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Suppose that there are two two-dimensional maps. For simplification, let's say one map is the temperatures in the 48 continental U. S. states, and the other map is the corresponding humidities. The data of temperature and humidity are collected at one specific moment at some equally-spaced (e.g., 1000) locations. If I directly calculate the correlation between the two maps, the coefficient would be inflated because the maps are spatially correlated? Suppose that I know the full width at half maximum (FWHM) that characterizes the spatial correlation, is there a way that I can adjust the correlation coefficient or its degrees of freedom?

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    $\begingroup$ It would be nice to have a clearer idea of what you mean by "inflated." This suggests a comparison, so one is moved to ask, inflated compared to what? If you mean the correlation coefficient of the distribution of all temperatures and humidities at that moment, we should pause for a moment to consider what relationship the sample correlation would have to it and why--if at all--it could be expected to be "inflated," and how--if at all--that would be related to "spatial correlation." I believe that if you clarify these points, you might be able to answer this question yourself. $\endgroup$
    – whuber
    Commented Dec 5, 2012 at 22:35
  • $\begingroup$ @whuber: Thanks a lot for the comment! The suspicion of inflation is not based on any empirical information (I don't have any), but comes from my intuition. In other words, my concern is that the data for temperature/humidity are not sampled independently, and that's why I thought the spatial correlation might provide a way to correct for any inflation, if existing. $\endgroup$
    – bluepole
    Commented Dec 6, 2012 at 16:37
  • $\begingroup$ In what sense are they not sampled "independently"? Consider the non-spatial situation where two related variables are observed: each measurement produces an ordered pair of numbers. How does your situation differ from that? Here's another way to think about it: imagine that $10^{1000}$ pairs of (temp, humidity) values could be recorded, effectively making an exhaustive census of the two fields. Taking a simple random sample of these pairs obviously has no problems with spatial correlation. Your sample does not differ a whole lot from that situation. $\endgroup$
    – whuber
    Commented Dec 6, 2012 at 18:35
  • $\begingroup$ Have you looked at Spacial Dynamic Factor Analysis? faculty.chicagobooth.edu/hedibert.lopes/scientific/pdf/… It appears to have been designed to solve exactly this problem. $\endgroup$
    – Mimshot
    Commented Dec 6, 2012 at 18:53
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    $\begingroup$ @whuber: You're right that the correlation coefficient is unbiased, but it's the significance of the correlation that I'm interested. And yes I thought about the adjustment about the DF, but just couldn't figure out how to make a proper adjustment for the DF based on the spatial correlation such as FWHM. Sorry I didn't bring up the real data context, but the essential issue in terms of spatial correlation remains the same. I hope that the temperature/humidity analogy captures the problem I'm facing. $\endgroup$
    – bluepole
    Commented Dec 6, 2012 at 22:17

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What does correlation between the maps mean? On the other hand, if you mean the correlation between the temperature and the humidity, we then have to ask what the population is that you are interested in and what the context of the question is. For example, the correlation between humidity and temperature in one location over time may differ from that in another location. The correlation between humidity and temperature over the entire geography of the US (assuming you can take a random sample from the US) during the summer may differ from that during the winter.

To even take a stab at answer those questions requires knowing why you are asking them.

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