Functions as models of data

From data to a formula

A scientist measures and collects numbers. A function turns the numbers into a rule, and the rule can then answer questions that nobody measured: what happens at $55\ \text{km/h}$, when we measured only $50$ and $60$?

A function that describes a real situation is called a model. Three families are enough for many problems.

Choosing the family

1. Plot the data. The picture often tells you the family.
2. A straight line means that equal steps in $x$ give equal steps in $y$.
3. A curve that bends upward and starts flat may be a parabola.
4. A curve that falls and flattens may be a reciprocal.
5. Think about the physics. The kinetic energy $\tfrac{1}{2}mv^2$ has a square of speed, so a parabola is natural for it.

Finding the best numbers

Suppose we choose a parabola $y = ax^2 + bx + c$. Which $a$, $b$ and $c$ are best? For each measurement, the model makes a small error, the difference between the model and the measurement. The best fit makes the sum of the squares of the errors as small as possible. This is called the method of least squares. We square the errors so that a positive error and a negative error cannot cancel each other.

You do not need to do the calculation by hand. In Python, numpy.polyfit(x, y, degree) does it. Degree $1$ gives a line, and degree $2$ gives a parabola.

The picture shows seven measurements of the stopping distance of a car at different speeds. The data are made up for this lesson, but they look like real ones. The best line misses the points at the two ends. The best parabola passes close to all of them. The reason is physical: the kinetic energy grows with the square of the speed, and a car must lose all of it.

Try $m = 2$ and $c = 0$. The bars almost vanish and the sum of squares drops near its smallest value. Then move $c$ a little: the sum grows again.

How good is the model?

Measure the typical error: the square root of the mean of the squared errors. For the data above, the best line is typically wrong by $1.9\ \text{m}$, and the best parabola by $0.5\ \text{m}$. The smaller error says that the parabola is the better model.

A warning about too much freedom. With $n$ points, a polynomial of degree $n - 1$ passes through every point, and the error is zero. But it wiggles wildly between the points, and it is useless for prediction. A good model is simple, and it follows the pattern of the data and not each small error.

A warning about the range. A model is built from data in some range, here $20$ to $80\ \text{km/h}$. Inside this range it is trustworthy. Outside it, nobody knows: at $300\ \text{km/h}$ the car would not even be on the road. Do not trust a model far outside the data.

Practise with new numbers

Try it in Python

This program fits a line and a parabola to the data, prints the typical error of each, and predicts the stopping distance at $55\ \text{km/h}$.

Remember these

• A model is a function that describes data. Plot the data first.
• numpy.polyfit(x, y, degree) finds the best polynomial by least squares.
• Judge a model by its typical error and by the physics, and prefer a simple model.
• Trust a model only inside the range of the data.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A taxi charges a fixed $\text{Rs } 50$ and $\text{Rs } 12$ for each kilometre. What is the fare for $7\ \text{km}$?

2. A journey of $120\ \text{km}$ is made at $40\ \text{km/h}$. How long does it take?

3. Why is it a bad idea to fit a polynomial of degree $6$ to $7$ measurements?

4. The readings $(1, 5)$, $(2, 8)$ and $(3, 11)$ were taken. Which simple model fits them exactly?