An original Markov process is described by a square matrix M (nxn) whose entries M i, j verify the following two conditions:
i- All inputs M i,j are real and belong to the closed interval [0,1]
ii-The sum of the entries for all columns (or all rows) is equal to 1.
However, since the Markov era, many attempts to improve M, and hence M-strings, have been suggested by adding one or two more conditions.
We guess the best improvement is to add one or two of the following physical conditions:
*iii- Constant remaining after each time jump dt, i.e.,
B i, i = RO , where RO is an element of [0,1] for all i = 1,2, ... n, which means that the main diagonal consists of constant entries.
For example if RO is equal to 0 , ie the matrix M is a null principal diagonal matrix which corresponds to the assumption of a null residue after each step or temporal jump dt for all the free nodes.
**iv- symmetry condition :
Mi,j=Mj,i for all i,j.
Condition-iv transforms the stochastic transition matrix M into a doubly stochastic transition matrix superior than the original Markov matrix which is just a single stochastic transition matrix.
A statistical Markov transition matrix is a matrix that describes the transition probabilities between different states of a system, usually in a discrete-time setting. A hyper-statistical transition matrix expands on that by considering more parameters that affect the transition probabilities. This could include factors such as temperature, pressure, or any other environmental variables that influence the system.
To transform a statistical Markov transition matrix into a hyper-statistical transition matrix, you would need to incorporate additional variables that influence the state transitions. This might involve collecting data on these variables and using statistical models to determine how they impact the Markov matrix. The resulting hyper-statistical transition matrix would then reflect the effects of these additional variables on the system's behavior
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