Interactive guide
The forecast settles down, no matter where it starts.
Probability of each weather, day by day, starting from the state below.
Sunny Cloudy Rainy steady-state
Drag a divider to change that row's transition probabilities.
Every row still sums to 100%, dragging one divider only trades probability with its neighbor.
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- Sunny
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- Cloudy
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- Rainy
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Try this. Click through all four "Start on day 0 as" buttons without touching the transition probabilities. The three lines take different paths to get there, but they all end up at the exact same place. That destination doesn't depend on today's weather at all, only on the transition probabilities do.
Tomorrow only depends on today
A Markov chain's one rule: tomorrow's weather depends only on today's weather, not on how you got there. Each row of the transition table above is "if today is this, tomorrow is this, with these odds", and every row has to sum to 100%, today's weather is always something.
To forecast day 2, multiply day 1's distribution by the same table again. Day 3, again. Do it enough times and the distribution stops changing, at that point, tomorrow's forecast is identical to today's, which is exactly what "steady-state" means.
Reading the chart
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1
Each line is one state's probability
Day 0 starts wherever you picked; day by day it's recomputed from the row above.
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2
Dashed lines are the steady-state
Solved directly from the transition table, not simulated, then drawn as a target the solid lines settle onto.
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3
"Sticky" states mix slowly
High diagonal probabilities mean today's weather is a strong predictor of tomorrow's, so convergence takes many days.
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4
The Cyclic preset overshoots on its way in
A strong preference to move to a specific next state makes the forecast swing past steady-state before settling.
This is the same "repeatedly multiply by a matrix" idea from the Matrix Transformations page, just aimed at a probability distribution instead of a grid. The steady-state here is exactly an eigenvector of the transition matrix, the one with eigenvalue 1, the single direction that repeated multiplication leaves completely unchanged. Everything the Eigenvalues & Eigenvectors page calls "a direction that only stretches" is here a forecast that's stopped changing.
The jargons
One table of odds, applied over and over, until it stops changing anything.
One of the possible conditions a system can be in, Sunny, Cloudy, or Rainy here.
The chance of moving from one state to another (or staying put) in a single step. Each state's outgoing odds sum to 100%.
The next state depends only on the current one, not on the history that led there.
How likely each state is right now, three numbers that always add up to 100%.
The distribution that, once reached, is unchanged by another step. Almost every starting point converges to it.
Roughly, how many steps it takes to get close to the steady-state. "Sticky" chains have a long one.