For example, suppose I have an observation network consisting of 10,000 gauges. It is easy to calculate the joint (Shannon) entropy of these 10,000 gauges, if these points are all present at the same time.
Now, if each of the single gauge within this network has a very low probability of presence (say p=0.01), which means that at each time we have an network consisting of about 100 gauges, and the ID of these 100 gauges is changing from time to time. If I know the probability of presence (p) of any of these 10,000 gauges, how would I calculate the (expected) joint entropy of this STOCHASTIC network?
Just a clarifying question for you. When you refer to "presence" are you referring to the existence of an observation at a location for a specific moment in time (e.g. the stage reading of a stream)? Or are you actually referring to the presence of some physical object (e.g. a species of fish) or a phenomena (e.g. a lightning strike) at a particular moment in time?
The distinction here being that in the case of the existence of an observation, one could infer that what was observed on some occasions could have been potentially observed on the other occasions, if only someone had bothered to make the observation (e.g. the stream had a stage, but it was unobserved/unrecorded). In the case of actual presence, one might infer that the species of fish was absent from the location that month, or that a lightning strike did not occur on that day.
Seems like a very interesting problem, and one that can be examined through a simulation experiment. One thing that would be helpful is knowing the arrangement of the observational network. For simplicity I will assume equally spaced observations in form of a grid (100X100). Within this grid you have 100 observations at any given time, each equally weighted to occur and therefore no location preference, thus preserving the randomness of the selection. Given enough time all locations will register at least one observation. If this time is short enough that the environment remain static then your answer is simply that of the easiest case--the joint entropy for all locations.
You can evaluate whether the environment is truly static by evaluating the variance of observations for locations with more than one observation throughout the length of time you conduct your experiment. If you have a map of cases with more than one observation then you can evaluate the spatial variability of your network to a good degree. The advantage of conducting these numerical experiments is that you can compare a case where all observations are available to only 100 observations, change the probability so that now you have 1000 instead of 100 at a given time, or change the frequency at which the observations switch to the next set of 100. Using these combination of techniques will allow you to calculate, to a fair degree, the joint entropy of a changing observational network with respect to time.
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