Modelling a dynamic system of permutations

In summary: At time t+1, the stocks continue to move around, but we will use the same permutation rule to rank them. This is an example of a Markov process.In summary, the conversation discusses a dynamic system with permutations and probabilities at different time instances. The speaker is seeking advice on what models would be suitable for studying this system and determining if it reaches a stable state after a certain period of time. The only information available is the permutations at each time instance, and the speaker mentions the possibility of a Markov process if the probability distributions are independent of time. However, without more specific details, it is difficult to determine the best model for this system.
  • #1
vdrn485
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Let us assume a dynamic system which has vector with 'n' components (which are non-negative integers from 1->n) at time t=1. In other words we have a permutation over 1->n at time t=1. Assuming time to be discrete, at any time time 't' , the system evolves such that there are 't' permutations with 'n' components in each. We do not know in advance which permutations in time 't' contributes to the birth of which permutation in time 't+1'. We can assume that each permutation in time 't+1' has a probabilty distribution over the permutation in time 't' for being its parent.
The only information available is the permutations at each time instance from 1 to T.

What kind of models can be used to represent such a system if we want to find out if the permutations attain a stable state with very less perturbations after certain time period?
 
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  • #2
vdrn485 said:
We can assume that each permutation in time 't+1' has a probabilty distribution over the permutation in time 't' for being its parent.

If the probability distributions are independent of time then you have a Markov process.

You haven't been specific enough to make any particular model a good candidate. Lots of different real life problems match the generalities you presented. For example, pick n stocks and name them 1,2,..n. Let t be the time in days. Let the permutation at time t be this list of stocks ordered from lowest closing price to highest closing price on that day. If there are ties, then break them by giving the stock with the highest market capitalization the higher rank.
 

1. What is a dynamic system of permutations?

A dynamic system of permutations refers to a mathematical model that describes the behavior of a system that changes over time. It involves analyzing the various combinations and arrangements of the elements within a system to predict how they will change and interact with each other.

2. How is a dynamic system of permutations different from a static system of permutations?

A dynamic system of permutations takes into account the time factor and how the elements within a system may change and influence each other over time. On the other hand, a static system of permutations only considers the possible combinations of elements at a specific point in time.

3. What are some real-world applications of modelling a dynamic system of permutations?

Dynamic systems of permutations have various applications in fields such as economics, biology, and physics. For example, it can be used to model the spread of diseases, predict the behavior of financial markets, or study the interactions between different species in an ecosystem.

4. What are some common tools and techniques used for modelling a dynamic system of permutations?

Some common tools and techniques used for modelling dynamic systems of permutations include computer programming languages such as MATLAB and R, as well as statistical methods like Markov chains and Monte Carlo simulations. These tools allow scientists to analyze and visualize the complex behavior of dynamic systems.

5. What are the limitations of modelling a dynamic system of permutations?

One limitation of dynamic systems of permutations is that they are based on assumptions and simplifications of real-world situations. Therefore, the predictions and results may not always accurately reflect what happens in reality. Additionally, the accuracy of the model depends on the quality of the data and variables used, which can be difficult to obtain in some cases.

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