Help! Matrix algebra people?
Oct. 1st, 2007 11:38 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
eve_primeI need help, please, to figure out why my data matrix is not "positive definite."
I'm using SPSS to do principal axis factoring (PAF). I have two data sets, each of which is a matrix with the same set of about 140 variables and people's responses to them. One data set has more than 300 responses, and the syntax to do the PAF works fine. The other data set has 96 responses, and (using the same syntax file) PAF won't work because the data matrix is not positive definite.
My first guess was that it won't work if there are more variables than cases. My advisor says that's not it. So now I'm trying to narrow down the problem.
The 140 variables can be broken down into, I think, 11 themes. Doing a PAF on each theme separately works okay, so I know the problem isn't in the data, it's in the combination of variables.
One reason that I wouldn't have a positive definite matrix is if one variable is a linear combination of other variables. (Note that response values range from 0 to 4). So my next thought was, narrow down the combinations of variables I'm using in my matrix until I have the smallest set of variables for which I can't get a positive definite matrix.
Okay, if I designate each theme as A, B, C, etc., and the variables then as A1, A2, etc., I can report that Matrix ABDEF fails (can't get a positive matrix). Therefore that's my starting point. However, dropping any theme from that matrix gets it to work, indicating that if it's a linear combination issue, then there are at least five variables involved, or at least one from each of those five themes.
So now I'm dropping individual variables within that set of five themes, trying to get the failing matrix as small as possible. I have spent four hours on this project.
However! ABDEF fails if I remove A1, but if I also remove A2, B1, D1, E1, or F1, it works. If I add A1 back in and remove B1, it fails.
And guess what? That is 97 variables! I bet if I remove one more at random the thing will work again!
Update - I had misunderstood my advisor. He said that principal components analysis works in this case. He's right; it does. Now I have to re-learn what on earth that is.
I'm using SPSS to do principal axis factoring (PAF). I have two data sets, each of which is a matrix with the same set of about 140 variables and people's responses to them. One data set has more than 300 responses, and the syntax to do the PAF works fine. The other data set has 96 responses, and (using the same syntax file) PAF won't work because the data matrix is not positive definite.
My first guess was that it won't work if there are more variables than cases. My advisor says that's not it. So now I'm trying to narrow down the problem.
The 140 variables can be broken down into, I think, 11 themes. Doing a PAF on each theme separately works okay, so I know the problem isn't in the data, it's in the combination of variables.
One reason that I wouldn't have a positive definite matrix is if one variable is a linear combination of other variables. (Note that response values range from 0 to 4). So my next thought was, narrow down the combinations of variables I'm using in my matrix until I have the smallest set of variables for which I can't get a positive definite matrix.
Okay, if I designate each theme as A, B, C, etc., and the variables then as A1, A2, etc., I can report that Matrix ABDEF fails (can't get a positive matrix). Therefore that's my starting point. However, dropping any theme from that matrix gets it to work, indicating that if it's a linear combination issue, then there are at least five variables involved, or at least one from each of those five themes.
So now I'm dropping individual variables within that set of five themes, trying to get the failing matrix as small as possible. I have spent four hours on this project.
However! ABDEF fails if I remove A1, but if I also remove A2, B1, D1, E1, or F1, it works. If I add A1 back in and remove B1, it fails.
And guess what? That is 97 variables! I bet if I remove one more at random the thing will work again!
Update - I had misunderstood my advisor. He said that principal components analysis works in this case. He's right; it does. Now I have to re-learn what on earth that is.