Setup
Between 2 and 12 is practical for quick editing.
Example: A, B, C. Names affect labels and exports.
Directly reads entry Pij from the matrix.
Transition matrix P (row‑stochastic)
Each row must sum to 1. Invalid rows highlight red; valid rows green.
Results
Next distribution π′ = πP
Selected one‑step probability
Diagnostics
Example data
Weather states example with three states.
| From \ To | Rainy | Sunny | Cloudy |
|---|---|---|---|
| Rainy | 0.60 | 0.20 | 0.20 |
| Sunny | 0.30 | 0.50 | 0.20 |
| Cloudy | 0.40 | 0.30 | 0.30 |
If starting in Sunny, the next distribution equals row two.
Formulas used
- P is row‑stochastic: for each row i, ∑j Pij = 1, and Pij ≥ 0.
- One‑step transition probability: Pij = Pr(Xt+1 = j | Xt = i).
- Distribution update: with row vector π, the next distribution is π′ = πP.
- Starting in a single state s: use unit vector es; then esP equals row s of P.
This tool focuses on one‑step behavior; multi‑step and stationary distributions are beyond scope.
Data: Row sums and dominant next states
Dominant next state is argmaxj P(i→j) for each row i.
Data: One‑step outcomes by starting state
Starting from es, the next distribution equals row s of P.
How to use
- Set the number of states, then confirm or edit names.
- Enter matrix P; each row should sum to one.
- Choose an initial distribution vector or a starting state.
- Click Compute one‑step to calculate π′ and P(i→j).
- Use Normalize rows to fix small row‑sum errors.
- Export results as CSV or PDF for sharing or records.
FAQs
What if rows do not sum to one?
Use Normalize rows to scale each row by its current sum.
Can I start in a single known state?
Yes. Select the single‑state mode; the next distribution is that row.
How do I import an existing matrix?
Click Import CSV and select a file with numeric entries.
Does the tool support multi‑step predictions?
This calculator covers one step. Multiply repeatedly for more steps.
How precise are the computations?
Values use double precision in JavaScript and are formatted consistently.
Can I reuse inputs later?
Yes. Names and matrix persist in your browser’s storage automatically.