(all the) directions perpendicular to hyperplane through p data points

Question

Hi people,

I have a simple question: given p points (non-collinear) in R^p i find the hyperplane passing by these points (to help clarify i type everything in R):

p<-2
x<-matrix(rnorm(p^2),p,p)
b<-solve(crossprod(cbind(1,x[,-2])))%*%crossprod(cbind(1,x[,-2]),x[,2])

then, given a p+1^th points not collinear with first p points, i find the direction perpendicular to b:

x2<-matrix(rnorm(p),p,1)
b2<-solve(c(-b[-1],1)%*%t(c(-b[-1],1))+x2%*%t(x2))%*%x2

That is, b2 defines a p dimensional hyperplane perpendicular to b and passing by x2. Now, my questions are:

The formula comes from my interpretation of this wikipedia entry ("solve(A)" is the R command for A^-1). Why this doesn't work for p>2 ? What am i doing wrong ?

PS: I have seen this post (on stakeoverflow edit:sorry cannot post more than one link) but somehow it doesn't help me.

Thanks in advance,

i have a problem implementation/understanding of Liu's solution when p>2:

shouldn't the dot product between the qr decomposition of the sweeped matrix and the direction of the hyperplane be 0 ? (i.e. if the qr vectors are perpendicular to the hyperplane)

i.e, when p=2 this

c(-b[2:p],1)%*%c(a1)

gives 0. When p>2 it does not.

---

Here is my attempt to implement Victor Liu's solution.

a) given p linearly independent observations in R^p:

p<-2;x<-matrix(rnorm(p^2),p,p);x
      [,1]       [,2]
[1,] -0.4634923 -0.2978151
[2,]  1.0284040 -0.3165424

b) stake them in a matrix and subtract the first row:

a0<-sweep(x,2,x[1,],FUN="-");a0
        [,1]        [,2]
[1,] 0.000000  0.00000000
[2,] 1.491896 -0.01872726

c) perform a QR decomposition of the matrix a0. The vector in the nullspace is the direction im looking for:

qr(a0)
          [,1]       [,2]
[1,] -1.491896 0.01872726
[2,]  1.000000 0.00000000

Indeed; this direction is the same as the one given by application of the formula from wikipedia (using x2=(0.4965321,0.6373157)):

       [,1]
[1,]  2.04694853
[2,] -0.02569464

...with the advantage that it works in higher dimensions.

I have one last question: what is the meaning of the other p-1 (i.e. (1,0) here) QR vector when p>2 ? -thanks in advance,

Answer 1

A p-1 dimensional hyperplane is defined by a normal vector and a point that the plane passes through:

n.(x-x0) = 0

where n is the normal vector of length p, x0 is a point through which the hyperplane passes, . is a dot product, and the equation must be satisfied for any point x on the plane. We can also write this as

n.x = p

where p = n.x0 is just a number. This is a more compact representation of a hyperplane, which is parameterized by (n,p). To find your hyperplane, suppose your points are x1, ..., xp. Form a matrix A with p-1 rows and p columns as follows. The rows of p are xi-x1, laid out as rows vectors, for all i>1 (there are only p-1 of them). If your p points are not "collinear" as you say (they need to be affinely independent), then matrix A will have rank p-1, and a nullspace dimension of 1. The one vector in the nullspace is the normal vector of the hyperplane. Once you find it (call it n), then p = n.x1. In order to find the nullspace of a matrix, you can use a QR decomposition (see here for details).