On fractal distributions.

nukapprentice

Hello, I had a question about data which is represented by a fractal distribution. I know that the linear regression lies in the plot of log(N) vs. log(x) for which the ratio represents the fractal dimension as the limit of x going to infinity. However, how would one get the representative fractal equation of such behavior? I am very interested since fractal equations are well known to be continuous everywhere but nowhere differentiable.

Stephen Tashi

A linear regression produces a linear equation and the graph of a linear equation is not a fractal.

Suppose we have some fractal object, say a coastline of an island and we define P(x) to be the perimeter of the island as measured by adding up measurements taken with a ruler of length x. Its an interesting question whether the graph of P(x) is itself a fractal. I don't know enough about fractals to say yes or no. The coastline itself and the graph of P(x) are two different things.

At any rate, you need to clarify your question. Explain what x and N are, and explain what "regression" you are asking about.

nukapprentice

Actually what I meant was if you plot the graph log(N) vs. log(x) where N is the probability density function and x is the data, such as the amount of long lines vs. small lines on a coastline you should get a straight line. If you take the slope of log(N) vs. log(x) you do in fact get a straight line, hence why I said linear regression. If one does get such an equation, I have read that it should be represented by a fractal equation such that it is everywhere continuous, but nowhere differentiable. I guess my question is, if you do find this sort of relationship, which is represented by a power law distribution, (such as in the case of fractals), how would one get the equation which represents the behavior of the system? I hope this clears it up.

Stephen Tashi

What probability density function are you talking about? And what is "the data"?

Is "the data" a random sample from "the probability function"?

nukapprentice

The data, as stated above, is the amount of long lines vs. short lines on a coastline (to use your example from before). You could also use data such as the times between episodes of the onset of rapid heart rate measured in patients (with a random sampling of people). Here there is no single average time that characterizes the times between these events. Most often the time between episodes is brief. Less often the time is longer. The PDF has a power law form that is a straight line on a plot of Log[PDF(t)] versus Log(t). That is the thing with fractal behavior, there is usually a large number of small samples, and smaller number of large samples.

Stephen Tashi

Are you saying that the mean value of a power law distribution does not exist?
Are you saying that he graph of a power law probability density is a fractal?

Fractals need not involve probability density functions or samples from them. You should explain what fractal you are talking about and what random variable you are talking about.

The relation of ruler length to coastline length usually discussed in books on fractals is that if you use a particular ruler length, then that ruler measures a particular coastline length. It doesn't give you a "randomly selected" coastline length.

nukapprentice

no, a mean value may exist for a power law distribution, however, for a fractal distribution, the data never converges on a mean value. Also, fractals obey a power law distribution, but not all power law distributions are fractal in nature.

For the context of my question the pdf's I am talking about do involve fractals. I guess what I was stating about the coastline analogy is that when you add up all the distances of the minicurves, you will end up with an infinite length. This is where the fractional dimension comes in with fractals since the curve described with this phenomenon has infinite length although it would normally be a 1-d object in this instance.

Stephen Tashi

You haven't explained what "the data" is.


If you want to talk about the pdf of a random variable, explain what random variable you're talking about.

I think you are using technical terms carelessly. And is "representative fractal equation" terminology that you have invented? What is its definition?