l.moransI {lctools}

Local Moran's I classic statistic for assessing spatial autocorrelation

Description

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.

Usage

l.moransI(Coords, Bandwidth, x, WType='Binary', scatter.plot = TRUE, family = "adaptive")

Arguments

Coords

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)

Bandwidth

a positive integer that defines the number of nearest neighbours for the calculation of the weights

x

a numeric vector of a variable

WType

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.

scatter.plot

a logical value that controls if the Moran's I Scatter Plot will be displayed (TRUE) or not. Default is TRUE.

family

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).

Details

The interpretation of the local I_i is similar to that of the global Moran's I.

Value

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).

ID

Numeric index from 1 to n

Ii

Classic lobal Moran's I_i statistic

Ei

The expected local Moran's I_i

Vi

The variance of I_i

Zi

The z score calculated for the randomization null hypotheses test

p.value

The p-value (two-tailed) calculated for the randomization null hypotheses test

Xi

The standardised value of the variable x

wXj

The standardised value of the lagged x (weighted some of nearest neighbours)

Cluster

The class each observation belongs based on the sign of Xi and wXj as well as the non-significant local Moran's I values

Note

Please note that the weights are row standardised.

Author(s)

Stamatis Kalogirou <[e-mail]>

References

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

Examples

data(GR.Municipalities)
l.moran<-l.moransI(cbind(GR.Municipalities$X, GR.Municipalities$Y),6,GR.Municipalities$Income01)