Moran's I is one of the oldest statistics used to examine spatial autocorrelation. This global statistic was first proposed by Moran (1948, 1950). Later, Cliff and Ord (1973, 1981) present a comprehensive work on spatial autocorrelation and suggested a formula to calculate the I which is now used in most textbooks and software: $$I = (n/W)*(\Sigma \Sigma w_{ij}*z_i*z_j/ \Sigma z_i^2)$$ where n is number of observations, W is the sum of the weights w_ij for all pairs in the system, \(z_i=x_i - mean(x)\) where x is the value of the variable at location i and mean(x) the mean value of the variable in question (Eq. 5.2 Kalogirou, 2003).
This function allows the computation of an number of Moran's I statistics of the same family (fixed or adaptive) with different kernel size. To achieve this it first computes the weights matrix using the w.matrix function and then computes the Moran's I using the moransI.w function for each kernel. The function returns a table with the results and a simple scatter plot with the Moran's I and the kernel size. The latter can be disabled by the user.
moransI.v(Coords, Bandwidths, x, WType='Binary', family='adaptive', plot = TRUE)Returns a matrix with 8 columns and plots a scatter plot. These columns present the following statistics for each kernel size:
an integer in the sequence 1:m, where m is the number of kernel sizes in the vector Bandwidths
the kernel size (number of neighbours or distance)
Classic global Moran's I statistic
The Expected Moran's I (E[I]=-1/(n-1))
The z score calculated for the resampling null hypotheses test
The p-value (two-tailed) calculated for the resampling null hypotheses test
The z score calculated for the randomization null hypotheses test
The p-value (two-tailed) calculated for the randomization null hypotheses test
a numeric matrix or vector or data frame of two columns giving the X,Y coordinates of the observations (data points or geometric / population weighted centroids)
a vector of positive integers that defines the number of nearest neighbours for the calculation of the weights or a vector of Bandwidths relevant to the coordinate systems the spatial analysis refers to.
a numeric vector of a variable
a string giving the weighting function used to compute the weights matrix. Options are: "Binary", "Bi-square", and "RSBi-square". The default value is "Binary".
Binary: weight = 1 for distances less than or equal to the distance of the furthest neighbour (H), 0 otherwise;
Bi-square: weight = (1-(ndist/H)^2)^2 for distances less than or equal to H, 0 otherwise;
RSBi-square: weight = Bi-square weights / sum (Bi-square weights) for each row in the weights matrix
a string giving the weighting scheme used to compute the weights matrix. Options are: "adaptive" and "fixed". The default value is "adaptive".
adaptive: the number of nearest neighbours (integer).
fixed: a fixed distance around each observation's location (in meters).
a logical value (TRUE/FALSE) denoting whether a scatter plot with the Moran's I and the kernel size will be created (if TRUE) or not.
Stamatis Kalogirou <stamatis.science@gmail.com>
The Moran's I statistic ranges from -1 to 1. Values in the interval (-1, 0) indicate negative spatial autocorrelation (low values tend to have neighbours with high values and vice versa), values near 0 indicate no spatial autocorrelation (no spatial pattern - random spatial distribution) and values in the interval (0,1) indicate positive spatial autocorrelation (spatial clusters of similarly low or high values between neighbour municipalities should be expected.)
Cliff, A.D., and Ord, J.K., 1973, Spatial autocorrelation (London: Pion).
Cliff, A.D., and Ord, J.K., 1981, Spatial processes: models and applications (London: Pion).
Goodchild, M. F., 1986, Spatial Autocorrelation. Catmog 47, Geo Books.
Moran, P.A.P., 1948, The interpretation of statistical maps, Journal of the Royal Statistics Society, Series B (Methodological), 10, 2, pp. 243 - 251.
Moran, P.A.P., 1950, Notes on continuous stochastic phenomena, Biometrika, 37, pp. 17 - 23.
Kalogirou, S. (2003) The Statistical Analysis and Modelling of Internal Migration Flows within England and Wales, PhD Thesis, School of Geography, Politics and Sociology, University of Newcastle upon Tyne, UK. https://theses.ncl.ac.uk/jspui/handle/10443/204
Kalogirou, S. (2015) Spatial Analysis: Methodology and Applications with R. [ebook] Athens: Hellenic Academic Libraries Link. ISBN: 978-960-603-285-1 (in Greek). https://repository.kallipos.gr/handle/11419/5029?locale=en
moransI.w, w.matrix
data(GR.Municipalities)
Coords<-cbind(GR.Municipalities@data$X, GR.Municipalities@data$Y)
#using an adaptive kernel
bws <- c(3, 4, 6, 9, 12, 18, 24)
moransI.v(Coords, bws, GR.Municipalities@data$Income01)
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