Need help to locate my mistake

Dear readers

I would like to make General Linear Model (GLM) for the
following data set http://louise.hoffman.googlepages.com/fuel.csv

The code I have written is

fuelData<-read.table('fuel.csv',header=TRUE, sep=',')
n<-dim(fuelData)[1]
xOnes<- matrix(1,nrow=n,ncol=1)
x<-cbind(xOnes,fuelData[,3])
y<-fuelData[,4]
theta<-((t(x)%*%x)^(-1))%*%t(x)%*%y

which gives

theta

            [,1]
[1,] 215.8374077
[2,]   0.1083491

When I do it in Matlab I get theta to be
79.69
0.18

which is correct. ~79 is the crossing of the y-axis.

Have I perhaps written theta wrong? The formula for theta is
(alpha,beta)^T = (x^T * x)^(-1) * x^T * Y

where ^T means transposed.

Unfortunately, x^(-1) is not the inverse of x:

x<-matrix(c(2,4,4,5),nrow=2)
x
#      [,1] [,2]
# [1,]    2    4
# [2,]    4    5

x^(-1)
#      [,1] [,2]
# [1,] 0.50 0.25
# [2,] 0.25 0.20

i.e. it is the matrix which you get by applying the operation
(...)^(-1) to each element of x.

In R, the inverse of a non-singular matrix is (somewhat obscurely)
denoted by solve(x):

solve(x)
#            [,1]       [,2]
# [1,] -0.8333333  0.6666667
# [2,]  0.6666667 -0.3333333

solve(x)%*%x
#      [,1]         [,2]
# [1,]    1 1.110223e-16
# [2,]    0 1.000000e+00
(Note the slight rounding error); whereas

(x^(-1))%*%x
#      [,1] [,2]
# [1,]  2.0 3.25
# [2,]  1.3 2.00

Best wishes,
Ted.

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E-Mail: (Ted Harding) <Ted.Harding at manchester.ac.uk>
Fax-to-email: +44 (0)870 094 0861
Date: 02-Mar-08                                       Time: 21:05:50
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