Maths IA · Statistics and probability · topic 25 of 40
Markov chains: what is the long-run weather? · IB Maths IA idea: guide, sample graph and step-by-step plan
stretching, but great at HLWill it rain tomorrow if it rained today? Where will you be on the board after 100 turns? Matrices can say, if you're careful about what they assume.
- ✓ you like matrices
- ✓ you enjoy a long-run question
- ✓ you want HL-level maths
The big idea
eigenvector = steady stateA Markov chain moves between states with fixed probabilities that depend only on the current state. Put those probabilities in a transition matrix and powers of it give the future; the long run settles to a steady state. The steady state is an eigenvector of the matrix, for the eigenvalue 1. You estimate the matrix from real records (daily weather, or a board game you simulate), predict the long run, and test the prediction against real data. The depth comes from the justifications: why powers of the matrix give n-step probabilities, why the chain converges, and how fast. Transition matrices and eigenvectors are AI HL content. AA has no matrices, so on AA you would be teaching yourself a whole topic.
Make it yours: pick one twist
Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.
Swipe for more twists →
Use daily records for a place you know and count transitions between states such as dry and wet.
→ The matrix comes from your data, and you test whether the Markov property is plausible with a goodness-of-fit check.Model a board game with snakes and ladders or a loop of squares and find the average number of turns.
→ You get a bigger matrix, absorbing states and expected game length: a more advanced use of matrices.Use the second eigenvalue to explain how quickly the chain forgets its start.
→ This links eigenvalues to a graph of probability against time and lifts the sophistication.Compare a one-day memory with a two-day memory, using pairs of days as states.
→ You test a model assumption head on and discuss whether the extra complexity helps.Estimate separate matrices for summer and winter and compare steady states.
→ Your model has to cope with a changing process, and you reflect on a constant-matrix assumption.Prove existence and uniqueness for your matrix, or at least for the 2-state case.
→ This provides the rigour HL criterion E asks for, instead of reading the answer off software.Your Maths IA research question: run the check
Write your own question first. Then make sure it ticks every box:
- Names the process and the states (how many, and how they are defined).
- Says where the transition probabilities will come from: data, a published source, or a simulation.
- Says what long-run quantity you will find and how you will check it against data.
- Is narrow enough that the matrix is no bigger than you can justify by hand.
- Allows for the Markov property failing, and says how you will test it.
The model
Each arrow is a transition probability. All arrows out of a state sum to 1. The matrix collects them so you can use algebra. The IB AI course writes the transpose (columns sum to 1): pick one convention and keep it.
How your Maths IA could be laid out
The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.
Why this process; what you expect about the long run; what the states are and why.
State space, transition matrix, why the n-step matrix is a matrix power. At HL, justify the link instead of quoting it.
Data source, how transitions were counted, a table of counts, the estimated matrix and its limits.
Method (solve πP = π, or eigenvectors), a worked example, and a check that probabilities sum to 1.
How fast the chain approaches the steady state, ideally linked to the other eigenvalue, with a graph.
Compare predicted long-run proportions with the real data or a simulation, with a number for the difference.
Test or argue the memoryless assumption, such as comparing one-day with two-day memory.
The long-run answer with its limits.
Assumptions, data size, constant matrix, and a next step with a reason.
Data and method sources.
What your Maths IA graph could look like
show the limit as a lineMade-up values for a two-state chain, to show the form of a strong graph. example only
- Both starting states drawn: the picture shows the chain forgetting its start.
- A horizontal line marks the steady state so the reader sees the limit.
- Simulated points have error bars from the number of simulated runs (state it).
- Axis titles say what is plotted: a probability, not a count.
- Another graph to consider: the distance from the steady state on a log scale, which is a straight line if one eigenvalue controls the decay.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 10 done
- Choose a processWeather in your town, or a board game. Decide on states you can observe without ambiguity.
- PredictWrite down what you think the long-run proportions will be, before calculating.
- Learn matricesMatrix multiplication, powers, eigenvalues and eigenvectors, if your course includes them. Do a 2-state chain by hand.
- Trial runTake 30 days of data (or 30 turns of the game). Count transitions. Are your states clear? Is any state too rare?
- Collect properlyGather enough transitions so each row of the matrix has a sensible count, and record the source.
- Build the matrix and the steady stateEstimate the probabilities, check rows sum to 1, find the steady state by two methods (algebra and a high matrix power).
- Study convergencePlot the probability against time. Link the speed to the matrix.
- Test the modelCompare predicted and observed long-run proportions. Test the Markov assumption.
- Reflect and extendState the model's limits. Pick one extension, such as memory length or seasons, and say why.
- Write, then cutShow one worked matrix calculation, then tables. Notation for matrices must be proper, not software syntax. Check before your teacher's draft.
Maths you'll need
Traps to avoid in this Maths IA
check your rows!- Rows of the matrix that do not sum to 1 because of rounding or a counting slip.
- Choosing states because they are easy to count, not because they make sense.
- Estimating from too little data, so a rare transition has probability 0 by luck.
- Reading the steady state from software without showing the working or checking it.
- Assuming a Markov chain without ever testing the memoryless property.
- At HL: only the 2-by-2 routine with no proof or comparison; criterion E needs sophistication and rigour.
- Mixing up row and column conventions part-way through.
More Maths IA ideas like this
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This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.