[R-sig-Geo] GWR : Confidence intervals on predictions
Jean-Paul Kibambe Lubamba
jean-paul.kibambe at uclouvain.be
Fri Oct 3 16:44:04 CEST 2008
Hello everybody,
I want to obtain confidence intervals on predictions when using gwr. I
received some matrix algebra to implement to do that, but I need the gwr
weight matrix and I do not know how to get it. Could anyone help me ?
Thanks in advance for any help !
JP
Here is below the matrix operations, using a latex notation:
The vector of regression coefficients at a given point is
\beta = (X^TWX)^(-1) X^TWy
Where the weight matrix W depends on the location of the point, as usual.
We can simplify this by writing
\beta = Cy where C = (X^TWX)^(-1) X^TW
Then
Var(\beta) = CC^T \sigma^2 - so far, this is in the GWR book - including
how to estimate \sigma^2
Now if we have a new vector of predictors x at the same point, then
predictor of the y-variable is
x^T \beta
And so its variance is
x^T C x \sigma^2
This gives the variance of the expected y-variable given a set of
predictors - but this is just the variance of the mean of its
distribution. The y-variable itself will be the sum of the predicted mean
plus an error term with mean zero and variance\sigma^2.
This has variance
x^T C x \sigma^2 + \sigma^2
or
(x^T C x + 1)\sigma^2
And so the standard error of the prediction is
sqrt(x^T C x + 1)\sigma
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