Page 148 - Data Science Algorithms in a Week

Regression

Fahrenheit and Celsius conversion - linear

regression on perfect data

For example, Fahrenheit and Celsius degrees are related in a linear way. Given a table with
pairs of both Fahrenheit and Celsius degrees, we can estimate the constants to devise a
conversion formula from degrees Fahrenheit to degrees Celsius or vice versa:

⁰F ⁰C

5 -15
14 -10

23 -5
32 0

41 5
50 10
Analysis from first principles:

We would like to derive a formula converting F (degrees Fahrenheit) to C (degrees Celsius)
as follows:

C=a*F+b

Here, a and b are the constants to be found. A graph of the function C=a*F+b is a straight
line and thus is uniquely determined by two points. Therefore, we actually need only the
two points from the table, say pairs (F1,C1) and (F2,C2). Then we will have the following:

C1=a*F1+b C2=a*F2+b

Now, C2-C1=(a*F2+b)-(a*F1+b)=a*(F2-F1). Therefore, we have the following:

a=(C2-C1)/(F2-F1)

b=C1-a*F1=C1-[(C2-C1)/(F2-F1)]

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