App Markov Chains

The purpose of the application is to provide convenient means for creating the transition matrix of the Markov chain and solving to determine the state vectors x(1), x(2),...,x(n),.. .

A Markov chain is a stochastic process that describes a sequence of possible events where the probability of each event depends only on the state attained in the previous event. This property is known as the Markov property or memorylessness. Markov chains simplify the study of many real-world processes by focusing on the current state and transition probabilities, making them a powerful tool in various scientific and engineering disciplines.

Suppose a physical or mathematical system undergoes a process of change such that at any moment it can occupy one of a finite number of states.

Suppose that such a system changes with time from one state to another and at scheduled times the state of the system is observed. If the state of the system at any observation cannot be predicted with certainty, but the probability that a given state occurs can be predicted by just knowing the state of the system at the preceding observation, then the process of change is called a Markov chain or Markov process.

If a Markov chain has k possible states, which we label as 1,2,...,k, then the probability that the system is in state i at any observation after it was in state j at the preceding observation is denoted by p(i,j) and is called the transition probability from state j to state i. The matrix P=[p(i,j)] is called the transition matrix of the Markov chain.

The state vector for an observation of a Markov chain with k states is a column vector x whose i- th component x(i) is the probability that the system is in the i - th state at that time.

Observe that the entries in any state vector for a Markov chain are non negative and have a sum of 1. Let us suppose now that we know the state vector x(0) for a Markov chain at some initial observation. The following statement will enable us to determine the state vectors x(1), x(2),...,x(n),... at the subsequent observation times. If P is the transition matrix of a Markov chain and x(n) is the state vector at the nth observation, then x(n+1) = P*x(n).

From the startup activity of Annex is launched Function for creation of New transactional matrix( button New), for Storage(button Save, save as) and delete( On the other hand, the Delete button matrix).

Transaction matrices are Stored in a database with the name MrkovChains.db and is of type SQlit. In creation of New transactional Matrix the Dialogue is being introduced the size of the Matrix which is square.

Any transactional Matrix names and its name is displayed in a drop-down list, when choosing Transactional matrix of list of its content is displayed in table and button appears Cаlculate, through which are calculated vectors of state x(k). In Press button Cаlculate in dialog enter k, numbers of the calculated vectors of state x(k). Annex It also has a function for formatting in a file ( named AppMarkovChains.txt) for printing on display Transactional matrix or Vectors shown state x(k). Formatted Files can to be stored in the file Directory of the device, which is as a tree-like structure. In Folder selection appears green storage button, By pressing is chosen by him from dialogue whether to perform Storage