Linearizing

12 September, 2006

eeps media

# Linearizing Data

When data are more or less linear, your life is easier. You can eyeball a line, or use some line of best fit to make the model. Furthermore, the parameters are easy to understand: the slope is a rate, the intercept some zero-point.

When you are faced with data that need a nonlinear model, you have at least two choices: make a model that itself is nonlinear, or transform the data so that they are linear, and use those easy linear techniques.

Let's talk about this second option, linearizing. What does it mean to transform the data so that they are linear?

Suppose we have these fake falling-rock data:

time height 0.27 s 0.3 m 0.33 s 0.5 m 0.48 s 1.0 m 0.57 s 1.5 m 0.64 s 2.0 m 0.88 s 4.0 m 1.19 s 7.0 m 1.41 s 10.0 m

The data are curved. No line of the form time = m * height + b fits very well. But if you square time (to get tsq) and plot that against height, you get a better line:

This means that there is a formula like tsq = m * height + b that works OK. In fact, since if you drop something from a height of 0.0 meters, it takes no time, b should equal zero. That makes sense from the graph: it looks as if the line would go through the origin. So we should have tsq = m * height; in this case, a good line is tsq = 0.2 * height.

Having found a good linear fit to tsq, the transformed data, we can now back-transform the equation for the line to get the equation for time.

Since

tsq = time ^2

and

tsq = 0.2 * height,

we can substitute to get

time^2 = 0.2 * height, or

time = sqrt ( 0.2 * height )

The resulting graph looks like this: