Dear Serge!

In general, Moran's I can also be interpreted as slope of a linear regression line with the dependent variable y and the independent variable (regressor) W*y, where y is the vector of observations and W is a row-standardized spatial weighting matrix. For instance, one could choose W as row-standardized matrix of W_tilde, where w_tilde_ij = 1 if the distance between region i and region j is between 0 and 1km. In that case, the first entry of the matrix product W*y would coincide with the average of all observations, for which the distance from the first region is smaller than 1km. The second entry would be the average of all observations closer than 1km from the second location/region, and so forth. To summarize, W*y are the average observations within a distance in the interval (0km, 1km]. Of course, one could also choose any other interval.

The idea of the plot, which you have attached, is to assume that the weighting matrix W depends on the distance interval, i.e., Moran's I is calculated for several "distance classes" (e.g., (0km,1km], (1km,2km], (2km,3km], ...). I am not familiar with the SAM software, so you should check, which distance intervals are assumed to calculate the Moran's I (red dots). The intervals could also be (0,1km], (0,2km], (0,3km], and so on. For instance, If the first interval is (60km, 65km], Moran's I indicates that the spatial correlation is 0.475 between all locations, which have a distance between 60km and 65km.

If the distance between all your locations can take only a discrete number of values (e.g., 60km, 170km, 300km, ...), it is not necessary to consider the "distance classes". In that case, Moran's I can be calculated for each of the possible distances (and I suppose, it is represented as red dot in the shown figure).

In order to check whether this spatial autocorrelation is significant, you should calculate confidence intervals. The mean and the variance of the Moran's I statistic can easily be calculated (cf. also: "On the Asymptotic Distribution of the Moran I Test Statistic with Applications", Kelejian and Prucha (2000)). Regarding your plot, I suppose that Moran's I differs significantly from zero, if the red dot is outside the red 95%-confidence band around zero.

A few things come to mind...

For one, it seems clear from this Moran's I correlogram that that residual spatial autocorrelation disappears after "distance units" of around 250 - though I don't know how that relates to kilometers.  If you maintain observations within that distance of each other, you can expect downward biased variance estimates for your regression model.

For two, since you really need to retain as many observations as possible, and you can identify the different sampling sources (i.e. museums, papers), then you may consider adjusting for the different sampling sources in your regression models.  The simplest approach is to include dummy covariables to identify which sampling source each observation came from.  This doesn't seem to warrant the treatment of sampling sources as a random effect, since the sampling sources don't represent some sample from a larger population (just a fixed set of sources).  I'd be curious to see if that reduces or eliminates residual spatial autocorrelation.  If not, then you may have to apply an appropriate "spatial" regression model if you want to retain all your observations.

Dear Philipp and Glen,

Thanks for your suggestions and detailed explanations! Now it seems clearer, though I am still not sure about certain moments...

Firstly, I need to find out what are the distance units that the program uses when it calculates intervals for my data. It seems plausible to assume that if I use "Geodesic" coordinate system, the distance is in meters. Right? If so, according to Nunez & Medley, 2011, I can use the distance where residual spatial autocorrelation ceases as a minimum distance between my presence points for spatial filtering. However, in this case it seems nonsensical, since I have no occurrence points closer than 1 km to each other (anyway, Maxen't excludes such points from the analysis) and the distance where SAC disappears is 293.521 meters(?!). Otherwise, if the distance in km, it will badly decimate the number of presence points. To give you better illustration of what are the settings that I use for calculation of Moran's values,  I've attached a couple of screenshots of the program.

Glen, I would be glad if you could help me with a simple algorithm/instructions for a layman on applying the dummy variable approach that you've suggested. It would be interesting to try this.

Kind regards,

Serhii 

Article Pine invasions: Climate predicts invasion success; something...

For adding a dummy variable, it may depend on the software, but most can simply accept the covariable "sources" as a categorical variable.  The software should then generate a linear coefficient for each source type (although it may not for one source type if it is treated as a reference value). 

Ayhow, if you can add a variable (one column) to your input dataset called something like sample_source and assign each observation (row) to its respective sampling source, you should be able to run a regression where you include "sample_source" as one of the covariables (predictors).  If it works, then any inference regarding your other covariables of interest will have adjusted for different sampling sources.

I suppose that your program is computing the Euclidean distance between two points. Depending on your coordinates, that distance does not necessarily has a meaningful interpretation. 

Glen, unfortunately the MaxEnt allows to use only 3 variables (viz. species name, decimal longitude and latitude). So, no dice, as the other SDM methods are beyond the scope of intended work (and beyond my abilities in modelling). Anyway, thanks for the interesting suggestion.

Philipp, it is strange that the program uses Euclidean distance while I am choosing the Geodesic system of coordinates (see screenshot). According to the tutorial the distances should be in km. And it seems that it worked perfectly for the others who applied similar approach (e.g. see Mateo-Tomás & Olea, 2010). Interestingly, the Moran's I plot in their supplementary file looks differently from what I am getting, though it was computed using SAM as well. Maybe I am doing something wrong...            Could you please also specify what did you mean saying that meaningfulness of the Euclidian distance can depend on my coordinates? And how I can figure out what's going on in my particular case? I will do some more research on this topic as well, but now I am apparently stuck...  

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2928263/

Hi Serhii,

based on screenshots you posted, I don't think you are doing something wrong in your settings. Distances in the correlogram are really in kilometres. You just indicated the form of input (coordinates) in the SAM settings - meaning you have a geodesic coordinate system (probably WGS84) and you know what are X(long) and Y(lat) axes. But it doesn't necessarily mean that calculated distances among points are in decimal degrees as well because they can be converted to metres/kilometres.  Moreover, from the practical point of view, it is much easier to understand distances in m/km rather than in decimal degrees (and SAM authors obviously knew it).

The are you are studying seem to be pretty big (assuming that your coordinates are in WGS84) and you have just around 300 samples. As noted by Philipp, the Moran's I in the correlogram is significant if the red dot (or red line) is outside the interval of confidence. So, your correlogram is might be indicating that records tend to cluster up to +/- 200 km (mutual population?) while it is rather dispersed from the distance of +/- 900 km (different populations?). But I agree with Glen that you may want to adjust your data based on the sources or try also other methods of  spatial patterns analyses (please see the link below - although it uses R, there is a comprehensive overview of methods and its interpretations). You might also want to compute more detailed correlogram.

Regards

Lukas

http://umaine.edu/computingcoursesonline/files/2011/07/SpatstatmodelingWorkshop.pdf

Hi Lukas,

Thanks for your input, it helps a lot. Unfortunately, I am not familiar with the R language, so it will take a great amount of time for me to learn how to apply the method you've suggested.

Yes, the area of study is large, I'm trying to model distribution of a Western Palearctic species. Moreover, the records span a considerable period of time (from 1913 to present). So, sampling bias is rather something inherent for such kind of data.

Now I am inclined to think that the distances calculated by SAM are actually in meters (see for example the screenshot of correlogram from Mateo-Tomás & Olea). Probably I don't have to deal with SAC in my input data by measuring Moran's I in the model's residuals. The initial idea of doing so was to correct for sampling bias because I thought that it would give me a better estimation at what distance I should filter my records. Alas, it didn't! So I need to dig more into the problem of dealing with sampling bias. Also it appears that the issue is still poorly addressed by MaxEnt users and quite a few studies simply ignore it. Or maybe it is acceptable to not address the sampling bias at all if the study is aimed at simply obtaining a species' potential distribution map and getting a first proxy of its ecology (as in case of my exercise)?

Also it will be great to hear some opinion from experienced MaxEnt users.

Kind regards,

Serhii

Look for the point that first reaches the I equal or less than 0.3. That's what you need (in km) to get rid of most of the autocorrelation. Don't forget to check "geodetic"!