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Leung, Mei, and Zhang have developed two tests for whether GWR is a better fit than OLS regression. Their paper is here, but it's behind a paywall if you don't have academic access. As for variograms, etc. I know that Bivand, et al. cover tools and mechanisms In their book. I know a pdf of this exists because I have it, but I forgot where I got it from. As ...

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No, they are not the same. Independent stationary increments is related to the index of a sequence (set) of random variables ($X_t-X_x\stackrel{d}{=}X_{t-s}$) while memorylessness relates to the same random variable and the events of interest ($P(X>t+s\vert X>t)=P(X>s)$).

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All Levy processes are Markov processes, a property of stochastic processes distinct, but related to memorylessness as presented in the Wikipedia article you link to, where it refers to a property of distributions. Note that the Markov property is sometimes also (informally) referred to as memorylessness. The Markov property of Levy processes is indeed a ...

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If L(observed) < L(expected), the pattern is more regular than expected, if L(observed) > L(expected) the pattern is clustered

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