# Interpolation

As described, a regression line was fitted through 30 data points in the trees30.rda data set. The formula of the line is found by:

fit<-lm(Mass~Girth,data=TreeGirthMass)
fit

Call:
lm(formula = Mass ~ Girth, data = TreeGirthMass)

Coefficients:
(Intercept)        Girth
-411.62         3.24

Therefore the equation of the regression line is:

Mass = 3.24×Girth -411.62

This is the equation of the line showing the relation between girth and mass of a tree. In the example, the tree had to be chopped down to measure its mass. It would be nice to estimate the mass of a tree by only measuring its girth (without chopping it down; so the profit margin could be estimated)! From the equation we can estimate the mass of a tree with a girth of 280 centimetres:

Mass = 3.24×280 -411.62 = 495.58 kg

The estimation we made is within the range we have measured. This process is called interpolation. Of course, it is also possible to let R do this directly using the predict function. The predict function takes the model (fit) as first argument and a data frame with the value(s) of the Girth:

predict(fit, data.frame(Girth = 280))

1
495.7158

The difference with the manual method above is due to rounding error.

Similarly, we can estimate the mass of a tree outside the range we have measured. This process is called extrapolation. For example, to estimate the mass of a tree with a girth of 500 cm:

Mass = 3.24×500 -411.62 = 1208.38 kg

Or in R:

predict(fit, data.frame(Girth = 500))

1
1208.622

Obviously, one has to be far more cautious with estimations found by means of extrapolation than with interpolation.

It is also possible to predict a range of value in R:

predict(fit, data.frame(Girth = c(280, 500)))

1                2
495.7158 1208.6223