RECOGNITION OF SEVERE CONGESTIVE HEART FAILURE USING PARALLEL CASCADE IDENTIFICATION

Transcription

1 RECOGNITION OF SEVERE CONGESTIVE HEART FAILURE USING PARALLEL CASCADE IDENTIFICATION by Yi Wu A thesis submitted to the Department of Electrical and Computer Engineering in conformity with the requirements for the degree of Master of Science (Engineering) Queen s University Kingston, Ontario, Canada (September, 2007) Copyright Yi Wu, 2007

2 Abstract In previous studies on heartbeat series, it has been proposed that the healthy heartbeat pattern represents complex nonlinear dynamics, and such cardiac nonlinearity may be used as a clinical indicator for the diagnosis of certain types of heart disease. However, it is still not quite clear whether there is any difference among the heartbeat series of patients with congestive heart failure (CHF), or whether cardiac nonlinearity represents a severe heart disease situation. In the present study, parallel cascade identification (PCI), which frequently requires only short stretches of data to obtain highly promising results, is used to distinguish severe congestive heart failure, a clinical situation associated with a high-risk of sudden death, from low-risk CHF. Parallel cascade identification is an accurate and robust method for identifying dynamic nonlinear systems. The PCI algorithm combined with a specified statistical test may be used as a severe congestive heart failure marker by comparing a nonlinear model with a linear model (more precisely, a first-order Volterra series). In this thesis, PCI is applied to distinguish R-R wave intervals of CHF patients who died from those of patients who survived in a 5-year study. The detection accuracy of the PCI detector is evaluated over a first set of 49 patients, and then over a larger set of a further 352 patients, and consistent results are obtained between the two sets. Over the larger set, Matthews' correlation coefficient of nonlinearity with unfavorable outcome (death) is , sensitivity for predicting unfavorable outcome is 98.35%, while the specificity is 77.06%. The R-R wave interval exhibits nonlinearity in patients who died during the 5-year study. However, typically nonlinearity cannot be detected in patients who survived during the study. These findings show that for patients with congestive heart failure, nonlinearity is associated with unfavorable outcome (death), while patients for whom nonlinearity cannot be detected ii

3 overwhelmingly have good outcomes. This is significant for clinical diagnosis and prognosis of severe congestive heart failure. iii

4 Acknowledgements I would like to particularly thank Dr. Michael Korenberg, for his supervision, advice and great help to my research, and his invaluable guidance for my thesis and others. I also want to thank Dr. Chi-Sang Poon of M.I.T. and Dr. Mark T. Kearney of University of Leeds for providing the congestive heart failure datasets used in this thesis. I thank my husband Tieli for all the love and support he gives to me. My parents: Yingliang and Chunyue, deserve a lot of appreciation for their emotional support and encouragement to let me finish my study. I would also like to express my appreciation to my friends for their generous advice and encouragement. Many thanks also go out to the Department of Electrical and Computer Engineering and the School of Graduate Studies and Research at Queen s University. iv

5 Table of Contents Abstract...ii Acknowledgements...iv Table of Contents...v List of Figures...vii List of Tables...viii List of Abbreviations...ix List of Symbols...x Chapter 1 Introduction Introduction Background Congestive heart failure Electrocardiogram Parallel cascade identification Thesis outline and contribution...6 Chapter 2 Literature Review Review of relevant literature Ho et al Voss et al Poon and Merrill PCI in the field of biology analysis Summary...10 Chapter 3 Parallel Cascade Identification Previous research Volterra series LNL parallel cascade model Parallel cascade identification Algorithm overview Basic parameters Dynamic linear element Static nonlinear element Requirement for accepting candidates Termination of parallel cascade development...21 v

6 3.3 PCI for detecting chaotic dynamics %MSE reduction and number of cascades accepted Wilcoxon signed-rank test MN-Wilcoxon signed-rank test Verification of nonlinear dynamics detection by PCI Ecological model NH3 experimental series and dwarf star experimental series Summary of some nonlinear examples Summary...32 Chapter 4 PCI in the Recognition of Severe Congestive Heart Failure Data source Study Samples in the smaller test dataset Study Samples in the larger test dataset Results Parameter selection Results for the smaller test set Results for the larger data set Accuracy of detection Summary...50 Chapter 5 Conclusions Conclusions of current work Significance of findings Recommendations for future research...53 Appendix A Results of the smaller test set (22 deceased patients)...59 Appendix B Results of the smaller test set (27 surviving patients)...60 Appendix C Results of the larger test set (121 deceased patients)...61 Appendix D Results of the larger test set (231 surviving patients)...64 Appendix E Characteristics of 121 deceased patients in the larger test set...68 Appendix F Characteristics of 231 surviving patients in the larger test set...70 vi

7 List of Figures Figure 1-1: Representation of normal ECG...4 Figure 3-1: Parallel cascade model...15 Figure 3-2: Structure of i -th cascade...16 Figure 3-3: Ecological model...26 Figure 3-4: I=2 and I=1 model comparison of ecological model without noise...27 Figure 3-5: I=2 and I=1 model comparison of ecological model with 50% correlated noise...28 Figure 3-6: I=2 and I=1 model comparison of an experimental time series of emission of an NH3 laser...30 Figure 3-7: I=2 and I=1 model comparison of an experimental time series of the intensity of a variable dwarf star...31 Figure 4-1: Representative R-R wave intervals from a deceased CHF patient and a surviving patient during the 5-year study...38 Figure 4-2: Variance of R-R wave intervals of all deceased patients in the smaller test set...39 Figure 4-3: Variance of R-R wave intervals of all surviving patients in the smaller test set...39 Figure 4-4: I=2 and I=1 model comparison from a deceased patient in the smaller test set...43 Figure 4-5: I=2 and I=1 model comparison from a surviving patient in the smaller test set...44 Figure 4-6: Results of MN-Wilcoxon test for the smaller test dataset...45 Figure 4-7: Results of MN-Wilcoxon test for the larger test dataset...47 Figure 4-8: A 2 2 contingency table...49 vii

8 List of Tables Table 3-1: Comparison of Wilcoxon test with m, n, and mn for ecological model...29 Table 3-2: Comparison of Wilcoxon test with m, n, and mn for NH3 experimental series...30 Table 3-3: Comparison of Wilcoxon test with m, n, and mn for dwarf star series...31 Table 3-4: Summary of the results of nonlinear examples (with MN-Wilcoxon test)...32 Table 4-1: Baseline characteristics of the smaller test set...35 Table 4-2: Characteristics of 22 deceased patients in the smaller test set...36 Table 4-3: Characteristics of 27 surviving patients in the smaller test set...37 Table 4-4: Baseline characteristics of the larger data set...40 Table 4-5: Accuracy of PCI with the MN-Wilcoxon test for the smaller and larger test sets...50 viii

9 List of Abbreviations ApEn CHF DFA EF ECG FOA HF HR HRV IHD LN LNL LF MI MSE PCI SD TP VWK VLF WGN Approximate entropy Congestive heart failure Detrended fluctuation analysis Ejection fraction Electrocardiograms Fast orthogonal algorithm High frequency Heart Rate Heart rate variability Ischemic Heart Disease Wiener model: dynamic linear followed by static nonlinear Dynamic linear, static nonlinear, dynamic linear Low frequency Myocardial infarction Mean squared error Parallel cascade identification Standard deviation Total spectral power Volterra-Wiener-Korenberg Very low frequency White Gaussian noise ix

10 List of Symbols a Polynomial coefficients defining the static nonlinear element im abcd,,, A p Cell frequencies on Fisher s exact test Values are chosen randomly in the rang 0, R C d The maximum number of cascades are permitted to be added to the model Degree of nonlinearity in VWK series gi ( j ) Discrete impulse response function of dynamic linear element h 0 zero th -order Volterra kernel h ( τ,..., τ ) i th -order Vloterra kernel I i K 1 i Degree of polynomial static nonlinearity The number of cascades are added to the model ka Embedding dimensions Kappa in VWK series M m The number of distinct terms in the Volterra series %MSE reduction m_ lin %MSE reduction of linear model m_ nl %MSE reduction of nonlinear model mn n N Product of %MSE reduction and the number of cascades accepted The number of cascades accepted The total number of patients in the study n _ lin The number of cascades accepted of linear model n_ nl The number of cascades accepted of nonlinear model p r R + 1 R e Th T + 1 ui ( n ) W p-value on Fisher s exact test and Yates -corrected Chi Square test Matthews correlation coefficient Memory length of dynamic linear element The Maximum number of cascades to be rejected before the termination of PCI Threshold to determine if the candidate cascade to be added to the model The total data used in PCI The output of dynamic linear element The sum of the signed ranks in Wilcoxon test x

11 x() t x( n ) Input of continuous time system Input of discrete time system X X Two matched pairs used in the Wilcoxon test A, B yt () yn ( ) Output of continuous time system Output of discrete time system yi ( n ) z zi ( n ) φxy i ( j ) 1 The residual after adding i th cascade to the model z-value in the Wilcoxon test Output of i th cascade First order cross-correlation of x(n) and y i-1 (n) δ ( j A) Discrete impulse function σ Standard deviation of the sampling distribution of the sum of rank in the W Wilcoxon test λ O i E i Coefficient of an ecological model An observed frequency An expected frequency xi

12 Chapter 1 Introduction 1.1 Introduction Because of the development of electrocardiogram (ECG) technology, great progress has been achieved in the understanding and analysis of the human heartbeat. Moreover, the efficiency of readily processing heartbeat data by computer analysis has dramatically improved our knowledge in this field, and the prognosis and management of heart disease. The present study concerns the use of parallel cascade identification (PCI), a particular algorithm of nonlinear system identification, in the analysis of congestive heart failure (CHF). The goal of this thesis is to distinguish severe (high-risk) congestive heart failure from low-risk CHF. Several techniques have been proposed to analyze heartbeat data successfully. Ho et al. [HO97] adopted time-domain measures, frequency-domain measures, approximate entropy (ApEn), and detrended fluctuation analysis (DFA) to analyze ambulatory ECG recordings and heart rate variability (HRV). They found that patients with CHF exhibit a breakdown of the physiological long-range (or fractal) correlations of heart rate. Poon et al. [POO97] used the Fast Orthogonal Algorithm (FOA) [KOR88] in studies of congestive heart failure patients and healthy subjects. They suggested that FOA with specific statistical tests [BAR96] is an efficient and effective tool for the identification of cardiac chaos in the normal heart and congestive heart failure. They also concluded that cardiac chaos is prevalent in the healthy heart, and such chaos is decreased in CHF. Based on previous studies, this thesis illustrates the efficiency of PCI in a successful application to congestive heart failure analysis. The PCI algorithm is employed to determine whether 1

13 heartbeat data exhibit any consistent difference, in a nonlinearity sense, between severe CHF and low-risk CHF patients. 1.2 Background In this section, congestive heart failure (CHF) is introduced, and then the electrocardiogram (ECG) is explained briefly. Finally, an introduction of parallel cascade identification (PCI) that is used in the present study is also reviewed Congestive heart failure The human heart is comprised of four contractile chambers that function to pump blood throughout the body [MOR05]. The upper chambers are called atria, and the lower chambers are called ventricles. The right atrium receives blood that has finished a tour around the body and is depleted of oxygen. This blood returns through the superior vena cava and inferior vena cava. The right atrium pumps this blood into the right ventricle, which pumps the blood into the blood vessels of the lungs. The lungs oxygenate the blood, and eliminate the carbon dioxide that has accumulated in the blood due to the body's many metabolic functions. Then the blood enters the left atrium, which pumps it into the left ventricle. The left ventricle then pumps the blood via the aorta back into the circulatory system of arteries and veins. The left ventricle has to exert enough pressure to keep the blood moving throughout all the blood vessels of the body. Heart failure [BRA04] characterizes the condition when the pumping action of the heart becomes less powerful. With heart failure, blood moves through the heart and body slowly, and pressure in the heart increases. The heart cannot pump enough oxygen rich blood as the body needs. The chambers of the heart have to stretch to hold more blood to pump through the body and become more stiff and thickened. Eventually, the heart muscle walls weaken and cannot pump strongly. As a result, the kidneys will cause the body to retain fluid and sodium. If fluid builds up in the 2

14 arms, legs, lungs or other organs, the body becomes congested. Congestive Heart Failure (CHF) is the condition that congestion happens in the blood vessels because of the reduced blood flow. When CHF happens, the heart's function as a pump to deliver oxygen rich blood to the body cannot meet the body's needs [MED07]. CHF may be caused by diseases that weaken the heart muscle, e.g., myocarditis. The diminished pumping ability of the ventricles due to muscle weakening is called systolic dysfunction [MED07]. A common measurement is the ejection fraction (EF) which is a calculation of how much blood is ejected out of the left ventricle divided by the maximum volume remaining in the left ventricle at the end of diastole or relaxation phase. Systolic heart failure has a decreased ejection fraction of less than 50%, while the normal ejection fraction is greater than 50%. After each ventricular contraction, the ventricle muscles relax, allowing blood from the atria to fill the ventricles, in a process called diastole. Diseases such as hemochromatosis can cause stiffening of the heart muscles, impairing the ventricles ability to relax and fill. This is diastolic dysfunction [MED07]. CHF can affect many organs of the body [MED07]. Weakened heart muscles cannot supply enough blood to the kidneys, which may cause retention of fluid in the body. The lungs may become congested with fluid, or fluid may accumulate in the liver, weakening its ability to get rid of toxins and to produce essential proteins. The intestines may be less able to absorb nutrients and medicines [MED07] Electrocardiogram The electrocardiogram (ECG) is a recording of the electrical activity of the heart [MOR05]. The ECG of a patient s heart can be readily obtained by using only a few electrodes on the surface of the patient s body. Furthermore, the R peak on the ECG of a heartbeat is high and narrow due to 3

15 the ventricular contraction. Therefore, it is easily detected and localized with high precision. For these two main reasons, the ECG is applied as a popular method of heartbeat analysis [ABS99]. ECG records [GED89] the propagation of electrical depolarization and repolarization over the atria and ventricles, which generates changing potential fields on the surface of the body. Figure 1-1 shows the representation of normal ECG. P wave on the ECG represents atrial depolarization. The QRS complex is a structure on the ECG that corresponds to the depolarization of the ventricles. Because the ventricles contain more muscle than the atria, the QRS complex is larger than the P wave. The QRS complex tends to look "spiked" due to the increase in conduction velocity. The T wave represents the repolarization of the ventricles. The R-R wave interval in seconds between two adjacent R peaks gives the heartbeat period, R-R wave interval indicates the duration of ventricular cardiac cycle. The succession of the R-R durations, the R-R series, is the ideal tool for measuring the cardiac rhythm. ECG analysis is a highly developed field in which pathologies or abnormalities of the heart can be identified from changes in the ECG [MOR05]. Figure 1-1: Representation of normal ECG Therefore, it is conceivable that valuable knowledge concerning congestive heart failure can be acquired through an analysis of R-R wave intervals of a CHF patient. In the present thesis, two groups of R-R wave intervals of CHF patients are studied to determine whether or not there is any 4

16 difference between the heartbeats of CHF patients who died and those who survived in a 5-year study. Thus, the ability to predict a CHF patient s outcome may serve a useful function for diagnosis and management of certain heart diseases Parallel cascade identification Although it has been discovered that cardiac chaos is prevalent in the healthy heart, and a decrease in such chaos is displayed in CHF [POO97], differences between severe CHF and lowrisk CHF patients are still not clearly known. In the present study, parallel cascade identification (PCI) with a certain statistical test is used for obtaining greater distinction between severe and low-risk CHF. Parallel cascade identification (PCI) was developed by Korenberg [KOR82] [KOR91] for nonlinear system identification. It has been applied in a number of biomedical studies [KOR00a] [KOR00b] [KOR02], and it has been shown that it is an efficient and strong method for such problems. PCI has several advantages that make it superior to some other techniques: First of all, PCI can be used to model a dynamic nonlinear system when given only the input and output of the system. Secondly, PCI frequently requires only very short data series to identify a nonlinear system, even when the data are heavily contaminated with noise. Moreover, the PCI method is not limited to inputs that are Gaussian [KOR91]. These characteristics allow PCI to be applied in a variety of nonlinearity detection problems, including problems in the biomedical field. In the PCI algorithm, a dynamic nonlinear system is modeled using a parallel arrangement of cascade models. The PCI algorithm fits the cascade models one at a time, achieving increased accuracy with each successive cascade. Parallel cascade identification can also be utilized as a useful approach to detect chaotic systems [KOR91]. This method is used in the present study. A delayed version of a given series data is 5

17 treated as the input to a system, and the output is set to represent the original signal [KOR91]. Using this input and output, PCI is applied to identify a parallel cascade approximation for this new system. If the signal is chaos, a smaller Mean Squared Error (see Equation 3-10 below) and more cascades accepted will result. 1.3 Thesis outline and contribution This thesis is divided into five chapters. In Chapter 2, the relevant literature and current methods are reviewed. Chapter 3 illustrates parallel cascade identification method in some detail. The use of PCI in detecting chaos is also reviewed. Finally, verification of the efficiency of PCI in detecting nonlinear dynamics is described. Chapter 4 describes the application of PCI to the analysis of congestive heart failure. This chapter begins with a description of the study samples. The results of PCI analysis obtained from the two groups of CHF patients are then summarized. General conclusions and the significance of findings are discussed in Chapter 5. Recommendations for future research are also given. In the present study, PCI was used to distinguish high-risk CHF from low-risk CHF successfully. Recognition of severe CHF may provide useful information for the diagnosis and therapy of severe CHF. 6

18 Chapter 2 Literature Review Congestive heart failure (CHF) occurs when the heart muscle fails to pump as much blood as the body needs [MED07]. Patients with congestive heart failure have a complex autonomic disturbance of the heart [PAR85]. The heartbeat data provide a noninvasive index of the neural activity of the heart. It is found that heart rate variability (HRV) decreases in CHF, i.e., there is decreased parasympathetic activity to the heart with CHF [CAS89]. In a previous study of electrocardiograms (ECGs) from healthy subjects and those with CHF, cardiac chaos is found prevalent in healthy hearts, and there is a decrease of chaos in hearts with CHF [POO97]. In this chapter, a brief review of relevant literature is discussed. Current methods that have been applied to analyze the heartbeat are also described. 2.1 Review of relevant literature In previous studies, the approaches that have been proposed for heartbeat analysis include temporal (e.g., the standard deviation used in [KLI89]) and frequential (spectral distribution of energy between high and low frequencies) [KLI87] [CAS89] measures. Other methods include detrended fluctuation analysis (DFA) [HO97, ABS99], the Fast Orthogonal Algorithm [KOR88] used for heartbeat analysis in [POO97], and symbolic dynamics and renormalized entropy approaches [VOS96] Ho et al. Ho et al. [HO97] analyzed ambulatory ECG recordings and HRV by time-domain measures (mean and standard deviation [SD] of heart rate), frequency-domain measures (power in the bands from to 0.01 Hz [VLF], 0.01 to 0.15 Hz [LF], and 0.15 to 0.5 Hz [HF] and total 7

19 spectral power [TP] over all three of these bands), and methods based on nonlinear dynamics (approximate entropy [ApEn], and detrended fluctuation analysis [DFA]). Their study samples consisted of 28 CHF patients and 41 sex- and age-matched control subjects. The authors concluded that there were statistically significant differences between the CHF patients and control subjects. The SD of the heart rate, VLF power, LF power, and the ratio of LF to HF power were lower in the CHF patients than in the control subjects. The DFA index, ranging between 0 and 1, with 1 indicating perfectly normal scaling behavior, was also lower in the CHF patients, indicating a lower amount of long-range correlations compared with the control subjects [HO97] Voss et al. Voss et al. [VOS96] recorded the ECG from 26 cardiac patients after myocardial infarction (MI) and from 35 healthy persons. The 26 cardiac patients were divided into two groups: low-risk group (10 patients) and high-risk group (16 patients). Standard measures in time and frequency domains, as well as measures from the nonlinear methods of symbolic dynamics and renormalized entropy, were applied to find differences between healthy subjects and cardiac patients. As a result, applying discriminant function techniques to HRV analysis yielded parameters of nonlinear dynamics that enabled differentiation between healthy persons and highrisk patients. Moreover, they classified three patients with apparently low risk into the same cluster as high-risk patients. The authors concluded that the methods of nonlinear dynamics describe complex rhythm fluctuations and separate structures of nonlinear behavior in the heartrate time series more successfully than time- and frequency-domain methods. The nonlinear dynamics methods may also improve discrimination between the heartbeats of healthy persons and high-risk patients. 8

20 2.1.3 Poon and Merrill Poon and Merrill [POO97] applied the Fast Orthogonal Algorithm [KOR88] to the problem of identifying electrocardiograms of a group of subjects with severe congestive heart failure and those of healthy subjects. They generated several Volterra-Wiener-Korenberg (VWK) series with different degrees of nonlinearity ( d ) and embedding dimensions Kappa ( Ka ) to produce a family of linear and nonlinear polynomial autoregressive models [BAR96] [WYS06]. The best linear model is obtained by varying the Ka value with d = 1 to minimize the Akaike information-theoretic criterion [AKA74]. The best nonlinear model is acquired by sequentially increasing Ka values with d > 1. Both models were then compared and the null hypothesis (linearity) was tested against the alternative hypothesis (nonlinearity) using parametric ( F -test) and nonparametric (Mann-Whitney) statistics. The null hypothesis was rejected if there was at least one nonlinear model that provided a significantly better fit to the data than linear autoregressive models of all dynamical orders [BAR96]. Poon and Merrill used the data sets of heartbeat intervals from 8 healthy subjects and 11 CHF patients (age range for all the subjects and patients: years old). The histograms of linear and nonlinear model selection for all 500-beat and 2000-beat data segments based on the statistical tests, in healthy subjects and CHF patients, showed high detection rates for chaos in the healthy group and relatively low detection rates in the CHF group. This result discovered that cardiac chaos is displayed in the healthy heart, and it is decreased in CHF. 2.2 PCI in the field of biology analysis The application of PCI to protein sequence analysis was reported in [KOR00a] [KOR00b]. Using PCI in a binary classifier configuration together with the mean test [KOR00b] and the Mean 9

21 Square Error (MSE) test [KOR00a], Korenberg classified proteins from three structural/functional families: globin, kinase, and calcium-binding. A PCI classifier was also used in a study to distinguish exon (coding) from intron (non-coding) human DNA sequences [KOR02]. The first exon and intron were used to construct a training input. As a result, the parallel cascade classifiers were able to achieve classification rates of 89% in a test set, and 75% in a blind test set. 2.3 Summary In this chapter, previous studies of heartbeat and CHF were reviewed. Ho et al. used time-domain and frequency-domain measures, and the DFA method for ECG analysis. They found that the SD of the heart rate, VLF power, LF power, the ratio of LF to HF power, and the DFA index were lower in CHF patients than in control subjects. Voss et al. utilized discriminant function techniques in HRV analysis for distinguishing 35 healthy subjects and 26 cardiac patients (16 high-risk patients and 10 low-risk patients). They differentiated healthy subjects and high-risk patients at 96%. Poon and Merrill applied the Fast Orthogonal Algorithm with certain statistical tests to process the ECG of healthy subjects and CHF patients. They concluded that there is a decrease of cardiac chaos in CHF. Moreover, some applications of PCI in the field of biology are also summarized. Following the highly promising results of analyzing protein sequences and human DNA sequences from using the PCI algorithm, PCI is utilized in this thesis to distinguish severe CHF from low-risk CHF. 10

22 Chapter 3 Parallel Cascade Identification Many biological systems show nonlinearity behaviour. Some nonlinear system techniques have been successfully applied, e.g., Volterra series and Wiener series approaches. Fast Orthogonal Algorithm (FOA) [KOR88] was applied in [BAR96] [POO97] [LEI06] [WYS06]. Parallel cascade identification (PCI), introduced by Korenberg [KOR82] [KOR91], is utilized in the present study. In previous research, PCI has been found to be surprisingly effective in classifying protein sequences [KOR00a] [KOR00b] and in the recognition of exon and intron DNA sequences [KOR02]. In this chapter, a detailed description of the PCI algorithm that is used in this thesis is provided, and then the verification of PCI in identifying nonlinear dynamics is also pursued. 3.1 Previous research Volterra series Before describing the PCI algorithm, it is necessary to introduce Volterra series. Volterra introduced mathematical models of nonlinear systems called Volterra series or Volterra functional expansions [VOL13] [VOL59]. Volterra indicated that the system may be represented by a sum of Volterra functionals. Equation 3-1 shows the Volterra series for a continuous-time system. 11

23 () ( ) ( ) ( ) ( ) ( ) y t = h + h τ x t τ dτ + h τ, τ x t τ x t τ dτ dτ +... or R R 0 0 R ( ) ( ) ( ) +... h τ,... τ x t τ... x t τ dτ... dτ I I 1 I 1 I 1 I R R () = = ( ) ( ) ( ) R R y t y ( t), y ( t)... h τ,... τ x t τ... x t τ dτ... dτ i i i 1 i 1 i 1 i i= (3-1) The right side of the upper equation is called a Volterra series of I th -order. h ( τ,... τ ) i is the i th - 1 i order Volterra kernel. h 0 is the zero th -order Volterra kernel and is constant. R is the memory length of the model. Both R and I may be infinite. The term R R is the i th -order Volterra functional. The i th - y ( t) =... h( τ,... τ ) x( t τ )... x( t τ ) dτ... dτ i i 1 i 1 i 1 i 0 0 order functional is homogeneous of degree i because if input x() t is replaced by c* x( t ), then the i th -order functional yi () t is multiplied by c i. Note that each Volterra kernel hi( τ1,..., τ i) may be assumed to be symmetric, i.e., invariant with respect to any permutation of τ 1,..., τ i without any loss in generality [VOL59]. Equation 3-1 is an I th -order Volterra series with memory length R. If I is finite, then the Volterra series is of finite order. If R is finite, then the Volterra series has finite memory. If both I and R are finite, then the series is said to be doubly finite. Most nonlinear systems are not analytic (systems for which certain functional derivatives of all orders exist [VOL59]), however, and cannot be exactly represented by a Volterra series. Frechet [FRE10] considered a finite-memory, causal nonlinear system whose output is a continuous 12

24 mapping of its input, in that small changes in the input produce small changes in the output. Then, over a uniformly-bounded equi-continuous set of input signals, the nonlinear system can be uniformly approximated, to an arbitrary degree of accuracy, by a Volterra series of sufficient, but finite, order [FRE10]. The discrete-time model was considered by Palm [PAL79]. As a direct result of the Stone- Weierstrass theorem [HIL57], Palm noted that a discrete-time causal finite-memory timeinvariant system, whose output is a continuous mapping of its input, may be approximated uniformly by a discrete-time Volterra series of sufficient, but finite, order. Equation 3-2 shows h,..., m j1 j m is the m th - the representation of a discrete-time Volterra series. In the equation, ( ) order Volterra kernel. ( ) ( ) ( ) ( ) ( ) ( ) y n = h + h j x n j + h j, j x n j x n j j= 0 j = 0 j = 0 j = 0 j = 0 1 R R R R 1 2 ( ) ( ) ( ) +... h j,..., j x n j... x n j m R m 1 m 1 m (3-2) Although this is of great theoretical use [KOR05], the required order of the Volterra series might need to be very large in order to accurately approximate a given nonlinear system. In practice, Volterra series are usually applied only to weakly nonlinear systems, with an order of nonlinearity less than or equal to three. Another problem is that, to find the Volterra kernels, a set of simultaneous linear equations must be solved, involving inversion of a matrix whose size grows rapidly with R. 13

25 3.1.2 LNL parallel cascade model A parallel cascade model that consists of a finite sum of dynamic linear, static nonlinear, and dynamic linear (LNL) cascades was proposed by Palm [PAL79] to uniformly approximate discrete-time systems that could be approximated by Volterra series. Palm showed that any system having a Volterra series representation with finite memory and anticipation could be uniformly approximated to an arbitrary degree of accuracy by a sum of a sufficient, but finite, number of LNL cascades. In Palm s proof, the static nonlinearities were exponential and logarithmic functions. Each of the parallel paths comprises a cascade, or a series connection of elements. The output of the first element (dynamic linear element) is the input to the second (static nonlinear element); the output of the second element is the input to the third (dynamic linear element) [KOR05]. Palm did not describe any procedure for identifying the model or building a parallel cascade approximation for a dynamic nonlinear system. 3.2 Parallel cascade identification Algorithm overview Based on Palm s promising proposal, Korenberg [KOR82] [KOR91] subsequently developed a parallel cascade model: each cascade consists of a dynamic linear (L) element and polynomial static nonlinear (N) element. Korenberg s parallel cascade model structure is illustrated in Figure 3-1. In the figure, each L is a dynamic linear element, and each N is a polynomial static nonlinear element. Such a LN structure is used in the present thesis. Korenberg also proposed an identification procedure for obtaining such a parallel LN model, given only input and output data, to approximate any discrete-time system which has a Wiener series representation to an arbitrary degree of accuracy in the mean-square sense. Wiener series [WIE58] were derived by applying 14

26 the Gram-Schmitt orthogonalisation process to the functionals in the Volterra series, for a particular white Gaussian input. Figure 3-1: Parallel cascade model L 1 N 1 y(n) x(n) L 2 N 2 + : : L K N K The parallel cascade identification algorithm is summarized below [KOR05]. A first cascade of dynamic linear and static nonlinear elements is found to approximate the dynamic nonlinear system to be identified. The residual, the difference between the system output and the cascade output, is calculated, and then is treated as the output of a second dynamic nonlinear system. A cascade of dynamic linear and static nonlinear elements is now found to approximate the second system. The new residual is calculated, and treated as the output of a third nonlinear system, and so on. Each succeeding cascade is fit in order to drive the cross-correlations of the input with the residual to zero. Consider a discrete-time dynamic nonlinear system [KOR05], which is sometimes called a black box because the only information known from the system is its input x( n) and output yn ( ), n= 0,..., T. Suppose that y ( n ) denotes the residual after adding the i -th cascade i to the model, and y ( n) y( n) 0 =. The variable z ( n ) denotes the output of the i -th cascade. The i 15

27 structure of the i -th cascade [KOR05] is shown in Figure 3-2, where gi ( n ) denotes the discrete impulse response function of the dynamic linear element, ui ( n ) is the output of this linear element, and the a im are the polynomial coefficients defining the static nonlinear element. Figure 3-2: Structure of i -th cascade x( n ) u ( n ) z ( n) gi () j i I m= 0 a im (.) m i Then for i 1, the i -th residual y ( n ), after adding the i -th cascade, is equal to the difference i between the previous residual, y ( ) i 1 n and the present cascade output zi ( n ). y i ( n) y ( n) z ( n) = 1 (3-3) i i Basic parameters Before identifying a parallel cascade model, a number of basic parameters must be specified first. R + 1 is the memory length of the dynamic linear element. I is the degree of the polynomial static nonlinearity that follows the linear element. C is the maximum number of cascades permitted in the model. R e is the maximum number of consecutive candidate cascades to be rejected before termination of the PCI algorithm. 16

28 Th is a threshold constant, for deciding whether a cascade s reduction of the mean squared error (MSE), defined below in Equation 3-10, justifies its addition to the model Dynamic linear element The discrete impulse response function, g ( j ), of the dynamic linear element can be defined i using a first-order cross-correlation, φ ( ) xy j i, or a slice of a cross-correlation of higher order P, 1 of the input x( n) with the latest residual, y ( ) i 1 n. The order P that will be used to define gi ( j ) can be selected at random. Equation 3-4 shows the discrete impulse response of a dynamic linear element when the first-, second-, third-, or fourth-order cross-correlation is employed. The maximum order set in the present study is 4. g ( j) is one of : φ φ i xy i 1 xxy xxxy i 1 ( j) ; ( ja, 1) ± D1δ ( j A1) ; ( ja, ) ( ) ( ) 1 1, A2 ± D1 j A1 ± D2 j A2 ; ( j A A A ) ± D ( j A ) ± D ( j A ) ± D ( j A ) φ δ δ i φ,,, δ δ δ. xxxxy i (3-4) In Equation 3-4, the discrete impulse function δ ( j A) = 1 if j = A, and equals zero otherwise. A is fixed at one of the values 0,..., R. The sign of the δ term is chosen randomly, and D is adjusted to tend to zero as the mean-square of the residual y ( ) 1 n approaches zero. D is set as shown in Equation 3-5. Because the nonlinear system to be identified is assumed to have finite memory lasting up to R lags, therefore, g ( j) = 0, j > R[KOR91]. i i 17

29 D ( n) ( n) 2 yi 1 =± (3-5) 2 y The overbar denotes time-average over the portion of the series from n= R to n= T. For cross-correlation orders P that are greater than 1, P-1 of the cross-correlation s arguments are fixed randomly at values A p in the range 0,..., R. Impulses are added or subtracted, as in Equation 3-4 at locations j = A, and the impulses are scaled by D. p The cross-correlations of the input with the residual are computed over the portion of the series extending from n = R to n= T as shown in Equations 3-6 in which j = 0,..., R, i = 1,...,4. i φ φ φ φ xyi 1 i 1 i 1 T R+ 1 n= R xxyi 1 xxxyi 1 xxxxyi 1 T 1 ( j) = y ( n) x( n j) = y ( n) x( n j) T 1 ( j, j ) = y ( n) x( n j ) x( n j ) 1 2 i T R+ 1 n= R T 1 ( j, j, j ) = y ( n) x( n j ) x( n j ) x( n j ) i T R+ 1 n= R T 1 ( j, j, j, j ) = y ( n) x( n j ) x( n j ) xn ( j) xn ( j) i 1 1 T R+ 1 n= R (3-6) After determining the impulse response gi ( j) of the linear element, the output, ui ( n ), of the dynamic linear component is calculated with Equation 3-7. Note that the convolution is calculated over the range n= R,..., T to avoid needing x( n ) for n < 0. u i R ( n) = g ( j) x( n j) j= 0 i (3-7) 18

30 3.2.4 Static nonlinear element The signal ui ( n ) is itself the input to a static nonlinear element in the cascade that is in the form of a polynomial. The output, zi ( n ), is shown below in Equation 3-8. Because each cascade consists of a dynamic linear element followed by a static nonlinearity, the output of the static nonlinear element is the cascade output. The polynomial coefficients a im defining the polynomial static nonlinearity are found by best fitting the output zi ( n ) to the current residual y ( ) i 1 n. z i I m ( n) = a u ( n) m= 0 i, m i (3-8) In this thesis, Fast-Orthogonal Algorithm (FOA) is applied to find a im. FOA [KOR88] uses an orthogonal approach that avoids the need to explicitly create the orthogonal basis functions. The details about FOA are described in [KOR88]. Thus, the polynomial coefficients a im minimize the mean-square of the new residual over n= R,..., T. Therefore, it can be shown that the mean square of the new residual is: y ( n) = y ( n) z ( n) (3-9) i i 1 i Once the parallel cascade model has been identified, the model MSE and %MSE, which are defined below in Equation 3-10, are calculated. In the equation, yn ( ) is the actual system output, yi ( n ) is the residual after adding the i -th cascade, zi ( n ) is the output of the i -th cascade. The %MSE is the MSE scaled by the variance of the original system output. The overbar still denotes a time average in the range n= R,..., T. %MSE is used to enable comparison between different 19

31 time-series data. But it is not essential when I=1 and I=2 models were compared over the same data interval. Suppose that the number of cascades accepted is K. Then MSE = [ y( n) z ( n)] K i= 1 2 i = y 2 K ( n), (3-10) % MSE = y 2 y 2 K ( n) ( n) ( y( n) ) 2 100% Requirement for accepting candidates Before accepting a given candidate for the i -th cascade, a cascade s reduction of the MSE, divided by the mean square of the current residual, must exceed the threshold constant Th divided by the number of output points T R+ 1 used to estimate the cascade. This requirement [KOR91] is shown in Equation Th is set at 4 in this thesis. 2 Th 2 zi( n) > yi 1( n) (3-11) T R + 1 This requirement helps to avoid selecting unnecessary cascades that are merely fitting noise, and Th = 4 corresponds to a confidence interval of about 95% [KOR91]. If the criterion is met, then the candidate cascade is accepted. The new residual yi ( n ) is subsequently calculated as shown in Equation 3-3, and a candidate for the ( i + 1) -th cascade is found. If a candidate cannot satisfy this requirement, then it is rejected and a new candidate must still be found. This process is repeated until the preset number of rejected cascades has been reached and the algorithm is terminated. 20

32 3.2.6 Termination of parallel cascade development Parallel cascade identification may be terminated when a specified number of cascades have been added. In this thesis, the maximum number C of cascades that can be added to the model is predetermined. The termination of parallel cascade development may be also made when the MSE has been made sufficiently small, or no remaining candidate cascade can cause a significant reduction in MSE [KOR91], or a preset maximum number R e of candidate cascades are consecutively rejected. 3.3 PCI for detecting chaotic dynamics A number of methods have been introduced to solve the problem of distinguishing between chaotic behaviour due to a deterministic system and noise due to random processes. Korenberg [KOR87] [KOR91] suggested that nonlinear modeling techniques, such as Volterra series estimation and PCI, could be well suited for distinguishing chaotic systems. The following sections discuss the use of PCI for detecting chaotic dynamics. Korenberg suggested [KOR91] that parallel cascade identification can provide a useful approach for detecting chaotic nonlinearity. This method is used in the present study. Suppose yn ( ), n= 0,..., T is given time-series data and it is to be determined whether yn ( ) is deterministic chaos or random noise. One approach is to first treat a delayed version of y as the system input, and the original (undelayed) signal y as the system output, then to identify a parallel cascade approximation for the created system. If the signal yn ( ) is chaos, then the identified parallel cascade array will have a smaller %MSE (as a percentage of the variance of y ), and more cascades will be accepted than if y represented independent noise [KOR91]. 21

33 3.3.1 %MSE reduction and number of cascades accepted If R + 1 is the memory length and I is the polynomial degree, then the number of distinct terms M in the Volterra series corresponding to the parallel cascade model is calculated according to Equation M ( R I)! = ( R + 1)! I! (3-12) To determine if nonlinearities are present in the given data, Korenberg [KOR06] suggested comparing the %MSE reduction and the number of cascades accepted by a linear model with that for a nonlinear model having the same number of terms using the Wilcoxon signed-rank test. The approach is explained in the following discussion. Suppose that the memory length is R + 1 and the polynomial degree is I. Using Equation 3-12, the total number of distinct terms may be calculated. The %MSE reduction and number of cascades accepted would be compared with that for a linear model with memory length M. This will give a pair of nonlinear %MSE reduction and linear %MSE reduction, and another pair of nonlinear number of cascades accepted and linear number of cascades accepted. Instead of a linear model, it's easier and more reasonable to fit a parallel cascade with I = 1 using the same threshold constant Th regulating the minimum MSE reduction required for a candidate cascade to be accepted as for nonlinear models, and the same number of candidates tested. In that case use memory length M 1 for I = 1 model, since there's also a constant term. Due to this constant term, for convenience the I = 1 model will henceforth be referred to as a first order Volterra series and sometimes as a linear model, but it is not in fact linear except in the unlikely event of the estimated constant equaling zero [KOR06]. Also, nonlinear models will 22

34 mean higher (i.e., I > 1) than a first-order ( I = 1) system, and sometimes we will use degree instead of order. These pairs can be made for a fixed I (say, I = 2, in this thesis) by varying R + 1. The difference between higher-order nonlinear and first order ( I = 1) models can be considered to determine if it is significant. Then, the process may be repeated for a different I, and in this way it may be determined for which values of I nonlinear models outperform I = 1 models. Alternatively, the pairs may be made up for fixed R + 1 by varying I, or by varying both R + 1 and I [KOR06]. If nonlinearities are important, then the nonlinear model should consistently have a larger %MSE reduction and more cascades accepted than the I = 1 model with the same number of terms. In particular, a Wilcoxon signed-rank test can be used to see if nonlinear models consistently have larger %MSE reductions, or number of cascades accepted, than I = 1 models with the same number of terms [KOR06]. Note that the nonlinear model, say with I = 2, is fit over the identical portion of the record as the model with I = 1, i.e., if the linear element s memory length is R + 1, then fitting the I = 1 model uses the R + 1 th to T + 1 th output points. So fit the I = 2 model using these same output points. The denominator T R+ 1 in Equation 3-11 refers to the number of output points used in the identification. It should be the same number when comparing the I = 1 model with the I = 2 model Wilcoxon signed-rank test Wilcoxon signed-rank test [VASSAR] is a non-parametric test for the significance of the difference between the distributions of two non-independent samples involving repeated measures or matched pairs X A, X B. The Wilcoxon test begins by taking the absolute value of 23

35 each instance of X A X B. The absolute values of the differences are then ranked from lowest to highest, with tied ranks included where appropriate. The positive or negative sign that had been removed from the X A X B difference is now attached to each rank. The sum, W, of the signed ranks, is then calculated. The standard deviation of the sampling distribution of W is equal to: σ w QQ ( + 1)(2Q+ 1) = sqrt 6 (3-13) where Q is the number of pairs X A, X B after discarding cases where X A =X B. The z -ratio for the Wilcoxon signed-rank test is: W 0.5 z = (3-14) σ W The table of critical values of z (for the unit normal distribution) can be used to see whether the observed value of z is significant beyond a specified level MN-Wilcoxon signed-rank test To see if nonlinear models for a given time series are significantly better than linear ( I = 1) models with the same number of terms, Korenberg [KOR06] suggests a way to incorporate the %MSE reduction, m, and the number of cascades accepted, n, into a single measure when using the Wilcoxon signed ranks test: simply use the product mn in place of m or n. This may consistently obtain a better level of significance. When PCI is applied to an input/output pair, with first I = 1 and then I = 2, two pairs of %MSE reduction ( m_ lin and m_ nl ) and the number of cascades accepted ( n_ lin and n_ nl) are computed. After calculating the product of m _ lin and n _ lin, and also the product of 24

36 m_ nl and n_ nl, the Wilcoxon signed-rank test is applied under the alternative hypothesis that the nonlinearity is more significant. Comparing the calculated z value with the critical z value provides the final decision on whether nonlinearity can be detected. Except where otherwise stated, the MN-Wilcoxon signed-rank test is used throughout this thesis as it obtains consistently better results for this application. Also, a delay of one lag is always used to create the input signal from the given time series, which served as the corresponding output in its undelayed form. 3.4 Verification of nonlinear dynamics detection by PCI In this section, the efficiency and robustness of PCI with Wilcoxon signed-rank test are verified on a wide variety of short time series, 512 points long. The original series data were treated as the output signal, and the signal was delayed by one point to form the input. To obtain nonlinear models, the nonlinear degree I was fixed at I = 2, and then the memory length R + 1 was varied over 1,...,18. The nonlinear degree I was then set at I = 1 and the memory length R + 1 was chosen to obtain a corresponding set of first-order Volterra series, each having the same number of distinct terms as one of the I = 2 models [KOR06]. All the codes for the verification in this section and the analysis of CHF in the next chapter were written and conducted with Matlab 7.0 in a computer with a 1.6GHz processor Ecological model Figure 3-3 shows an ecological model with λ = 118 [BAR96] [CAZ92] whose chaotic component is obscured by strong periodicity (see Equation 3-15). The trajectory evolves in an attractor comprised of fractals in several disconnected domains and visits each of them in a periodic manner [BAR96]. The periodicity makes detection of the chaotic component difficult [BAR96]. 25