spatial effects: how to find and measure them? some examples...

Spatialeffects...

TomaszKossowski

Spatial effects: How to find and measure them?Some examples on Visegrad Countries

Tomasz Kossowski

Adam Mickiewicz University, Poznań, Poland

April 22, 2013

Tomasz Kossowski Spatial effects...

Spatialeffects...

TomaszKossowski

Introduction

The spatial effects are the core of consideration in geographyand geosciences. What are the spatial effects? We have twokinds of them:

1 spatial dependence and

2 spatial heterogeneity.

Geography as a science is trying to explain spatial dependenceand spatial heterogeneity. The spatial dependence and spatialheterogeneity lead us to spatial processes. The relationshipsbetween points in space establish spatial structure.Our idea is modeling both spatial processes and spatialstructures. But how can we do it? I will try to say...


Spatialeffects...

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Spatial dependence

Spatial dependence is the form of interaction where theprocess (or error term) in one spatial unit depends on theprocess (or error term) in an another one (substantive).

yi = D (y1, . . . , yi−1, yi+1, . . . , yn) .

Spatial dependence can be a result of mismatch betweenspatial boundaries of the processes under study andadministrative boundaries used to organize the data (nuisance).


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Spatial dependence – nuisance

AB

C


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How to measure spatial dependence

For data collected at particular geographic localities as themeasure of their dependence is often used an autocorrelationwhich in that case is called spatial autocorrelation.What do we mean when we think spatial autocorrelation?Like autocorrelation, spatial autocorrelation means thatadjacent observations of the same phenomenon are correlated.Spatial autocorrelation is about proximity in (two-dimensional)space. Spatial autocorrelation is more complex thanautocorrelation because the correlation is two-dimensional andbi-directional.


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Spatial heterogeneity

In the literature of Regional Science we have evidence for lackof uniformity of the effects of space. Factors such as centralplace hierarchies, effects in urban growth argue for modelingstrategies, that take into account the particular properties ofeach location. In addition, spatial units of observation are faraway form homogeneous. This situation we name spatialheterogeneity.Spatial heterogeneity reflects a general instability of abehavioral or other relationship across observational units inspace (Anselin 1988).


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Spatial heterogeneity

The spatial heterogeneity usually results as an instabilityparameters or functional form of relationship over space:

yi = fi(xi, βi, εi),

where i is an index of spatial observation.We usually make an assumption that the spatial effects arerestricted to first-order neighbors.An existence of spatial heterogeneity leads us to models whichtake into account this kind of spatial effects like GWR. Insituation of spatial dependence, we use spatial regressionmodels.


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How to represent space?

In the study of spatial patterns and processes, we may logicallyexpect that close observations are more likely to be similar thanthose far apart (The First Geography Law formulated byTobler). But how to represent space?The answer is: A SPATIAL WEIGHTS MATRIXA spatial weights matrix is an algebraic construction whichincludes some information about true space.


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Spatial weights matrix

It is usual to associate a weight (cij) to each pair (xi, xj)which quantifies this. In the simplest form, these weights willtake values 1 for close neighbors, and 0 otherwise. We setcii = 0. We can consider two versions of spatial matrices:

1 a spatial contiguity matrix,

2 a nearest neighbour matrix.

In the simplest form, the spatial weights matrix is binary andsymmetric.


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An example of spatial weights matrix

C =

0 1 0 0 0 01 0 1 0 0 00 1 0 1 0 00 0 1 0 1 00 0 0 1 0 00 0 0 0 1 0

(1)

represents1 2 3 4 5 6


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Another example for regular tessellations


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Spatial weights matrix

UsuallyW is defined as:

1n

s0C, where n denotes number of spatial units, s0 is sum

elements of the contiguity matrix C,

2 W is the row-standardized matrix C.

As reported by Getis (2007), at least 12 different spatialweights matrix definitions are used in Regional Science papers.


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Spatial weights matrix – coding schemes

The most popular matrix coding schemes are (Getis, Aldstadt2004, Getis 2007):

1 inverse of distance between observations powered to fixedpower 1

dnij,

2 length of common border divided by an parameter |∂i∩∂j|p ,

3 distance limited to n neighbor,

4 ordered distances,

5 weights limited by an parameter a,

6 all centroids of spatial units in distance smaller the criticalcut-off d,

7 n nearest neighbors,

8 using local spatial statistics (more advanced with somebackground theory).


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Moran’s I

Moran (1950) formulated an index I (the most commonlyused) to measure the amount of spatial autocorrelation in ageoreferenced data set. The index shows whether and to whatextent the particular observations influence each other via thestructure of the network.


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Moran’s I

Moran’s I is defined as

• I =

n∑i=1

n∑j=1

wij(xi−x̄)(xj−x̄)

n∑i=1

(xi−x̄)2,

• wij =n

s0cij ,

• cij are elements of a matrix of spatial contiguousnessC = (cij),

• xi are observations,

• n is a number of spatial units,

• s0 is the sum of all elements of a matrix of weights C.


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Moran’s I versus Person’s r

The homology between Pearson’s r and Moran’s I is moreobvious for a slightly modified and rewritten its version Is –

with the use of weighted averages of neighbors. x̃i =n∑j=1

wijxj

(elements of a spatial lag vector of x),

wij =cijn∑j=1

cij

.


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Moran’s I versus Pearson’s r

Is =

n∑i=1

n∑j=1

(xi − x̄)(x̃j − x̄)√n∑i=1

(xi − x̄)2

√n∑i=1

(xi − x̄)2

,

Pearson’s correlation coefficient beetwen variable x and x̃:

rx,x̃ =

n∑i=1

n∑j=1

(xi − x̄)(x̃j − ¯̃x)√n∑i=1

(xi − x̄)2

√n∑i=1

(x̃i − ¯̃x)2

.


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Interpretation of I

If value of Moran’s I is:

• I >−1n− 1

we have positive spatial autocorrelation,

• I =−1n− 1

no spatial autocorrelation,

• I <−1n− 1

negative spatial autocorrelation.

Let Cs = ns0C, and zi = xi − x. We can write

I =z′Cszz′z

.


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Moran’s I and Pearson’s r

:

Moran’s I: 0.681

Pearson’s r Pearson’s r(0.422) (0.422)

n: 37Mean: 1.838

Variance: 0.514

Pearson’s r(0.422)

Moran’s I: 0.386 Moran’s I: -0.186


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Some further properties of Moran’s I

I =y′ECsEyy′y

,

where E = I− 1n11′.

λmin(ECsE) ¬ I ¬ λmax(ECsE).

n

s0λmin(ECE) ¬ I ¬ n

s0λmax(ECE).

Moran’s I is not standardized like Person’s r. Values of Idepends on structure of matrix C.


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Singular tessellations

∀n, cij = 1, for i 6= j, I = − 1n−1

n = 3x1 = 1 x2 = 5 x3 = 3 c12 = c21 = 1 I = −1, 5x1 = 3 x2 = 3 x3 = 1 c12 = c21 = 1 I = 0, 5

n = 4x1 = 1 x2 = 3 x3 = 2 x4 = 2 c12 = c21 = 1 I = −2x1 = 1 x2 = 1 x3 = 2 x4 = 2 c12 = c21 = 1 I = 1


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Regular tessellations


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Regular tessellations

Boots, Tiefelsdorf (2000)

Tessalation n Feasible rangeTriangles 64 [−1.0725, 1.0330]

256 [−1.0477, 1.0375]1024 [−1.0273, 1.0246]

Squares 64 [−1.0739, 0.9747]256 [−1.0485, 1.0216]1024 [−1.0276, 1.0206]

Hexagons 64 [−0.5519, 1.0065]256 [−0.5330, 1.0403]1024 [−0.5186, 1.0306]


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Tessellations – a generalization


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0 10 20 30 40

0

5

10

15

20

25

30

35

40

n = 40, Imin = −1, 1042, Imax = 1, 0451


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0 10 20 30 40 50 60 70 80 90

0

10

20

30

40

50

60

70

80

90

n = 97, Imin = −1, 0978, Imax = 1, 0901


Spatialeffects...

TomaszKossowski

An empirical example

n = 45, s0 = 212, Imin = −0, 64, Imax = 1, 20Tomasz Kossowski Spatial effects...

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Moran's Index = 0,273830Expected Index = -0,022727

Variance = 0,004755Z Score = 4,300785

LegendSYMPOP204

36460 - 4972649727 - 7768877689 - 123615123616 - 279540279541 - 557962


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Variance = 0,000896Z Score = 5,021713

LegendSYMPOP204

36460 - 6341463415 - 8992989930 - 166971166972 - 279540279541 - 1557962


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Variance = 0,007911Z Score = 0,255527

LegendSYMPOP204

36460 - 4972649727 - 7166171662 - 108668108669 - 166971166972 - 279540


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TomaszKossowski


Moran's Index = -0,320241Expected Index = -0,022727

Variance = 0,007858Z Score = -3,356167

LegendSYMPOP204

0 - 4313243133 - 6780367804 - 108668108669 - 166971166972 - 279540


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Geary coefficient

Another measure for spatial autocorrelation is Geary coefficient,defined as follow (Geary 1954)

c =n− 12s0

n∑i=1

n∑j=1

cij(zi − zj)2

n∑i=1

z2i

.

For c < 1 we have positive spatial autocorrelation, c = 1 nospatial autocorrelation and c > 1 negative spatialautocorrelation.


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Geary coefficient

Using matrix notation we obtain

c =n

n− 1

[n

s0· z′ diag(ci.)zz′z

− I].

Moran’s I is a measure of global spatial autocorrelation, whileGeary’s c is more sensitive to local spatial autocorrelation.


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Global and local spatial statistics

We have two global spatial statistics: Moran’s I and Geary’s C.The global statistics let us to find spatial dependence overstudied area. The global statistics are synthetic characteristicsof spatial dependence. But are not fragile for local deviationsglobal autocorrelation pattern.We have „local” versions of global statistics. They are: localMoran’s and local Geary’s. These statistics consist LISA (LocalIndicators of Spatial Association).


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LISA properties

The properties of local spatial statistics are as follow:

1 identification of deviations from global autocorrealationpattern,

2 values of local statistics are computed for every spatialunit,

3 show us similar spatial units,

4 identification of Hot Spots and outliers.


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LISA

The formulations of local spatial statistics are:

Ii = zin∑j=1

wijzj ,

Ci =n∑j=1

wij (zi − zj)2 ,(2)

where zi, zj are deviations from mean.The local spatial statistics are proportional to global statistics.The sum of local Moran’s Ii (or Geary’s Ci) over studied set ofspatial units is equal to global statistics.


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LISA and other methods

The local Moran’s statistics can be used for identification ofagglomeration effects. Using local Moran’s statistics we havepossibility to identify spatial clusters of high and low values.The local Geary’s statistics identifies spatial differences andsimilarities, and shows mean of differences between regions andits neighborhood.We can use another local spatial statistics as G and G∗introduced by Getis and Ord.


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An (detailed!) example on spatial dependence and clusteranalysis: distribution of own incomes in Poland


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Motivation

An important element of the political and socio-economictransformation initiated in Poland in 1989 was the restorationof local government. In place of local organs of publicadministration and a centralized system of financing them,there appeared decentralized structures pursuing their ownfinancial policy independently of the central authorities. Theindependent financial policy was guaranteed by a system ofown incomes.


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Own incomes

In the Polish legislation and publications in the fields offinancial law and economy, many classifications of communeincome have been presented. The criteria of those divisions arenot always precise, which leads to the separation of variousgroups.Of fundamental importance here is article 167, section 2 of thePolish Constitution, under which the income of territorialself-government units can be divided into:

1 their own income,

2 a general subvention from the state budget,

3 target-oriented subsidies from the state budget,

4 income from foreign sources, especially from the EuropeanUnion.


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Own incomes

The most important source of money of territorialself-government units is their own income. This categorydenotes earnings flowing to them, whole and unconditionally,from sources located in their area, and the local authoritieshave a statutory right to determine their level. Own income isdecisive for the financial self-reliance of the units and theirindependence from the state budget. Own income includesprimarily local taxes, local charges, and receipts from property.Communes receive a proportion of the personal income tax andthe corporate income tax transferred to them by the state.

Tax 1995 2006PIT 15% 35.95%CIT 5% 6.71%


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Expenditures

Public expenditures are divided in to two groups:

1 for the financing of current commune activity, includingthe maintenance of public facilities, social security, andwages (running),

2 investment expenditures.

Investment expenditures are intended to add to the property ofterritorial self-government, e.g. through an expansion oftechnical and social infrastructure. Investment expendituresimprove the quality of services provided by territorialself-government and contribute to socio-economicdevelopment. They are financed primarily from its ownincome, credit, communal obligations, and EU means.


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Results of previous research

1 Studies of own incomes have been carried out from thevery start of the systemic transformation at the national,regional (NUTS 2) and local (NUTS 4 and NUTS 5)levels.

2 The studies have revealed a relationship between the levelof socio-economic development and the communes’(NUTS 5) own incomes at each level of data aggregation(NUTS 2 and 4). The relationship is proved by estimatedclassic regression models (Motek 2006, 2007, andSwianiewicz 1996, 2004) employing own incomes andindexes characterizing the development level.

3 The most important indexes include: proportion ofpopulation of the working age, employment in industry,population density, employment in market services, andnumber of companies.


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Research problems

1 What are the relationships between communes’socio-economic structure and the spatial distribution oftheir own income and investment expenditures?

2 What are the factors affecting the spatial distribution ofcommunes’ own income and expenditures?

3 What is the spatial distribution of clusters of communes’own income and investment expenditures?

4 What changes have taken place in the distribution ofclusters of communes’ own income and investmentexpenditures?


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Own incomes

Share in receipts from personal income tax(PIT)

Communes' own incomes

Share in receipts from corporate income tax(CIT)

Local taxes, e.g.real estate tax,agricultural tax,conveyance tax

Local charges, e.g. stamp duty, administrative charge

Property income

Other income

- institutional determinants,e.g. legal rules concerning income, - GDP - socio-economic structure - wealth of society - economic activity - legal regulations - socio-economic and financialpolicies of the state - socio-economic and financialpolicies of commune authorities- efficiency of state and local--government administration- infrastructure- local-government property- tourist attractiveness- location

Basic determinantsof own incomes


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Expenditures

Communes' investmentexpenditures

- level of communes'own income,- access to the financial market,e.g. credit, communal obligations,- level of debt,- level of socio-economicdevelopment,- socio-demographic structure,- access to EU means,- institutional determinants,- deficiencies of technicaland social infrastructure, and- investment policy oflocal government authorities

Basic determinantsof investmentexpenditures


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Spatial autocorrelation

We detected positive spatial autocorrelation of own incomes. Itgrew from 0.05 to 0.16 between 1995 and 2006.

Spatial autocorrelation

0

0,02

0,04

0,06

0,08

0,1

0,12

0,14

0,16

0,18

0,2

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

Year

Mor

an's

I


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Univariate LISA. Clusters in 1995

Warsaw

Upper Silesia

Wroclaw

Poznan

Legnica-Lubin

3CityBaltic

Clusters of own incomes in 1995Not significantHigh-HighLow-LowLow-HighHigh-Low

±

I = 0.0548, p = 0.0024Tomasz Kossowski Spatial effects...

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Warsaw

Upper Silesia

Wroclaw

Poznan

Legnica-Lubin

3CityBaltic

Lodz

Clusters of own incomes in 2006Not significantHigh-HighLow-LowLow-HighHigh-Low

±


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WarsawPoznan

West Pomerania

Lower Silesia

Upper Silesia

Konin

Clusters of expendintures in 1995Not significantHigh-HighLow-LowLow-HighHigh-Low

±


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Warsaw

Poznañ

West Pomerania

Lower Silesia

Warmia

Clusters of expendintures in 2006Not significantHigh-HighLow-LowLow-HighHigh-Low

±


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Conclusion

The research on communes’ own income and investmentexpenditures in the years 1995-2006 reported here allows thefollowing conclusions to be formulated:

1 What affect the level of communes’ own income aremainly factors connected with economic activity asmeasured by the number of economic entities and the levelof personal incomes.

2 The level of a commune’s investment expendituresdepends largely on its own income as well as itsinfrastructure.

3 The testing of the spatial autocorrelation of residuals fromthe classical regression models allowed the adoption of aspatial approach. Spatial effects were found that influencedboth, communes’ own income and their investmentexpenditures, even though the effects were not too strong.


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Conclusion

4 The use of the LISA analysis made it possible to find areasof deviations from the global pattern of spatialautocorrelation. Statistically significant clusters of ownincome and investment expenditures were determined forthe years 1995 and 2006. Over that period, the distributionand range of the clusters underwent serious changes. In thecase of communes’ own income, there was an expansion ofhigh-value clusters, concentrating mainly in the westernpart of Poland, and a stabilization of the low-value ones inthe eastern part. For investment expenditures, it is hard toidentify any tendency in the period 1995-2006. Thisresults from the low stability of clusters of both high andlow values over the time interval studied.


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Some further examples


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People 18-64

I = 0.4890, p = 0.0000


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People 18-64


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People 0-18, 65-

I = 0.5877, p = 0.0000


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People 0-18, 65-


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Migrations per 1000

I = 0.2347, p = 0.0000


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Migrations per 1000


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Unemployment

I = 0.7426, p = 0.0000


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Unemployment


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Usable floor space per capita

I = 0.4318, p = 0.0000


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Usable floor space per capita


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New houses and apartments per 1000

I = 0.3245, p = 0.0000


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New houses and apartments per 1000


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Private companies for 1000

I = 0.4240, p = 0.0000


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Private companies per 1000


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Average salary per capita

I = 0.6981, p = 0.0000


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Average salary per capita


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Beds in hospitals per 10000

I = 0.3245, p = 0.0000


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Beds in hospitals per 10000


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Water demand per capita

I = 0.4318, p = 0.0000


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Water demand per capita


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The end

Thank you for your attention


spatial effects: how to find and measure them? some examples...

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