HOME

TheInfoList



OR:

In
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, Moran's ''I'' is a measure of
spatial autocorrelation Spatial analysis or spatial statistics includes any of the formal techniques which studies entities using their topological, geometric, or geographic properties. Spatial analysis includes a variety of techniques, many still in their early dev ...
developed by
Patrick Alfred Pierce Moran Patrick Alfred Pierce Moran FRS (14 July 1917 – 19 September 1988) was an Australian statistician who made significant contributions to probability theory and its application to population and evolutionary genetics. Early years Patrick ...
. Spatial autocorrelation is characterized by a correlation in a signal among nearby locations in space. Spatial autocorrelation is more complex than one-dimensional
autocorrelation Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...
because spatial correlation is multi-dimensional (i.e. 2 or 3 dimensions of space) and multi-directional.


Global Moran's ''I''

Global Moran's ''I'' is a measure of the overall clustering of the spatial data. It is defined as : I = \frac N W \frac where * N is the number of spatial units indexed by i and j; * x is the variable of interest; * \bar x is the mean of x; * w_ is a matrix of spatial weights with zeroes on the diagonal (i.e., w_ = 0); * and W is the sum of all w_ (i.e. W = \sum_^N \sum_^N ).


Defining weights matrices

The value of I can depend quite a bit on the assumptions built into the spatial weights matrix w_. The matrix is required because, in order to address spatial autocorrelation and also model spatial interaction, we need to impose a structure to constrain the number of neighbors to be considered. This is related to
Tobler's first law of geography The First Law of Geography, according to Waldo Tobler, is "everything is related to everything else, but near things are more related than distant things." This first law is the foundation of the fundamental concepts of spatial dependence and spati ...
, which states that ''Everything depends on everything else, but closer things more so'' - in other words, the law implies a spatial
distance decay Distance decay is a geographical term which describes the effect of distance on cultural or spatial interactions. The distance decay effect states that the interaction between two locales declines as the distance between them increases. Once the d ...
function, such that even though all observations have an influence on all other observations, after some distance threshold that influence can be neglected. The idea is to construct a matrix that accurately reflects your assumptions about the particular spatial phenomenon in question. A common approach is to give a weight of 1 if two zones are neighbors, and 0 otherwise, though the definition of 'neighbors' can vary. Another common approach might be to give a weight of 1 to k nearest neighbors, 0 otherwise. An alternative is to use a distance decay function for assigning weights. Sometimes the length of a shared edge is used for assigning different weights to neighbors. The selection of spatial weights matrix should be guided by theory about the phenomenon in question. The value of I is quite sensitive to the weights and can influence the conclusions you make about a phenomenon, especially when using distances.


Expected value

The expected value of Moran's ''I'' under the null hypothesis of no spatial autocorrelation is : E(I) = \frac The null distribution used for this expectation is that the x input is permuted by a permutation \pi picked uniformly at random (and the expectation is over picking the permutation). At large sample sizes (i.e., as N approaches infinity), the expected value approaches zero. Its variance equals : \operatorname(I) = \frac - (E(I))^2 where : S_1 = \frac 1 2 \sum_i \sum_j (w_+w_)^2 : S_2 = \sum_i \left( \sum_j w_ + \sum_j w_\right)^2 : S_3 = \frac : S_4 = (N^2-3N+3)S_1 - NS_2 + 3W^2 : S_5 = (N^2-N) S_1 - 2NS_2 + 6W^2 Values of ''I'' usually range from −1 to +1. Values significantly below -1/(N-1) indicate negative spatial autocorrelation and values significantly above -1/(N-1) indicate positive spatial autocorrelation. For statistical hypothesis testing, Moran's ''I'' values can be transformed to z-scores. Moran's ''I'' is inversely related to Geary's ''C'', but it is not identical. Moran's ''I'' is a measure of global spatial autocorrelation, while Geary's ''C'' is more sensitive to local spatial autocorrelation.


Local Moran's ''I''

Global spatial autocorrelation analysis yields only one statistic to summarize the whole study area. In other words, the global analysis assumes homogeneity. If that assumption does not hold, then having only one statistic does not make sense as the statistic should differ over space. Moreover, even if there is no global autocorrelation or no clustering, we can still find clusters at a local level using local spatial autocorrelation analysis. The fact that Moran's ''I'' is a summation of individual
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
s is exploited by the "local indicators of spatial association" (LISA) to evaluate the clustering in those individual units by calculating Local Moran's ''I'' for each spatial unit and evaluating the statistical significance for each . From the equation of Global Moran's ''I'', we can obtain: : I_i = \frac \sum_^N w_ (x_j-\bar x) where: : m_2= \frac then, : I= \sum_^N \frac is the Global Moran's ''I'' measuring global autocorrelation, is local, and is the number of analysis units on the map. LISAs can for example be calculated in
GeoDa GeoDa is a free software package that conducts spatial data analysis, geovisualization, spatial autocorrelation and spatial modeling. It runs on different versions of Windows, Mac OS, and Linux. The package was initially developed by the Spatial ...
, which uses the Local Moran's ''I'', proposed by
Luc Anselin Luc Anselin (born December 1, 1953) is one of the developers of the field of spatial econometrics. Life and contributions Luc Anselin was previously the Regents' Professor, Walter Isard Chair and Director of the School of Geographical Sciences an ...
in 1995.


Uses

Moran's ''I'' is widely used in the fields of
geography Geography (from Greek: , ''geographia''. Combination of Greek words ‘Geo’ (The Earth) and ‘Graphien’ (to describe), literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, and ...
and geographic information science. Some examples include: * The analysis of geographic differences in health variables. * Characterising the impact of
lithium Lithium (from el, λίθος, lithos, lit=stone) is a chemical element with the symbol Li and atomic number 3. It is a soft, silvery-white alkali metal. Under standard conditions, it is the least dense metal and the least dense solid el ...
concentrations in public water on mental health. * In
dialectology Dialectology (from Greek , ''dialektos'', "talk, dialect"; and , ''-logia'') is the scientific study of linguistic dialect, a sub-field of sociolinguistics. It studies variations in language based primarily on geographic distribution and their assoc ...
to measure the significance of regional language variation. * Defining an objective function for meaningful terrain segmentation for geomorphological studies


See also

*
Concepts and Techniques in Modern Geography ''Concepts and Techniques in Modern Geography'', abbreviated CATMOG, is a series of 59 short publications, each focused on an individual method or theory in geography. Background and impact ''Concepts and Techniques in Modern Geography'' were p ...
*
Distance decay Distance decay is a geographical term which describes the effect of distance on cultural or spatial interactions. The distance decay effect states that the interaction between two locales declines as the distance between them increases. Once the d ...
*
Geary's C Geary's ''C'' is a measure of spatial autocorrelation or an attempt to determine if adjacent observations of the same phenomenon are correlated. Spatial autocorrelation is more complex than autocorrelation because the correlation is multi-dimension ...
*
Indicators of spatial association Indicators of spatial association are statistics that evaluate the existence of clusters in the spatial arrangement of a given variable. For instance, if we are studying cancer rates among census tracts in a given city local clusters in the rates ...
*
Spatial heterogeneity Spatial heterogeneity is a property generally ascribed to a landscape or to a population. It refers to the uneven distribution of various concentrations of each species within an area. A landscape with spatial heterogeneity has a mix of concentra ...
*
Tobler's first law of geography The First Law of Geography, according to Waldo Tobler, is "everything is related to everything else, but near things are more related than distant things." This first law is the foundation of the fundamental concepts of spatial dependence and spati ...


References

{{reflist Spatial analysis Covariance and correlation