Group Theory – Direct Products

One of the ways of producing (potentially) new groups from old is called the direct product.Let me show you the direct product of two groups by example, first.

Let me invoke the abstract algebra package and set it to “groups”.

C2 x C2

I want two copies of C2, the cyclic group of order 2…

… and I am going to ask for their “direct product”.

What did we get?

First of all, we see that the elements of the direct product are just ordered pairs – the first component from H, the second component from K.

Regression 1: Archer Daniel Midlands (polynomials) – 1

Now I want to illustrate another problem, this time with the powers of x. The following comes from Draper & Smith, p. 463, Archer Daniel Midlands data; it may be in a file, but – with only 8 observations – it was easier to type the data in. Heck, I didn’t even look to see if it was all in some file somewhere.

raw data

I have chosen to divide the years by 1000; in the next post I will do something else.

The output of the following command is the given y values… I typed integers and then divided by 100 once rather than type decimal points.

Regression 1: eliminating multicollinearity from the Hald data

I can eliminate the multicollinearity from the Hald dataset. I’ve seen it said that this is impossible. Nevertheless I conjecture that we can always do this – provided the data is not linearly dependent. (I expect orthogonalization to fail precisely when X’X is not invertible, and to be uncertain when X’X is on the edge of being not invertible.)

The challenge of multicollinearity is that it is a continuum, not usually a yes/no condition. Even exact linear dependence – which is yes/no in theory – can be ambiguous on a computer. In theory we either have linear dependence or linear independence. In practice, we may have approximate linear dependence, i.e. multicollinearity – but in theory approximate linear dependence is still linear independence.

But if approximate linear dependence is a continuum then it is also a continuum of linear independence.

So what’s the extreme form of linear independence?

Orthogonal.

What happens if we orthogonalize our data?

The procedure isn’t complicated: use the Gram-Schmidt algorithm – on the design matrix. Let me empahsize that: use the design matrix, which includes the columns of 1s. (We will also, in a separate calculation, see what happens if we do not include the vector of 1s.)

Here we go….