What is the representative fractal equation for a power law distribution?

In summary, the conversation discusses the use of linear regression to represent fractal dimension as the limit of probability density function and data as x approaches infinity. The question is raised about how to obtain a representative fractal equation for such behavior, as fractal equations are known to be continuous but not differentiable. The conversation also touches on the concept of fractal behavior and power law distributions, and clarifies that not all power law distributions exhibit fractal behavior. There is also a discussion about the existence of mean values in fractal distributions and the difference between fractals and power laws. The conversation concludes with a request for clarification on the use of technical terms and definitions.
  • #1
nukapprentice
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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.
 
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  • #2
nukapprentice said:
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?.

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.
 
  • #3
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.
 
  • #4
nukapprentice said:
where N is the probability density function and x is the data.

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

Is "the data" a random sample from "the probability function"?
 
  • #5
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.
 
  • #6
nukapprentice said:
Here there is no single average time that characterizes the times between these events.

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?

That is the thing with fractal behavior, there is usually a large number of small samples, and smaller number of large samples.

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.
 
  • #7
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.
 
  • #8
nukapprentice said:
no, a mean value may exist for a power law distribution, however, for a fractal distribution, the data never converges on a mean value.

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


For the context of my question the pdf's I am talking about do involve fractals.

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?
 

1. What is a fractal distribution?

A fractal distribution is a type of distribution that exhibits self-similarity, meaning that the overall pattern is repeated at different scales. This means that the distribution looks similar regardless of the scale at which it is observed.

2. How is a fractal distribution different from a normal distribution?

A normal distribution is characterized by a bell-shaped curve, while a fractal distribution does not have a defined shape. Normal distributions also have a finite variance, while fractal distributions have an infinite variance.

3. What are some real-world examples of fractal distributions?

Some examples of fractal distributions include the distribution of coastlines, river networks, and even the distribution of galaxies in the universe. These natural phenomena exhibit self-similarity and can be described using fractal geometry.

4. How are fractal distributions used in science?

Fractal distributions are used in various fields of science, such as physics, biology, and economics. They can help scientists understand complex systems and patterns, as well as model and predict the behavior of these systems.

5. What are the limitations of using fractal distributions in scientific research?

One limitation is that fractal distributions are often only approximations of real-world phenomena, and may not fully capture all aspects of the system being studied. Additionally, the complexity of fractal distributions can make them difficult to analyze and interpret.

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