The local Moran's I proposed by Anselin (1995). The formula to calculate the local \(I_i\) which is now used in most textbooks and software is: $$I_i = ((x_i - mean(x))/m_2)*(\Sigma w_{ij}*z_j)$$ where \(n\) is number of observations, \(w_{ij}\) are the weights, \(z_j = x_j - mean(x)\), \(x\) being the value of the variable at location \(i\) and \(mean(x)\) being the mean value of the variable in question, and \(m_2 = (\Sigma (x_i - mean(x))^2) / n\). This function calculates the local Moran's I values for each observation along with goodness of fit statistics, it classifies the observations into five classes (High-High, Low-Low, Low-High, High-Low, and Not Significant) and optionally plots a Moran's I Scatter Plot.
l.moransI(Coords, Bandwidth, x, WType='Binary', scatter.plot = TRUE, family = "adaptive")Returns the calculated local Moran's I and a list of statistics for the latter's inference: the expected Ei, the variance Vi, the Xi scores and the p-values for the randomization null hypotheses. It also returns the standardized value and the standardized lagged value of the variable to allow creating the Moran's I scatter plot and the classified values for creating the cluster map similar to those available in GeoDa (Anselin et al., 2006).
Numeric index from 1 to n
Classic lobal Moran's I_i statistic
The expected local Moran's I_i
The variance of I_i
The z score calculated for the randomization null hypotheses test
The p-value (two-tailed) calculated for the randomization null hypotheses test
The standardised value of the variable x
The standardised value of the lagged x (weighted some of nearest neighbours)
The class each observation belongs based on the sign of Xi and wXj as well as the non-significant local Moran's I values
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 positive integer that defines the number of nearest neighbours for the calculation of the weights
a numeric vector of a variable
string giving the weighting scheme used to compute the weights matrix. Options are: "Binary" and "Bi-square". Default 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.
a logical value that controls if the Moran's I Scatter Plot will be displayed (TRUE) or not. Default is TRUE.
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).
Stamatis Kalogirou <stamatis.science@gmail.com>
The interpretation of the local \(I_i\) is similar to that of the global Moran's I.
Anselin, L.,1995, Local Indicators of Spatial Association-LISA. Geographical Analysis, 27, 93-115.
Anselin, L., Syabri, I. and Kho., Y., 2006, GeoDa: An Introduction to Spatial Data Analysis. Geographical Analysis 38(1), 5-22.
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
data(GR.Municipalities)
l.moran<-l.moransI(cbind(GR.Municipalities$X, GR.Municipalities$Y),6,GR.Municipalities$Income01)
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