Derivation of correlation dimension from spatial autocorrelation functions

Background Spatial complexity is always associated with spatial autocorrelation. Spatial autocorrelation coefficients including Moran’s index proved to be an eigenvalue of the spatial correlation matrixes. An eigenvalue represents a kind of characteristic length for quantitative analysis. However, if a spatial correlation process is based on self-organized evolution, complex structure, and the distributions without characteristic scale, the eigenvalue will be ineffective. In this case, a scaling exponent such as fractal dimension can be used to compensate for the shortcoming of characteristic length parameters such as Moran’s index. Method This paper is devoted to finding an intrinsic relationship between Moran’s index and fractal dimension by means of spatial correlation modeling. Using relative step function as spatial contiguity function, we can convert spatial autocorrelation coefficients into spatial autocorrelation functions. Result By decomposition of spatial autocorrelation functions, we can derive the relation between spatial correlation dimension and spatial autocorrelation functions. As results, a series of useful mathematical models are constructed, including the functional relation between Moran’s index and fractal parameters. Correlation dimension proved to be a scaling exponent in the spatial correlation equation based on Moran’s index. As for empirical analysis, the scaling exponent of spatial autocorrelation of Chinese cities is Dc = 1.3623±0.0358, which is equal to the spatial correlation dimension of the same urban system, D2. The goodness of fit is about R2 = 0.9965. This fractal parameter value suggests weak spatial autocorrelation of Chinese cities. Conclusion A conclusion can be drawn that we can utilize spatial correlation dimension to make deep spatial autocorrelation analysis, and employ spatial autocorrelation functions to make complex spatial autocorrelation analysis. This study reveals the inherent association of fractal patterns with spatial autocorrelation processes. The work may inspire new ideas for spatial modeling and exploration of complex systems such as cities.


Introduction
One of the keys to the method of data analysis is covariance, which reflects the joint variability of two random variables.In statistics, covariance is defined as the mean value of the product of the deviations of two random variables from their respective means.The application of covariance is extended to two directions.One is correlation coefficient, which can be treated as standardized covariance, and the other is correlation function, which can regarded as generalized covariance.A number of measures have been derived from correlation coefficient, including autocorrelation coefficient, partial correlation coefficient, autocorrelation function, partial autocorrelation function, and spatial autocorrelation coefficient.The typical spatial autocorrelation coefficient is Moran's index (Moran, 1948).Correlation function is associated with spectral analysis.Spectral analysis includes the methods of power spectrum for time series and wave spectrum for spatial series (Chen, 2008a;Chen, 2010;Liu and Liu, 1994;Stein, 2000;Stoica and Moses, 2005).Today, correlation function is linked to multifractal analysis because the global fractal dimension is based on Renyi entropy and generalized correlation function (Chen, 2013;Feder, 1988;Grassberger, 1983;Grassberger, 1985;Halsey et al, 1986;Hentschel and Procaccia, 1983;Vicsek, 1989).In theory, the spatial analyses based on correlation coefficients and those based on correlation functions should reach the same goal by different routes, and thus can be integrated into a logical framework.
However, how to establish the relationships between spatial autocorrelation coefficients and spectral density and fractal dimensions is still not clear enough.
Where geographical research is concerned, spatial data analyses rely heavily on spatial correlation, including autocorrelation and cross-correlation.The precondition of using traditional statistical methods to analyze spatial data is that there is no correlation between spatial sampling points.
Otherwise, the probability structure of spatial samples is not determinate, and thus the conventional statistical methods such as regression analysis and principal component analysis will be not credible.
In this case, spatial autocorrelation modeling is always employed to make data analysis (Anselin, 1995;Cliff and Ord, 1973;Cliff and Ord, 1981;Griffith, 2003;Haggett et al, 1977).The common spatial autocorrelation measures include Moran's index (Moran, 1948;Moran, 1950), Geary's coefficient (Geary, 1954), and Getis-Ord's index (Getis, 2009;Getis and Ord, 1992).However, in the process of spatial analysis, we encounter a paradox.This paradox may suggests the uncertainty principle of spatial correlation.If there is no spatial autocorrelation among a group of spatial elements, the spatial autocorrelation coefficient is reliable and equal to zero.On the contrary, if there is spatial autocorrelation, the values of spatial autocorrelation indicators such as Moran's index will be incredible.The calculation of the spatial correlation coefficient depends on the mean or even the standard deviation (Chen, 2013).The mean is based on the sum of observational values.Spatial autocorrelation implies that the whole is not equal to the sum of its parts, and therefore the mean and standard deviation are not affirmatory.As a result, the value of spatial autocorrelation coefficients will significantly deviate from the confidence values.One way to solve the above problem is the integration analysis of multiple correlation measures.Today, there are many measurements can be used to make spatial correlation analysis.Among various spatial correlation statistics, Moran's index and spatial correlation dimension are important ones.In order to integrate these different correlation measures, we had better reveal the logic relations between them.This paper is devoted to deriving the inherent association of spatial autocorrelation coefficient with spatial correlation dimension.In Section 2, the concepts and models of spatial correlation functions and spatial correlation dimension are clarified, and the then spatial correlation dimension is derived from spatial autocorrelation functions based on Moran's index.In Section 3, to verify the theoretical results, the derived models are applied to the Chinese cities.In Section 4, the related questions are discussed.Finally, the discussion is concluded by summarizing the main points of this work.

Spatial correlation dimension
Correlation functions can be divided into two types: correlation density function and correlation sum function.The former is based on density distribution function, and the latter is based on cumulative distribution function.In urban science, spatial correlation density function is also termed density-density correlation function, which can be expressed as follows where c(r) refers to the density correlation, ρ(x) denotes city density, x is the location of a certain city (defined by the radius vector), and r is the distance to x and it represents spatial displacement parameter.In terms of equation (1), if there is a city at x, the probability to find another city at distance r from x is c(r).The correlation function based on integral is useful in theoretical deduction.
In application, the continuous form should be replaced by discrete form, which can be expressed as 1 ( ) ( ) ( ) where S denotes the area of a geographical unit occupied by a city.The other symbols are the same as those in equation ( 1).If we can find the relationship between the correlation function c(r) and the spatial displacement r, we can make a spatial analysis of cities. Equation ( 1) is the discrete expression of density-density correlation function.Through integral, it can be transformed into a correlation sum function as below (Chen, 2008b; Chen and Jiang, 2010): where C(r) is called correlation integral or correlation sum (Williams, 1997).The density correlation is a decreasing function, while the mass correlation is an increasing function.Correlation density functions are susceptible to random perturbations.In contrast, cumulative function has strong anti-noise ability, and thus can better reflect the spatial regularity.
In practice, if we use the categorical (nominal) variable to substitute the metric variable, the correlation sum function can be further simplified.Based on spatial nominal variable, equation ( 3) can be rewritten as which r refers to the yardstick, N denotes city number, N(r) is the number of the cities have correlation, dij is the distance between city i and city j (i, j=1,2,3,…,N), and H(▪) is the Heaviside function.The property of Heaviside function is as below This implies that r forms a distance threshold by the Heaviside function.If the relationship between correlation sum and the distance threshold follow a power law such as 1 () we will have a scale-free correlation, and Dc is the correlation dimension coming between 0 and 2.
In equation ( 6), C1 refers to the proportionality coefficient.In empirical analyses, the correlation sum C(r) can be replaced by correlation number N(r) to determine fractal dimension.Obviously, the correlation number is Then equation ( 6) should be substituted with the following relation where N1= C1N 2 denotes the proportionality coefficient.Replacing the correlation function C(r) with the correlation number N(r) has no influences on the value of the spatial correlation dimension, Dc.
In this this, equation ( 8) is actually equivalent to equation ( 6) in geographical spatial analysis.

Spatial autocorrelation function based on Moran's I
Generalizing spatial autocorrelation coefficients yields corresponding spatial autocorrelation functions.Spatial autocorrelation coefficients are determined by size measures and spatial proximity measures.A spatial proximity matrix, which is a spatial distance matrix or a spatial relation matrix, can be converted into a contiguity matrix as follows The spatial contiguity can be defined by a relative step function as below where dij refers to the distance between locations i and j, r denotes a variable distance threshold.If dij=0 suggests vij(r)=0, then we will have On the other, if dij=0 suggests vij(r)=1, then we will have 1 ( ) ( ) Obviously, the difference between M * (r) and M(r) is a unit matrix E, that is The sum of the elements in the contiguity matrix is as follows Define an one vector e=[1, 1, …, 1] T , we have Apparently, N=e T Ee.Thus the number of non-zero elements in the matrix According to equation (7), N(r) is just the correlation number of cities.In order to unitize the spatial contiguity matrix, define * 11 ( ), ( ) 0 ( ) ( ) Thus we have With the preparation of the above definitions and symbolic system, we can define the spatial autocorrelation function.Based on standardized size vector z and global unitary spatial weight matrix W, Moran's index of spatial autocorrelation can be expressed as (Chen, 2013 Replacing the determined unitary spatial weight matrix W by the variable unitary spatial weight which is a spatial autocorrelation function based on Moran's index. The conventional spatial autocorrelation coefficient, Moran's I, is obtained by analogy with the temporal autocorrelation function in the theory of time series analysis.For time series analysis, if time lag is zero (τ=0), the autocorrelation coefficient reflect the self-correlation of a variable at time t to the variable at time t.In this case, the autocorrelation coefficient must be equal to 1, and thus yields no information.As a result, the zero time lag is not taken into account in time series analysis.
The diagonal elements of the space contiguity matrix correspond to the zero lag of the time series.
Accordingly, the values of the diagonal elements of the spatial contiguity matrix is always set as 0.
As a matter of fact, the diagonals represent the self-correlation of spatial elements in a geographical system, e.g., city A with city A, city B with city B. This kind of influence cannot be ignored in many cases.If we consider the self-correlation of geographical elements, Moran's index can be generalized to the following form In the spatial weight matrix W * (r), the values of the diagonal elements are 1.

Derivation of correlation dimension from spatial autocorrelation function
If a geographical process of spatial autocorrelation has characteristic scales, we will have certain values of Moran's index.In this case, the spatial correlation function is not necessary.On the contrary, if a geographical correlation process bear no characteristic scale, the spatial autocorrelation function suggests scaling in the geographical pattern.Scaling is one of necessary conditions for fractal structure.Thus, maybe we can find the fractal properties in spatial autocorrelation.Based on the concepts of spatial correlation functions and spatial autocorrelation functions, the relations between Moran's index and fractal dimension can be derived.Based on matrixes, the expression of the spatial autocorrelation function based on Moran's index can be decomposed as in which the total number of all elements in a given geographical system can be expressed as (Chen, 2013) TT Thus, equation ( 24) can be rewritten as (26) The two sides of equation ( 26) divided by the correlation number N(r) at the same time yields This suggests that the autocorrelation function based on the generalized Moran's index can be decomposed as follows From equation ( 27) it follows Substituting equation ( 8) into equation ( 29 With the increase of r, N/M0(r) approaches 0. Thus we have approximate expression as below: where denotes the difference between I * (r) and I(r).The spatial correlation function can be approximately expressed as

Discussion
Based on the relative step function of distance, spatial autocorrelation coefficients has been generalized to spatial autocorrelation functions.The typical spatial autocorrelation coefficient is Moran's index.The spatial autocorrelation function on the basis of Moran's index can be expressed as equation ( 22).Taking into account the self-correlation of geographical elements, the standard spatial autocorrelation can be generalized to the form of equation ( 23).Equations ( 22) and ( 23 where The ideas from correlation are important in the research on city fractals and fractal cities.As indicated above, one of fractal dimension definition is based on correlation functions.Spatial correlation can be divided into four types based on equation (10) (Figure 1).If r is a constant, we have a correlation based on fixed scale, which is used to define the common spatial autocorrelation coefficient; is r depends on the size of geographical elements, we have correlation based on characteristic scales; if r is a variable but i or j is fixed to a certain element, we have a local scaling correlation, which can be used to define radial dimension of cities; if r is a variable and i and j are not fixed to a certain element, we have a global scaling correlation, which can be used to define spatial correlation dimension derived above.The local correlation is termed one point correlation or central correlation, while the global correlation is termed point-point correlation or density-density correlation (Chen, 2013).The former reflects the 1-dimensional correlation, while the latter reflect the 2-dimensional correlation.Spatial correlation is one of approaches to estimating fractal dimension of cities (Batty and Longley, 1994;Frankhauser, 1994;Frankhauser, 1998).A number of interesting studies have been made to calculate fractal dimension of urban form, and the method can be combined with dilation method (De Keersmaecker et al, 2003;Thomas et al, 2007;Thomas et al, 2008;Thomas et al, 2010;Thomas et al, 2012).The spatial correlation can be integrated into the percolation analysis to model the complex evolution of urban growth (Makse et al, 1995;Makse et al, 1998;Stanley et al, 1999).The above results form a bridge between spatial correlation of urban patterns and spatial autocorrelation of geographical processes by means of the concepts from fractals and scaling.
The shortcoming of this work lies in two respects.First, the empirical analysis for the spatial autocorrelation functions based on Geary's coefficient and Getis-Ord's index are not made for the time being.Although different spatial autocorrelation functions proved to be equivalent to one another, it is still necessary to make case studies to verify the theoretical inferences.However, limited to the space of a paper, the related empirical studies are not implemented.Second, the empirical analysis is only based on the observational data of Chinese cities.If we can obtain the spatial dataset of other countries, maybe we can make a comprehensive positive studies.
Unfortunately, due to the limitation of observed data, the work remains to be done in the future time.

Conclusions
For the complex spatial systems, the spatial autocorrelation coefficients should be replaced by spatial autocorrelation functions.One of simple and important approach to constructing spatial autocorrelation functions based on spatial autocorrelation coefficients is to make use of the relative step function based on variable distance threshold.Thus, we can derive the spatial correlation dimension from the spatial autocorrelation functions.The main conclusions of this study can be reached as follows.First, the spatial correlation dimension can be calculated by means of the relationships between the standard spatial autocorrelation function and the generalized spatial autocorrelation function.The spatial autocorrelation coefficients are not enough to reflect the complex dynamics process of geographical evolution.Spatial autocorrelation functions can be employed to characterize the spatio-temporal dynamics of geographical systems, but the measurement procedure and quantitative description are complicated.Using spatial correlation dimension, we can condense sets of spatial parameters into a simple number, and thus it is easy to make spatial analyses of geographical processes.Second, the spatial correlation dimension reflect both the spatial autocorrelation and spatial interaction.Moran's index is a spatial correlation coefficient, Geary's coefficient is a spatial Durbin-Watson statistic, while Getis-Ord's index proved to be equivalent to the potential formula under certain conditions.Moran's index and Geary's coefficient reflect the extent and property of spatial autocorrelation, while Getis-Ord's index reflect both the spatial autocorrelation and spatial interaction.All these spatial statistics are associated with the spatial correlation dimension.In this sense, the spatial correlation dimension contain two aspects of geographical spatial information: spatial autocorrelation and spatial interaction.Third, the scaling ranges of spatial correlation dimension suggests the geographical scope of spatial autocorrelation and interaction.In theory, the spatial correlation dimension is absolute, but in practice, the spatial correlation dimension is a relative measure and is always valid within certain range of measurement scales.By means of log-log plots, the scaling range can be approximately identified visually.The scaling range corresponds to the scope of positive autocorrelation reflected by the generalized spatial autocorrelation function based on Moran's index.This implies that the scaling range represents a quantitative criterion of spatial agglomeration of geographical distributions.
) which gives the mathematical relationships between the spatial autocorrelation function, I(r), the generalized autocorrelation function, I * (r), and the spatial correlation dimension, Dc.Considering equation (4), C(r)=N(r)/N 2 , we have derived the exact and approximate relationships between spatial correlation dimension and spatial autocorrelation function.The spatial correlation function comprises a series of spatial autocorrelation coefficients based on Moran's index.Using observational data, we can testify the main relations derived from the theoretical principle of spatial correlation processes.
)proved to be equivalent to the reciprocal of spatial correlation The spatial correlation dimension Dc can be derived from the standard spatial autocorrelation function I(r) and the generalized spatial autocorrelation function, I * (r).Thus, the mathematical relationships between fractal dimension, autocorrelation coefficients, and spatial correlation dimension have been brought to light.The spatial correlation dimension can be associated with Geary's coefficient and Getis-Ord's index.The relationship between Moran's index and Geary's coefficient can be demonstrated as

Figure 1 A
Figure 1 A sketch map of spatial correlation which fall in four types