Spatial Autocorrelation - Science topic

Use multiple linear regression where nominal variables will be classified as dummy variables. I hope I helped.

Great job! Keep up the good work. This information is super helpful. I'm really looking forward to reading this book.

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Dear William,

it is really simple, just two lines of code. If you have loaded the "raster" package, the RasterLayer containing the predictions and the RasterLayer containing the occurrence (presence/background) data, then do the following:

1) calculate the difference between the predictions and the occurrence (i.e. the residuals). Please note that this calculation assumes that all non-presence raster cells (=background cells) are absences, which is not a valid assumption in the case of a presence-only dataset.

2) calculate the global Moran's I measure of the residuals. Please note that Moran's I, by definition, should be calculated using the rook's case weight matrix, hence, you should change the w parameter of function Moran() from its default value, which is queen's case.

Here is a sample code:

library(raster)

predictions = raster("file_name_of_predictions.tif")

occurrences = raster("file_name_of_occurrences.tif")

residuals = predictions - occurrences

Moran(x = residuals, w = matrix(data = c(0, 1, 0, 1, 0, 1, 0, 1, 0), nrow = 3, ncol = 3))

HTH,

Ákos

Spatial autocorrelation can have different implications and interpretations depending on the specific context and characteristics of the species being studied. It is not always considered a problem in habitat suitability modeling, but rather a phenomenon that needs to be understood and accounted for appropriately.

In the case of highly mobile species like raptors, their ability to disperse and explore larger areas may result in a more random or dispersed distribution of presence points. Spatial autocorrelation may still be present, but it might be weaker compared to immobile species due to their greater mobility. In this situation, the presence of raptors in close proximity could indicate suitable foraging or nesting areas, rather than clustering due to limited dispersal.

On the other hand, for immobile species like plants, their dispersal is often limited, and their distribution can exhibit spatial autocorrelation and clustering. This clustering can be an indicator of suitable habitat conditions, as certain environmental factors may favor the growth and survival of plants in specific locations. The limited dispersal range of plants can lead to localized colonization and establishment, resulting in spatially clustered occurrences.

When modeling habitat suitability for immobile species, it is important to consider the potential influence of spatial autocorrelation. Techniques such as spatial filtering, spatial weighting, or spatially explicit modeling approaches can be used to account for spatial autocorrelation and ensure that the modeling process appropriately captures the relationships between environmental variables and species occurrences. These techniques can help mitigate the impact of spatial autocorrelation and provide more accurate estimations of habitat suitability.

The presence of spatial autocorrelation in habitat suitability modeling should be assessed in relation to the characteristics and mobility of the species being studied. While spatial autocorrelation can be informative and indicative of suitable habitat conditions for immobile species, it may have different implications for highly mobile species. Understanding these differences and tailoring the modeling approach accordingly is crucial for accurate and meaningful habitat suitability assessments.

Chandan Roy

The main difference between Moran I in ArcMap 10.4 and GeoDa is the way that they calculate the spatial lag. In ArcMap 10.4, the spatial lag is calculated using a queen contiguity matrix, while in GeoDa, the spatial lag can be calculated using a variety of different matrices, including queen contiguity, rook contiguity, and k nearest neighbors.

The queen contiguity matrix is a binary matrix that indicates whether or not two cells are adjacent to each other. The rook contiguity matrix is also a binary matrix, but it only indicates whether or not two cells are horizontally or vertically adjacent to each other. The k nearest neighbors matrix is a weighted matrix that indicates the distance between each cell and its k nearest neighbors.

The choice of spatial lag matrix can affect the results of the Moran I test. For example, if the spatial lag is calculated using a queen contiguity matrix, then the Moran I test will be more sensitive to spatial autocorrelation that is present in the data. However, if the spatial lag is calculated using a rook contiguity matrix, then the Moran I test will be less sensitive to spatial autocorrelation that is present in the data.

In addition to the choice of spatial lag matrix, the results of the Moran I test can also be affected by the size of the spatial window. The spatial window is the area around each cell that is used to calculate the spatial lag. The larger the spatial window, the more likely it is that the Moran I test will detect spatial autocorrelation. However, the larger the spatial window, the more likely it is that the Moran I test will detect spurious spatial autocorrelation.

It is important to choose the spatial lag matrix and spatial window carefully when conducting a Moran I test. The choice of these parameters can have a significant impact on the results of the test.

Here are some additional tips for conducting a Moran I test:

Maybe of minor concern, but let me note that it is kind of misleading to call Ripley's K and Moran's I methods of 'cluster analysis'. I know that ArcGIS does so, but still. Traditionally, 'cluster analysis' means making groups of initially ungrouped observations within a sample. In contrast, Ripley's K and Moran's I analyse the spatial structure of the sample. They do not create groups, but describe/test aggregation of a point pattern or spatial autocorrelation of a variable.

Ripley's K and Moran's I serve different purposes. Ripley's K describes the expected number of neighbours of any point in a point pattern across different radii. You can use it to analyse point patterns in the space, whether they are aggregated, random, or systematic. The only information you use is the coordinates (or distances between them). Moran's I analyses how a measured variable is structured in space. It uses a spatial neighbourhood matrix between the observations and a measured variable for each observation. It has a global and a local variant; with the latter, you can step across different distance classes to reveal scale-dependency in the spatial correlation. If reasonable null-hypotheses are provided, both can be used for hypothesis testing.

Anyway, it would be interesting to outline a real clustering method based on the Ripley's K or Moran's I. Maybe such a method exists already.

The spatial error model is significant and the lag is not. This means you should run a spatial error model

More than likely you have an error in your program.. I doubt that you have overwhelmed the memory. Check your code. Best wishes David Booth

Hi,

There are so many ways, yet I would prefer R (Rstudio) software for this.

More methods could be found here: -

doi: 10.1111/j.2007.0906-7590.05171.x

Cheers,

Dear Marco,

That test will give you a coefficient (-1 to +1), as you aware of it. So, there is no need to a special test. You can compare them. Also, for this matter you can take Z-scores into account.

Convert each polygon from vector to raster making sure that the aspect is retained. Then do an autocorrelation analysis of the aspect of each cell.

Bilateral trade flows traditionally have been analysed by means of the spatial interaction gravity model. Still, (auto)correlation of trade flows has only recently received attention in the literature. This paper takes up this thread of emerging literature, and shows that spatial filtering (SF) techniques can take into account the autocorrelation in trade flows. Furthermore, we show that the use of origin and destination specific spatial filters goes a long way in correcting for omitted variable bias in an otherwise standard empirical gravity equation. For a cross-section of bilateral trade flows, we compare an SF approach to two benchmark specifications that are consistent with theoretically derived gravity. The results are relevant for a number of reasons. First, we correct for autocorrelation in the residuals. Second, we suggest that the empirical gravity equation can still be considered in applied work, despite the theoretical arguments for its misspecification due to omitted multilateral resistance terms. Third, if we include SF variables, we can still resort to any desired estimator, such as OLS, Poisson or negative binomial regression. Finally, interpreting endogeneity bias as autocorrelation in regressor variables and residuals allows for a more general specification of the gravity equation than the relatively restricted theoretical gravity equation. In particular, we can include additional country-specific push and pull variables, besides GDP (e.g., land area, landlockedness, and per capita GDP). A final analysis provides autocorrelation diagnostics according to different candidate indicators.

Hey, Iraida Redondo García sorry for the delay.

I think that the unit will depend on which unit is the coordinate, no? If it's decimal degrees, then the distance is in decimal degrees.

Peicheng Zhuang you need to select an appropriate initial 1D velocity model in cases of non-existence of a priori geophysical information.

You need to separate two questions, first there is the number and spatial pattern of the data locations used in estimating and modeling the variogram. Secondly there is the number and spatial pattern of the data locations used in applying the kriging estimator/interpolator. These are two entirely different problems. The system of equations used to determine the coefficients in the kriging estimator only requires ONE data location but the results will not be very useful or reliable. Now you must decide whether to use a "unique" search neighborhood to determine the data locations used in the kriging equations or a "moving" neighborhood. Most geostatistical software will use a "moving" neighborhood, if you use a moving neighborhood then about 25 data locations is adequate, using more may result in negative weights and larger kriging variances. Depending on the total number of data locations and the spatial pattern there may be interpolation locations where there are less than 25 data locations. Using a "unique" search neighborhood will likely result in a very large coefficient matrix to invert.

With respect to estimating and modeling the variogram you must first consider how you are going to do this. Usually this will include computing empirical/experimental variograms but for a given data set the empirical variogram is NOT unique. It will depend on various choices made by the user such as the maximum lag distance, the width of the lag classes and whether it is directional or omnidirectional. An empirical variogram does not directly determine the variogram model type, e.g. spherical, gaussian, exponential, etc. It also does not directly determine the model parameters such as sill, range.

Silva's question may seem like a reasonable one to ask but it does NOT have a simple answer. Asking it implies a lack of understanding about geostatistics and kriging.

1991, Myers,D.E., On Variogram Estimation in Proceedings of the First Inter. Conf. Stat. Comp., Cesme, Turkey, 30 Mar.-2 April 1987, Vol II, American Sciences Press, 261-281

I am a PhD student in geography, specifically using spatial analysis in environmental geochemistry. For me I would consider that the data size that you collected and the research design about the fish species (like you said you drove back and back). The spatial autucorrelation should be a index that measured by both of these two parameters, so if your results are not significant. Maybe it need to be improved of the sampling design, or maybe it should be called spatial dependence. Here is a paper that I read for you, about the correlated spatial autocorrelation in ecology: . And another paper written by my supervisor abou the sample size on the statistical significance: . And you can find how we interprete the results of a significant z-score of std. residures in a case that we do not need (for residures it should be spatial non-autocorrelation): .Hope this help!

HI

These files can help you

Dear Lubna,

I think that the command in Stata is the following

In xsmle the spatial weighting matrix can be

a Stata matrix

a spmat object

In both cases the matrix can be standardized or not.

e.g.

a Stata matrix can be created using matrix define, imported from Mata using st matrix(”string scalar name”, real matrix) or imported from GIS softwares like GeoDa using spwmatrix gal using path to gal file, wname(name of the matrix) spmat objects are created by spmat spmat import name of the object using path to file .

the command in Stata is the following:

* spmat dta W W*, replace

The spmat dta command allows users to store an spmat object called W in the Stata memory. Notice that, to fit a model using xsmle, one must use the spatial weight matrix as a Stata matrix or an spmat object. The following spmat entry allows users to easily summarize the W object:

* spmat summarize W, links Because xsmle does not make this transformation automatically, the next step consists in the row-normalization of the W object. This can easily be performed using the following:

* spmat dta W W*, replace normalize(row)

Hello,

"geepack" of R has limited features. Instead you can use SAS GENMOD. See user guide ;

https://support.sas.com/documentation/onlinedoc/stat/131/genmod.pdf

What is asked to you is to create a spot map showing the areas with significant correlation. You can use specific color for the areas to show whether they have significant r or not? Say, all areas with significant r to shown as red while those areas with non-significant r may be shown as green. I presume you know the calculation of r and testing of it for significance?

Dear Rui,

As Cédri says, you have three libraries in R.

- McSpatial,McMillen (2013)

- spatialprobit,Wilhelm and Godinho de Matos (2013). It is a bayesian approach.

- ProbitSpatial, Martinetti and Geniaux (2015).

Best regards

The use of temporal autocorrelation models is not a satisfactory solution to taper data. This is because the errors have a certain structure: for example the residual for height 1 meters is negatively correlated with residual at 2 meters, and the residual for height 1.3 meters is always zero and in general, residuals below breast height are negatively correlated with residuals above breast height. We have some discussion and references about this in chapter of 12 Mehtätalo & Lappi (2020).

GWR is a good method to take into account heterogeneity at the local level. I have used GWR quite a bit since it provides local regression coefficients. Try Geoda program also developed by Prof Anselin.

As mentioned above, most solutions are relatively straightforward. If the serial correlation is in the cross-section (clustering), most software will readily calculate cluster-adjusted standard errors. If the serial correlation is in the time dimension, my advice is usually to get rid of as much of the time dimension as possible (i.e. collapse to a cross section), but there may be alternative solutions. Do not try and apply the weird adjustments used in time-series analysis.

Moran's I can provide spatial autocorrelation which is not as useful as kernel density which can provide "heat maps" of problem areas. Please look at some of my publications in RG where there is a presentation on traffic crashes in a state as well as kernel density applications in other work.

The principal components (PCs) with the largest Eigen value are the most important and explain the larger variation in the data set. The eigen value is the measure of importance of the PCs. The first and second PC usually accounted for the total variance in the entire data set. You can equally consider the eigen vectors of the variables under the PCs to determine and evaluate the parameter loading/significance.

Kindly go through the attached article for your reference purpose.

Hi Lorrayne Miralha ,

You might want to check the following out - the first is an interesting article/tutorial that covers how to use R-INLA for performing the spatial and spatiotemporal modelling in R.

The second is an extremely useful textbook you should definitely get a hold off. You will find what you need in chapter 6 and 7. Beneath it is a link to the data and codes used in all chapters.

Datasets and code: https://sites.google.com/a/r-inla.org/stbook/r-code

Most examples are epidemiological but highly applicable to your area of expertise.

Anwar.

Dear Mouni, thanks for your comment: it makes me realize that I reversed SAC in the residuals and SAC in the response.

Let start from the basic assumption of the OLS.

A basic assumption of OLS is that E(e_i|X) = 0, for all i.

When residuals are spatially autocorrelated this assumption is not satified.

So, SAC in residuals makes the OLS biased and unconsistent.

When the response is spatially autocorrelated, no basic assumption are violated, but a desirable one is: that of uncollinearity among explanatories, since X and WY are likely correlated.

So, when you have SAC in the response, OLS is not biased, but you may have wrong standard errors.

All in all, reviewer is partially right: ignoring SAC in the residuals (when you have it) is a big mistake, ignoring SAC in the response (when you have it) is a moderate mistake.

About the Moran I, you might look to their p-values, the Moran Index is not informative, per se.

I hope this helps,

Best, Rodolfo

for checking for spatial autocorrelation between two variables...lookup bivariate Moran's I..in Geoda..its very easy to do. You can check our publication which uses it. http://mires-and-peat.net/pages/volumes/map26/map2604.php

For any spatial data, you can calculate Moran's I, which determines the presence of spatial autocorrelation. This can be done in ArcGIS, Geoda, and R software.

You may go through the below link for determining Moran's I using Geoda: http://geodacenter.github.io/workbook/5a_global_auto/lab5a.html#the-moran-scatter-plot

Dear Angeliki, in your analysis spatial auto-correlation refers to the data set from each of your individual cameras and temporal auto-correlation refers to the set of data measurements made at a given point in time from all cameras. The term auto-correlation indicates the process of statistical analysis of a valid data set who's quantifying variables are a defining constant.

Clearly in a given subset of the GPS points bear No. is a defining constant and so forth. As far as I understand the statistics in question, the SDM (Squared Distance Mean) may be used in your analysis. The concept of auto-correlation would indeed help if the bears were interacting territoriality.

There are different ways of incorporating spatial autocorrelation into regression models through the error term. One method for instance that works with linear regression models is Simultaneous Autoregressive Model (SAR).

You can use Geoda in R for getting a correction for the regression coefficients in the case of spatial autocorrelation using Lagrange multiplier tests.

Also, although written for beginners to spatial statistics, this chapter and the references there may help you:

Richard John Smith,

I think you'll find these papers interesting:

Dear Somnath,

I suggest to check the space-time pattern mining tool of Arc Map software.

Best,

Behzad

Hi everyone,

after a little research, I finally came up with the answer I was looking for.

when using Global Moran's I index (I) with incrementally increasing distance searches (thus, changing the weight matrix at every iteration), only the the z-values are independent from both weight matrices and variable intensity variations, thus, they are comparable across multiple analyses.

The I in Moran's I statistics is not comparable across analyses, i.e, if with distance of 10m I=0.3 and distance 15m I=0.6, we cannot say that with a distance of 15m the clustering strength is double.

We could only say that in both cases there is a positive (sign of the I) spatial autocorrelation.

For the strengths, we use the z-values.

That is why ESRI plots distances in the x-axis and z-values in the y axis, indicating significant (p-value < than specified signification level) peaks as interesting distances.

For more information, it is clearly explained during a class that Luc Anselin in this Global Autocorrelation class, given in 2016 in Chicago University.

https://www.youtube.com/watch?v=d1WJNBwXfgo&list=PLzREt6r1Nenkr2vtYgbP4hs44HO_s_qEO&index=4

follow from minute 38 when he talks about the permutation approach.

Enjoy!

Hi Haibin

ESRI ArcGIS has autocorrelation analysis, you will get the interpretation result in *html format.

Have a look at

which covers both panel data (serial auto-correlation ) and spatial dependence. As you are an academic in the UK, you can access MLwiN for free. You can stay in the R environment and use R2MLwin to use the power and flexibility of the bespoke software.

http://www.bristol.ac.uk/cmm/software/r2mlwin/

http://www.bristol.ac.uk/cmm/software/mlwin/mlwin-resources.html

and on panel data per se

I should say that I generally agree with Dr. Fernandes with 2 caveats:

1

If the ones you describe as continuous are not times series I think the answer is yes. Check my suggestions on references but then you probably need a real time series person which you can find on this site. Best wishes, David

No. Local Moran and Global Moran are used for different purposes and it depends on the assumptions you are using on your research. Global Moran implies that one single statistic can account for all of your data. Local Moran will return local clusters that may or may not be correlated.

Thanks for answers,

Actually I do not consider the values which are below a cutoff (2pct by mass S) and extremely high values for analysis. In this case, the plot seems to have a linear trend. When I apply a linear trendline result is SulphateSulphur(pct)= 0.8xSulphur(pct) - 1. It says 80% us total S belongs to sulphate molecules but what does fixed value 1 mean here? Another problem is number of S measurements are very high around lower values. I think they dominate the trendline.

For spatial distribution, I am wondering If I can plot the distribution of (SSpct+1)/(Spct) ratios over an area and to classify possible domains using statistical tools. Thats why i am aligned to choose linear regression.

Other than that, If you suggest a better statistical approach, it will be very much appreciated

Best regards

Thank you very much for your return. Thank you for this exchange, for links and documents too.

cordially

The best solution is using a spatial GLM, otherwise use dbMEM as a covariate.

Regards,

So interesting Question.

Thank you

The number of lags depends on the kind of autocorrelation pattern you wish to test for. One lag if AR(1) errors and so on. Wooldridge's Introductory Econometrics is a very good reading.

Are you sure your time series are stationary? That goes before checking for autocorrelation.

Hi Rafael Rios Moura

Depending on the structure of your data and your design, a way to account for spatial autocorrelation is to average the predictions from several submodels built on random subsets of your data.

See for instance this paper by Parisien & Parks 2014: " An analysis of controls on fire activity in boreal Canada: comparing models built with different temporal resolutions"

Hope it helps.

Matthew Kerry

Thanks for your excellent suggestions.

I'm sorry that I have not made the question clear. I want to predict some biological responses from climate variables (e.g. annual mean temperature). The climate variables may be some kind of time series (https://www.stat.pitt.edu/stoffer/tsa4/tsa4.pdf) . This means that the climate variables may be time dependent with the value in year t can to some extent be predicted from the value in year t - 1. However, the biological responses may not be time series (I'm not sure). In such condition, will the independent assumption of linear models be violated?

Dear All,

thank you for taking part in this discussion. I hope I will finf optimal tool.

Best wishes

Washim, Thank you for taking the time to answer my question and provide the most valuable references, Sincerely, John Wheeldon

The insignificant LM tests on your SAR and SEM models indicate that there is no further spatial autocorrelation (SA) in these models. The significant rho and lamda values suggest that there was SA in the error terms of the OLS non-spatial model. To test this, you should use the Moran I test on the error terms of your OLS model. I would expect that it is significant. Your next hurdle is to choose between the SAR and SEM model. In principle, this should be based on theoretical arguments. For instance, is there a physical or economic justification for the dependent variable to be spatially autocorrelated? If there is not, the SEM model is preferable. You might also test the GSM (General Spatial Model), that combines the SAR and SEM.

Dear Diego, Although I don't know the answer to your question, I suggest you to post it to the r-sig-geo mailing list, where it will surely be answered.

HTH,

Ákos

Hi Diego,

Sorry my bad I forgot about the fact the two different species could occur in a same point.

Try this approach

require(dplyr)

require(reshape2)

species_data <- dataset %>%

select(LONGITUDE, LATITUDE, SPECIES )

species_data <- dcast(LONGITUDE + LATITUDE ~ SPECIES, data = especies_data)

species_data <- especies_data %>%

group_by(LONGITUDE, LATITUDE)

summarise_all(.,sum(.))

envir_var <- dataset %>%

select(-c(SPECIES, CLASS, FAMILY))

envir_var <- envir_var %>%

group_by(LONGITUDE, LATITUDE, OCEAN) %>%

summarise_all(.,mean(., na.rm = TRUE))

data <- plyr::join(envir_var, species_data)

coordinates(data) <- ~ LONGITUDE + LATITUDE

delau <- rgeos::gDelaunayTriangulation(data)

neib <- poly2nb(delau)

This should solve your problem. Could be some typo errors in my code, but I hope that u can get the idea how to solve your problem.

Regards,

Moreno

Autocorrelation of lag k is the correlation between Xt and Xt+k where the time series is {Xt}. The partial autocorrelation of lag k is the conditional correlation of Xt and Xt+k given the values of Xt+1, Xt+2, ..., Xt+k-1. There lies the difference.

Hello Mohsen,

Generally, yes … all variables being binary. However, I would argue (as you have in your opening question) that you need to transform your (logit) coefficients into probabilities. This helps in discussing and comparing the parameters in a way that the reader can understand. It’s next to impossible to communicate different types of non-linear parameters (like logit, cubic, probit, etc) in a write-up and make sense of them.

So, I think you are on the right track with transforming the regression parameters into probabilities. Keep in mind, that you don’t really have to break them down into all possible groupings. Each transformed parameter, is the increase/decrease in probability for the DV for that independent parameter =1 (assuming your binary variables are 0 or 1), HOLDING all other variables in the equation equal.

However, because these predictors have covariance (as one would expect), the probabilities are not simply additive (e.g., the probability for x & y =1 is not the same as adding up the probabilities of when only x=1 and separately when y=1; in your example at the top).

The odds ratio is a better way of showing the magnitude of for each independent parameter.

I have added a spreadsheet for doing the logit conversions, in case it is helpful.

Wishing you well,

In the new version of ArcMap (10.5 and 10.6) and in ArcGIS Pro a new tool is included in the Spatia Statistics Toolbox "Similarity Search". You ca find how it works in the Online Help of the program.

http://desktop.arcgis.com/en/arcmap/10.3/tools/spatial-statistics-toolbox/similarity-search.htm.

Dear Bipin,

Yes, spatial autocorrelation and spatial non-stationarity are different concepts.

SPATIAL AUTOCORRELATION

If you measure something over space, for example the household income, it is likely that two observations that are close to each other in space are also similar in measurement. This assumption is also known as the First Law of Geography: "everything is related to everything else, but near things are more related than distant things" (Tobler, 1970). In this sense, Spatial Autocorrelation is a measure of similarity (correlation) between nearby observations. It describes the degree two which observations (values) at spatial locations (whether they are points, areas, or raster cells), are similar to each other. A commonly used statistic that describes spatial autocorrelation is Moran’s I, Geary’s C and, for binary data, the join-count index.

SPATIAL NON-STATIONARITY

Spatial nonstationarity is a condition in which a simple “global” model cannot explain the relationships between some sets of variables. The nature of the model must alter over space to reflect the structure within the data (Brundson et al., 1996). For example, you are trying to model the relationship between two variables, number of cars per person and income, in a given city. Using a global linear regression, you find that neighborhoods with higher incomes have more cars per person. However, in some neighborhoods where public transport is better this relationship changes, and even with a high average income, people tend to use public transportation. Thus, the relationship you are modeling is non-stationary throughout space. The Geographically weighted regression (GWR) is a technique mainly intended to indicate where non-stationarity is taking place on the map, that is where locally weighted regression coefficients move away from their global values.

Tobler W., (1970) "A computer movie simulating urban growth in the Detroit region". Economic Geography, 46(Supplement): 234-240.

Best,

Sacha

GeoDa includes tests for non-normality that may be used to determine which model would be more robust.

See the chapter on the Spatial Lag model:

http://www.csiss.org/clearinghouse/GeoDa/geodaworkbook.pdf

Usually 5% significance level is used. However, this test is bit different from others. You need to get both lower limit and the upper limit. Please use the attach guideline to see how it is performed.

Kind regards and good luck with your research.

Thushara

Pretty old question, but maybe it's still worth adding to the comments given so far:

Type of weights: Spatial weights form a crucial part in estimating spatial autocorrelation. The choice of weights depends on a range of factors like the type of spatial behaviour you assume in the analysed process, the type of data you have at hand, the scale of your process, etc. Thus, there is no easy answer regarding an appropriate choice of weigths and it is ultimately up to a specific analytical task.

Software/different p-values: I'm not familiar with ape but have used spdep. spdep allows you to calculate distance-based weights. The function dnearneigh gives you binary weights indicating whether neighbours are located at a disc of radius dmax (dmin is also possible). If you need something like IDW weights, then you need to do some (not too extensive) R programming indeed, but it is still possible. In addition, spdep also provides functionality for creating k-nearest-neighbour weights through knearneigh.

Inference about Moran's I: There are two ways of drawing inference about Moran's I. One is called the Normal assumption (N), the other one is based on Randomisations (R). The N assumption is based on the so-called Pitman-Koopmans theorem that roughly states that Moran's I is independent of its denominator under iid normal variates. This assumption allows you to evaluate p-values of Moran's I from a normal distribution regardless of sample size, but under the (somewhat strict) assumption of normality. This is obviously a very efficient way of assessing p-values. In contrast, the R assumption is based on a permutation argument that assumes random reallocations of your values within the fixed set of spatial units. Therefore, the R assumption is a non-parametric one. While N is based on sampling with replacement, R corresponds to sampling without replacement, causing statistical power to be lower (because your decision is based on less information content). In the latter case you would construct a bootstrap from recurrently calculating Moran's I a certain number of times. The (pseudo) p-values are then assessed from that bootstrap distribution. However, even under R, Moran's I converges to normality rather quickly (though this is also influenced by the layout of your spatial units/geographic region). A valuable source for this is the series of seminal books published by Cliff and Ord (1973, 1981).

Thank you Elia, the subsequent question is: how to implement random factors in RDA (in R vegan)? By partial-RDA?

Andrew, RDA per se doesn't account for spatial non-independence, I suppose. Probably there are specific techniquest to deal with it in RDA. I am wondering how...

Several observations

1. Morans"I is an empirical measure of spatial correlation and is based on "randomization" (in contrast with a spatial autocorrelation function)

2.It is not a question of whether the "samples" are independent, first of all independence is a theoretical property of random variables (not of data)

3. Don't mix up confidence level and significance in a test of hypotheses. Significance is the probability of a Type I error (incorrect rejection of the null hypothesis) whereas confidence level is the related to whether the true but unknown parameter (the one being estimated) is inside the confidence interval (a confidence level is not a probability but is related to the probability that the estimatOR value is inside of an specified interval) You also have to be careful about the distinction between a one sided confidence interval and a two sided confidence interval.

4. The phrase " the correlation between variable, X, and the “spatial lag” of X formed by averaging all the values " doesn't make any sense. It is completely wrong

Hello Zhang!

As you know that, "every thing related to every thing else but near thing are more alike than farther" (Tobler). Here, the question is, how strong they related. The strength of relation can be measured by spatial autocrrelation. Moran-I is of of such index used to measure the spatial autocrrelation.

In your case Moran-I with 0.8 represents high clustering of housing prices, that means housing prices in New York cannot be random or dissimilar. Instead, they follow a spatial pattern i.e., the highest housing prices are mostly near by higher housing prices and higher are near by high like that. This pattern is natural and logical too. You can justify the natural observed phenomena by your results.

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As you said, "I know that spatial autcorrelation of spatial variables violates the assumption of classic statistics, i.e., independence of sample, so we have to consider it when analyzing influence of spatial phenomena or spatial modeling."

Explanation: Yes you are right! we need to take it account in modeling because the high spatial dependency of spatial variable make models biased. It means estimated model may not give us the right information. It doesn't mean, the spatial auto correlation among spatial variable is a problem. It is a natural phenomena and should exist.

Now, we will re-frame our approach, we are using model to understand the natural phenomena. Since model are abstraction of reality. They cannot be perfect, now the question is 'to what extent can we trust our model?' This generally done by test the spatial autocorrelation among standard residuals from our model. And we assume standard residuals of a good model are randomly related in space. in other words, the spatial auto-correlation among the residuals are null (Moral-I is 0).

regards

"semivariograms and distance",there are two problems with Fararini's comment.

With only data values you can only estimate and model the variogram (using an empirical variogram). This is not an exact process, i.e. there is no way to be able to claim that you have chosen "the" model and have exactly estimated the parameters of the model. Moreover for a given (finite) data set the empirical variogram is not unique, you must choose the lag spacings and also you must distinguish between an omnidirectional empirical variogram and directional empirical variograms.

Secondly, the variogram is only indirectly related to correlation (unlike a covariance function or (auto)correlation function). A variogram model need not have a true range, e.g. the exponential and Gaussian models do not, moreover the Power models do not even have a sill. It is common to refer to a range for both the exponential and Gaussian models but this is not a theoretical result, rather it is the distance at which the value of the variogram model is 95% of the sill (95% is an arbitrary choice). If the variogram has a geometric anisotropy then the range depends on the direction (in 3D there are two direction parameters)

However the suggestion of using the empirical variogram to choose a maximum distance is not a bad one but recognize that you are choosing a distance, you are certainly NOT determining the "exact distance at which the autocorrelation stops/starts". It is critical to distinguish between an empirical variogram and a theoretical model.

Finally, "semivariogram" is long out dated terminology (since about 1988 when G. Matheron stopped using it)

To Chris Broekhoven, since you are using R packages, you might want to check references on the author of those packages. Do a search in Google for "R package dnearneigh", you will find a lot of documentation and some tutorial examples as well as for "R package knearneigh"

Both dnearestneighbor and knearestneighbor pertain to data sets where the interest is mostly or completely in the locations of the data points, variograms pertain to data sets where there is a value (for some variable, e.g. hydrologic parameter) at each data location and the interest is in how that value relates to the position coordinates.

Dear Firoz Ji,

Please check the following attached pdf file, hope you will find these Articles helpful.

Regards

Amit

Dear All

From my analysis in city of bangalore,found that planning emphasis is high for industry than housing,however traditional forward groups as well elite have been able to win in layout allocation.With not adequate coordination among developing agents has caused high congestion.Both within as well in urban sprawl.

Hi Hoegun,

Are you referring to the spatial correlation among the variables used in Maxent? Or the spatial bias in occurrence points?  If you are talking about the latter, maybe you may want to use only one point per grid when modeling.

Cheers!

Eric

If you want to make the statistical comparison between two functions it is possible to use the covariance functions. This is a procedure that will allow you to find statistical similarities between two time series. For this there is the theoretical definition from the integration operations and also several available computing algorithms. I recommend using the Matlab application which has built-in almost all covariance functions.

You expect HWE in a panmictic population. If there is population structure (IBD), why should you test for HWE?

Simple!

1) After fitting the logistic regression, estimate residuals from the model

2) Export the sample ID and residual to a text file with X and Y

3) Create a shape file from the text file of point no.2 in arc gis

4) Now you can use "Spatial Auto correlation (Moran's I)" tool under Analyzing patterns of Spatial statistic tool box in arc tool box, to estimate the Global Moran-I.

Hope this help

You may want to have a look at R package "spatstat" which can be very handy if one wants to perform statistical analyses on spatial data sets. Please follow the link:

Also, here you will find how to compute spatial auto correlation in R:

Hope this helps.

Dear Ngo,

Normality is not a big issue in ARDL because even if normality is solved the estimates does change that much. You know this of course from your elementary statistics that JB is for a large sample n=100 so if you try to add dummy using standardized residual to identify the outlying data point and the results do not improve just go ahead a use Newey-west  estimator (ignore mentioning JB) since all the nonspherical errors have been contained