Questions tagged [transition-matrix]

A transition matrix is a square matrix used to describe the transitions of a Markov chain.

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Markov chain: inferring transition rates from equilibrium

I feel like I have been having a very dumb week trying to solve/research this problem and that I am missing an easy solution or that it is not possible. Given an equilibrium distribution (say from a ...
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How to prove a sequence of random variables dependent on different random variables is a homogeneous Markov chain?

In Pierre Brémaud's book, Markov Chains - Gibbs Fields, Monte Carlo Simulation and Queues, exercise 2.6.9 is stated as follows: Let $\{Z_n\}_{n \geq 1}$ be an IID sequence of geometric random ...
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Coin flipping (Markov chain)

Suppose that coin 1 has probability 0.7 of coming up heads, and coin 2 has probability 0.6 of coming up heads. If the coin flipped today comes up heads, then we select coin 1 to flip tomorrow, and if ...
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How to decide $X_n, n \in [0,\infty)$ is a Markov chain and how to compute transition probability matrix?

Three white and three black balls are distributed in two urns in such a way that each contains three balls. We say that the system is in state i, i = 0, 1, 2, 3, if the first urn contains i white ...
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Is this transition probability matrix correct?

A pensioner receives 2 (thousand dollars) at the beginning of each month. The amount of money he needs to spend during a month is independent of the amount he has and is equal to i with probability $...
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Interpreting Chart of Transition Densities from MSBVAR Gibbs Sampling

I've run the MSBVAR package on 3 time series, and then did a Gibbs Sampling and used the plotregimeid function based on the elements of the transition matrix Q. I'm trying to get an understanding of ...
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When using Markov models for attribution modeling,what information does the transition matrix have,that causes the steady state vector not to be used?

I've just finished my msc thesis in attribution modeling, comparing higher order Markov models and the heuristic approaches. The professor's question is what information does the transition matrix ...
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How to map a sequence to a transition matrix

I have the following transition matrices, one for Maria and one for Anna: ...
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How to analyze a time series of categorical data?

I have some data that look like this: ...
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What are the transition functions for RNNs

From what I understand, the hidden states of RNNs are equivalent to the deterministic probability distribution over hidden states in for example a Hidden Markov Model. Thus, just as probabilistic ...
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R CODE Transition Matrix having zeroes for some transitions

I have been trying to create a Transition Matrix using the data from 2000 entities over 40 observations (Years). I have ranked the data into percentiles, for example the highest value entity in a ...
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Estimating transition probabilities of a Markov chain with bayesian approach

I am trying to model heteroskedasticity in time series data,and the volatility $\left(\sigma_{t}\right)$ is taken as a Markov chain with two values $\sigma_{h}>\sigma_{l}>0$ with transition ...
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Approximating a countable-state (infinite) Markov model with a finite-state one

TL;DR—in a nutshell I have a countable-state Markov model (with a countably infinite number of states) in which the probability of transitioning to states $S_{i>k}$ for large $k$s are practically ...
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R MSM: unexpected identical transition rate matrices for different values of covariate; why?

I am fitting a continuous-time Markov model to a panel dataset using the R package MSM. Because I am interested in sex-differences in transition rates, I fit the ...
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What is the relationship between 2 covariance matrices of 2 variables?

Assume variable $\textbf{x} = [x_1, x_2]^\top$ with mean $\mathbf{\mu} = [\mu_1, \mu_2]^\top$ and covariance matrix: \begin{equation} \Sigma_{\textbf{x}} = \begin{bmatrix} \sigma_1^2 & \text{cov}(\...
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Estimating transition probability from a state to itself with the MSM package

For a given panel dataset I have used the MSM package to estimate transition probabilities between states. Using the pnext.msm function I can obtain the ...
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Transition Matrix for a non-trivial example

So I've just been introduced to Transition Matrices; and I was wondering what one for look like the following example: "Symmetric random walk on the integers" $S$ $=$ $\mathbb{Z}$, $\forall$ ...
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"Forcing" equal probabilities in the matrix exponential of a Markov intensity matrix

I have an upper-right triangular transition intensity matrix $Q$ for a 7-state Markov model (with states $X_1,X_2,...,X_7$), from which I numerically calculate the matrix exponential to derive a ...
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How to generate a realization from a transition matrix?

Consider a Markov chain of 4 states described by the transition matrix, $$ T_{ij} = \begin{bmatrix} 0.40 & 0.56 & 0.03 & 0.01\\ 0.45 & 0.51 & 0.04 & 0.00\\ 0.25 & 0.25 &...
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Probability of doing a specific Path in a Markov Chain

My problem is the following: I have this graph, representing a Markov Chain: For example, if I am in state 1, the probability of going in state 2 or 4 is $\frac{1}{2}$. So I'm saying that the ...
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How to derive transition matrix in this stochastic process?

I am new to stochastic processes and trying to solve a question related to finding a transition matrix of some experiment. The question is a A sequence of experiments is performed, in each of which ...
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Why doesn’t two ways of calculating stationary distribution result in the same answer?

I use matpow=function(M,n){ ans=M for(i in 1:(n-1)){ ans=ans%*%M } ans } to set the matpow function. then I enter the transition matrix ...
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Number of states in HMM

I am testing a HMM model by generating data from a 3x3 transition matrix and 3x4 emission matrix and then trying to train a HMM model against this data with different initializations. When I plot the ...
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How to maximize the steady state transition probability for a state in a Markov chain by altering that state's outgoing transition probabilities?

Let's say we have a transition matrix of which can be solved to come up with steady state transition probabilities of Alpha: 34.9% Beta: 24.0% Gamma: 16.9% Delta: 24.2% Now, imagine Alpha, Beta, ...
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Simulate discrete state space CTMC from generator matrix

Consider a generator matrix $Q\in\mathbb{R}^{h\times h}$ for a discrete state space $\{1,...,h\}$. I want to determine the probability of a single transition of a stochastic process $X(t)$ with $X(0)=...
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How to calculate the probability Matrix (Alpha) for Regular Markov chains

Pardon me for being a novice here. In the image attached, eq 3.1 represents the transition matrix (it's pretty clear). I am not able to comprehend the eq 3.2, alpha*P = alpha, as well as the further ...
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$\pi_i P^n_{i, j} =$ long-run proportion of time the chain is in $i$ and will be in $j$ after $n$ transitions?

I am currently studying the textbook Introduction to Probability Models by Sheldon M. Ross. Chapter 4.4 Long-Run Proportions and Limiting Probabilities says the following: Because $\pi_i$ is the long-...
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generate a realization from a transition matrix

Consider a markov chain of 4 states $\{S_1, S_2, S_3, S_4\}$ described by the transition matrix $$ A = \begin{bmatrix} .25 & .20 & .25 & .30 \\ .20 & .30 & .25 & .30 \\ ....
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Calculating limit law for matrix

I my notes on Markov chains, I am presented with the following matrix: $$\mathcal{P} = \begin{bmatrix} 0.97 & 0.03 & 0 & 0 \\ 0.008 & 0.982 & 0.01 & 0 \\ 0.02 & 0 & 0....
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Example where unique stationary law, which is an occupation law, but no limit law exists

I am currently learning about the balance equations, mass equation, limit law, occupation law and stationary law in Markov models. The following example is presented: Example 2: $$\mathcal{P} = \...
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Computation of balance equation example in Markov model

I am studying some examples of balance equations for Markov models. I am presented with the following example: $$\mathcal{P} = \begin{bmatrix} 0.2 & 0.3 & 0.5 \\ 0.1 & 0 & 0.9 \\ 0.55 ...
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Markov models and occupation time

I'm presented with the following explanation and proof: Let $(X_n)$ be a Markov chain, and fix a state $j \in S$. Define indicator variables: For $n = 0, 1, \dots$, let $$I_n(j) = \begin{cases} 1 &...
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Calculating 7-step transition matrix for example

In my notes on Markov processes, I am presented with two related examples: Example 1: Classify daily weather for some region as Sunny (state $1$), Cloudy (state $2$), or rainy (state $3$). ...
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Proof that the $n$-step transition matrix is the $n$th power of $\mathcal{P}$

I am presented with the following theorem in the context of Markov chains and stochastic systems: The $n$-step transition matrix is the $n$th power of $\mathcal{P}$: $$\mathcal{P}^{(n)} = P^n.$$ And ...
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Transition probability matrix

A gambler tosses a coin and a tetrahedron at each stage. If $H$, he receives the amount appearing at the face of the tetrahedron. If $T$, he pays the amount. The tetrahedron is fair, but probability ...
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Find transition probability matrix

A box contains 3 balls. Each is either white or red. The game is to draw a ball from each period. If red is drawn, a white is replaced. But if white is drawn, all the balls in the box is replaced by ...
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Number of stationary distributions of a Markov chain

How do i determine the number of stationary distributions that a Markov chain has if it is not irreducible or regular. The transition matrix is ...
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Estimating model for transition probabilities of a Markov Chain

Suppose that I have a Markov chain with $S$ states evolving over time. I have $S^2\times T$ values of the transition matrix, where $T$ is the number of time periods. I also have $K$ matrices $X$ of $T\...
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How to show that the transition probability is equal to $\overline p_{ij} = \frac{P_{ij}}{\sum_{k\neq i}p_{ik}}$

(No new answers needed) I would like to award @whuber for his good answer with my bounty! Suppose that $(X_n)_{n≥0}$ is Markov$(λ, P)$ but that we only observe the process when it moves to a new ...
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Rows and columns of the one-step transition probability matrix

I am currently studying the textbook Introduction to Modeling and Analysis of Stochastic Systems, Second Edition, by V. G. Kulkarni. In a section on discrete-time Markov chains, the author introduces ...
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Singular state transition probability matrix in David Silver's UCL Lesson 2

I'm studying David Silver's second lesson on reinforcement learning: https://www.youtube.com/watch?v=lfHX2hHRMVQ&list=PLqYmG7hTraZDM-OYHWgPebj2MfCFzFObQ&index=2 and the state transition ...
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Specifying a multi-state model with unobservable terminal state

Suppose there is a multi-state process with three states, listed below and labelled as terminal/non-terminal and observable/unobservable: Initializing: non-terminal observable Active: non-terminal ...
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Make sense of plotting a transition matrix

I'm studying statistics and I'm trying to understand markov chain topic. I'm using the package "markovchain" in R to obtain the stationary distribution. From this transition matrix $M$: <...
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Measure the distance between two probability transition matrices

I have a probability transition matrix $P$ that contains some values very close to zero. I want to sparsify this matrix by taking the k largest values for each row and setting the others to zero. For ...
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Long run proportion of transitions in a Markov chain

Let $S$ be a set of states for a Markov chain and let $S^C$ be the remaining states. Explain the identity $$\sum_{i\in S}\sum_{j\in S^C}\pi_iP_{ij}=\sum_{i\in S^C}\sum_{j\in S}\pi_iP_{ij}$$ I know ...
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Irreducible (communicating) classes [closed]

The Markov chain $(Xn; n\geq)$ has state-space $S = (0, 1, 2, . . .)$, with $p_{i,0} = \frac{1}{4}$ and $p_{i,i+1} = \frac{3}{4}$ $\forall i \geq 0$, so that the transition matrix is P =$\...
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Biased coins and Markov processes

Good day, I am attempting an optional exercise and I am finding it hard to interpret the problem in terms of matrices and vectors. Coin 1 has probability 0.4 of coming up heads, and coin 2 has ...
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Formulating a Transition matrix for Markov Process

I am dealing with a medical process which is as follows. There are 10000 Veterans who are enrolled in this study. All 10000 have medical condition called onychocryptosis which is a fancy term for ...
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What is the difference betwen a time non-homogenous Markov Chain and a non-linear Markov Chain? Example

A time non-homogenous Markov Chain is one in which the transition probabilities are not constant over time. A non-linear Markov Chain is a model that is not linear in parameters and satisfies the ...
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Irreducible Markov chain and transition matrix

We know that a matrix is irreducible if it is not similar via a permutation to a block upper triangular matrix. Is the transition matrix of a irreducible Markov chain irreducible?