topic Re: Ripley's K Confidence Envelope doesn't follow the blue expected line in Spatial Statistics Questions

Ripley's K Confidence Envelope doesn't follow the blue expected line

Mareike_TabeaScheller — Thu, 15 Sep 2011 12:36:07 GMT

Hello!
I'm working for the first time with the Spatial Statistics tools.
I have point data of birds on an island. I want to calculate the K-Function, but the result graph always shows a Confidence Envelope that is under the blue expected line. Why is that? Shouldn't the confidence envelope follow the expected line?
I would be glad if anyone had a suggestion what mistake I might have made here.

Mareike

Re: Ripley's K Confidence Envelope doesn't follow the blue expected line

Mareike_TabeaScheller — Tue, 20 Sep 2011 14:59:55 GMT

I'm working in Iceland, so I use the ISN_2004 Projected Coordinate System. In the results window of the K-function, it always shows that the tool used NAD_1927_to_NAD_1983_NADCON in the field "Geographic transformations". This just appears automatically. Could this be the "Error" here? What are these geographic transformations for and what kind of transformation should I use?
I would be really really glad if someone had a suggestion! :confused:

Re: Ripley's K Confidence Envelope doesn't follow the blue expected line

LaurenRosenshein — Tue, 04 Oct 2011 23:39:11 GMT

Hi Mareike,

I'm really sorry that you're having trouble! Would it be possible for you to attach a copy of the graphical output from K-Function? That would help us start to figure out exactly what's going on.

Hopefully we'll be able to figure this out and get you up and running!

Lauren Rosenshein
Geoprocessing Product Engineer

Re: Ripley's K Confidence Envelope doesn't follow the blue expected line

Mareike_TabeaScheller — Mon, 10 Oct 2011 13:16:24 GMT

Hi!

Thanks for your answer. Here is my graph. I noticed if I switch the Boundary Correction Method (I used "Simulate Outer Boundaries") off, the confidence envelope follows the observed line, which is of course also wrong. If I want to use the other two methods I get an error. So I'm still confused..

Mareike

Re: Ripley's K Confidence Envelope doesn't follow the blue expected line

JeffreyEvans — Mon, 10 Oct 2011 16:24:45 GMT

My guess is that your point process is in fact inhomogeneous (nonstationary). One of the underlying assumptions of point pattern analysis is that your point pattern is representing a stationary random field. Some test of this assumption are closed-space distance, Morishita or Fry plots. The null that your are testing against in the K statistic is a CSR (Complete Spatial Randomness) process. If the underlying random field is not CSR but conditional on the intensity of the measured process the null is invalid and the expected will not follow Gibbs randomization used to generate the simulation envelope. Unfortunately, inhomogeneous PPA statistics are not widley available and are very computational expensive. There is a inhomogeneous K available in Spatstat. As an alternative I would recommend fitting an empirical point process model using covariates. This has the potential of detrending a variable may be conditioning the intensity. An option without covariates is to fit a 2nd order polynomial to detrend the point process. However, this is assuming that the nonstationarity is the result of 1st order spatial variation and in ecological process rarely is this the case.

Re: Ripley's K Confidence Envelope doesn't follow the blue expected line

LaurenScott — Fri, 21 Oct 2011 23:00:33 GMT

Hi Mareike,
Sorry you�??re having trouble with this!
The K Function works by simply counting feature pairs: the tool �??visits�?� each feature in the dataset, selects all features within a specified distance of the target feature, and counts the number of feature pairs among the selected features�?�feature pair counts are accumulated as the tool visits every feature. The distance is then increased and the counting repeated�?� and this process continues however many times you�??ve specified for the Number of Distance Bands parameter.

These accumulated counts (one for each distance) are converted to an index and plotted on a line graph. When your points tend to be clustered, the accumulated counts are higher, and the index falls above the blue diagonal expected line. When the points tend to be dispersed, counts are lower and the index falls below the expected line.

To decide if the clustering or dispersion is significantly different from what you would get if the points were randomly distributed in your study area, the tool uses simulation. The tool randomly pitches your points into your study area 9, 99, or 999 times and for each simulation, it performs the whole distance/counting thing. From all the simulations, it remembers (for each distance) the most clustered index obtained from the random process of pitching your points into the study area, and it remembers the most dispersed index obtained. These extreme values form the confidence envelope, and they show you (given X number of points and the peculiarities of your study area), what is the range of possible indices you can obtain from a random process.

For a weighted K function, the confidence envelope follows the observed line and the simulation process is a bit different than I described above. From the graphic you sent, my guess is you are using the unweighted K Function, but please let me know if I�??ve guessed incorrectly.

For the unweighed K function, if the study area has a very simple shape (circle, rectangle) the confidence envelope will enclose the expected line. When the study area isn�??t simple (there are peninsulas, or you are working with an �??L�?� shape, for example) then the study area itself can force randomly placed features to be far away from each other, so the confidence envelope appears below the expected line (more dispersed).

Okay, so why might someone run the unweighted K function? The K function provides a kind of spatial �??fingerprint�?� of how spatial clustering among your point features changes across multiple scales (across increasing distances). Why is this interesting? Whenever we see clustering in the landscape, we are seeing evidence of underlying spatial processes at work. Statistically significant peaks or dips of the observed index are evidence that spatial processes are operating at the associated spatial scale. Sometimes knowing something about these statistically significant spatial scales provides clues about the underlying processes at work. Comparing the spatial �??fingerprints�?� for two different point datasets within the exact same study area can tell you if their spatial patterns are being influenced by the same or different spatial processes.

Some questions for you: You indicated you are analyzing birds on an island. Are you providing a study area polygon when you run the K function? If so, might that polygon be forcing a structure on the simulations that would explain why the confidence envelope falls below the expected line?

Do the points you have reflect a sample of bird sitings, or do they represent ALL possible data (like ALL bird nests on the island)? Sampled data, especially when the samples might be biased by observer behavior or the sampling scheme, are not good candidates for the K Function�?� there is the risk that you will model observer behavior rather than bird behavior.

You mentioned a projection/transformation warning message or error�?� that sounds like a problem. If possible, I�??m hoping you can send me your data so that we can figure out exactly why you are getting the unexpected results. Please contact me directly at LScott@Esri.com if that might be possible.

Again, I�??m sorry you are having problems with the K Function. I hope this information is helpful to you. If anything is unclear, please contact me or reply here and I will do my very best to clarify.
Lauren

Lauren M Scott, PhD
Esri
Geoprocessing, Spatial Statistics

Re: Ripley's K Confidence Envelope doesn't follow the blue expected line

EllenKersten — Tue, 25 Oct 2011 21:20:30 GMT

I am having the same problem with the confidence intervals. I have a rectangular study area and no projection problems.
Also, in the table output, the ExpectedK values appear to just be the distance thresholds that are evaluated (5, 10, 15, 20, etc.) and are not actually calculated ExpectedK values. Why is this?

Re: Ripley's K Confidence Envelope doesn't follow the blue expected line

LaurenScott — Wed, 26 Oct 2011 16:28:46 GMT

Hi Ellen,
I will look at the data you sent to me. Thank you.
With regard to the Expected K values being exactly equal to the Distance values, that is what you will always get. The reason is because we are using a transformation that converts the Expected K value to be equal to distance.

For more information on this, please see:
Getis, A. Interactive Modeling Using Second-Order Analysis. Environment and Planning A, 16: 173�??183. 1984.

The actual formula for the L(d) transformation is given in:
http://help.arcgis.com/en/arcgisdesktop/10.0/help/index.html#/How_Multi_Distance_Spatial_Cluster_Analysis_Ripley_s_K_function_works/005p0000000s000000/

I do have a bug in for myself to improve the K Function documentation. Sorry for the confusion! (I can't believe I include the L(d) formula and then don't actually tell you what it does... my very bad! So sorry!).

Thank you for your post and for sending me the data. More soon.
Lauren

Lauren M. Scott, PhD
Esri
Geoprocessing, Spatial Statistics

Re: Ripley's K Confidence Envelope doesn't follow the blue expected line

EllenKersten — Wed, 26 Oct 2011 18:18:24 GMT

Thanks. I understand what the L transformation does, so if that is indeed the formula that is being used then it seems like the output.dbf should describe the fields as ExpectedL(Distance) and ObservedL rather than saying K. The .dbf fields differ from the results window box, which labels the output fields Distance and L(d).

My reading on the L transformation suggests that it is used to make graphical interpretation of results more straightforward. In that case, it would be more helpful if the graphical output displayed L(d)-d on the y axis so that the expected line (which from my understanding represents complete spatial randomness) is equal to y=0 rather than a line with a slope of 1. Also, the legend of the graphic should say ExpectedL and ObservedL to be consistent with the formula that is used.

I look forward to hearing your response for why the confidence intervals at some distances do not include the expected value for L (CSR).

Re: Ripley's K Confidence Envelope doesn't follow the blue expected line

Mareike_TabeaScheller — Sat, 29 Oct 2011 14:25:40 GMT

Thank you very much for your answer, Lauren. It is right, that I have a very complicated study area, because it is the polygon of the whole island, so that explains the problems with the confidence envelope! That´s good to know. The points reflect bird sightings. As you suggested, it might be the reason why Ripley´s K shows strange results?! I used the Spatial Autocorrelation tool also to analyse the clustered areas. The results from this tool where right, I suppose, but the Ripleys K-Function showed completely different results. So I used Spatial Autocorrelation in my report in the end. I think I first had some problems with the projection/transformation because I had two coordinate systems in the ArcMap Document, ISN1993 and ISN2004.

Thanks for your help,
Mareike

Re: Ripley's K Confidence Envelope doesn't follow the blue expected line

LaurenceCulot — Tue, 21 Feb 2012 16:18:19 GMT

Thanks. I understand what the L transformation does, so if that is indeed the formula that is being used then it seems like the output.dbf should describe the fields as ExpectedL(Distance) and ObservedL rather than saying K. The .dbf fields differ from the results window box, which labels the output fields Distance and L(d).

My reading on the L transformation suggests that it is used to make graphical interpretation of results more straightforward. In that case, it would be more helpful if the graphical output displayed L(d)-d on the y axis so that the expected line (which from my understanding represents complete spatial randomness) is equal to y=0 rather than a line with a slope of 1. Also, the legend of the graphic should say ExpectedL and ObservedL to be consistent with the formula that is used.

I look forward to hearing your response for why the confidence intervals at some distances do not include the expected value for L (CSR).

I have exactly the same problem with the Ripley's k function confidence envelope. I tried to run the analysis with and without edge correction but always obtained a confidence envelope going below the expected line from some distance. Is there a response/solution for this problem?

Re: Ripley's K Confidence Envelope doesn't follow the blue expected line

LaurenScott — Tue, 06 Mar 2012 15:33:39 GMT

Hi Laurence,
Thanks for your question. There are a couple different reasons that the confidence envelope may not follow the expected line.

1) Differences between weighted and unweighted K function.

When you run K function just on your point features (no weight field), the confidence envelope will tend to follow the Expected Blue line. The confidence envelope is created by taking your point features and (conceptually) throwing them down into your study area (a rectangle if you select minimum enclosing rectangle, otherwise the polygon feature you provide). It repeats this random process of throwing down your points, letting them fall where they may within the study area, for 9, 99, or 999 times. Each time it computes the K function value for all distances and the lower confidence line is derived from the lowest observed L(d) values; the upper confidence line is derived from the largest L(d) values. If the study area is simple (rectangle, circle), the confidence envelope will enclose the expected line (but see #2 and #3 below).

When you run the K function with a Weight Field, the confidence envelope will tend to follow the Observed L(d) line (the red line). In this case the confidence envelope is created by throwing down the feature values (the weights) onto the existing feature locations. The locations themselves remain fixed, only the weights associated with the features are randomly re-distributed for 9, 99, or 999 permutations. Because the spatial distribution of your points restrict where the values can land, the confidence envelope follows the observed L(d) line showing you the range of outcomes given the fixed location of your features.

2) Boundary correction.

The K function works by counting all feature pairs within a given distance of each feature. When you specify NONE for the Boundary Correction method, this counting process is biased near the edges/boundaries. Imagine a circle representing the distance where pairs will be counted. When that circle overlays a point/feature near an edge, a portion of the circle will fall outside the study area where there are no points.... the counts will be smaller because there are fewer pairs within the circle. If there really are no points/features outside the study area, this drop in clustering at increasing distances is valid. If the boundaries are an artifact, you should correct for this undercounting bias by selecting a Boundary Correction method.

3) Study area size.

The K Function is one of two tools in the Spatial Statistics Toolbox that is VERY (VERY) sensitive to study area size (the other tool is Average Nearest Neighbor). Imagine a cluster of points enclosed by a very, very tight study area... with that configuration, the pattern appears dispersed. Now imagine that same cluster of points enclose by a very large study area (so the cluster is at the middle with vast space all around it)... now the points would definitely appear clustered. For a graphic, please see: http://help.arcgis.com/en/arcgisdesktop/10.0/help/index.html#/Multi_Distance_Spatial_Cluster_Analysis_Ripley_s_K_Function/005p0000000m000000/ (About the 12th usage tip that starts: "The k-function statistic is very sensitive to the size of the study area.").

4) Study area shape.

In #1 above, I described how the confidence envelopes are constructed. In essence, features are pitched onto your study area, each feature landing where it may. When you have a very convoluted study area, this can impact where features are allowed to land. Hmmm... okay imagine a square study area with two long skinny arms, two long skinny legs, and a head 🙂 Features that fall into the arms and legs will have fewer neighbors because the study area itself doesn't allow many features to fall into the skinny parts... (does that make sense)?

But this kind of thing can also happen if you elect the Minimum Enclosing Rectangle study area when your features aren't very rectangular. Imagine a set of features randomly distributed into a circle. Then imagine a rectangular study area around it. In the corners of the study area there will be no features. When the K function starts counting pairs near those corners, the pair counts will drop. This can result in a drooping confidence envelope for weighted K function.

I hope this helps. If you still have questions, please feel free to contact me. I am happy to look at your data and evaluate the results to see why you might be seeing the drooping confidence envelope even when you apply a boundary correction method.

Best wishes,
Lauren

Lauren M Scott, PhD
Esri
Geoprocessing, Spatial Statistics
LScott@esri.com

Re: Ripley's K Confidence Envelope doesn't follow the blue expected line

LaurenScott — Tue, 13 Mar 2012 22:14:50 GMT

A bit more... we have identified a bug in the Multi-Distance Spatial Cluster Analysis (Ripley's K Function) tool, for ArcGIS 10.0 only, when you select Simulate Outer Boundary Values for the Boundary Correction method and also elect to Compute a Confidence Envelope (sorry!). In this circumstance, you will notice that the observed L(d) values (the red line on the K Function graph) will have the appropriate (accurate) correction, but that the confidence envelope lines (the gray lines on the graph) will continue to droop because no correction is applied. I'm very sorry that we didn't catch this problem sooner!

Fortunately, since almost all of the tools in the Spatial Statistics toolbox are written using Python, you have our source code and can correct the bug if you so choose. Below are instructions for making the correction (it involves changing one word in the source code). If you are not comfortable making this change, but need this fix, please contact me and I'm happy to send you the corrected Python script file.

To make the correction yourself:
1) Navigate to the Scripts folder and locate the KFunction.py script: <ArcGIS>\Desktop10.0\ArcToolbox\Scripts
2) Create a backup copy of this script file (name the copy something like KFunctionSave.py)... this is just in case something goes wrong.
3) Open KFunction.py with any text editor (like Notepad, for example). Alternatively, from within ArcMap you can also just right click on the K Function tool (via the Catalog or the ArcToolbox pane) and select Edit to access the source code.
4) Locate the following section of code (at about line 517) and make the change indicated below (shown in red):

#### Resolve Simulate Points ####
if self.simulate:
[INDENT]simTable = GAPY.ga_table()
tempN = len(newTable)
simID = self.maxID + 1
for i in xrange(tempN):
[INDENT]row = newTable
id = row[0]
x,y = row[1]
simTable.insert(id, (x,y), 1.0)
if near[id] <= self.stepMax:
[INDENT]nearX, nearY = nearXY[id]
dX = nearX + (nearX - x)
dY = nearY + (nearY - y)
point = (dX, dY)
inside = UTILS.pointInPoly(point, self.studyAreaPoly, tolerance = self.tolerance)
if not inside:
[INDENT]newTable.insert(simID, point, 1.0) <-- change "newTable" to "simTable" on this line: simTable.insert(simID, point, 1.0)
newSimDict[simID] = id
simID += 1
[/INDENT][/INDENT][/INDENT][/INDENT]

Again, my sincere apologies for this error. Please contact me (or contact Tech Support) if you have any questions or concerns.
Lauren

Lauren M Scott, PhD
Esri
Geoprocessing, Spatial Statistics
LScott@Esri.com

Re: Ripley's K Confidence Envelope doesn't follow the blue expected line

VladimirLisitsin — Thu, 19 Apr 2012 06:15:32 GMT

Hi Laurence,
Thanks for your question. There are a couple different reasons that the confidence envelope may not follow the expected line.

1) Differences between weighted and unweighted K function.

When you run K function just on your point features (no weight field), the confidence envelope will tend to follow the Expected Blue line. The confidence envelope is created by taking your point features and (conceptually) throwing them down into your study area (a rectangle if you select minimum enclosing rectangle, otherwise the polygon feature you provide). It repeats this random process of throwing down your points, letting them fall where they may within the study area, for 9, 99, or 999 times. Each time it computes the K function value for all distances and the lower confidence line is derived from the lowest observed L(d) values; the upper confidence line is derived from the largest L(d) values. If the study area is simple (rectangle, circle), the confidence envelope will enclose the expected line (but see #2 and #3 below).

When you run the K function with a Weight Field, the confidence envelope will tend to follow the Observed L(d) line (the red line). In this case the confidence envelope is created by throwing down the feature values (the weights) onto the existing feature locations. The locations themselves remain fixed, only the weights associated with the features are randomly re-distributed for 9, 99, or 999 permutations. Because the spatial distribution of your points restrict where the values can land, the confidence envelope follows the observed L(d) line showing you the range of outcomes given the fixed location of your features.

2) Boundary correction.

The K function works by counting all feature pairs within a given distance of each feature. When you specify NONE for the Boundary Correction method, this counting process is biased near the edges/boundaries. Imagine a circle representing the distance where pairs will be counted. When that circle overlays a point/feature near an edge, a portion of the circle will fall outside the study area where there are no points.... the counts will be smaller because there are fewer pairs within the circle. If there really are no points/features outside the study area, this drop in clustering at increasing distances is valid. If the boundaries are an artifact, you should correct for this undercounting bias by selecting a Boundary Correction method.

3) Study area size.

The K Function is one of two tools in the Spatial Statistics Toolbox that is VERY (VERY) sensitive to study area size (the other tool is Average Nearest Neighbor). Imagine a cluster of points enclosed by a very, very tight study area... with that configuration, the pattern appears dispersed. Now imagine that same cluster of points enclose by a very large study area (so the cluster is at the middle with vast space all around it)... now the points would definitely appear clustered. For a graphic, please see: http://help.arcgis.com/en/arcgisdesktop/10.0/help/index.html#/Multi_Distance_Spatial_Cluster_Analysis_Ripley_s_K_Function/005p0000000m000000/ (About the 12th usage tip that starts: "The k-function statistic is very sensitive to the size of the study area.").

4) Study area shape.

In #1 above, I described how the confidence envelopes are constructed. In essence, features are pitched onto your study area, each feature landing where it may. When you have a very convoluted study area, this can impact where features are allowed to land. Hmmm... okay imagine a square study area with two long skinny arms, two long skinny legs, and a head 🙂 Features that fall into the arms and legs will have fewer neighbors because the study area itself doesn't allow many features to fall into the skinny parts... (does that make sense)?

But this kind of thing can also happen if you elect the Minimum Enclosing Rectangle study area when your features aren't very rectangular. Imagine a set of features randomly distributed into a circle. Then imagine a rectangular study area around it. In the corners of the study area there will be no features. When the K function starts counting pairs near those corners, the pair counts will drop. This can result in a drooping confidence envelope for weighted K function.

I hope this helps. If you still have questions, please feel free to contact me. I am happy to look at your data and evaluate the results to see why you might be seeing the drooping confidence envelope even when you apply a boundary correction method.

Best wishes,
Lauren

Lauren M Scott, PhD
Esri
Geoprocessing, Spatial Statistics
LScott@esri.com

Hi Lauren,

I also have a problem with the Ripley's confidence envelopes, almost regardless of the shape of the study area. It is easier to illustrate by an example. There are 11 points, quite obviously arranged in a band, within a quasi-rectangular study area: [ATTACH=CONFIG]13657[/ATTACH]
On the Ripley's K graph (unweighted, study area defined, 99 permutations; ArcGIS 9.3.1), the observed line plots above the expected line (as expected for a clustered pattern). However, the confidence envelope closely follows the observed line. Curiously, the envelope converges to a single horizontal line which exactly coincides with the expected line at the distance equal to the distance between the furthermost sample points:
[ATTACH=CONFIG]13658[/ATTACH]
This appears to suggest that the permutations are not based on true random sets of points, with each point randomly placed within a study area. The convergence to a horizontal line indicates that the 'random' permutations are reproducing essentially the same pattern, almost precisely maintaining the same maximum point separation. ArcGIS outputs in this example are not due to weighting, or the study area shape or size, or the boundary effects - 999 permutations using a much larger precisely rectangular area with the points in the middle produce the same results. The only way I could coax ArcGIS' Ripley's K function to confirm the existence of statistically significant clustering was by adding one or more 'fake' data points significantly removed from the cluster.

I would appreciate it if you could advise me how to produce more reliable confidence envelopes.

Thank you.

Regards,
Vladimir

Re: Ripley's K Confidence Envelope doesn't follow the blue expected line

VladimirLisitsin — Fri, 20 Apr 2012 05:17:35 GMT

Just to clarify the point on the convergence between the confidence envelope and the observed line from the previous post. Theoretically, they should indeed converge at the MAX(L(t)) for a given study area size and the number of points - but only at a distance of at least half of the maximum dimension of the study area. And in cases of geometrically simple study areas and unweighted K simulations, the confidence envelopes should not deviate too much from the Expected line (apart from the usual boundary-effect drop-off at larger distances) �?? as Lauren has repeatedly mentioned. The problem is, unweighted simulations sometimes seem to behave similar to the weighted ones�?�

Re: Ripley's K Confidence Envelope doesn't follow the blue expected line

LaurenScott — Mon, 21 May 2012 20:55:01 GMT

A bit more again.
I found another problem in the Ripley�??s K function tool associated with the confidence envelope. It is most apparent when the study area is much larger than the points being analyzed, but could also show up with the Minimum Enclosing Rectangle option if the distribution of the points is not very rectangular. Unfortunately, I found this bug too late to get the fix into 10.1 (not yet released) or into 10.0 service pack 5.

Consequently, I'm attaching a file that fixes this problem for ArcGIS 10.0 (it also fixes the issue described earlier relating to "simTable"). This fix will only work for ArcGIS 10.0. Here are the instructions for installing the fix:

1) Navigate to your <ArcGIS>\Desktop10.0\ArcToolbox\Scripts folder.
2) Rename the KFunction.py file (to something like KFunctionOrig.py <-- this is just in case �?�)
3) Copy the attached KFunction.py into that same Scripts folder
4) Run the K function as usual.

Please feel free to contact me if you have any questions or concerns.

My sincere apologies,
Lauren

Lauren M. Scott, PhD
LScott@esri.com
Esri
Geoprocessing, Spatial Statistics

Re: Ripley's K Confidence Envelope doesn't follow the blue expected line

boonejardot — Fri, 01 Jun 2012 20:31:37 GMT

Yup the tech pretty much nailed it. Ripleys K and the blue line/ confidence envelope. This stuff is kind of advanced just so ya know. The blue line is based on a rectangular area. Therefore if the study area is not rectangular, I E an Island/Circular jagged etc, the points will only fall within your study area. therefore the blue line is irrelevant if you are using a confidence envelope. Run your analysis as if the confidence envelope IS the blue line. Any divergence from that, shows something that may not be random.

Re: Ripley's K Confidence Envelope doesn't follow the blue expected line

AprilNewlander1 — Wed, 17 Oct 2012 20:53:33 GMT

I too have been having the same problem with the expected line falling outside the CI. See attachments... pattern analysis shows the study area and points I am using. Each point represents a 1x1 m grid cell that contains a 1-4 m plant (classified as 1 for plant size class) as determined by LiDAR data. The area is divided into 2 study areas: above the road and below the road. Above unweighted shows results of UW ripley's; the figure Above is the weighted (all weights =1; represent a 1x1 m grid cell that has a plant 1-4 m tall in it) observation plotted with the unweighted CI and the expected line adjusted to zero. Notice the drooping CI and that the pattern changed dramatically from that of the unweighted observation...from clustered across all distance scales (UW) to small scale clustering and large scale dispersion. The figure 'below Unweighted' and 'below' show the analysis below the road. I used the 'simulate outer boundary values' for edge correction; and user defined study area. I am aware that maybe the study area size could impact my results (clustering is an artifact). Does the weighted analysis seem appopriate to use to represent the pattern?

Re: Ripley's K Confidence Envelope doesn't follow the blue expected line

VitaBakker — Mon, 19 Feb 2024 08:14:56 GMT

Has this issue been resolved? When I apply Ripley's K, I simulate outer boundaries and have a polygon, and the study area is a feature class. The upper and lower confidence level lines still don't follow the expected K.

Best,

Vita