LoCoH: Homerange Tutorial

Introduction

So you spent the past six months holed up in various blinds in the Andes squinting through the pouring rain for a rare sighting of that ever reclusive mammal known as the Llama. Or, if you are the kind of researcher who has a budget, you have been sipping a latte at the 'El Llame Grande' while reams of radio collar Llama locations streamed into your computer. Either way, you are now the proud owner of a stack of animal location data from which you need to tease out important statistics before that last dime of grant money trickles through your grasping fingers. You ponder for a second, what to do, what to do, then it hits you: construct a homerange.

Homerange Techniques

A number of methods have been developed for locating the homerange of an animal:

The Minimum Convex Polygon Method The grandaddy of homerange analysis techniques, the MCP method is still the most popular method around. The homerange generated by it is the area bounded by the smallest convex polygon that contains all your data points. In essence, a rubber band snapped around the data. The MCP is incredibly simple to implement and easy to understand conceptually, which explains its continued popularity even after much better methods have been developed. Unfortunately, this method generates horrible results as can seen in this example of wolf sightings in Yellowstone. The MCP is in red and clearly does not accurate represent where the wolves (black dots) spend their time. MCP is also crippled by its lack of the ability to generate density distributions.

The Alpha-Hull Method Proposed by Burgman and Fox in 2003, this method is based on using alpha-hull constructions, which involves producing Delauney triangulations of the data, and then removing all sides that are a times longer than the median of the original sides. This method has several defects. For one, it is often too conservative as it leaves points outside the bounds of the homrange. Secondly, it does not provide any density distribution.
The Kernel Method Kernel methods are a powerful group of techniques for generating homeranges. By taking weighted sums of local parametric distributions (e.g. bivariate normal kernels) centered on each point in the data set being modeled, kernel methods produce suprisingly accurate homeranges. Kernel methods have some important downsides, however. For one, there is no boundary on the homerange, the homerange expands to infinity. Often, tricks are employed such as bounding the homerange with the MCP, but this is less the ideal. More troubling, is that as data is added to the data set, the kernel method generates increasingly more 'bumpy' homeranges, the inverse of what one desires. It also performs poorly when dealing with data where there are sharp boundaries such as fences or lakes. An example of a kernel analysis of the wolf data follows.

Clearly these methods have some downsides, but fortunately for you, there is one last family of methods we have not yet discussed: the LoCoH family. LoCoH stands for Localize Convex Hulls and inlcudes three distinct methods: Fixed k LoCoH, Fixed r LoCoH, and Adaptive LoCoH. Fixed k LoCoH is also known by the moniker k-NNCH for k-Nearest Neighbors Convex Hulls and will be the focus of this discussion. First described in Getz and Wilmers (2004), Fixed k LoCoH is a powerful method that creates homeranges which accurately reflect animal locations while simultaneously creating density indicators. For a complete description please read the Getz and Wilmers article, but basic procedure is as follows:

For each data point locate its k-nearest neighbors.
Form a convex polygon hull from these neighbors.
Successively merge these hulls together from smallest to biggest.
The merged hulls form what is known as isopleths with the 10% isopleth (high density) containing 10% of the points, the 100% isopleth (low density) containing all the points.

Walkthrough

More important than the math, however, are the results. This tutorial will take you on a brief walkthrough on how to successfully analyze a homerange using the LoCoH web application.

Our first step is to select what data we want to use. In this tutorial, we will be using the demo data built into the application. Just check the 'Use demo data' button in the first section of the main page. If you want to, you can also upload your own data. The data must to be in ESRI's point shapefile format which is compatible with both ArcView, ArcMap, and a number of other programs. It is ArcView's native data format, but you can also tell ArcMap to export a shapefile, if you have Layer file of points. If you are uploading your data, you will need to upload three interconnected files: the '.shp', '.shx', and '.dbf' files are all required.

Now that we selected our data, lets just skip down to the button of the page and tell LoCoH to analyze our data using the default settings. The page will change to inform you that your data is being analyzed and that you should please be patient. Please take its advice to heart. This process can take up to half an hour. Just let your computer do its thing and go get a cup of coffee and the graduate student cafe. Do not reload the page or hit the back button. Just be patient.

Now our data has been analyzed, let's look at the results. For the moement we can confine ourselves to the first section: "Homerange Coverage." Those pretty pictures you see are a graphical representations of your homerange with and without animal locations. The dots are the location of animal sightings, and the colored areas are the homerange where a hotter area means a higher density of animal sightings (aka a higher isopleth).

The JPEG image is very nice by itself, but LoCoH also allows you to access your results in more usable forms. For instance, you are able to download the image in a high-resolution PDF form, which can easily be inserted into articles. You may also download the isopleths as an ESRI polygon shape file for use with ArcMap or Arcview.

Looking at the homerange, you will probably come to the conclusion that it looks a little sparse or skinny. "It should be fuller," you mutter to yourself darkly under your breath. This problem is become we are using too small a number for k. The higher the value of k, the larger the convex hulls. In fact, if k is equal to the number of points in the dataset, the LoCoH homerange is the same as that generated by the Minimum Convex Polygon method. In other words, as k increased, the homerange converges to the MCP! So to expand our homerange we should increase k.

Scroll down to the bottom of the page, to the "Repeat Analysis of Data" box. Here we have the ability to re-analyze our data using a different value of k. Values of 15, 20 or 25 might be a good test; let's try them all. Now we could run LoCoH once for each value of k, but that would be long and tedious. Instead, we can tell LoCoH to try all values at once. To do this, we We separate the values of k we want to have analyzed by commas. The expression that would serve our purposes for k is "10, 15, 20, 25". Enter this value and click "Analyze."

The processing message will appear again, and the wait will be longer; four times as long in fact. Now is a good time to make two important points:

The analysis are not being run on your computer, they are being run on our server.
Our server is limited.

What this means is that you should limit the number of k's you use and the size of your data sets. A data set of 500 points is good, a data set of 1000 points is doable, a data set of 10,000 will crash our server. Similarly, limit the number of k's you use to the number of fingers you have on one hand. If you have larger datasets or need more k's, you should download the R script that powers LoCoH and run it on your own computer.

Learn About the R Script

Once your data has been analyzed, you will be presented with four different homerange graphs. These graphs represent the homeranges generated for each of the values of k's. It is rather hard to see the homerange on such a small image, so why don't we download the image as a pdf and open it up in a Adobe Acrobat. Now, we can zoom in and see an accurate image of the homerange.

At first glance, I like the value of k=20, but I am not sure if I think that is really the best value. Fortunately, LoCoH has one more tool that helps you choose an accurate value of k. Scroll down the window, until you can see the "Area Covered vs k" box. This box graphs how the area covered by the homerange changes as we change k. Each line represents a different isopleth, so in our case, the top line is the 100% isopleth and the second to top line is the 90% isopleth. Our graph is rather sparse because we only used four values of k, but you can download the R script yourself and run it with as many k's as you want on your own computer. Important points are those just before and after a big jump in area. This jump means the homerange has suddenly started to cover a large portion of previously uncovered area. You might want to try generating the homerange with the k before and after the jump, and studying it in detail.

On our graph there is a big jump, between 15 and 20, reexamining these points on the homerange graph I think that maybe k=15 is the best choice for the homerange, as k=20 covers too much area. Let's generate a new homerange just with the k value of 15.

Great! We have created our first LocCoh homerange. Wasn't that easy? Now a good move would be to download the shapefile images of the homerange and open them up in your GIS program to carry out additional analyses. Additionally some useful information is also located in the "Summary" box at the bottom of the page, such as the areas of the isopleths.

Now before you go off and start analyzing your Llama data, I want to give one last boost to the R script. This script powers LoCoH and is very flexible. It has many more features and capabilities than the LoCoH website gives you access to, and is a great tool for any homerange analysis you need to carry out. It also has an R GUI and can be installed in ArcMap as a toolbox.

Check Out the R Script