I have a Cartesian grid (100x100) in which some of the points are known (30 out of 10,000) and the rest are unknown. I want to use the known points and estimate the other cells. Is there any philosophies behind the way that I must use the known points to estimate the unknown cells? In other words, does it matter that where I should start the estimation? Generally speaking, I personally prefer to start from the locations where I have the maximum amount of the information rather than a random location that probably I might have one known data. However, there is another school says that I should start from a random location in order to prevent any bias. I am more interested to know your ideas about any differences between these methods. Thanks !

  • $\begingroup$ What is the relationship between the data that you can reconstruct/estimate the uknown from the known data? $\endgroup$ – nicoguaro Apr 10 '15 at 19:41

The important question is "What do you know how the values at points where you did not measure behave, relative to surrounding points?". In other words, what prior knowledge do you have about how the values behave?

If you know that values at each point are completely independent of their neighbors, but that they are typically small, then the best you can do is to set unknown values to a small value. If you know that the value at each point is typically correlated with the values of its neighbors, then you will likely want to choose the value at unknown points to be "similar" to the value at known or inferred neighbors, where "similar" depends on the correlation statistics you know.

I would suggest you read the book "Statistical Inverse Probles" by Kaipio and Somersalo that explains the process and the thought that goes into modeling your prior knowledge in great detail, with well chosen examples.


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