What is the initial state vector?

Additionally, a Markov chain also has an initial state vector, represented as an N x 1 matrix (a vector), that describes the probability distribution of starting at each of the N possible states. Entry I of the vector describes the probability of the chain beginning at state I.

What is initial probability vector?

ii) Initial probability vector is the vector that contains the current state before transition. iii) Equilibrium state is the state that a system settles on in the long run. iv) Absorbing state is one in which cannot be left once entered. It has a transition probability of unity. to itself and of zero to other states.

Does an initial state vector have any influence on long term behavior?

If P is an n × n regular stochastic matrix, then P has a unique steady state vector v. Further, if x0 is any initial state and xk+1 = Pxk for k = 0,1,2,··· , then the Markov chain {xk} converges to v. Remark. The initial state does not affect the long time behavior of the Markv chain.

What is a state vector?

A state vector (geographical) specifies the position and velocity of an object in any location on Earth’s surface. Orbital state vectors are vectors of position and velocity that together with their time, uniquely determine the state of an orbiting body in astrodynamics or celestial dynamics.

What is initial probability vector in Markov chain?

Markov Models and Social Analysis Let p (0) s be the probability that an individual is in the initial state s at t=0, s=1, …, S and write p (0) as the S×1 vector whose elements are p (0) s . The matrix P is called a transition probability matrix (tpm) and p (0) is the initial probability vector.

What is the transition matrix of a Markov process?

The Transition Matrix and its Steady-State Vector The transition matrix of an n-state Markov process is an n×n matrix M where the i,j entry of M represents the probability that an object is state j transitions into state i, that is if M = (m.

How do you find the eigenvectors of Markov processes?

Finally, Markov processes have The corresponding eigenvectors are found in the usual way. After a sufficient number of iterations, the state vector will nominally equal its steady-state vector. The eigenvector associated with the eigenvalue “1” determines the steady-state vector.

What is the transition probability of a transition matrix?

Definition: If a system featuring “n” distinct states undergoes state changes which are strictly Markov in nature, then the probability that its current state is ” ” given that its previous state was ” ” is the transition probability, ” “. The nxn matrix ” ” whose ij th element is is termed the transition matrix of the Markov chain.

What is Markov chain or Markov process?

And suppose that at a given observation period, say period, the probability of the system being in a particular state depends on its status at the n-1 period, such a system is called Markov Chain or Markov process.