Random function coupled to a non-random function question.

In summary, the conversation discusses the formulation of a theory for elastodynamics and finding a relationship between a non-random function and a random function, specifically the covariance. The problem arises when one function is non-random and continuous, and the search for a connection between the two becomes difficult. Suggestions are given, including using a binary characteristic function and evaluating the correlation function numerically. The conversation concludes with the understanding that the correlation functions will have to be determined numerically.
  • #1
KayBox
6
0
Hello and thank you in advance for anyone taking time to respond.

I working on formulating a theory for elastodynamics, but my statistics is admittedly weak. I'm trying to find a relationship between a non random function and a random function, for example, the covariance.

<A(x)B(y)>=some 2 point probability for 2 random functions.

These are discrete random variables, so the result is usually calculated numerically or by experimental observation.

The problem I'm having is what if B(y) is non-random (ie deterministic) and continuous (not discrete)? I understand there is a discrete and integral formulation for discrete and continuous functions respectively, but I keep getting stuck. Its like I'm trying to find a connection that shouldn't be there, but they always come up in the formulation. Can anyone give me an idea of where to start?
 
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  • #2
If B(y) is not random, then <A(x)B(y)> = B(y)<A(x)>.
 
  • #3
Ah, not what I wanted to hear (<A(x)>=0 centered fluctuations).

What if B(y) is a binary characteristic function that is 1 inside a specified volume size within the total volume(though its position is not known) and zero otherwise?
 
  • #4
KayBox said:
Ah, not what I wanted to hear (<A(x)>=0 centered fluctuations).

What if B(y) is a binary characteristic function that is 1 inside a specified volume size within the total volume(though its position is not known) and zero otherwise?

You need to be specific about the relationship between y and x.
 
  • #5
y and x are both chosen at random. When A(x) and B(y) correspond to random material property fluctuations, the correlation function is either assumed to be of the form exp(r/L) where r is the absolute distance between the points and L is the average grain size for the polycrystalline material, or is evaluated numerically. When evaluated numerically using a Voronoi tesselation, randomly tossing line segments in the model, then calculating the product produces the numerical result, which is then fitted to some function similar to the assumed form above. Its a measure of the likelihood that the two points are in the same grain.

For this case, x and y are still chosen at random, but only one varies randomly in space. The other is deterministic, in that its center is located in the inspection volume with a its own volume occupied by the foreign material, while the other varies randomly. Its basically an inclusion type problem in a polycrystalline medium. Hope this helps.
 
  • #6
Since y is random B(y) is random (even if B is deterministic), so you cannot take it outside the bracket.
 
  • #7
Thanks for your help. Looks like I will have to determine the correlation functions numerically.
 

1. What is a random function?

A random function is a mathematical function that generates a random output or result each time it is used. It is commonly used in statistics, probability, and computer science to simulate randomness and make predictions.

2. What is a non-random function?

A non-random function is a mathematical function that produces a predictable output or result for a given input. It is used to model relationships between variables and is often used in engineering, economics, and other fields.

3. How are random and non-random functions coupled?

Random and non-random functions can be coupled by using the output of one function as the input for the other. This can create dynamic and unpredictable relationships between the two functions, allowing for more complex and realistic modeling.

4. What are some real-world applications of coupling random and non-random functions?

Coupling random and non-random functions is commonly used in fields such as finance, biology, and physics to model complex systems and make predictions. For example, in finance, random and non-random functions may be coupled to simulate stock prices and market trends.

5. Are there any limitations to coupling random and non-random functions?

Yes, there are limitations to coupling random and non-random functions. The accuracy of the model depends on the complexity of the functions and the quality of the data used. Additionally, it may be difficult to interpret and explain the results of a coupled function model.

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