LoCoH: A k-NNCH Implementation

There are three different LoCoH Algorithms:

• Fixed k LoCoH: In this algorithm the convex hull for each point is constructed from the point and the (k-1) nearest neighbors to the point.
• Fixed r LoCoH: Here, the convex hull is constructed from all the points within a fixed distance r, from the root point. This algorithm is similar to fixed kernel algorithms.
• Adaptive LoCoH: In this last case, k is set, for each point, to one plus the greatest number of nearest neighbors such that the sum of their distances from the root point is less than or equal to a distance variable a.

In the Fixed r LoCoH algorithm, hulls are constructed from all points within distance r from the root point. Thus r acts as a radius around the root point for hull creation. It should be related to the basic clumping size of your data.

You may enter multiple r's at once by separating the individual r's by commas.

WARNING: Our server is limited, so do not enter too many r values on a large dataset. If you do, your process will fail, your data will not be analyzed, and, you might upset us. If you have a lot of data to analyze, download the R script and run it yourself.

In Adaptive LoCoH, hulls are created from the maximum number of nearest neighbors such that the sum of their distances from the root point is less than or equal to a. a should scale with both the spatial scale of your data set, and the sample size of your data set.

You may enter multiple a's at once by separating the individual a's by commas.

WARNING: Our server is limited, so do not enter too many a values on a large dataset. If you do, your process will fail, your data will not be analyzed, and, you might upset us. If you have a lot of data to analyze, download the R script and run it yourself.

In Fixed r and Adaptive LoCoH, there is not guarantee that every point will be included in a hull. If the k-value for any given point is less than minimum k, than that point's k value is set to minimum k. If you require that every point be part of at least one hull, you can set minimum k to a value of three or more.

k is an important value for constructing a k-NNCH utilization distribution. k represents the total number of points from which local hulls are constructed, including the original point. Thus if k=5, a local hull will be constructed for each point and its 4 nearest neighbors. The lowest possible value for k is 3, meaning all local hulls will be triangles. The largest possible value for k is the number of data points you have. In this case, the total k-NNCH homerange (100% isopleth) will be equivalent to the homerange generated by the Minimum Convex Polygon method.

It is worthwhile to experiment with a number of different k's to find the one that appears to generate the most accurate homerange. Small values of k will result in many 'holes' in the homerange, while large values of k will result in large swaths of habitat included where no animals were observed. As a rule of thumb, good values of k should be around the square root of the number of data points.

LoCoH allows you to analyze multiple values of k at once to aid this purpose. To try multiple discreet values, separate your k's with commas. So '4, 5, 10' would try k values of 4, 5, and 10.

WARNING: Our server is limited, so do not enter too many k values on a large dataset. If you do, your process will fail, your data will not be analyzed, and, you might upset us. If you have a lot of data to analyze, download the R script and run it yourself.

In animal location data, especially when data are generated with a radio or GPS collar at regular time intervals, there are often duplicate points. This can cause problems when constructing local hulls because you need at least three unique points to create a hull. There are a few options for how LoCoH handles duplicate points.

If the 'displace' option is selected, duplicated points will be displaced in a random direction by the amount specified. The displacement distance should represent an approximate radius of habitat used by the animal when stationary (e.g., a patch size or the distance they monitor for their 'safety zone').

If the 'include duplicates in nearest neighbor searches' option is selected, duplicate points will be included when searching for the k-1 nearest neighbors. And a hull might be formed with less that three unique points, resulting in a zero-area hull.

The 'delete' option will simply exclude duplicate points from hull creation and nearest neighbor searches. For example if your dataset has 429 points, but 5 of those points lie on top of each other, then 4 of the 5 duplicates will be basically excluded from the analysis.

The isopleth levels tells LoCoH for what percentages it should generate isopleths. Isopleths contain a certain number of the data points and are sorted by density. Thus the 10% isopleth is the merger of the smallest hulls that contain 10% of all the original data points. The 100% isopleth will contain all the points. If you're just trying to see which value of k works best for your data, then you probably only need to view the 100% isopleth. The format for data entry is the same as that for the value of k, multiple values are separated by commas.