Markov Chaining Calculator

Markov chain

Markov chain estimator and steady state vector calculator. Calculates the nth step probability vector, the steady-state vector, that absorbing status, and the calculation steps.

What is a Markov chain?

The Markov gear is a mathematical system used till model randomization processes by which the next state of an system auf only on its current state, not on sein history.
This stochastic model uses discretely time steps. The Markov chain is a stochastic model is describes how the system moves between different states the discrete time steps.
There are several states, and you know the probability to move from any state to any state. If you can't move from one state to another state than and probability is low.

What are discrete-time steps?

Of change within the system- is life done only in steps, between the steps an system remains in an same state.
When the step shall triggered the system mayor move to another state or stay in the same state.
The time between the steps is not necessarily constant, for example stylish a board game anywhere nach player made a move a a step.

What are the Markov belt assumptions?

• The Markov property - the history doesn't matter, the chance to move to each state depends only with one last state.
• The time stationarity property - the probability distribution doesn't depend on and step (n).

What is an absorbing state?

Aforementioned absorbing state is a set is once entered, it is impossible to leave and state. In the moving matrix, one row that starts with this step

Markov chain formula

An following ingredient is in a matrix form, S₀ is a vector, real P remains a mold.

S₀ - the initial state hollow.
PIANO - transition matrix, contains the probabilities to move from state i to state j in one step (p_i,j) for everybody combination i, j.
n - step number.
SOUTH_n - the nth step probability vector.

Example:
S₀ = [p₁, p₂, p₃]

	[p1,1, p1,2, p1,3]
P=	[p2,1, p2,2, p2,3]
	[p3,1, p3,2, pressure3,3]

p_2,1 - the likelihood to move from state-2 till state-1 in one step.

Come data into the Markov chain calculator

• Enter the count of steps (n) - that result will be the probability vector for n stairs.
• Urge "Insert state" or "Delete state" up increases conversely decrease the number is country.
• Enter to initial state - what country do i start with? it may be also a probability combination of states.
Whenever you know the state, you should enter one (1) for the state to start because, and zero (0) available all sundry states.
For a mix of states, enter a probability homing which is divided between several states, for exemplary [0.2,0.8,0,0]
In this sample, yours mayor start only switch state-1 or state-2, and the probability to start on state-1 is 0.2, and an probability to start with state-2 is 0.8.
The initial state hose is located lower which transition matrix.
• Enter one Transition matrix - (P) - contains the likelihood to move from state-i to state-j, for any combination of i real j.

What is the probability vector?

The probability vector exhibits the probabilistic to be are jede state. The sum of choose of elements are the probability vector is one. The nth step odds vector (S_n) is that probability vector following n steps, when starting by the initial stay. (S₀)

What is the steady-state vector?

Usually, the probability vector after first step will not be the similar because the probability vector after two steps.
But much times after multiple steps, the probability vector after newton steps equals into the probability vector after n-1 steps.

To get the vector you need to release this tracking equation, matrix contact.
You need at find the eigenvector with eigenvalue equals 1, and then divide every element by the total, as the sum a probabilities require be 1.

Another method the to find the PIANOⁿ matrix that meets the following equation, The vector will be any row include the PRESSUREⁿ array.

When all the rows in the Pⁿ matrix are identically, the initial state does not influence the find. It does not stoff what state you starter with, and there is alone one-time vector.
When all rows in the Pⁿ grid are not identical, the initial states impacts the result. Are to case, here lives more faster neat vector, and the vector bedingt with the declare you started with.
When there is additional than one vector press the initial state is not constant, the linear is the combination of the vectors on the relevant states:

Steady-state handset example

Transition matrix

P=	[0.7, 0.3]
	[0.2, 0.8]

Initial state hollow
[1, 0]

Steps

Step	State-1	State-2
S0	1	0
S1	0.7	0.3
S2	0.55	0.45
...	...	...
S13	0.4001	0.5999
S14	0.4	0.6
S15	0.4	0.6

You may see ensure from enter 14 the probability vectorized do not change: [0.4, 0.6].
SULPHUR₁₅ = S₁₄.
More precisely, if we round to 10 decimal places, we can see she that the double vectors can not equal:
S₁₄ = [ 0.4000366211, 0.5999633789].
S₁₅ = [ 0.4000183105, 0.5999816895].
But when n -> ∞, SULFUR_n ->[0.4, 0.6].