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

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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

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

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This can be linearized by plotting distance wrt time 2

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A Cubic relation…

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The Logarithm – a great “power flattener” If…

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Any power law will get flattened…

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

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

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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

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

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

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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

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Logarithms Properties and Uses. Some background Read pp. 469-473 in Agresti and Finlay. A logarithm (generally called the “log” of a number) is the “power.

Logarithms Properties and Uses. Some background Read pp. 469-473 in Agresti and Finlay. A logarithm (generally called the “log” of a number) is the “power.

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