# Inferring from a combination of uncertain and certain data

I am trying to estimate the surface (isochrone), zi(x) for which T(x,z)=0 from noisy measurements of T(x,z) everywhere and 3 almost noise free control points:

Instead of using the T(x,z) data directly I first estimate the gradient: $T_x$ and $T_z$ by finite differences on T(x,z). I now have two pde's to solve:

$\frac{\partial}{\partial x}T=T_x, \frac{\partial}{\partial z}T=T_z$

Using finite differences again I express the derivation operators as matrices:

$\ D_x T=T_x, D_z T=T_z$

where $T_x$ and $T_z$ are now column vectors made by concatenating all the columns of their respective matrices vertically, and $D_x$ and $D_z$ are matrices that when multiplied with the T vector gives the partial derivative with respect to x and z.

In addition to these pde's I also have equations for the 3 control points:

$T(xc_i,zc_i)=0$

(Weighted) Least Squares Error (LSE) is a framework that allows me to mix certain and uncertain data.

My (weighted) linear equations are now:

$\ D_x T=T_x$

$\ D_z T=T_z$

$\ \alpha T(xc1,zc1) = 0$

$\ \alpha T(xc2,zc2) = 0$

$\ \alpha T(xc3,zc3) = 0$

where I have introduced $\ \alpha$: a parameter that decides how much the control points are honored relative to the measurements of T(x,z).

Is there some way to estimate $\alpha$ or alternatively express it as a more intuitive expression?

So far I have just tuned this parameter and compared the offsets from the control points iteratively. It would be good to have some other way to set this, e.g. max allowed z offset from control point to resulting isochrone.

Do statisticians have some other way than LSE of combining certain and uncertain data?

Is Bayes such a method? If so what is the a priori in my example?

-
Hi, Andy. I've tried to do some very mild editing of this post, but to me it still reads like it could use a second pass from you. It is a little hard to tell what you are asking and also a little difficult to read your notation. Looking at the help on MathJax $\LaTeX$ formatting (or peeking at some other examples on the site) may help with the second issue. :) –  cardinal Jul 15 '12 at 21:49
I think what you are after is weighted least squares estimation, which takes into account the individual errors of the data points. See for example en.wikipedia.org/wiki/Least_squares#Weighted_least_squares. This corresponds to what a Bayesian would do, given certain assumptions. I haven't posted this as an answer because the question isn't sufficiently clear, so I don't know if I'm answering the right question. If this is what you are after I can flesh this out to a full answer (or someone else might). –  Bogdanovist Jul 15 '12 at 22:37
Ahh I see, I re-read the question and understand a bit better what you want. You are already doing weighted least squares, but don't know what the weights should be. In that case there is no real answer. You assert that these points have a much smaller error than the others, but then want the data to tell you by how much. In the situation you describe there is no way to do this. From a Bayesian perspective, you want to apply your prior knowledge that these points are know with greater certainty that the others. But you need to specify that information quantitatively. You can't get it from –  Bogdanovist Jul 15 '12 at 23:26
the data, because by definition the prior knowledge is what you knew before you took the data. –  Bogdanovist Jul 15 '12 at 23:28
@cardinal: thanks for the editing. I also added a bit more text in order to explain more in detail what I am trying to do. –  Andy Jul 16 '12 at 7:40