!!!UPDATE!!!Since this post is still getting a lot of views, some of you might be interested in the outcomes of my experiments with the cross K-function. I used the function in 2 recent papers. Links to the articles are found on the Publications page. Juhász, L. and Hochmair, H. H. (2017). Where to catch ‘em all? – a geographic analysis of Pokémon Go locations. Geo-spatial Information Science. 20 (3): pp. 241-251 Hochmair, H.H., Juhász, L., and Cvetojevic, S. (2018). Data Quality of Points of Interest in Selected Mapping and Social Media Platforms. Kiefer P., Huang H., Van de Weghe N., Raubal M. (Eds.) Progress in Location Based Services 2018. LBS 2018. Lecture Notes in Geoinformation and Cartography (pp. 293-313) Berlin: Springer.

One of the research papers I’ve submitted recently (yes, about Pokémons!) dealt with spatial point pattern analysis. Visually it seemed that two of my point sets prefer to cluster around each other, in other words I suspected that Pokéstops have a preference of being close to Pokémon Gyms. Check the map below to see what I mean. Pokémon locations (cyan dots) are all over the place as opposed to Pokéstops (orange) that almost exclusively appear to be in the proximity of gyms (red).

To confirm what’s obvious from the map, I used the bivariate version or Ripley’s K-function (a.k.a. the cross-K function) that can help us characterize two point patterns. As it turns out, it’s not as easy to interpret as I though it would be (at least with real world data) and I was trying to get my head around it for quite some time. As a result, I came up with a simple interactive visualization of this function to illustrate what it really means. If you’re anything like me and try to understand your stats instead of just reporting the results, you might want to read on more for some musings about the cross K-function.

### Ripley’s K and cross-K functions

In spatial statistics, Ripley’s K-function can be used to characterize point patterns (i.e. if they’re clustered, dispersed or randomly distributed). Basically it tells you whether you observe more/less points within a given radius than what would be expected under complete spatial randomness. The bivariate version of the same function does the same for marked point patterns (or for two point patterns). It reports the number of type *j* events within a given radius of type *i* events. It is calculated as follows:

(see Dixon 2002 for details). Simply put, we’re inserting circles with growing radius on type *i* events (~controls) and check how many type *j* events (cases) are nearby. Theoretically, there could be two different cross K-functions but *K _{ij} = K_{ji}* therefore, it is enough to compute

*K*If events are a result of a random spatial process (i.e. points are drawn from a homogeneous Poisson distribution), we can simplify the K function to

_{ij. }*K(r) = πr*. This theoretical function represents complete spatial randomness and can be used to compare our observed cross-K function to.

^{2}### Making sense of the function

And now the fun part begins. First step is looking at the observed cross K pattern and the theoretical Poission K curve (under CSR). These two curves allow you to assess the relationship between type *j* (remember Pokéstops?) and *i* (gym) events. An observed cross K curve well above the theoretical K curve means that type *j* points prefer to cluster around type *i* points. In simple words (referring back to the Pokémon example) this scenario means that Pokéstops are attracted to Pokémon gyms. Or we can further tweak the words and say type *j* events are closer to type *i* events than it would be expected under complete spatial randomness. Either interpretation works. Similarly, if the observed cross K-function is below the the Poisson-K, type *j* events prefer to disadvantage type *i* events (~Pokéstops would be further from gyms). Unfortunately for the bivariate case of K function, it is not easy to test whether the deviation from the theoretical curve is meaningful or not (like if the the curves differ significantly) so in this step it’s enough to rely on your instincts. Just get a sense of the the observed pattern, keep in mind what would be expected under CSR and you can interpret your results easily.

### Testing statistical significance

We were taught to report statistical significance as this is how we can support our findings. The original question I asked was whether Pokéstops and Pokémon Gyms are *SIMILARLY* clustered or not. Statistical inference of difference can be tested with a Monte Carlo simulation with random labeling. Namely, we can randomly assign points as type *i* and *j* (keeping the original rations) and compute the same cross K-function. The simulation mean and the established simulation envelopes tells us whether the observed between type pattern is statistically significant or not. An observed curve within the confidence envelopes means that no matter how we group your points into categories, the pattern we identified in the previous step (by checking on the observed and theoretical values) doesn’t change when randomly assigning events into categories.

### Example patterns

Below is an interactive visualization of the these things that helped me understand how these functions works. I started off having 25 type j* *(blue) and 5 type *i* (red) events as a result of a random spatial process within a 10 unit radius circle. The observed cross K function (grey line) and the theoretical Poission K-function (red dashed line) are shown in the second plot. The third plot shows the results of a Monte Carlo simulation (99 steps) of random labeling of events (simulation mean – red dashed line; confidence envelope – yellow area). By clicking on the radio buttons, you can change the point pattern into a few predefined scenarios and assess how that affects the cross K function. It helped me a great deal to understand the principles. I hope that it will help you as well to make sense of the cross K function and you can use this tool to interpret your own results.

*** Select a point pattern***

#### A simple interpretation of Point Pattern 2

Type* i* events (blue points) seem to cluster around type *j* events (red). It appears that the overall point pattern is clustered (could be confirmed with an univariate K-function). Calculating the cross K-function (2nd plot) reveals that there’s an interaction between types as the observed curve is above the Poission-curve (what would be expected under randomness). In other terms, it means blue points tend to be attracted to red points as we observe more blue points in the proximity of red ones than if their positions were random. For larger distances (> 4) the pattern is closer to random but that’s just the effect of their spatial distribution (the study area is small and using larger radius eventually means selecting most of the points).

As for the Monte Carlo simulation, the grey line running close to the simulated mean would indicate that these two point patterns cluster similarly. However, we can even tell more! For shorter distances, it appears that blue points are even closer to red ones (i.e. observed cross K above the high simulation envelope) than it is the case when randomly re-labeling them so their attraction is even stronger than expected.

To learn more about the cross K-function, check out this material:

Dixon, Philip M. 2002. “Ripley’s K function.” In Encyclopedia of Environmetrics 1796– 803. Wiley Online Library.

The computations and simulation steps in this post were conducted in R with the spatstat package. The visualization is D3.js. To reproduce this experiment, check out the data and code (even the visualization) on my GitHub.

MABEL KATSOwhat a simple and straight forward explanation, thank you.

jleventePost authorGlad you like it. I might write more of these posts later.

M KThanks for this post man.

Just one question, do you know who developed the theory cross-K function? The literature I have seen isn’t so clear. I need to know if there is proper documentation on the cross-K. I’ve read Dixon’s paper and its also a bit unclear. It might be Diggle but I am not too certain too.

jleventePost authorIt was Ripley, actually. In the same original paper he described the K-function, he also mentioned that it’s possible to investigate interactions between two point sets and basically defines the cross K-function. See pages 264-265 in [1]. He doesn’t call it “cross K-function” but he also doesn’t call the original function “Ripley’s K” for obvious reasons, right? :)

[1]: Ripley, B.D., 1976. The second-order analysis of stationary point processes. Journal of applied probability, 13(2), pp.255-266.

PDF available e.g. from here: https://www.jstor.org/stable/3212829?seq=1#page_scan_tab_contents

jleventePost authorIf you’re looking for more recent examples and applications, you might want to take a look at the papers below, too. The 2nd one is not open access, so shoot me an email if you’re locked out and want to read it.

Juhász, L. and Hochmair, H. H. (2017). Where to catch ‘em all? – a geographic analysis of Pokémon Go locations. Geo-spatial Information Science. 20 (3): pp. 241-251 (http://dx.doi.org/10.1080/10095020.2017.1368200)

Hochmair, H.H., Juhász, L., and Cvetojevic, S. (2018). Data Quality of Points of Interest in Selected Mapping and Social Media Platforms. Kiefer P., Huang H., Van de Weghe N., Raubal M. (Eds.) Progress in Location Based Services 2018. LBS 2018. Lecture Notes in Geoinformation and Cartography (pp. 293-313) Berlin: Springer (http://dx.doi.org/10.1007/978-3-319-71470-7_15)

M KThanks for getting back to me and sorry for the late response.

I’ve seen that paper but the bivariate K is not as detailed as I’d like but no worries, I guess I’ll just take what I can from all these different papers.

Thanks again, and for the article recommendations too. Much appreciated.

KeithVery helpful explanation!

jleventePost authorAwesome, glad you like it!

Laura B.Hi there,

I’m looking at your data.csv and I’m confused why each A or B points has four different x,y coordinates. Thanks!

jleventePost authorHi Laura,

The four different sets of coordinates denote the 4 different cases I have in the plots (i.e. 4 different point patterns). So if you look at x_1 and y_1 (or geom_1.x and geom_1.y in data.json), they refer to Point Pattern 1. Similarly, point pattern 2 will be x_2 and y_2, etc. It just made no sense to use different files for each point pattern since I am using the same number of points.

Hope this helps.

Cheers,

Levente

Michael SinclairHi Levente,

I have to say, this explanation and the website in general are excellent. Your example has helped me grasp this more fully. That said, I assume the significant testing using the Monti Carlo simulation does not return a measure of significance and the results can only be interperated based on line fit in relation to the envelope?

I am doing similar research with social media in environmental research. Until now I have not come accross your papers but I downloaded a few of them now and am having a read. They are of interest to my own PhD which seems to share some similarities with the work you are doing.

This is an open access systematic review I published this year on the topic I am dealing with, it may be of interest to you. https://www.sciencedirect.com/science/article/pii/S0959378018309920

All the best and keep the content flowing!

jleventePost authorHey Michael,

I’m glad to hear that you found my post useful. You can determine empirical significance levels with Monte-Carlo simulation by comparing the observed value to the upper/lower tails of the null distribution. The easiest way to do this is with the pointwise approach. For a given r you can sort simulated values and take the mth lowest and mth highest values. For example m=1 would compare the observed value with the max and min simulated values (at r distance). You reject the null hypothesis if the observed value is outside the simulated “envelope”. You then can express p = (2 * m)/(N + 1) where N is the number of simulations – for a given r.

Another possibility is to calculate the max absolute difference between the observed values and the theoretical values for each simulation over a range of r. This gives you a bunch of deviation values which you can sort as above to define your envelopes. Here p = m / (1 + N) where N is the number of simulations and m is the rank. This is for any r over that range you used.

Hope it helps. Good luck with your research.

Cheers,

Levente

SylvainExcellent and helpful post, thank you Levente. Do you think it is possible to compare the cross-K across time, if the count of points in the area changes according to some independent production process (e.g. changes in the number of Pokémon players)? The goal would be to analyze whether clustering increases/decreases across time, independently from this production process. Presumably, the increase in point count mechanically reduces the distance between them, complicating the interpretation of clustering. Might it be possible by somehow normalizing the cross-K according to point count?

MSGamaHow we can apply this method for different cities with different populations

GuillemThank you SO much! This is the best explanation and visualization I have found on the Internet. It really helps me with my data about the spatial ecology of plants and their biotic interactions with insects (used as marks).