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Suppose one has $N$ sensors labeled $1,2,3\dots$ arranged in an equally spaced grid.

For each point you have a measurement that can vary in time $X^{i}(t), i \in {1,2,3\dots}$. The point $A$ needs to be classified with ON/OFF labels. If, at time $t_1$, $X^i(t)$ varies, it might also influence the measurements of the neighbor points in the grid according to an unknown distribution.

Also, consider I have not got a training set; anyway, I might force a situation in which all the points are classified as OFF and the measurements in that case would be close(d) together, i.e. all of them inside a sphere.

When in OFF state, the point measurement might vary according to a gaussian noise if state OFF holds (the sphere radius might be related to the std deviation of the noise); otherwise, if a threshold is crossed the state might be ON, i.e. the measurement goes outside the sphere.

Anyway, suppose all points are OFF and they all detect this transition to ON: in that case the next state should be OFF, since this must be considered a case where all measurements have been biased by a noise affecting all the grid, i.e. the sphere containing all the measurements has shifted its position. This requires the algorithm to be resiliant to a common noise which might affect all measurements.

Based on that, a threshold approach may be considered as a solution, but there may be a case in which all points are labeled as ON but the sphere which determines the OFF state has moved, and in that case, it would be impossible to return to the OFF state based on the following measurements.

A toggling approach might be a solution for the last case: when the measurements differs over a certain threshold the state is toggled. But, here is another problem: a point $B$ which is closed to $A$ may change its current state to ON, and this could affect the measurement for $A$, causing the state of $A$ to toggle. This means, I need something which filters out the fact that measurements from neighbours are correlated.

Question: Finally, depending on the previous label for point $A$, the labels and measurements of the neighbor points, and the new and old measurement at point $A$, is there any smart algorithm I can use to classify the new status for point $A$?

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  • $\begingroup$ I tried to format this so it is somewhat clearer what you are asking, but you might want to take a look at this further, see if you can remove some redundant background information, or phrase things more concisely, it might increase the chances that people understand quickly what you are looking for and thus know how to help you. Also, you might want to start with as simple of a model as possible, before adding additional properties (like Gaussian noise). $\endgroup$ – Chill2Macht Mar 20 '17 at 20:12
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I do not quite understand as you said that you "have not got a training set", but ... you have "previous label for point A, the labels and measurements of the neighbor points, and the new and old measurement at point A".

If you have the a set of training points and its labels, I guess SVM is a suitable technique for solving this binary classification.

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