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


1 Answer 1


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.


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