Choosing and Validating a Function Model

You’ll be able to

Select a model family from the pattern in a data table
Use residuals to judge whether a model fits, rather than relying on a correlation value
Explain why a better fit is not the same as a better model

Read the differences, not the numbers

For equally spaced x values, the pattern in the differences identifies the family. Constant first differences means linear. Constant second differences means quadratic; constant third differences, cubic. Constant ratios between successive values means exponential — the values are multiplied rather than added. A repeating rise and fall means sinusoidal. This single diagnostic covers nearly every modeling question on the exam.

Identifying a model from a table

constant 1st differences → linear · constant 2nd → quadratic · constant nth → degree n · constant ratios → exponential · periodic → sinusoidal

Requires equally spaced inputs. With unequal spacing, differences and ratios are not comparable and the test does not apply.

Worked example

Which family fits each table? A: (1, 4), (2, 12), (3, 36), (4, 108). B: (1, 5), (2, 11), (3, 21), (4, 35).

1.Table A first differences: 8, 24, 72 — not constant. Ratios: 12/4 = 3, 36/12 = 3, 108/36 = 3 — constant.
2.Constant ratio 3 means exponential with base 3. Working back one step from (1, 4) gives an initial value of 4/3, so y = (4/3)·3ˣ.
3.Table B first differences: 6, 10, 14 — not constant. Second differences: 4, 4 — constant.
4.Constant second differences means quadratic.

Answer: A is exponential, y = (4/3)·3ˣ. B is quadratic. Checking ratios and differences before committing is what separates the two cases — both tables are increasing and both are concave up, so the shape alone does not decide it.

Residuals are the real test

A residual is observed minus predicted, computed for each data point. Plot the residuals against x. If the model captures the structure, the residuals should look like featureless noise scattered about zero. If they show a pattern — a run of positives followed by a run of negatives, a U shape, a fan that widens — then the model is missing something systematic, and no amount of parameter tuning within that family will fix it. The residual plot diagnoses the shape of the failure, which is what tells you which family to try instead.

Watch out

A high r or R² does not validate a model. A curved relationship can produce r near 0.99 while the residual plot shows an obvious arch. Always look at the residuals; the summary statistic can hide the very defect you are testing for.

Checkpoint

A linear model is fit to data and the residual plot shows a clear U shape — positive at both ends, negative in the middle. What does this indicate?

Overfitting: why more parameters is not better

A polynomial of degree n can be made to pass exactly through any n + 1 points. So a high enough degree will always fit your data perfectly — and will usually be worthless, because it has fitted the noise along with the signal, and it typically swings wildly between and beyond the data points. Fit quality therefore cannot, on its own, justify a model. The justification has to come from the mechanism: is the quantity changing by a constant amount (linear), by a constant factor (exponential), or cycling with a natural period (sinusoidal)?

Worked example

Bacteria are counted hourly: 100, 150, 225, 338, 507. Choose a model and state one limit on its use.

1.First differences: 50, 75, 113, 169 — increasing, so not linear.
2.Ratios: 150/100 = 1.50, 225/150 = 1.50, 338/225 ≈ 1.502, 507/338 ≈ 1.500 — essentially constant.
3.Constant ratio 1.5 means exponential: P(t) = 100·(1.5)^t, with t in hours.
4.The mechanism supports it: each cell divides independently, so growth is proportional to the current population.
5.But no real culture grows exponentially forever — food runs out and waste accumulates.

Answer: P(t) = 100·(1.5)^t fits well and is mechanistically justified for early growth. It should not be extrapolated far: at t = 48 it predicts roughly 1 × 10¹⁰ cells, about a hundred million times the starting count, which no finite culture can support. Real growth turns over into a logistic curve once resources bind.

On the exam

When a question asks you to "justify the choice of model," a numerical fit is not enough for full credit. Name the structural feature — constant ratio, constant second difference, fixed period — and, where the context allows, the reason that feature makes sense.

Checkpoint

A degree-8 polynomial passes exactly through all nine data points, while a linear model has visible residuals. Which is the better model?

Answer the 2 checkpoints as you read.