and Depression subjects are between years of age, where as the subjects of group one and two are between years of age. Two different methods are used to derive the HRV (RR interval) data required for the HRV analysis for different subjects mentioned above. The first method acquired ECG using Power Lab. HRV (RR interval) data is directly derived from the Power Lab instrument. In the second method HRV (RR interval) data is directly acquired from Physiobank for CHF subjects and AF subjects [104] Data Acquisition: Methodology I The HRV (RR interval) data required for the analysis is obtained from MLS360/ 6 ECG Analysis Module of Power lab data acquisition systems (ADI Instruments). ECG data of all healthy subjects (15 male and 10 female), between years of age are recorded 35

3 from ECG lead I with a sampling rate of 500 Hz. Similarly the ECG data of Five Hypothyroid and five Depression subjects (2 male and 3 female) of the age between years is recorded. The data has been recorded for about 15 to 20 minutes with the subjects in the sitting position. The Power lab generated data is free from artifacts. Data recorded in the last fifteen minutes of the recording is obtained for the analysis from the data base of Ved Vignan Maha Vidya Peeth Research Center, Bangalore. Out of the 25 RR interval data generated for healthy subjects, only five healthy subjects (N17, N18, N19, N20 and N21), whose age is between years (2 male and 3 female subjects) has been selected for the analysis Data Acquisition: Methodology II The required RR interval data of five Congestive Heart Failure (CHF) subjects and five Atrial Fibrillations (AF) subjects recorded for long duration are obtained from [104]. The data analysis has been limited to the the HRV data collected for five subjects in each category of diseased and healthy subjects as the scope of the present work is to establish the behaviour on nonlinear parameters for healthy subjects, cardiac and non-cardiac diseased subjects. The results of our analysis are limited to selected sample of data to arrive at definitive conclusions. 3.2 Correction of Data Anomalies The data acquired from the methods discussed in the previous section 3.1 contained intermittent entries too high or too low values to be considered for data. These intermittent values are called outliers. The presence of outliers in the data causes negative effects on the entropy calculations. Because the presence of outliers increases the standard deviation of the RR interval data which in turn increases the tolerance r, and reduces the entropy values. We removed the outliers manually. 36

4 3.3 Programs for Data Analysis Computer programs used to compute Approximate Entropy (ApEn), Sample Entropy (SampEn), Symbolic Entropy (SymbEn), Spectral Entropy (SE), and Correlation Dimension (CD) are written in MATLAB. The computations of all entropies and CD are based on the algorithms described in this section Approximate Entropy Algorithm ApEn measures the logarithmic probability that runs of patterns that are close and remain close on the next incremental comparisons. The concept of ApEn is introduced by Pincus. ApEn quantifies regularity of the data without having previous knowledge about it [105]. Larger values of ApEn indicate more random data. Smaller values of ApEn indicate the regularity in data [106]. The ApEn technique is applied for the HRV data analysis and is presented the procedure of computing it. The RR interval (HRV) data series is considered for the estimation of ApEn in this thesis work. Let the original HRV data be h(n) = h(1), h(2). h(n), where N is the total number of data points. The parameters m the embedding dimension and r the threshold must be indicated for the computation of ApEn. The algorithm is described below [107]. 1. Using the original HRV data, form m-vectors H (1) to H (N-m+1) defined by H (a) = [h (a), h (a+1).h (a+m-1)] ; a=1, N-m+1 (3.1) 2. Define the distance between H(a) and H(b), d[h(a), H(b)], as the maximum absolute difference between their corresponding scalar elements (3.2) 3. For a given H(a), find the number of for b=1, N- m+1. And let the number be N m (a) 37

5 Therefore Then compute Where ; for a=1, N-m+1 (3.3) 4. Take the natural logarithm of each And average it for all a : (3.4) 5. Increase the dimension m to m+1 and repeat steps 1-4 and find 6. Theoretically, the approximate entropy is defined as ApEn (3.5) In general the number of data points N is finite. (3.6) Algorithm to estimate Considering the same HRV data {h (n) = h(0), h(2).h(n)} and embedding dimension m, the method of computing is described below[107] 38

6 1. The absolute difference between any two data points (a, b = 1, N) is calculated and entered into the corresponding position d(a,b) of NXN distance matrix D.D is a symmetrical matrix D(a,b)= D(b,a), and elements on its main diagonal da(a=1,n) are equal to zero. 2. Compare each element (a, b) of the difference matrix of step1 with the threshold r resulting S matrix. The elements of S matrix are formed according to the following rule. s (a,b) = s(b,a) s(a,a ) = 1 a=1 to N Find for m=2. When a is specified a th and (a+1) st rows of matrix S can be used to find through the following steps Find by using Then Find 39

7 Where 3. Find for m=3. And (3.12) Where (3.13) 4. Find and for a=1, N m then find using equation (3.4) with m=2 and m=3 respectively. And Sample Entropy Algorithm The concept of Sample entropy proposed by Richman.et.al [108] is to give the entropy value independent of data length. The SampEn is analogous to the ApEn, there exists a minor computational difference between SampEn and ApEn. In ApEn comparison is made between template vector and other vectors. The template vector is also compared with itself. This assures that probabilities C m r(a) is not zero. Thus it is possible to take a 40

8 logarithm of probabilities. The comparison of template vector with itself results lower ApEn values leading to the understanding of lower regularity of the signals than their actual regularity. In case of SampEn, the self-match of the template vector to itself is removed. The algorithm for SampEn is described below [108]. Consider Implementation of ApEn and SampEn To implement the ApEn and SampEn algorithms presented in section and 3.3.2, it is required to derive optimal values for the embedding dimension m, which is the length of the pattern vector and the tolerance / threshold r and N is the length of HRV (RR interval) data. From the published literature [84-88,112] it has been observed that with small values of m, and with r between 10% - 20% of the standard deviation (SD) of the data gave good results. The consideration of SD in r allows the indirect normalisation of the data. Values of r less than 0.1xSD results in poor conditional probability and r greater than 0.2SD leads to loss of detailed information Where Standard Deviation SD is expressed as 41

9 3.3.4 Symbolic Entropy algorithm Symbolic Entropy is calculated for short length data. The number of RR intervals used for estimation of Symbolic Entropy is 52. The algorithm developed is described below [109] 1. Data of length N has to be considered. The mean of these data points is estimated and is taken as the threshold 2. Each and every element of the data series is compared with the threshold. If the data element value is greater than the threshold, then the data element is rewritten as 1 else the data element is rewritten as The symbols obtained in step 3 are grouped to form words Word (j) = Sj, Sj+1, Sj+2 for j=1,n-2 4. Find the probability of occurrence of each word P w 5. Estimate the Shanon s Entropy of the words formed and computed symbolic entropy using. 42

10 Where P w is the probability of the occurrence of the word w and n is the number of words formed. For the words formed with three symbols n=8.so as to generate symbols easily and even to do estimation manually Spectral Entropy algorithm Spectral Entropy (SE) indicates the spectral complexity of time series data at frequency f. Sequence of operations to be performed for Computation of SE is described below [110,111]. 1. Transformation of HRV data into power spectrum by applying FFT 2. Computation of the Power Spectral Density (PSD) 3. Normalisation of the PSD \ 4. Computation of SE using Shannon s entropy as given below Where is the Probability Density Function (PDF) at frequency f. Application of Shannon s entropy gives an estimate of SE Correlation Dimension Algorithm Correlation Dimension (CD) is another measure of complexity is computed from the RR interval data using Grass Berger Procassica algorithm. Grass Berger Procassica algorithm is based on determining the relative number of pairs of points in the phase-space set that are separated by a distance of less than r[100,103]. CD is computed using equation (3.22). 43

11 where C(r) is the correlation integral C (r) the correlation integral measures the probability that the arbitrary points Ai and Aj of the phase space will be separated by a distance r. The algorithm to compute CD is explained below. Step 1: Consider each element of the HRV (RR interval) data and estimate its difference with every other element. Step 2: Find the minimum difference and maximum difference out of the calculated values and designate them as r min and r max. In the range from r min - r max a number of points are to be considered at fixed intervals. Step 3: The logarithmic values for the points between r min - r max need to be calculated. Then using the formula find Correlation Dimension (CD) values. m = (2 CD) + 1 (3.24) Different values of m return different CD values. Where m = embedding dimension and ranges from 3-12 Having known the CD values the Correlation Integral C(r) is estimated for the r values computed in step 3 using the equation (3.22) 44

12 Step 4: Step 5: Plot between log r versus log C(r) has to be plotted. The log r value should be selected from the linear region. From this log r the actual r value can be estimated. With the estimated value of r, Correlation integral and Correlation Dimension values are computed. For the selected r value and an m ranging from 3-12 the following formula is evaluated Consider the HRV data vectors represented by A. where N ref = 0.25 N, =1 (3.26) N = no. of data being processed and is the Heavyside function i.e. Step 6: For each m the corresponding CD is calculated by using equation (3.22) Step 7: A plot of m versus CD is drawn and the amplitude value to which the plot saturates gives actual value of CD. The value of m for which the CD saturates gives the embedding dimension value 45