Calculator Input
Enter a row-stochastic matrix. Each row should represent probabilities from one state to all possible next states.
Example Data Table
This sample describes a simple weather model with three states.
| From State | To Sunny | To Cloudy | To Rainy | Row Sum |
|---|---|---|---|---|
| Sunny | 0.70 | 0.20 | 0.10 | 1.00 |
| Cloudy | 0.30 | 0.40 | 0.30 | 1.00 |
| Rainy | 0.20 | 0.30 | 0.50 | 1.00 |
Formula Used
A Markov chain uses a transition matrix P. Each value Pij is the chance of moving from state i to state j.
n-step transition matrix: P(n) = P^n
Future distribution: v(n) = v(0) × P^n
Steady state: π = πP, with all probabilities in π adding to one.
Row condition: every row should satisfy Σ Pij = 1.
How to Use This Calculator
- Select the number of states in your Markov model.
- Enter state names separated by commas.
- Enter the transition matrix with one row per state.
- Enter the initial distribution vector.
- Choose the forecast steps and decimal precision.
- Enable normalization if your rows are proportional counts.
- Press the calculate button to view probabilities and charts.
- Use CSV or PDF buttons to save the report.
Markov Chain Transition Matrix Guide
What This Calculator Does
A Markov chain studies movement between states. The next state depends only on the current state. This tool turns that idea into a clear calculation. You enter a transition matrix. Each row shows where a state can move next. The calculator checks row totals, applies matrix powers, and forecasts the probability distribution after many steps.
Why the Matrix Matters
The transition matrix is the center of the model. A valid row-stochastic matrix has non-negative values. Each row must add to one. This means every possible next move is covered. If your rows contain counts, you can normalize them. That converts counts into probabilities. This is useful for traffic models, weather movement, customer behavior, ranking systems, genetics, and queue analysis.
Forecasting Future States
The calculator uses the starting vector and multiplies it by the transition matrix power. For example, five steps means the tool evaluates the starting vector against P raised to five. The final vector shows the chance of being in each state after those transitions. The graph also shows how the distribution changes at each step.
Steady State Meaning
A steady state is a long-run balance. When it exists and converges, repeated transitions stop changing the distribution by much. The equation is simple. The steady vector multiplied by the matrix gives the same vector back. This calculator estimates that vector by iteration. It reports whether the tolerance was reached.
Advanced Interpretation
The tool also checks absorbing states, closed recurrent classes, possible transient states, periodic behavior, and regularity. These details help you understand the structure behind the numbers. An absorbing state traps the process. A closed class keeps probability inside a group. A regular chain eventually spreads positive probability across all states. These labels make the output easier to audit.
FAQs
1. What is a Markov chain transition matrix?
It is a square matrix showing probabilities of moving from each current state to each next state. Every row usually adds to one.
2. What does row-stochastic mean?
Row-stochastic means each row contains non-negative probabilities, and the row total equals one. It describes all possible next movements.
3. What is the initial distribution?
The initial distribution is the starting probability across states. It tells the calculator where the process begins before transitions occur.
4. What does P raised to n mean?
P raised to n gives transition probabilities after n steps. It shows how likely each state is after repeated transitions.
5. What is a steady state?
A steady state is a long-run probability vector that remains unchanged after another transition. It satisfies the equation π equals πP.
6. What is an absorbing state?
An absorbing state has probability one of staying in itself. Once the chain reaches it, the process cannot leave that state.
7. Can I enter counts instead of probabilities?
Yes. Enter non-negative counts and enable row normalization. The calculator will convert each row into probabilities before analysis.
8. Why does my steady state not converge?
Some chains are periodic, reducible, or slow to settle. Increase iterations, adjust tolerance, or inspect state classification for structural issues.