Closed-system assumptions and model limitations | Year 12

01 / LEARN

The idea, step by step

Consistency is not the same as accuracy

A closed transition matrix assumes all individuals remain within its stated categories. The column totals of 1 preserve the population. Constant matrix powers also assume the same transition behaviour for every period. If entry, exit, ageing or different subgroups matter, add states or explicit inputs rather than silently ignoring them.

A valid stochastic matrix is not automatically a good forecast. Data from a small or unusual sample can give uncertain transition estimates. Predictions are expected distributions; real observations need not match every predicted entry. State definitions, time interval and the population used to estimate the matrix are all part of the model.

Use fresh evidence to evaluate a forecast

Compare predictions with observations not already used to fit the matrix. Inspect errors for every state and the total. Small differences may be sampling variation; large or repeated patterns may justify changing the matrix or redefining categories. A signed total error of zero can conceal large redistributions between states.

Revise a model for a reason, such as a new route or policy, not merely to force one observation to agree. Keep full numerical precision until reporting. Distinguish an open-system problem from changing transition rates and explain which additional data would help.

A common trap

Using column sums alone to prove a model predicts well.

Instead: Column sums check internal closed-system consistency; data and context test usefulness.

02 / WORKED EXAMPLES

See the thinking

Example 1

A total reveals an omitted process

A closed model predicts (60,40), but observed counts are (58,47). Identify the first warning.

The predicted total is 100 and the observed total is 105.
Transfers alone cannot add five members.
Check entry, exit or counting/state-definition changes before adjusting transfer proportions.

The closed-system assumption needs checking.

Example 2

Compare two forecasts

Observed counts are (72,28). P predicts (70,30); Q predicts (60,40). Compare maximum absolute state error.

P has errors (2,2), maximum 2.
Q has errors (12,12), maximum 12.
P fits this observation better by the stated measure; one observation does not prove lasting superiority.

Maximum error: P 2, Q 12.

Check your understanding

Observed totals rise from 500 to 540. Can a closed transition alone explain it?

Show answer

No.

A column-stochastic matrix preserves the total.

03 / WORKSHEET

Your turn

6 questions · 18 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Name two assumptions behind a six-month forecast $T^{6} x_{0}$ .
3 marks
A matrix has columns totalling 1 but an entry −0.1. Why is it invalid as a transition-probability matrix?
3 marks
Predicted counts are (120,80), observed (112,88). Find absolute errors and their sum.
3 marks
The same 30 travellers estimate a matrix and are then used to claim its forecast is perfect. Explain a weakness.
3 marks
A model predicts proportions (0.35,0.65) for 1000 members.
3 marks
1. (a)
  Find expected counts.
  1 mark
2. (b)
  Does observing (340,660) alone prove failure? Explain.
  2 marks
A new bus route changes behaviour halfway through a year. How should the model respond?
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Name two assumptions behind a six-month forecast $T^{6} x_{0}$ .
3 marks
The periods must also be equal and state labels consistent.
A constant monthly transition rule and a closed set of states/population.
A matrix has columns totalling 1 but an entry −0.1. Why is it invalid as a transition-probability matrix?
3 marks
Each entry must be between 0 and 1.
A probability cannot be negative.
Predicted counts are (120,80), observed (112,88). Find absolute errors and their sum.
3 marks
Signed errors are (−8,8).
Take absolute values before adding.
(8,8), sum 16.
The same 30 travellers estimate a matrix and are then used to claim its forecast is perfect. Explain a weakness.
3 marks
Use later or separate observations; the small original sample also limits precision.
The claimed check is not independent of the fitting data.
A model predicts proportions (0.35,0.65) for 1000 members.
3 marks
a) Multiply each proportion by 1000.
b) One difference of 10 in each state does not establish a changed underlying rule.
1. (a)
  Find expected counts.
  1 mark
  Multiply each proportion by 1000.
  (350,650).
2. (b)
  Does observing (340,660) alone prove failure? Explain.
  2 marks
  One difference of 10 in each state does not establish a changed underlying rule.
  No; assess variation, sample uncertainty and repeated prediction errors.
a) (350,650).; b) No; assess variation, sample uncertainty and repeated prediction errors.
A new bus route changes behaviour halfway through a year. How should the model respond?
3 marks
Keep compatible state labels and check whether the population remains closed.
Estimate a new transition rule from suitable later data and apply old/new matrices in the correct chronological order.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 General Mathematics matrix-models option. This is the offered fifth-topic option, alongside the four compulsory topics; a school-designed Open Topic is an alternative, not prescribed content in this chapter.

Official curriculum source ↗

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