Transforming Relationships or How to Flatten a Function! …subtle twists on regression

Not all data sets are linearly related… Many relations between data sets are not linear, but… Some classes of functions can be “linearized” Relations (functions) that follow a simple power law y = ax p are easy to linearize

Example.. In the Physics Lab Students measure the distance and object falls as a function of time… Get Excel spreadhseet of this

This can be linearized by plotting distance wrt time 2

A Cubic relation…

The Logarithm – a great “power flattener” If…

Any power law will get flattened…

Why do we need this? Logarithms flatten AND compress data Example…Question 2.105

By taking logs of both variable the scatterplot looks like this!

Exponential Growth When something grow exponentially, it grows by a constant multiple in each equal time period: –Bacterial growth (early) –Radioactive decay –Many biological processes –Some economic processes

Exponential growth can be easily transformed to linear growth… This produces a line when log Y is plotted against x

Example… Moore’s Law Question 2.92: –The number of transistors on an IC doubles every 18 months (Gordon Moore – Intel 1965)

In conclusion… Most important things in this section: –Logarithms flatten and compress –Exponential growth Read summary on page 203 Try 2.95, 2.102, 2.109