Regression with a Single Predictor

Let us construct a correlation for the property y of members of a sample population S, relating them to a single predictor x. We would like to know whether x and y are very closely related, so that a knowledge of x enables one to calculate y with reasonable accuracy. One use of such a correlation is easy retrieval of the values of y within S but it is much more useful if this correlation established in S can be extrapolated to a bigger population P where the property y has not been measured. 

It is seen that the boiling points and most of the candidates for a predictor are monotonic increasing with the period P, with the exception of the ionization energy. 

Tabie 5.5 Boiiing points and predictors of nobie gases  

Let us propose a correlation for the boiling points of the six noble gases of the form 

This function has one predictor x and two arbitrary coefficients Co and Ci. Since the noble gases produce n = 6 sets of data and there are two arbitrary coefficients, the remaining degrees of freedom is 6 — 2 = 4. The principle of linear regression is to minimize the sum of the squares of the error term 

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