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MathidiaMathematics Academy
Mathematical modelling II
Lesson 68 of 75
Standard practice

Model selection

Objective. Compare models against purpose and evidence.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Model selection weighs competing mathematical descriptions of the same phenomenon against their purpose, predictive accuracy and available evidence, rather than choosing the most complex option by default. A simpler model with fewer parameters may generalise better (lower risk of overfitting) even if it fits existing data slightly worse, judged using criteria like residual error balanced against parameter count. Example: given population data, a linear model may fit short-term trends adequately, while a logistic model better captures long-term saturation; choosing between them depends on whether the question concerns short-term rates or long-run behaviour. Good model selection matches model complexity to the question being asked and the data actually available.

Worked example

Two models fit population data: a linear model with 2 parameters and residual sum of squares 40, and a cubic model with 4 parameters and residual sum of squares 36. Using model-selection reasoning that penalises extra parameters, which model is preferable if the improvement in fit is marginal relative to the added complexity?

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Bar chart of the values 2, 40, 4, 36
  1. 1Compare residual sums of squares: cubic (36) is only slightly lower than linear (40)
  2. 2Note the cubic model uses two more parameters than the linear model
  3. 3A small gain in fit from extra parameters suggests overfitting risk rather than real structure
  4. 4Model-selection reasoning favours the simpler linear model here

Answer: The linear model

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Practice

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Common mistake. Students default to the model giving the smallest fitting error on existing data, ignoring that an overly flexible model may simply be overfitting noise rather than capturing genuine structure.

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