1
$\begingroup$

I have previously studied the problem of defining tolerance limits for normal distributions for a given small sample of observations.

Now I would like to take into consideration the fact that the observations are not only just a few but they are also not independent, since they are spatially correlated.

But I still need to evaluate a tolerance interval.

I know that this problem is very complicated, can anyone give me suggestions on where to look to solve such kind of problems? I was looking into the theory of random fields, but I'm starting to feel that that is the wrong direction...

EDIT: I'll try to be more clear:

a) Let's say I have a surface made of a given material.

b) Let's also say that this material has a property X that is not constant and it changes from point to point on the surface in a continuous way.

c) I can take a small number of samples (10-20 max) from the surface and measure the property X.

d) I can't take the samples where I would like to, because of practical reasons. So I may have clusters of samples on certain locations whereas on others no samples at all. Neighbouring samples may have a strong spatial correlation.

e) Now I want to evaluate what is the value of X under which I expect to find no more than 10% of the population (values assumed by X on the surface) with, say, 95% confidence, given the observations I have measured.

PS. I don't even know if this problem can actually be solved

$\endgroup$

1 Answer 1

0
$\begingroup$

This was meant to be a comment, but it is too long.

I'm not entirely sure what you want to accomplish but maybe a multivariate normal distribution $y \sim N(X\beta, \Sigma)$ can be helpful. Here $y$ is a vector of observations ($y(s_1), y(s_2), y(s_3),..., y(s_n)$) where $s_i$ represents a spatial location. $X$ are independent variables and $\Sigma$ is the covariance matrix that describes the spatial relationship between points.

This is closely related to Gaussian Processes with the following predictive distribution:

$y_*|X_*, X, y \sim N(K(X_*, X)K(X,X)^{-1}y, K(X_*,X_*)-K(X_*,X)K(X,X)^{-1}K(X,X_*))$

where $y_*$ are predictions to be sampled, $K$ is a covariance function and $X$ are covariates for your predictions. This gives you confidence intervals for free due to the covariance matrix of this multivariate normal distribution.

The canonical reference for Gaussian Processes is Rasmussen - Gaussian Processes for Machine Learning.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.