There are a couple choices that you could research.

1) you could subsample you data until the Moran's I (or variogram) indicates that there is no longer spatial correlation.

2) you could group the data so that locations within a certain distance of each other belong to a group variable. You would then specify the intercept of your models as a random effect across groups.

3) Or you could model the spatial correlation explicitly.

Take a look at this website. It describes things fairly well and has example code...

http://rstudio-pubs-static.s3.amazonaws.com/9687_cc323b60e5d542449563ff1142163f05.html

Hello,

This will help you: "Methods to account for spatial autocorrelation in the analysis of species distributional data: a review" by Dormann et al (2007).

http://www2.unil.ch/biomapper/Download/Dormann-EcoGra-2007.pdf 

Cheers,

FMestre

if you want to combine modelling spatial dependency and the smooth non-linearity of  a Gam have a look at the free software Bayes X and its fairly extensive  manual that specifically covers this case

http://www.stat.uni-muenchen.de/~bayesx/bayesx.html

You  can also have a look at this package in R

http://www.maths.bath.ac.uk/~sw283/igam/

ans Simon Wood's home page

http://www.maths.bath.ac.uk/~sw283/

Thank you all for very useful suggestions. But the last response from Dr Kelvyn is exactly what I need. linear and nonlinear relationships with continuous and categorical variables in case of spatial autocorrelation. 

Bayes X seems able to model such relationships. 

To account for potential spatial autocorrelation you can include the geographical position, i.e. the latitude-longitude interaction term in the analyses (Wood, S.N., 2006. Generalized additive models: an introduction with R. London: Taylor & Francis, CRC Press, 384 pp. ISBN, 978-7-58488-474-3.)

Hi Levan,

Perhaps this paper might be of use to you.  It has a recipe for rigorously testing spatial autocorrelation (spline correlograms are better than straight Moran's I in my opinion), and a recipe for testing whether you need to bother with spatially explicit models. Good luck!

Joe

Hi Levan,

I have been using marginal models fit with generalized estimating equations (GEEs). This are helpful if your data was collected repeatedly for the same subjects. These can be fit in R using the geepack library https://cran.r-project.org/web/packages/geepack/geepack.pdf . You can find good details on this approach and useful books here https://wildlifesnpits.wordpress.com/2014/10/24/dealing-with-ugly-data-generalized-estimating-equations-gee/ :) I hope everything goes well!

Hi Levan, I applied the Moran's eigenvector mapping methods to GLMS in my paper "Spatial interactions in saithe and hake distribution". This technique is might at first be difficult to handle but at then it is really efficient. There is a review for Dormann 2007 that deals and compare several methods to do that (including MEM).

Have a look in the R-INLA package, which solves a large class of statistical models using the integrated nested laplace approximation and the latent Gaussian field approach to lead with spatial autocorrelation.

http://www.r-inla.org/

Hello Levan,

One option for linear e non-linear GLM is to account for spatial autocorrelation using spatial eigenvectors, specifically Moran Eigenvectors, which will transform the GLM into a Spatial Eigenvectors GLM. I recently published a paper using this approach and you can have a look if you want. Also, the review published by Dormann is really useful.

Best luck,

One option is past 3.18 multivariate ordination PCA, PCoA, MDS, CCA, DCA , LDA.

Hammer, Ø. 2017. PAST: Reference manual. Paleontological Statistics Software Package for Education and Data Analysis. Palaeontologia Electronica . 259pp.

Try using VAST (https://github.com/James-Thorson-NOAA/VAST) and CARBayes (https://cran.r-project.org/web/packages/CARBayes/index.html)

Hello Levan,

Moran’s I test can detect spatial autocorrelation using R-package ‘spdep’. Spatial autocorrelation can be minimized by removing closely occurring points or evenly spreading occurrence points across the area of interest. Moran's I test should be performed after each treatment until satisfying results obtained. You can see the detailed method of evenly distributing occurrence points to minimize spatial autocorrelation in the supplementary material of the paper listed below.

https://www.researchgate.net/publication/261184763_Ensemble_forecast_of_climate_suitability_for_the_Trans-Himalayan_Nyctaginaceae_species

DOI: 10.1016/j.ecolmodel.2014.03.003

Also, suggested reading the following paper -

Carl, G., Kühn, I., 2007. Analyzing spatial autocorrelation in species distributions using Gaussian and logit models. Ecol. Model. 207, 159–170.

All the best

Sailesh

I suggest to follow the explanations provided in this excellent paper, by Dormann et al. 2007

https://biol609.github.io/Readings/DormannEcography30_57164.pdf

You have different solutions.

Best,

Federico

First of all, you should check whether you have a general spatial trend (with a global Moran's test) or whether you data points are spatially autororrelated (made with you GLMM residuals fitted previously) with a LOCAL Moran's test, which makes randomisations with increasing increment (distance for correlation tested), so if the autocorrelation is only at the distance 0, then it's fine. Package "ncf" or "spdep" handles that.

If you have a significant autocorrelation in your data, then you should add a spatial covariance matrix in your model, to control for autocorrelation. Package glmmTMB propose several solutions :

https://cran.r-project.org/web/packages/glmmTMB/vignettes/covstruct.html

Dormann et al. (2007) is a nice review to deal with autocorrelated data. GLMs and GAMs with autocorrelated data are types of mixed models that account for non-independece among sampling units. As several pointed out, there are several good packages for this purpose including nlme, lme4, MASS, glmmTMB. I also recommend you Zuur's book "Mixed effects models and extensions in ecology with R" for clear and easy step-by-step routines.

To the previous answers you can add some approaches from Russel Gray, rjgrayecology.com

There are several good statistical packages are available. R-INLA is one of them. Dormann et al. have published a review to deal with autocorrelated data

https://www.researchgate.net/deref/https%3A%2F%2Fbiol609.github.io%2FReadings%2FDormannEcography30_57164.pdf

One option for GAM in the R package 'mgcv' is to explicitly model latitude and longitude as a spatially autocorrelated smoothing term (similar to a random effect) using a term like this:

+ s(longitude, latitude, bs="gp", k= 100, m=2)

There are lots of great resources from Wood, the creator of mgcv that shows just how powerful GAMs are for these kinds of problems. Check out for example this page on the different types of smoothing terms you can use and why:

https://stat.ethz.ch/R-manual/R-patched/library/mgcv/html/smooth.terms.html

I think that this method allowing you to test bayesian approches. You Can apply GLM and The CAR model can be written as (Keitt et al. 2002): Y = Xb +rW (Y -Xb)o.

simultaneous, GLS modeling directly co variance using parametric function.

You can check this publication

Article Generalized additive models with delayed effects and spatial...

One option could be to use a mixed-effects model, randomizing the spatial location of the data.