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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.

Gabriel
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Aengus
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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
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    @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](http://dx.doi.org/10.1111/j.1467-9469.2011.00752.x) for one example. They have implemented their corrections in the [spatstat](http://cran.r-project.org/web/packages/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
  • The best edge correction for 1D Ripley's functions seems to be to weight by dividing the counted points by the proportion of the interval that was actually sampled. For a 2t interval around i that gets out of the transect start (zero) it will be w=(i+t)/2t; on the other side, w=(t-i+[top limit])/2t. So this "inflates" the actual counted points within 2t to compensate for the "uncounted" (though expected under stationarity) that were out of bounds [Sorry, I can't find the proper reference now, but will add it if it eventually finds me again] – FairMiles Sep 02 '15 at 16:06
  • As promised: the reference for my previous comment is Fortin, M-J & MRT Dale (2005) Spatial analysis: a guide for ecologists (pp.75-79) – FairMiles Mar 02 '20 at 16:25

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Have you seen Gavin's K1D code?

In his vignette, synchronicity is $K(t) > 2t$ and asynchrony between $2t$ and $0$. I believe the $2t$ arises from the evaluation of the pair in both ways which is necessary as Gavin explains that sometimes the edge correction $w(t_i,t_j)\neq w(t_j,t_i)$.

Timothy Pollington
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