Does this type of process have a name?

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Discussion Overview

The discussion revolves around a mathematical expression related to a potential stochastic process, specifically exploring whether it can be classified as a Markov process or if it aligns with an autoregressive moving average (ARMA) model. The scope includes theoretical considerations of stochastic processes and their properties.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents an equation for the expected value of a sequence of random variables and seeks information about the type of process it represents.
  • Another participant suggests that the equation resembles a generalization of a Markov process.
  • A question is raised regarding the Markov property when considering states as vectors of past states.
  • A participant emphasizes the need for definitions regarding the independence of the random variables involved to determine if the process can be classified as Markov.
  • It is noted that the equation does not define a stochastic process, as it lacks information about the distribution of the random variables.
  • A suggestion is made that the equation might fit an autoregressive moving average (ARMA) model, provided certain conditions are met.
  • One participant expresses satisfaction with the ARMA model suggestion, indicating it aligns with their inquiry.

Areas of Agreement / Disagreement

Participants express differing views on whether the process can be classified as a Markov process, with some emphasizing the importance of independence among random variables. The discussion remains unresolved regarding the classification of the process without further definitions.

Contextual Notes

The discussion highlights limitations in defining the stochastic process, particularly concerning the independence of random variables and the lack of distribution specifications.

euroazn
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E(Xn+1+i)=Ʃn+ij=i+1 cj-iXj, where
Ʃnj=1 cj = 1.
n is a fixed constant here, c is a fixed set of n coefficients.

Can anyone tell me anything about such a process?
 
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It looks like some sort of generalization of a Markov process.
 
Is it markov if we consider states to be vectors of the past n states?
 
euroazn said:
E(Xn+1+i)=Ʃn+ij=i+1 cj-iXj, where
Ʃnj=1 cj = 1.
n is a fixed constant here, c is a fixed set of n coefficients.

Can anyone tell me anything about such a process?

You haven't defined a stochastic process. I assume the X_k are random variables. The equation you gave only specifies the expected value of X_k in terms of the realizations of some other X_j. It doesn't specify any distribution for X_k. It doesn't specify whether the distribution of X_k is or is-not independent of other random variables that don't appear in the sum.
 
Oh, I am well aware that it doesn't DEFINE a stochastic process, but if it were to satisfy these conditions, do we know anything nifty about the process?

Edit: The random X's form a time series if that's not clear.
 
Last edited:
You have to say something about the independence (or lack of it) between different X's before you can determine if the process can be viewed as a Markov process. It is perfectly OK to use a vector of values as a "state" in a Markov process. The "states" can even be vectors of different lengths.

An "auto-regressive moving average" model might fit your equation. Make the additive noise term have zero mean.
 
Yes, this is perfect. I think ARMA is exactly what I'm looking for. Thanks a bundle!
 

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