"Science, Faculty of"@en . "Zoology, Department of"@en . "DSpace"@en . "UBCV"@en . "Yuan, Yeh"@en . "2009-03-21T22:05:04Z"@en . "1997"@en . "Master of Science - MSc"@en . "University of British Columbia"@en . "When analyzing short term heart rate variability (HRV) signals using the FFT technique,\r\nthe linear trend appears to be perfect 1/f signals. The non-linear trend in short term HRV\r\nsignals produces the regression 1/f components. De-trending the data using a moving\r\naverage is an effective technique for removing the 1/f components. However, though\r\nremoving the trend may sometimes produce clearer spectral pictures of respiratory sinus\r\narrhythmia or the breathing frequency, it has little impact otherwise. The linear trend is\r\nfractal, but not chaotic. The 1/f components in the signal does not imply the signal fractal;\r\nnor does a fractal signal implies it chaotic. No evidence in the present research suggests\r\nthat the non-linear trend is fractal, nor is there evidence to suggest that the HRV signal is\r\nfractal, or chaotic."@en . "https://circle.library.ubc.ca/rest/handle/2429/6320?expand=metadata"@en . "1663916 bytes"@en . "application/pdf"@en . "1/F COMPONENTS in SHORT TERM HEART RATE VARIABILITY SIGNALS by YEH YUAN B.Sc. University of Science and Technology of China, 1985 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Zoology) We accept this thesis as conforming jtathe required standard THE UNIVERSITY OF BRITISH COL UMBIA May 1997 \u00C2\u00A9 Yeh Yuan, 1997 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freefly available for reference and study.. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of ^ . Q o t 0 ^ The University of British Columbia Vancouver, Canada Date T ^ e \\u00C2\u00B0[ > f,^?7' DE-6 (2/88) Abstract When analyzing short term heart rate variability (HRV) signals using the FFT technique, the linear trend appears to be perfect 1/f signals. The non-linear trend in short term HRV signals produces the regression 1/f components. De-trending the data using a moving average is an effective technique for removing the 1/f components. However, though removing the trend may sometimes produce clearer spectral pictures of respiratory sinus arrhythmia or the breathing frequency, it has little impact otherwise. The linear trend is fractal, but not chaotic. The 1/f components in the signal does not imply the signal fractal; nor does a fractal signal implies it chaotic. No evidence in the present research suggests that the non-linear trend is fractal, nor is there evidence to suggest that the HRV signal is fractal, or chaotic. ii TABLE OF CONTENTS Abstract ii Table of Contents iii List of Figures iv Acknowledgment vi INTRODUCTION 1 METHODS 5 RESULTS 7 1) The 1/f Components due to the Linear Trend 7 2) The 1/f Components due to the Non-linear Trend 10 DISCUSSIONS 14 1) The Linear Trend 14 2) The Non-linear Trend 19 3) 1/f Components in HRV Signals, Fractals and Chaos 24 4) The Mistakes of CGSA 29 5) The Significance of the Current Study 34 CONCLUSIONS 35 References 36 Appendix 39 iii List of Figures Fig. 0-1 to 0-2 Perfect 1/f spectra 2 Fig. 0-3 to 0-4 Regression 1/f spectra 3 Fig. 3-1 to 3-18 Linear trends in a 1 minute FfRV signals 8, 9 Fig. 3-19 to 3-27 Non-linear trend in a 2 minute FIRV signal 12 Fig. 3-28 to 3-36 Non-linear trend in a 4 minute FfRV signal 13 Fig. 4-1 to 4-3 The linear trend and its spectra when Af = y 16 Fig. 4-4 to 4-6 The linear trend and its spectra when Af = ^ ^ 17 Fig. 4-7 to 4-9 Aliasing in the FFT of the linear trend 18 Fig. 4-10 to 4-11 The frequency response of de-trending by the method of moving average of the 2 minute FfRV signal 20 Fig. 4-12 to 4-13 The frequency response of de-trending by the method of moving average of the 4 minute FfRV signal 20 Fig. 4-14 to 4-19 Removing 1/f components from all frequency ranges 22 Fig. 4-20 to 4-21 The frequency response of removing 1/f components from all frequency ranges 23 iv Fig. 4-22 to 4-23 Fractal Examples from Mandelbrot 25 Fig. 4-24 to 4-32 1/f signals with different phases 28 Fig. 4-33 to 4-35 The signals in CGSA 30 Fig. 4-36 to 4-38 The power spectra in CGSA 31 Fig. 4-39 The phase spectrum of &*\u00E2\u0080\u00A2 in CGSA 33 Fig. 6-1 to 6-9 Non-linear trend in a 40 minute HRV signal 37 v Acknowledgment I thank Dr. David R. Jones for his guidance and support of the research presented in this thesis. It was his expectations and encouragement, his patience and trust, that made this thesis possible. Also, I thank my committee members, Dr. John M. Gosline and Dr. James F. Carolan for their valuable comments pertaining to this research. I would also like to thank Dr. David R. Jones and Russell Andrews for serving as subjects by allowing me to record their heart rate variability signals, and also Dr. Pierre Signore for setting up the equipment and assisting in data acquisition. I would like to thank Dr. Doug Syme for his careful review of the thesis and his very helpful comments, and, finally, all my colleagues and friends who have given their care, encouragement, support and help to me during the course of my studies. I thank especially my wife, Cynthia, for her interest and enthusiasm in my studies; I thank her for her help in performing the analytical Fourier transform of the linear trend using MATHEMATICA; I thank her for her technical support with respect to the computer at home and in my office; I thank her for her inspirational discussions with me; I thank her for her genuine love and unshakable faith in me that made all the hard work and struggles worthwhile. INTRODUCTION Using spectral analysis to investigate the frequency characteristics of heart rate variability (HRV) signals dates back to 1973, when Sayers published his paper 'Analysis of Heart Rate Variability'(l)- In this paper spectral analysis was used to determine blood pressure fluctuations as well as those of respiration. In the early 1980s, spectral analysis coupled with pharmacological blockade (2,3,4) and direct nervous stimulation (5) revealed that peaks of short-term HRV spectra >0.15 Hz were mediated almost entirely by the vagus, whereas those <0.15 Hz could be mediated by both vagal and cardiac sympathetic nervous inputs to the sinoatrial node (4). These results give a rationale for investigating short-term HRV spectra in cardiovascular research as well as in clinical situations (6). Many investigators using spectral analysis of HRV signals came across a phenomenon which they named the 1/f components of the HRV signals. In 1982, Kobayashi and Musha reported that the HRV signal had a 1/f power spectrum in the frequency range slower than 0.02 Hz, when the heartbeat period was analyzed over time scales longer than 50s (7). Earlier, Mandelbrot and Ness (1968) had proposed the concept of fractional noise to describe the origin of the 1/f components(14). Later Saul et. al. (1988) used 5-minute data segments and calculated that at frequencies between 0.00003 and 0.1 Hz, the power spectrum of the HRV signal was fundamentally 1/f (8). Finally, Yamamoto and Hughson (1991) proposed a Coarse-Graining Spectral Analysis (CGSA) to selectively remove the 1/f or fractal components(6) to improve the quality of the power spectral diagrams. l The 1/f spectrum was first reported in 1925 for an electric current passing through a vacuum tube(9). Later, the 1/f spectrum was reported in the cellular membrane potential(lO), frequency fluctuation of the alpha brain wave(ll), concentration modulation of action potential impulses propagating in the giant axon of a squid(12), as well as highway traffic current fluctuations(13). There are a lot of terms to describe this phenomenon, such as 1/f fluctuation, 1/f noise, 1/f spectrum, 1/f-like power spectrum, ... etc. They describe a spectrum in which the power is cc inversely proportional to its frequency. In mathematical form: P(f) = -yj , where P is power, f is frequency, a and B are positive real constants. Two cases illustrating the 1/f spectrum follow: cc cc Case 1: Fig. 1-1 shows P(f) when a=\ and/?=2. Fig.1-2 showsP(/) when it is plotted in double log form log(P) = log(a) - /?log(/). It is a straight line with a negative slope. Fig. 1-1 Fig. 1-2 Frequency (Hz) log(Frequency) 2 Case 2: Usually, a perfect 1/f spectrum like those of Case 1 will not be seen. Instead, the more usual cases are those illustrated in Fig. 1-3 and Fig. 1-4. Fig. 1-3 Fig. 1-4 Frequency (Hz) log(Frequency) The spectrum is said to have 1/f components because there is a 1/f-like spectrum in the lower frequency range (0+ to 0.1+ Hz) of Fig. 1-3 and the regression line of the double log spectrum (Fig. 1-4 ) has a negative slope. A perfect 1/f signal which produces the power spectra illustrated by Case-1 has never been observed before. This study identified its source as the linear trend of HRV. The origin of the regression 1/f spectra has been proposed (16), yet has not been identified before. This study shows it to be the non-linear trend of HRV. The CGSA technique (6) of removing the 1/f components is ineffective and mathematically incorrect. This study shows that de-trending by the method of moving average is quite effective for removing the lower frequency range 1/f components. An effective method for removing all 1/f components is also proposed. 3 What is the significance of knowing that the origin of the 1/f components is the trend? It was thought that the 1/f phenomenon had major implications with respect to the nature of HRV signals, in that the 1/f spectra indicated fractal noise. Further, these fractal components have been assumed to indicate chaos. This study clarifies that it is not the case: the 1/f component is nothing but the trend. This study also indicates that although the linear trend is fractal, it is by no means chaotic; clear evidence that being fractal does not imply chaos. No evidence in our research suggests that the non-linear trend is fractal, or the FfRV signal is fractal, or the FfRV signal is chaotic. What is the significance of removing the 1/f components? The work of this thesis suggests that it is of little or no significance. Removal of the 1/f components may sometimes produce a clearer spectral picture of the breathing frequency, otherwise it is not necessary to remove the 1/f components as some used to believe (6). 4 METHODS Data collecting: Electrocardiograph (EKG) signals were recorded from 3 subjects (DRJ, age 55; RA, age 32; YY, age 34) resting in an armchair using three electrical leads attached to the subjects' lower chest, upper chest and wrist. The leads were connected to an isolation unit which was connected to a Gould universal amplifier and filter. The EKG signals were sampled at 200 Hz on a computer (Lab Tech Notebook) and were stored as beat-to-beat or RR-intervals, also called heart rate variability (HRV) signals. Data pre-processing: RR-intervals were linearly interpolated at 4 Hz to ensure equidistant sampling. This procedure inevitably introduced some higher frequency components into the signal. Therefore, these signals were filtered by a low pass filtering procedure at a cut-off frequency of 0.5 Hz. Then the signals were re-sampled at 1 Hz. This procedure reduces the data set size for calculation economy, yet keeps the information of interest intact. Since the highest frequency in the signals is 0.5 Hz, and the sampling frequency is 1 Hz, which is twice as much as the signal's highest frequency, and according the Nyquist sampling theorem (21), no aliasing should occur. The linear interpolations were done using the AWK program of the MKS TOOLKIT (Mortice Kern Systems Inc.). The filtering and re-sampling were done using MATLAB (Math Works, Inc. Cochituate Place, 24 Prime Park Way, Natick, Mass. 01760, USA). 5 Linear trends were calculated by the method of least squares. Non-linear trends were calculated by moving averages of 24 second periods in the case of 2 minute HRV signals, and 38 second periods in the case of 4 minute HRV signals. All spectral analyses were done using MATLAB on PC. Analytical Fourier Transform of the linear trend was done using MATHEMATIC A. Courtesy of UBC TeleCom. 6 RESULTS 1) The 1/f Components due to the Linear Trend: Fig.3-1 is a one minute HRV signal segment. Its power spectrum (Fig.3-2) and its double-log plot (Fig.3-3) show that it has 1/f components. It has a linear trend (Fig. 3-4). The power spectrum of the linear trend is a perfect 1/f spectrum (Fig.3-5). This is also shown in its double-log plot (Fig.3-6). When the trend is removed from the signal (Fig.3-7), the power spectrum and its double-log plot show that no 1/f components remain (Fig.3-8 and Fig. 3-9). Note: 1. The tangent of the regression line of Fig.3-3 is -1.4242; whereas that of Fig.3-9 is 0.0471; 2. Changing the frequency range of interest will change the tangent of the regression line; 3. The peak at 0.1667 Hz in Fig.3-2 and Fig.3-8 correspond to the breathing frequency of the subject. Fig.3-10 is another one minute HRV signal segment. Its power spectrum (Fig.3-11) and its double-log plot (Fig.3-12) show that it has 1/f components. It has a linear trend (Fig.3-13). The power spectrum of the linear trend is a perfect 1/f spectrum (Fig.3-14). This is also shown in its double-log plot (Fig.3-15). When the trend is removed from the signal (Fig.3-16), the power spectrum and its double-log plot show no 1/f components remain (Fig.3-17and Fig. 3-18). Note: 1. The tangent of the regression line of Fig.3-12 is -1.4115; whereas that of Fig.3-18 is 0.0579; 2. Changing the frequency range of interest will change the tangent of the regression line; 3. The peak at 0.1667 Hz in Fig.3-11 and Fig.3-17 correspond to the breathing frequency of the subject. 7 Signal Power Double-log Original signal Fig. 3-1 Fig. 3-2 Fig. 3-3 Time (min) Frequency (Hz) log(Frequency) Trend Fig. 3-4 Fig. 3-5 Fig. 3-6 Time (min) Frequency (Hz) log(Frequency) Trend-free signal Fig. 3-7 Fig. 3-8 Fig. 3-9 Time (min) Frequency (Hz) log(Frequency) 8 Signal Power Double-log Original signal Fig. 3-10 Fig. 3-11 Fig. 3-12 Time (min) Frequency (Hz) log(Frequency) Trend Fig. 3-13 Fig. 3-14 Fig. 3-15 Time (min) Frequency (Hz) log(Frequency) Trend-free signal Fig. 3-16 Fig. 3-17 Fig. 3-18 Time (min) Frequency (Hz) log(Frequency) 9 2) The 1/f Components due to the Non-linear Trend: Fig.3-19 is a two minute HRV signal segment. Its power spectrum (Fig.3-20) and its double-log plot (Fig.3-21) show that it has 1/f like components. It has a non-linear trend (Fig.3-22). The power spectrum of the non-linear trend is not a perfect 1/f spectrum (Fig.3-23). This is also shown in its double-log plot (Fig. 3-24). However, the regression line of the double-log plot is 1/f (Fig.3-24). When the trend is removed from the signal (Fig.3-25), power spectrum and its double-log plot show little remaining 1/f component (Fig.3-26 and Fig. 3-27). Note: 1. The tangent of the regression line of Fig.3-12 is -1.2651; whereas that of Fig.3-18 is -0.0032; 2. Changing the frequency range of interest or the length of the moving average period will change the tangent of the regression line; 3. The peak at 0.1667 Hz in Fig.3-20 and Fig.3-26 corresponds to the breathing frequency of the subject. Fig.3-28 is a four minute HRV signal segment. Its power spectrum (Fig.3-29) and its double-log plot (Fig. 3-30) show that it has 1/f like components. It has a non-linear trend (Fig.3-31). The power spectrum of the non-linear trend is not a perfect 1/f spectrum (Fig.3-32). This is also shown in its double-log plot (Fig.3-33). However, the regression line of the double-log plot is 1/f (Fig.3-33). When the trend is removed from the signal (Fig.3-34), the power spectrum and its double-log plot show no 1/f components remain (Fig.3-35 and Fig. 3-36). 10 Note: 1. The tangent of the regression line of Fig.3-30 is -0.8936; whereas that of Fig.3-36 is 0.0295; 2. Changing the frequency range of interest or the length of the moving average period will change the tangent of the regression line; 3. The peak at 0.1667 Hz in Fig.3-29 and Fig.3-35 corresponds to the breathing frequency of the subject. 11 Signal Original signal Fig. 3-19 Time (min) Trend Fig. 3-22 Time (min) Power Fig. 3-20 0 0.05 Dt 015 01 Frequency (Hz) Fig. 3-23 \u00E2\u0080\u0094\u00E2\u0080\u00A2 OH 0 COS 01 015 02 Frequency (Hz) Trend-free signal Fig. 3-25 Fig. 3-26 Time (min) Frequency (Hz) Double-log Fig. 3-21 log(Frequency) Fig. 3-24 log(Frequency) Fig. 3-27 o , \u00E2\u0080\u009E i \u00E2\u0080\u0094 , \u00E2\u0080\u0094 . \u00E2\u0080\u0094 , \u00E2\u0080\u0094 , \u00E2\u0080\u0094 , \u00E2\u0080\u0094 . \u00E2\u0080\u0094 i 4 -i -35 -3 -3.5 -2 -15 log(Frequency) 12 Power Fig. 3-29 O a* O Double-log Fig. 3-30 Time (min) Frequency (Hz) log(Frequency) Trend Fig. 3-31 Time (min) o OH Fig. 3-32 Frequency (Hz) Fig. 3-33 log(Frequency) Trend-free signal Fig. 3-34 a. Fig. 3-35 o O H o Fig. 3-36 Time (min) Frequency (Hz) log(Frequency) 13 DISCUSSION 1) The Linear Trend Why does the linear trend appear to be a perfect 1/f signal? Every linear trend can be expressed by the equation: y(t) = a-t + b (0n) = \ a \u00E2\u0080\u0094 + b 2 1 a + iba>n e~iTa\" {a + iboo n + iaTco n) 1 co \u00E2\u0080\u009E oo \u00E2\u0080\u009E , w = 2 , 3 , 4 . . . J V / 2 conj[Y(o)N_n+l)] ,n = {N 12 + \)XN 12 + 2),...,N This is because in the analytical form, the period F = lim 1 0 dt 00 14 whereas in digital form F At When in the FFT format where Af = y F T N =\u00E2\u0080\u0094 = \u00E2\u0080\u0094. Af At = 2n-(n-\)Af = 2*-(n-l) .y Tco\u00E2\u0080\u009E = T-2n-{n-\)-= 2n(n - 1 ) T e \" = cos(Tco \u00E2\u0080\u009E) - i \u00E2\u0080\u00A2 sin (To \u00E2\u0080\u009E) = cos[2^-(\u00C2\u00AB - 1)] - /' \u00E2\u0080\u00A2 sin[2;r(\u00C2\u00AB - 1 ) ] = 1 Y(a) ) = 1 [ a + i b ( D \" i e~iTm (a + i b o } n + i a T \u00C2\u00AE J j T co \u00E2\u0080\u009E2 co \u00E2\u0080\u009E2 1 a + ibco\u00E2\u0080\u009E 1 \u00E2\u0080\u00A2 (a + ibco \u00E2\u0080\u009E + iaTco \u00E2\u0080\u009E) ~ 2 ' 2 J T co \u00E2\u0080\u009E co \u00E2\u0080\u009E = / ft? ^ K ) = |y(ftO| a Substituting 2 n f n for <2> M , we obtain: 15 P(fn) < a ^ (a I2n)2 f 2 J n Let a = (a I 2 k ) 2 , and 6 = 2, a P(fn) = fn P n = 2,3,4... Nl 2 Therefore the linear trend is a perfect 1/f signal when expressed in the FFT format. The graphs shown below are of the trend equation y(t) = -y t + 800 (0 < t < 120s) expressed in the FFT format: Signal Power Fig.4-1 Fig.4-2 en s o CL, o OH O 1.5 -1 -0.5 Time (min) Frequency (Hz) log(Frequency) 16 When in the DFTformat of Af = * Then N = 10*T F 10*T 4/ At , v l r a+ibcon e~lTa\" (a +ibco \u00E2\u0080\u009E +iaTco \u00E2\u0080\u009E)n . . . . . and Y(a> \u00E2\u0080\u009E)\u00E2\u0080\u00A2=\u00E2\u0080\u0094[ ^ + * f \"\u00E2\u0080\u0094] ,n = 2,3,4... N 12 1 CO \u00E2\u0080\u009E CO \u00E2\u0080\u009E Signal Power Double-log Fig.4-4 Fig.4-5 Fig.4-6 Time (min) Frequency (Hz) log(Frequency) Here, it does not appear to be a perfect 1/f signal anymore. However, the base line of the power and the double-log form is what we have seen earlier in the FFT format (Fig.4-2 and Fig.4-3). So strictly speaking, the linear trend is not a perfect 1/f signal; it only appears to be one when its power spectrum is expressed in the FFT format where Af =-^. This is usually the case in spectral analysis. 17 Aliasing in the FFT of the linear trend. When using FFT at the sampling rate of 1 FIz to calculate the power and then the double-log of the power spectrum, we obtain Fig. 4-8 and Fig.4-9 which are very similar to those of Fig.4-2 and Fig.4-3. However, there are some differences. The major differences being that the end of the line is tipped upward (Fig.4-9). It is due to aliasing. From formula (2) we know that the linear trend has infinite frequency components. The FFT's sampling rate here is set at 1 Hz. According to the Nyquist sampling theorem (21), the sampling rate should be at least twice that of the signal's highest frequency, otherwise, aliasing occurs. In the case of the linear trend, however, there is no way that one can have a sampling frequency twice that of its highest frequency, since that highest frequency is infinite. Therefore, aliasing is bound to occur. Fortunately, the power of the higher frequencies is usually very small compared with those of the lower frequency ones, so aliasing is negligible. Signal Power Double-log Fig.4-7 Fig.4-8 Fig.4-9 Time (min) Frequency (Hz) log(Frequency) 18 2) The Non-linear Trend The frequency response of the de-trending procedure by moving average. Fig.4-10 and Fig.4-11 show the frequency response of the transfer function which converts the signal of Fig.3-19 to the signal of Fig.3-25. Fig.4-12 and Fig. 4-13 show the frequency response of the transfer function which converts the signal of Fig.3-28 to the signal of Fig.3-34. From Fig.4-10 and Fig.4-12 it can be seen that the de-trending procedure attenuates the amplitude of the lower frequency range of the signal, having little effect on the higher frequency range. From Fig.4-11 and Fig.4-13 it is can be seen that it also has little effect on the phase of the signals over all of the frequency range. De-trending by moving average only removes the lower frequency range of the 1/f components. Removal of the 1/f components over the entire frequency range needs a special technique which is discussed in the following section. 19 Amplitude Response 2-minutes Fig.4-10 0 0.1 0.2 0.3 04 0.5 Frequency (Hz) Phase Response Fig.4-11 08 0.6 0.4 -0.B 0 0.1 0,2 0.3 0.4 0.5 Frequency (Hz) 4-minutes Fig.4-12 a) 10 CD 0. Fig.4-13 Frequency (Hz) Frequency (Hz) 20 The procedure for removing 1/f components over the entire frequency range. A brief explanation of how this is accomplished is given here. Fig.4-14 is the same 4 minute signal as Fig. 3-28. Fig.4-15 is its power spectrum. Fig.4-16 is the power spectrum plotted in the double log form with its regression line (tangent =-1.7732), the 1/f component. The 1/f component free power, in the double log form with its regression line (tangent = 0) (Fig. 4-19), is obtained by subtracting the components of the regression line from the components of the double log power spectrum then adding the results to the mean value of the components of the double log power spectrum. The anti-log is taken to obtain the 1/f component free power spectrum (Fig.4-18). The phases are not changed in the procedure. The 1/f component free signal (Fig.4-17) is obtained by inverse Fourier transform. 21 Fig.4-14 Fig.4-15 Fig.4-16 Time (min) Frequency (Hz) log(Frequency) Fig.4-17 Fig.4-18 Fig.4-19 Time (min) Frequency (Hz) log(Frequency) 22 The frequency response of the transfer function. Figs.4-20 and 4-21 describe the frequency response of the transfer function which converts the signal of Fig.4-14 to that of Fig.4-17. From Fig.4-20 we can see that the procedure attenuates the amplitude of the lower frequency range of the signal, and amplifies that of the higher frequency range. From Fig.4-21 we can see that the procedure does not affect the phase of the signals. This procedure removes all the 1/f components over the entire frequency range. Amplitude Response Phase Response Fig.4-20 Fig.4-21 Frequency (Hz) Frequency (Hz) 23 3) 1/f Components in HRV Signals, Fractals and Chaos Fractal is a word coined by Mandelbrot in 1975 from the Latin fractus, which describes a broken stone. A fractal object looks the same when examined from far away or nearby - it is self-similar (20). Mandelbrot himself gave two examples of the fractal: the cauliflower (Fig.4-22) (20)and the Sierpinski gasket (Fig.4-23) (20). The fractal of the cauliflower is self-similar and irregular, whereas that of the Sierpinski gasket is self-similar and regular. So whether it is irregular or not, if the object is self-similar, then it is fractal. A chaotic system, as its name implies, is inherently unstable. It has two important properties: 1) it is sensitive to its initial point, i.e. small changes in input cause big changes in the output; 2) its behavior is unpredictable. A well-known example is a swinging pendulum with a iron bob that is attracted equally to two magnets positioned below it. When the bob moves near to a point midway between the magnets, it is affected almost equally by the force from each magnet. Its future motion becomes extremely sensitive to small changes in its present position and velocity, and therefore its motion is chaotic. If we assume the sensitivity is so great that the error in measuring the position of the bob increases by 10 times in one swing between the magnets, which is not at all exceptional, prediction of its position to within a centimeter after one swing entails measuring its position at any point in the swing to within a millimeter. For a prediction with the same degree of precision after four swings, its position would have to be 24 Fig.4-22 This cauliflower, a variety called c. Romanesco, is an example of a natural fractal. Fig.4-23 The Sierpinski gasket - a simple fractal produced by breaking up a triangle into successively smaller ones. 25 measured to within the size of a bacterium, and after nine swings, to within less than the size of an atom. The pendulum obeys Newton's deterministic laws, but any attempt to predict its future behavior over long times is impossible (20). Chaos and fractals are fascinating new ideas, which have impinged on many fields across the scientific spectrum. Of special interest to physiologists is the role played by fractals in the spectrum of heart rate variability (HRV) signals: a 1/f spectrum suggests a fractal signal, and hence, a chaotic process (Goldberger, 1991) (17). One of the objectives of this thesis is to show that this hypothesis is not true: 1) Fractal does not imply chaos: According to the rule set by Mandelbrot (Fig.4-23), the linear trend (Fig.4-24) is fractal, because any part of the signal is similar to each other and to the whole signal, yet it is obviously not chaotic because it is totally predictable and its power spectrum is not sensitive to the signal's initial point. Therefore, this is evidence that being fractal does not imply being chaotic; 2) 1/f does not imply fractal: Fig.4-24 shows a linear trend; Fig.4-25 shows its 1/f power spectrum; Fig.4-26 shows the phase spectrum (Note: It was calculated by FFT. The phases shift from 0.5 n to 1 n from 0 to 0.5 Hz is due to aliasing). However, the same 1/f power spectrum (Fig.4-28 ) with random phases (Fig.4-29) has a completely different time domain signal (Fig.4-27 ); Fig.4-28 is a signal which has the same 1/f power spectrum yet the phases of the 2-min HRV signal of Fig.3-19. The same 1/f power spectrum does not have a linear trend. The signals of Fig.4-27 , Fig.4-30 and any other signal with the 26 same 1/f power spectrum yet different phases are not necessarily all fractal. A 1/f spectrum does not implies fractal components in the original signal. Therefore, the Hypothesis that a 1/f power spectrum suggests a fractal signal, and hence, a chaotic process, is not valid. In the case of the non-linear trends, its double-log regression power spectrum is 1/f. Is it chaotic? Is it fractal? As a special case, the double-log frequency domain regression power spectrum of the linear trend is certainly 1/f as well, but it is not chaotic. Therefore, the 1/f double-log regression power spectrum does not imply a chaotic signal. Is it fractal? The work of this thesis suggests that not every non-linear trend is self-similar, therefore, the 1/f double-log regression power spectrum does not imply a fractal signal. As for the HRV signals, my experiments show that not every HRV signal is self-similar, therefore, not every HRV signal is fractal; not every HRV signal's power spectrum is sensitive to small changes of the initial point of the signal, therefore, not every HRV signal is chaotic. 27 Signal Linear Trend Fig.4-24 1/f Power Spectrum Fig.4-25 o CL, Time (min) Frequency (Hz) Phase Spectrum Fig.4-26 a) 01 0.3 Frequency (Hz) Signal with Random Phases Fig.4-27 Time (min) Fig.4-28 a "Thesis/Dissertation"@en . "1997-11"@en . "10.14288/1.0087907"@en . "eng"@en . "Zoology"@en . "Vancouver : University of British Columbia Library"@en . "University of British Columbia"@en . "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en . "Graduate"@en . "1/F components in short term heart rate variability signals"@en . "Text"@en . "http://hdl.handle.net/2429/6320"@en .