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Last time I explained that Fractals are self similar geometric objects with fractional dimension. The fractional dimension can be determined by the formula;
N = SD
where N is the number of miniature pieces found in object, S is the scaling factor, and D is the dimension (See Fractals - Part I).
This entry I'd like to go through some of the practical applications of fractals. The problem is that fractals pop up in all areas of science. To simplify the explanation of how fractals can be used practically, while still providing a broad scope of the applications fractals can be used for, I will provide a detailed example of fractal analysis and then I will provide a few abstracts that are from papers that use fractals in many different fields.
Human Heartbeats - A Detailed Example of Fractal Analysis
The idea here is that heartbeat patterns over time can be modeled by fractals. Healthy and unhealthy heartbeat signals can be quantified in terms of their fractal dimension, which then, if different enough from each other, can be used as an early warning diagnostic tool for heart failure.
In order to determine whether fractal analysis of a heartbeat signal, or any signal is worthwhile, we must first determine if a signal is self similar. After all, it only makes sense to use a fractal to approximate a fractal like pattern, just as one would use a sphere, not a pyramid, to model the mass distribution of the Earth. You could use a pyramid if you wanted, but it wouldn't give very accurate results. In the same way, its absolutely essential we make sure that the signal in question does in fact look like a fractal and furthermore that we can somehow quantitatively determine its characteristic fractal dimension. Guessing here does no good, we need a mathematical approach.
To start, we should see how to test and characterize a time varying pattern as self similar. We can use the following;
To put the above discussion into mathematical terms: A time-dependent process (or time series) is self-similar if
where  means that the statistical properties of both sides of the equation are identical. In other words, a self-similar process, y(t), with a parameter  has the identical probability distribution as a properly rescaled process,  , i.e., a time series which has been rescaled on the x-axis by a factor a ( ) and on the y-axis by a factor of  ( ). The exponent is called the self-similarity parameter.
In practice it is impossible to determine whether two processes are statistically identical, because this strict criterion requires their having identical distribution functions (including not just the mean and variance, but all higher moments as well). Therefore, one usually approximates this equality with a weaker criterion by examining only the means and variances (first and second moments) of the distribution functions for both sides of Eq. 1. Although not perfect, this at least gives us a high probability of certainty.
Figure: Illustration of the concept of self-similarity for a simulated random walk. (a) Two observation windows, with time scales  and  , are shown for a self-similar time series y(t). (b) Magnification of the smaller window with time scale . Note that the fluctuations in (a) and (b) look similar provided that two different magnification factors,  and  , are applied on the horizontal and vertical scales, respectively. (c) The probability distribution, P(y), of the variable y for the two windows in (a), where  and  indicate the standard deviations for these two distribution functions. (d) Log-log plot of the characteristic scales of fluctuations, s, versus the window sizes, n.
So now that we know how to characterize a time varying signal, lets apply it to heartbeats and see what we find.
Human Heartbeats
The following is from a paper on human heartbeats, the link is provided at the bottom.
Analysis of data from patients with congestive heart failure is likely to be particularly informative in assessing correlations under pathologic conditions since these individuals have abnormalities in both the sympathetic and parasympathetic control mechanisms that regulate beat-to-beat variability. Previous studies have demonstrated marked changes in short-range heart rate dynamics in heart failure compared to healthy function, including the emergence of intermittent relatively low frequency ( cycle/minute) heart rate oscillations associated with the well-recognized syndrome of periodic (Cheyne-Stokes) respiration, an abnormal breathing pattern often associated with low cardiac output. Of note is the fact that patients with congestive heart failure are at very high risk for sudden cardiac death.
Figure: Plot of  vs  for three interbeat interval time series: healthy young subject, elderly subject, and a subject with congestive heart failure. Compared with the healthy young subject, the heart failure and healthy elderly subjects show different patterns of altered scaling behavior (see text).
Figure 6 compares a representative result of fractal scaling analysis of representative 24-hour interbeat interval time series from a healthy subject and a patient with congestive heart failure. Notice that for large time scales (asymptotic behavior), the healthy subject shows almost perfect power-law scaling over more than two decades ( ) with  (i.e., 1/f noise), while for the heart failure data set,  (closer to Brownian noise). This result indicates that there is a significant difference in the scaling behavior between healthy and diseased states, consistent with a breakdown in long-range correlations.
A relevant question regarding these new measurements is: Does fractal analysis, such as the DFA method, have clinically predictive value, independent of conventional statistical indices? To answer this question, we have studied the predictive power of the DFA exponent in comparison with multiple conventional measures based on mean, variance and spectral analysis. We analyzed two-hour ambulatory ECG recordings of 69 participants (mean age  years) in the Framingham Heart Study--a prospective, population-based study. The study population consisted of chronic congestive heart failure patients, and age- and sex-matched control subjects. Importantly, we found that this fractal measurement carried prognostic information about mortality not extractable from traditional methods of heart rate variability analysis. Subsequent studies have confirmed and extended these observations, suggesting that fractal scaling measures may have a practical use in bedside and ambulatory monitoring.
http://www.physionet.org/tutorials/fmnc/index.shtml
What does all that mean?
Sometimes scientific papers can read like a legal contract or Olde-English literature and the meaning of whats going on is lost (I'm talking to ye, Beowulf!) . I'll try a quick explanation of the heartbeat paper above. Basically, heartbeats are noisy, but it turns out they are noisy in a very predicable way. The randomness of the noise is self similar over varying time frames. In other words, they behave like a fractal over time. Fractals can have different dimensions, as we established in part one. This paper is saying that healthy hearts and unhealthy hearts have signals of different fractal dimension. What this means is that a doctor can now take what was once disregarded as noise and use it to categorize the health of a heart!
Here's an analogy with traditional physics. The flight of a projectile in time is approximated by a parabola (I say approximated because in the real world there is varying air resistance, temp, etc.). We can use the type of parabola that approximates a particular flight of a projectile to determine the initial velocity, distance travelled, etc., of that projectile.
So all that is going on is we are using fractals instead of parabolas to model how something behaves in time (In the heartbeat example). It doesn't have to be in time though, it can be how things are distributed in space, in temperature, in anything really. Just as a sphere or a square or a parabola can be used in all branches of physics and engineering to approximate the real world (The Earth isn't really a sphere, but we can learn things if we approximate it to be one), so too can fractals approximate systems in all disciplines.
Examples of Fractal Analysis in Various Fields
The abstracts below cover a large variety of subjects. The thing they all have in common is they involve the use of fractals to categorize physical systems. Links are provided to the papers for those who are interested. Registration will likely be required at many of the stories because they are peer reviewed journals.
Homogeneity of the Universe
The assumption that the Universe is homogeneous and isotropic on large scales is one of the fundamental postulates of cosmology. We have tested the large-scale homogeneity of the galaxy distribution in the Sloan Digital Sky Survey Data Release One (SDSS-DR1) using volume-limited subsamples extracted from the two equatorial strips that are nearly two-dimensional. The galaxy distribution was projected on the equatorial plane and we carried out a 2D multi-fractal analysis by counting the number of galaxies inside circles of different radii, r, in the range 5–150 h 1 Mpc centred on galaxies.
http://www.blackwell-synergy.com/doi/abs/10.1111/j.1365-2966.2005.09578.x
Fractal Analysis of Microscopic Teeth Wear
They used a technique that he and colleagues pioneered to examine the microscopic wear on teeth using modified fractal analysis software and a state-of-the-art laser scanning microscope. The pits and grooves in animals' teeth point to different dietary preferences. A pit-laden texture indicates consumption of hard, brittle foods, such as nuts or woody plants. A scratched texture indicates the shearing of food, such as grasses.
http://www.physorg.com/news82911778.html
Fractal Analysis of Turbulent Flows
Fractal analysis methods are applied to derive a new model of turbulence for use in numerical flow simulations. The implementation of a turbulence model based on this equation is outlined, and selected (previously published) results are presented in graphs for: (1) plane and round jet flows; (2) near-wall flows; (3) the three-dimensional flowfield of a centrifugal compressor; and (4) sudden-expansion swirling flows. The present turbulence model is found to give predictions in better agreement with experimental data than the commonly used k-epsilon and Baldwin-Lomax models.
http://adsabs.harvard.edu/abs/1990icas....2.2254G
Fractal Dimension of Dielectric Breakdown
It is shown that the simplest nontrivial stochastic model for dielectric breakdown naturally leads to fractal structures for the discharge pattern. Planar discharges are studied in detail and the results are compared with properly designed experiments
http://prola.aps.org/abstract/PRL/v52/i12/p1033_1
Fractal Analysis of Permeabilities for Porous Media
A fractal analysis of permeabilities for porous media, both saturated and unsaturated, is presented based on the fractal nature of pores in the media. Both the fractal-phase permeability and the fractal relative permeability are derived and found to be a function of the tortuosity fractal dimension, pore-area fractal dimension, phase fractal dimension, saturation, and microstructural parameters. The proposed models for permeabilities - both the phase permeability and the relative permeability - do not contain any empirical constant. The validity of the present analysis is verified by a comparison with the existing measurements, and excellent agreement between the model predictions and experimental data is found.
http://www3.interscience.wiley.com/cgi-bin/abstract/107063727/ABSTRACT?CRETRY=1&SRETRY=0
Structure of the Internet
We introduce and use k-shell decomposition to investigate the topology of the Internet at the AS level. Our analysis separates the Internet into three sub-components: (a) a nucleus which is a small (~100 nodes) very well connected globally distributed subgraph; (b) a fractal sub-component that is able to connect the bulk of the Internet without congesting the nucleus, with self similar properties and critical exponents; and (c) dendrite-like structures, usually isolated nodes that are connected to the rest of the network through the nucleus only.
http://arxiv.org/abs/cs.NI/0607080
Fractal Analysis for Brain Tumor Detection
Magnetic resonance (MT) images typically have a degree of randomness associated with the natural random nature of structure. Thus fractal analysis is appropriate for MR image analysis. The purpose of this study is to apply fractal analysis to identify the presence of tumor in brain MR images. For tumor detection in MR brain images, the authors propose three different fractal-based methods.
http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=901528
Gravitation theory in a fractal space-time
Assimilating the physical space-time with a fractal, a general theory is built. For a fractal dimension D=2, the virtual geodesics of this space-time implies a generalized Schrödinger type equation. Subsequently, a geometric formulation of the gravitation theory on a fractal space-time is given. Then, a connection is introduced on a tangent bundle, the connection coefficients, the Riemann curvature tensor and the Einstein field equation are calculated.
http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JMAPAQ000047000005053503000001&idtype=cvips&gifs=yes
Thats all for now, thanks to Google Scholar for helping me find the papers. Till next time.
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