# Edge correction of Ripley's K-function for two 1D point processes?

I am just beginng an investigation involving characterizing the dependence between two 1D stochastic point processes $x$, $y$. The natural approach seems to involve Ripley's K-function:

$$K(t) = \frac{T}{n_xn_y} \sum_{i=1}^{n_x} \sum_{j=1}^{n_y} w(x_i,y_j) I[d(x_i,y_j)<t]$$

where $n_x$ is the number of observations in $x$ and $n_y$ is the number of observations in $y$ across the interval $T$. Deviation from $K(t)=t$ is an indication of correlation between the two point processes.

However, it's not clear to me how to estimate the edge correction $w(x_i,y_j)$ the 1D case. Some papers refer to Hani Doss' 1989 papers, but in the JSTOR paper he explicity states that ' this edge correction will not concern us.' FWIW-I am currently using a correction weight of 2, but intuitively that seems excessive in my case.

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I am confused by the apparent contradiction between "1D" (one-dimensional?) and "bivariate" (two-dimensional, of course). Would you mind explaining what you mean by these? –  whuber Aug 22 '12 at 17:05
@whuber: there seems to be various ways to refer this situation (bivariate, cross, multivariate, etc.) and, with apologies, I tried to hit them all. Hopefully the edit will clear up the confusion. I believe you have referred to it as 'cross-K' in a slightly related posting. –  Aengus Aug 22 '12 at 19:35
You may be interested in some more recent work that looks at Ripley's K for point processes on 1-D road networks. See Ang et al. 2011 for one example. They have implemented their corrections in the spatstat package for R. –  Andy W Aug 23 '12 at 12:34
@Andy: Thanks much for the reference. I had not seen this; there's some good ideas in here and, while not directly applicable to the immediate problem, will apply to some related work. –  Aengus Aug 23 '12 at 13:39