De Rerum Natura

Mathematical objects have dimensions associated with them. The temperature outside my house is measured as a single number or scalar. It is a one-dimensional quantity. This fact can be observed in how mercury thermometers are built: they are a long tube or line. Thermometers are never built as squares.

The position of house in a city is an example of a two dimensional quantity. It requires two numbers to specify and is therefore two-dimensional. This fact is obvious in that maps of cities are usually printed in a sheet of paper not a very thin strip of paper. The position of the house is expressed as a vector. This vector can be expressed as an ordered series of numbers of the form $(v_1,v_2,\dots ,v_n)$. Another way to represent the vector is just with the single symbol $v_i$. You tell me the value of $i$ and I go down the list and read off the component $v_i$.

Generalizing further, we can have matrices like the precision error covariance matrix I have been going on and on about all these months. This matrix can be represented by the symbol $m_{i,j}$. You now have to tell me two numbers, $i$ ad $j$, for me to read off the correct entry in the matrix.

We can keep playing this game forever. It is possible to invent mathematical quantities of the form $m_{i,j,k}$. Three “indices” need to be specified to read off an entry. You can think of this as a cube of numbers.

Precision error covariance matrices can also be generalized to precision error tensors. Instead of just asking how are the errors between two models correlated, we can ask how are the errors of three models correlated. We can have a cube of cross-correlations between the different model errors!

I submitted my paper on autonomous precision error estimation in 3-D models to the 2008 International Conference on Machine Learning yesterday. One week early, too, a first for me! The format for the paper is the standard double column format and this makes it very hard to have complex equations in the paper. One mathematical object that is hard to display are the covariance matrices for the DEM errors that I keep talking about in these posts. These are nxn matrices of real numbers. One particular example I use comes from images of a desert terrain in the Twenty-Nine Palms area in California. We have four photographs and can therefore produce $12=4*3$ DEMs. Because of mistakes, two of the DEMs have to be dropped so I end up with 10 DEMs. The resulting covariance matrices are then 10×10 matrices — a hard thing to display in the double-column format since now you have to present 10 numbers in row. So I have hit upon a simple graphical way to present them that saves space but also ends up being more informative to the reader (or me) about the structure of the matrix.

The idea is to turn the 10×10 matrix into a 10×10 pixel image. Each pixel is now a shade of gray. The highest value in the matrix gets the darkest shade, the lowest gets the lighest. Here is an example that illustrates our correlated-pair error model The only terms that are “turned on” are those along the diagonal. In contrast, here is the covariance matrix when you do ${\ell }_1$-minimization and do not assume beforehand that certain DEMs are uncorrelated with each other. So the correlated-pair error model is close to the actual covariances but we see that there are some cross-correlations off the diagonal that are on, albeit weaker than those on the block-diagonal defined by the asymmetric DEM pairs.

I apologize for the strange layout of the mages relative to the text of this post but my WordPress instalation does not save changes that I make to the img tag to identify it as requiring it to have text flow around it.
In any case, I hope this illustration makes clear some of the more abstract ideas I have been discussing about errors in DEMs.