· We investigate the long-run historical pattern of R&D-outlays by reviewing aggregate growth rates and historical cases of particular R&D projects. Government of pakistan ministry of finance, economic affairs, statistics and revenue (revenue division) ***** islamabad, the 31st december, 2011. An Introduction to Portfolio Theory John Norstad [email protected] http:// April 10, 1999 Updated: November 3, 2011 Abstract. A basic income (also called basic income guarantee, Citizen's Income, unconditional basic income, universal basic income (UBI), or universal demogrant) is a form of. Procedia Economics and Finance 15 ( 2014) 1619 – 1631 Available online at www.sciencedirect.com 2212-5671 2014 The Authors. Published by Elsevier B.V.Blackboard Learn. Student Help. Library Computing Hub. Email: hub@mail. http: //library. Phone: 6. 19- 5. 94- 3. Faculty Help. Instructional Technology Services. Phone: 6. 19- 5. 94- 3. ![]()
Sincerely, SDSU Blackboard Support Team. Estimating the Hurst Exponent. The Hurst exponent occurs in several areas of applied mathematics. Hurst exponent estimation has been applied in. Estimation of. the Hurst exponent was originally developed in hydrology. However. the modern techniques for estimating the Hurst exponent comes from. The mathematics and images derived from fractal geometry exploded into. It is difficult to think of an area of. Along with. providing new insight in mathematics and science, fractal geometry. Nature is full. of self- similar fractal shapes like the fern leaf. A self- similar. shape is a shape composed of a basic pattern which is repeated at. An example of an artificial. Sierpinski pyramid shown in Figure 1. ![]() Figure 1, a self- similar four sided Sierpinski pyramid(Click on the image for a larger version). From the Sierpinski. Pyramid web page on bearcave. More examples of self- similar fractal shapes, including the fern leaf. The. Dynamical Systems and Technology Project web page at Boston. The Hurst exponent is also directly related to the "fractal. The. fractal dimension has been used to measure the roughness of. The relationship between the fractal. D, and the Hurst exponent, H, is. There is also a form of self- similarity called statistical. Assuming that we had one of those imaginary. Statistical. self- similarity occurs in a surprising number of areas in engineering. Computer network traffic traces are self- similar (as shown in Figure 2). Figure 2, a self- similar network traffic. This is an edited image that I borrowed from. I've misplaced the reference. I apologize to the author. Self- similarity has also been found in memory reference traces. Congested networks, where TCP/IP buffers start to fill, can show. The self- similar structure observed in. Other examples of statistical self- similarity exist in cartography. Figure 3, A White Noise Process. Estimating the Hurst exponent for a data set provides a measure of. Another way to state this is that a random process with an underlying. When. the autocorrelation has a very long (or mathematically infinite). Gaussian process is sometimes referred to as a. Processes that we might naively assume are purely white noise. Hurst exponent statistics for long. One example is seen in. We might expect that network traffic would. Following this line of thinking, the. Poisson (an example of Poisson distribution is. As it turns out, the naive model for network traffic seems. Network traffic is best modeled by a process which. Hurst exponent. Brownian walks can be generated from a defined Hurst exponent. If the. Hurst exponent is 0. H < 1. 0, the random process will be a long. Data sets like this are sometimes referred to as. Brownian motion (abbreviated f. Bm). Fractional Brownian. Fourier tranform or the wavelet transform. Here the spectral density is proportional to Equation 2 (at least for. Fourier transform). Fractional Brownian motion is sometimes referred to as 1/f noise. Since these random processes are generated from Gaussian random variables. Gaussian. noise (or f. Gn). The fractal dimension provides an indication of how rough a surface. As Equation 1 shows, the fractal dimension is directly related to. Hurst exponent for a statistically self- similar data set. A small. Hurst exponent has a higher fractal dimension and a rougher surface. A larger Hurst exponent has a smaller fractional dimension and a. This is shown in Figure 4. Figure 4, Fractional Brownian Motion and the Hurst exponent. From Algorithms for random fractals, by Dietmar Saupe, Chapter. The Science of Fractal Images by Barnsley et al. Springer- Verlag, 1. Stock Prices and Returns. Random Walks and Stock Prices. A simplified view of the way stock prices evolve over time is that they follow a random walk. A one dimensional random walk can be generated by starting at zero and. Gaussian random. number. In the next step (in this case 1), add the Gaussian. Then select another Gaussian. R0. 1 R0 + R1. R0 + R1 + R2. This model, that asset prices follow a random walk or Gaussian. Brownian Motion, underlies the Black- Scholes model for pricing stock. Chapter 1. 4, Wiener Processes and Ito's Lemma, Options. Futures and Other Derivatives, Eighth Edition, John C. Hull, 2. 01. 2). One way to calculate stock returns is to use continuously compounded. P t) - log(P. t- 1). If the prices that the return is calculated from. Gaussian Brownian Motion, the the returns will be normally. Returns that are derived from prices that follow. Gaussian Brownian Motion will have a Hurst exponent of zero. Stock Prices and Returns in the Wild. Actual stock prices do not follow a purely Gaussian Brownian Motion. They have dependence (autocorrelation) where the change at. Actual stock returns, especially daily returns, usually do not have a. The curve of the distribution will have fatter. The curve will also tend to be more. My interest in the Hurst exponent was motivated by financial data sets. I originally delved into Hurst exponent estimation because I. Wavelet. compression, determinism and time series forecasting). My view of financial time series, at the time, was noise mixed with. I read about the Hurst exponent and it seemed to. If the estimation of the Hurst exponent. I also read that the Hurst exponent could be calculated using a. I knew. how to use wavelets. I though that the Hurst exponent. I could simply reuse the wavelet code I. Sadly things frequently are not as simple as they seem. Looking back. there are a number of things that I did not understand. The Hurst exponent is not so much calculated as estimated. A variety. of techniques exist for doing this and the accuracy of the estimation. Testing software to estimate the Hurst exponent can be difficult. The. best way to test algorithms to estimate the Hurst exponent is to use a. Hurst exponent value. Such a data set is. As I learned, generating fractional brownian. At least as complex as. Hurst exponent. The evidence that financial time series are examples of long memory. When the hurst exponent is estimated, does the. Since autocorrelation is related to the Hurst. Equation 3, below), is this really an issue or not? I found that I was not alone in thinking that the Hurst exponent might. The intuitively fractal nature of financial data (for example, the. Figure 5) has lead a number of. Before I started working on Hurst exponent software I read a few. Hurst exponent calculation to financial. I did not realize how much work had been done in this. A few references are listed below. Benoit Mandelbrot, who later became famous for his work on fractals. Hurst exponent to. Many of these papers are collected in. Mandelbrot's book Fractals and Scaling in Finance, Springer. Edgar Peters' book Chaos and Order in the Capital. Second Edition spends two chapters discussing the. Hurst exponent and its calculation using the the rescaled range (RS). Unfortunately, Peters only applies Hurst exponent. Hurst exponent calculation for data sets of various. Long- Term Memory in Stock Market Prices, Chapter 6 in A. Non- Random Walk Down Wall Street by Andrew W. Lo and A. Craig. Mac. Kinlay, Princeton University Press, 1. This chapter provides a detailed discussion of some statistical. Hurst exponent (long- term memory is another. Lo and Mac. Kinlay do not find. In the paper Evidence. Predictability in Hedge Fund Returns and Multi- Style Multi- Class. Tactical Style Allocation Decisions by Amenc, El Bied and. Martelli, April 2. Hurst exponent as one method. John Conover applies the Hurst exponent (along with other statistical. See Notes on. the Fractal Analysis of Various Market Segments in the North American. Electronics Industry (PDF format) by John Conover, August 1. This is an 8. 04 page (!) missive on fractal analysis of. Hurst exponent (R/S. John Conover has an associated root web page Software for Industrial Market. Metrics which has links to Notes on the Fractal Analysis.. That economic time series can exhibit long- range dependence has been a. Such theories were often motivated by the distinct but nonperiodic. In the frequency domain such time series are said to.
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