spatial and regional economic analysis mini-course

Vicente Rios

Spatial and Regional Economic Analysis

Mini-Course:

Author: Vicente Rios

Vicente Rios

Practice: Spatial Weight Matrices for European

Regions

Testing Spatial Autocorrelation

Global Spatial Autocorrelation

Local Spatial Autocorrelation

Practice: Unemployment Rate Spatial

Autocorrelation Analysis

Lecture 2: Introduction Spatial Analysis

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We are going to put in practice some of previous ideas/concepts by

developing W matrices to describe connectivity among European

NUTS-2 regions

NUTS-2 regions (Nomenclature of Territorial Units for Statistics)

NUTS2 is the territorial unit most commonly used in the literature

on regional analysis in the EU

NUTS2 regions are the basic unit for the application of cohesion

policies in the EU

(http://ec.europa.eu/regional_policy/sources/docoffic/official/repo

rts/cohesion7/7cr.pdf)

Spanish Autonomous Communities, French regions, etc..

NUTS-x refers to a level of data aggregation (3 is provinces, 0 are

countries, 1(is intermediate between country aggrregation and

regional aggregation))

Practice (W matrices)

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http://ec.europa.eu/regional_policy/sources/docoffic/official/reports/cohesion7/7cr.pdf

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In Italy there are 20 NUTS-2 regions

NUTS-2 regions

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In Germany this is the current NUTS-2 regional division:

NUTS-2 regions

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Switch to Matlab “Tutorial1_W_Matrices.m”

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Goals

In a first place, we are going to use latitude and longitude

coordinates of each NUTS2 region to obtain a matrix of

distances 𝑫𝒊𝒋 in kilometers for all European regions.

Second, we will use this distance matrix 𝑫𝒊𝒋 to derive different

spatial weight matrices W using some of the spatial functional

forms we have seen before.

In particular we will derive (i) the gravity W matrix, (ii) the 5 and

25 nearest neighbors and (iii) an exponential matrix with cut-offs

at the first quartile of the distribution of distances.


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Software: Matlab

We are going to use some already built in functions in this exercise:

1) distance.m→ calculates distances (in degrees) between points in a sphere

2) deg2km.m → converts degrees into kilometers

3) normw.m→ row-normalizes the matrices

4) make_nnw.m→ finds “k” nearest neighbors and builts W based on the specified k

5) spy.m→ plots the sparsity pattern of a matrix

6) vec.m→ vectorizes a matrix

7) quantile.m→ finds quantiles of the distribution of the supplied variable

8) gplot.m→ creates a network representation of a matrix


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Code

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(code continued)

Practice part 1

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(code continued)


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(code continued)


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(code continued)

Practice Day 1

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An interesting thing to do is to visualize how sparse or how dense is

our matrix.

In applied analysis matrix sparsity greatly accelerates computations.

Sparsity carries a number of substantive and computational advantages:

Dense matrices are noisy and contain a potentially large number of

irrelevant connections.

Dense matrices will bias downward indirect effects of a change in

observation j (the individual weights of non-zero entries in row-

standardized weight matrices will be smaller).

Dense matrices can be computationally intensive to the point that even

simple matrix operations are infeasible


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Some computational facts on W sparsity:

With n = 65K

Dense Matrix = 31.9 GB of storage vs Sparse Matrix = 0.01 GB

With n = 208K


With n = 8M (big data)


Ways to increase efficiency: ordering north-south or east-west

Concentrates nonzero elements around the diagonal

Log determinant of W with n=62K more than 12GB and almost infeasible in time for

most of machines

The same operation for a geographically ordered matrix takes less than a minute


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K-nearest neighbor matrices


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K-nearest neighbor matrices


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(code continued)


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The gravity matrix


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Exponential decay matrices with cut-offs


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5-nearest neighbor representation of NUTs-2 European regions


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Indicator of spatial association

Global Autocorrelation

Local Autocorrelation

Global Autocorrelation:

It is a measure of overall clustering in the data. It yields one statistic to

summarize the whole study area.

Captures departures from random spatial distribution

Famous/Commonly employed statistics:

Moran’s I

Gery’s C

Getis and Ord’s G(d)


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Real distribution of unemployment rates in European regions (2011) vs a

spatially random distribution with the same parameters

𝑈 ~ (𝜇 = 9.4, 𝜎 = 5.25)

Real world vs spatial randomness

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Moran’s I

This statistic is given by:

𝐼 =𝑛σ𝑖=1

𝑛 σ𝑗=1,𝑗≠𝑖𝑛 𝑤𝑖𝑗(𝑥𝑖 − ҧ𝑥)(𝑥𝑗 − ҧ𝑥)

𝑆 σ𝑖=1𝑛 𝑥𝑖 − ҧ𝑥 2

where 𝑆 = σ𝑖=1𝑛 σ𝑗=1

𝑛 𝑤𝑖𝑗 and 𝑤𝑖𝑗 is an element of the spatial weight matrix

that measures spatial distance or connectivity between regions i and j.

In matrix form:

𝐼 =𝑛

𝑆

𝑧′𝑊𝑧

𝑧′𝑧where 𝑧 = x − ҧ𝑥. If the W matrix is row-standardized, then:

𝐼 =𝑧′𝑊𝑧

𝑧′𝑧because S=n.

Values range from -1 (perfect dispersion) to +1 (perfect correlation). A zero

value indicates a random spatial pattern


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Note that:

መ𝛽𝑂𝐿𝑆 =σ𝑖=1𝑛 𝑥𝑖 − ҧ𝑥 𝑦𝑖 − ത𝑦

𝑥𝑖 − ҧ𝑥 2

Therefore

𝐼 =𝑛σ𝑖=1

𝑛 σ𝑗=1,𝑗≠𝑖𝑛 𝑤𝑖𝑗(𝑥𝑖 − ҧ𝑥)(𝑥𝑗 − ҧ𝑥)


I, is equivalent to the slope of coefficient of a linear regression of the

spatial Wx on the observation vector x, measured in deviation from their

means. ෪𝑊𝑋 = 𝛽1 ෨𝑋

where the tilde represents the variable is in deviation from the mean


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A very useful tool for understanding the Moran’s I test is the

Moran’s scatterplot:


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𝑥 = 𝜌𝑊𝑥 + 𝜖 → 𝑥 = (𝐼 − 𝜌𝑊)−1𝜖

Example with 𝜌 = 0.99


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𝑥 = 𝜌𝑊𝑥 + 𝜖 → 𝑥 = (𝐼 − 𝜌𝑊)−1𝜖

Example with 𝜌 = 0.00


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𝑥 = 𝜌𝑊𝑥 + 𝜖 → 𝑥 = (𝐼 − 𝜌𝑊)−1𝜖

Example with 𝜌 = −0.99


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Many spatial econometric based studies use Moran’s I to motivate the

employment of spatial econometrics to analyze a specific variable

It can be complemented by means of spatially conditioned stochastic

kernels estimates of 𝑔 𝑦|𝑊𝑦 :

𝑔 𝑦|𝑊𝑦 =𝑓(𝑦,𝑊𝑦)

𝑓(𝑦)

where 𝑔 𝑦|𝑊𝑦 is the density of y conditional on the neighbor’s values of y

The kernel 𝑔 𝑦|𝑊𝑦 is a conditional density and shows the probability that a

given region has a specific value or state of y, given their neighbor’s values.

To estimate 𝑔 𝑦|𝑊𝑦 one first estimates the joint density of f 𝑦,𝑊𝑦 and

then the marginal density 𝑓 𝑦 is calculated by integrating over y.


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Visual inspection of the kernel shape g(y|Wy) allows to observe if spatial patterns

are strong or weak.

If probability density flows along the main diagonal it means y would have had similar

values even if it have had different neighbor’s (space not relevant)

If probability density flows in paralallel to the y axis it means spatial effects are

important drivers of the observed distribution of y. (space is relevant)


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Moran’s I calculation (example of Kosfeld lecture notes)

Assume we have the following spatial arrangement of units:

and the following geo-referenced variable


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Unemployment

Region 1 8

Region 2 6

Region 3 6

Region 4 3

Region 5 2

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The spatial scheme:

can be represented by the following 5 x 5 contiguity matrix:

𝑊 =

0 1 1 0 01 0 1 1 0100

110

010

101

010

We will work first with the non-standardized W


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𝐼 =𝑛σ𝑖=1

𝑛 σ𝑗=1,𝑗≠𝑖𝑛 𝑤𝑖𝑗

∗ (𝑥𝑖 − ҧ𝑥)(𝑥𝑗 − ҧ𝑥)


Table 1: Cross products (𝑥𝑖 − ҧ𝑥)(𝑥𝑗 − ҧ𝑥)


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UR

Region 1 8

Region 2 6

Region 3 6

Region 4 3

Region 5 2

Avg 5

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Unstandardized weights

Table 2: Unstandardized weights (𝑤𝑖𝑗∗ )


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Table 3: weighted cross products: σ𝑖=1𝑛 σ𝑗=1,𝑗≠𝑖

𝑛 𝑤𝑖𝑗∗ (𝑥𝑖 − ҧ𝑥)(𝑥𝑗 − ҧ𝑥)


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Table 4: Sum of squared deviations σ𝑖=1𝑛 𝑥𝑖 − ҧ𝑥 2

Moran’s I (unstandardized weight matrix):

𝐼 =𝑛

𝑆

𝑧′𝑊𝑧

𝑧′𝑧=

5

12

18

24= 0.3125


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Opening Matlab you can reproduce these tedious calculations by

typing:

n=length(x);

xm=mean(x);

xmv=ones(n,1)*xm;

cprod1 = (x-xmv)'*W*(x-xmv); % this is Table 3

xsqr=(x-xmv)'*(x-xmv); % this is Table 4 (part of the denominator)

Wv=vec(W,2);

S=sum(Wv); %needed for the calculation of the numerator

I = [n/S]*[cprod1/xsqr]; % this is the Moran’s I value

res.I = I;


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Calculation of Moran’s I with standardized weight matrix:

𝐼 =σ𝑖=1𝑛 σ𝑗=1,𝑗≠𝑖

𝑛 𝑤𝑖𝑗(𝑥𝑖 − ҧ𝑥)(𝑥𝑗 − ҧ𝑥)

σ𝑖=1𝑛 𝑥𝑖 − ҧ𝑥 2

The cross-products are the same as in our previous case: (𝑥𝑖 − ҧ𝑥)(𝑥𝑗 − ҧ𝑥)


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The row-normalized W, however, is different:

Table 5: Standardized weights 𝑤𝑖𝑗

Table 6: Weighted cross-products𝑤𝑖𝑗(𝑥𝑖 − ҧ𝑥)(𝑥𝑗 − ҧ𝑥)


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Again, the sum of squared deviations is σ𝑖=1𝑛 𝑥𝑖 − ҧ𝑥 2 = 24

Thus, the Moran’s I (for the standardized W) is slightly higher:

𝐼 =σ𝑖=1𝑛 σ𝑗=1,𝑗≠𝑖

𝑛 𝑤𝑖𝑗 (𝑥𝑖 − ҧ𝑥)(𝑥𝑗 − ҧ𝑥)

σ𝑖=1𝑛 𝑥𝑖 − ҧ𝑥 2

=11

24= 0.4583

Check in Matlab that by inputing the standardized W you actually

get this value.

This was the standardized W matrix


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Moran’s I statistical significance

To analyze significance of the observed spatial correlated pattern we need a Z statistic that allows

us to calculate p-values

It is given by:

𝑍 𝐼 =𝐼 − 𝐸(𝐼)

𝑉𝐴𝑅(𝐼)~𝑁(0,1)

Expected value 𝐸 𝐼 = −1

𝑛−1

Variance of I (normal aprox)

𝑉𝐴𝑅 𝐼 =𝑛2𝑆1+𝑛𝑆2+3𝑆0

2

(𝑛−1)𝑆02 − 𝐸(𝐼) 2

where:

𝑆0 =

𝑖=1

𝑛

𝑗=1

𝑛

𝑤𝑖𝑗 ; 𝑆1 =1

2

𝑖=1

𝑛

𝑗=1

𝑛

(𝑤𝑖𝑗 + 𝑤𝑗𝑖)2 ;

𝑆2 =

𝑖=1

𝑛

𝑗=1

𝑛

𝑤𝑖𝑗 +

𝑗=1

𝑛

𝑤𝑗𝑖

2


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In the example of 5 regions (n=5) we have calculated a value of Moran’s I of

0.4583 on the basis of the standardized weight matrix

Despite the small sample, we use the proportional approximation of the significance

of the Moran’s I test for illustrative purposes.

Expected value 𝐸 𝐼 = −1

𝑛−1= −0.25

𝑆0 =

𝑖=1

𝑛

𝑗=1

𝑛

𝑤𝑖𝑗 = 5


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𝑆1 =1

2

𝑖=1

𝑛

𝑗=1

𝑛

𝑤𝑖𝑗 +𝑤𝑗𝑖2

= [ 𝑤12 + 𝑤212 + 𝑤13 +𝑤31

2 + 𝑤14 +𝑤412 + 𝑤15 +𝑤51

2

+ 𝑤21 +𝑤122 + 𝑤23 +𝑤32

2 + 𝑤24 +𝑤422 + 𝑤25 +𝑤52

2

+ 𝑤31 +𝑤132 + 𝑤32 +𝑤23

2 + 𝑤34 +𝑤432 + 𝑤35 +𝑤53

2

+ 𝑤41 +𝑤142 + 𝑤42 +𝑤24

2 + 𝑤43 +𝑤342 + 𝑤45 +𝑤54

2

+ 𝑤51 +𝑤152 + 𝑤52 + 𝑤25

2 + 𝑤53 + 𝑤352 + 𝑤54 + 𝑤45

2

= [ 𝑤12 + 𝑤212 + 𝑤13 +𝑤31

2 + 𝑤14 +𝑤412 + 𝑤15 +𝑤51

2

+ 𝑤21 +𝑤122 + 𝑤23 +𝑤32

2 + 𝑤24 +𝑤422 + 𝑤25 +𝑤52

2

+ 𝑤31 +𝑤132 + 𝑤32 +𝑤23

2 + 𝑤34 +𝑤432 + 𝑤35 +𝑤53

2

+ 𝑤41 +𝑤142 + 𝑤42 +𝑤24

2 + 𝑤43 +𝑤342 + 𝑤45 +𝑤54

2

+ 𝑤51 +𝑤152 + 𝑤52 + 𝑤25

2 + 𝑤53 + 𝑤352 + 𝑤54 + 𝑤45

2

= 𝑤12 +𝑤212 + 𝑤13 + 𝑤31

2 + 𝑤21 +𝑤122 + 𝑤23 +𝑤32

2 + 𝑤24 +𝑤422 +

𝑤31 + 𝑤132 + 𝑤32 + 𝑤23

2 + 𝑤34 + 𝑤432 + 𝑤42 + 𝑤24

2 + 𝑤43 + 𝑤342 +

𝑤45 + 𝑤542 + 𝑤54 + 𝑤45

2


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= 𝑤12 +𝑤212 + 𝑤13 +𝑤31

2 + 𝑤21 + 𝑤122 + 𝑤23 +𝑤32

2

+ 𝑤24 + 𝑤422 + 𝑤31 + 𝑤13

2 + 𝑤32 + 𝑤232 + 𝑤34 + 𝑤43

2

+ 𝑤42 + 𝑤242 + 𝑤43 + 𝑤34

2 + 𝑤45 + 𝑤542 + 𝑤54 + 𝑤45

2

Recall that the standardized W is given by:

=1

2+1

3

2

+1

2+1

3

2

+1

3+1

2

2

+1

3+1

3

2

+1

3+1

3

2

+1

3+1

2

2

+1

3+1

3

2

+1

3+1

3

2

+1

3+1

3

2

+1

3+1

3

2

+1

3+ 1

2

= 4.5


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𝑆2 =

𝑖=1

𝑛

𝑖=1

𝑛

𝑤𝑖𝑗 +

𝑖=1

𝑛

𝑤𝑗𝑖

2

=

𝑖=1

𝑛

𝑤𝑖∗ + 𝑤∗𝑖2

𝑖=1

𝑛

𝑤𝑖∗ +𝑤∗𝑖2 =

= 𝑤1∗ + 𝑤∗12 + 𝑤2∗ + 𝑤∗2

2 + 𝑤3∗ + 𝑤∗32 + 𝑤4∗ + 𝑤∗4

2 + 𝑤5∗ + 𝑤∗52

= 1.16

Suming up we have:

• 𝑆0 = 5

• 𝑆1 = 4.5

• 𝑆2 =21

18= 1.16

• 𝑉𝐴𝑅 𝐼 =𝑛2𝑆1+𝑛𝑆2+3𝑆0

2

(𝑛−1)𝑆02 − 𝐸 𝐼

2= 0.0745

• 𝑍 𝐼 = 2.5955 → 𝑝 = 0.0095


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MATLAB code for the test on the statistical significance of the Moran’s I

EI=-(1./(n-1)); % expected value

S0=S; s1=zeros(n,n);

for i =1:n % loop over i

for j =1:n % loop over j

s1(i,j) = (W(i,j) + W(j,i)).^2;

end

end

s1v=vec(s1,2); S1=0.5*sum(s1v); %vectorize s1 and sum

s2=zeros(n,1);

for i= 1:n

s2(i) =([sum(W(i,:))+ sum(W(:,i))].^2);

end

S2=sum(s2);

NumVarI=(n^2)*S1-n*S2 + 3.*(S0).^2; DenVarI=(n^2-1)*(S0).^2;

VARI=(NumVarI./DenVarI)-EI^2;

Z=(I-EI)./(sqrt(VARI)); prob = norm_prb(Z);


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If you copy previous sequence of code to calculate Moran’s I and to

calculate its significance in the Matlab editor and save it as a

function, each time you provide the function a variable x and a W

matrix, Matlab will calculate for you the corresponding values.


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Moran scatterplot (5 regions)


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The Moran’s scatterplot is a special form of a bivariate scatterplot which

makes use of the standardized values of the pairs (X,WX)

Augmented with the regression line it can be used to asses to the degree of

fit and to identify outliers

Table: Types of local spatial association

Positive spatial association: HH and LL

Negative spatial association: LH and HL (spatial outliers in case of + spatial

autocorrelation


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Spatially lagged geo-referenced variable

(WX)

High Low

Geo-referenced

variable (X)

High Quadrant I (HH) Quadrant IV(HL)

Low Quadrant II (LH) Quadrant III (LL)

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Goals

Draw a Moran’s Scatterplot using of the unemployment rate in

European regions using the W matrices created previously. We will use

the 5-nearest neighbors and the exponential with cut-off at the first

quartile.

Run a OLS regression of X (unemployment rates) and WX (the

spatial lag of unemployment rates) using the two matrices with the

variables expressed in deviations with respect the sample average (or

use the originals with a constant)

Check the slopes obtained in the regression with the results of the

MoranI.m function we have coded. Is the Moran’s I statistically

significant at the 5% level ?

Practice (Global Spatial Autocorrelation)

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Switch to Matlab“Tutorial2_Global_Spat_Autocorr.m”

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Matlab functions

MoranI.m → function that calculates Moran’s I statistic and its

significance level

OLS_demo.m → function to perform basic OLS regression

prt.m → prints the results into the command window of a regression

structure results.

Practice (Part Global Spatial Autocorr)

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Moran’s scatterplot of unemployment rates in European NUTS-2

regions


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Main message: spatial correlation is NOT universal like time

correlation. Depends on our defenition of space and interdependence.

Finding the true W is of key importance to model real spatial associations


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Global spatial correlation summarizes the strength of spatial

dependencies by a single statistic.

LISA focuses on heterogeneity of spatial association over space.

LISA provide detailed information of spatial clustering.

LISA for each observation tells how values around that observation are

clustered.

Useful for detecting local clusters and spatial outliers.

A local cluster is characterized by a concentration of high values of an

attribute variable X. A spatial clustering of high value contiguous

region is called hot spot whereas a spatial clustering of low value

contiguous regions is called cold spot.

Spatial outliers are regions with reversed orientation compared to the

predominant global one

Local Indicators of Spatial Association (LISA)

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We deal with one of two may well-known LISA:

The Local Moran’s I statistic

Useful for detecting spatial outliers and general clustering

formations (we focus on this one)

The Getis-Ord G statistic

Better suited for identifying specific cluster formations


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Local Moran statistic I(i) detects local spatial autocorrelation. The I(i)’s are

indicators of local instability. They decompose the Moran’s I into contributions for

each location.

According to this property, Local Moran’s I can be used for two purposes:

Indicators of local spatial clusters

Diagnostics for outliers in global spatial patterns

Local Moran’s I statistic:

𝐼𝑖 =(𝑥𝑖− ҧ𝑥) σ𝑗=1

𝑛 𝑤𝑖𝑗 𝑥𝑗− ҧ𝑥

1

𝑛σ𝑗=1𝑛 𝑥𝑗− ҧ𝑥

2

Numerator: determines the sign of I(i)

+ if both the i-th region and neighbors have above or below average values in the geo-

referenced variable X and

(-) if the i-th region as an above(below) neighboring regions have a below(above) average

values of X

Denominator: standardization of the cross-product by the variance of the georeferenced

variable X


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Below is a plot of Local Moran Z-scores for the 2004 Presidential

Elections. Higher absolute values of z scores (red) indicate the presence of

hot spots", where the percentage of the vote received by President Bush was

significantly different from that in neighboring counties.


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Example of Local Moran’s I:

We calculate Local Moran statistics with the standardized weights wij:

The sum of squared deviations from the mean has already been calculated with

the Moran’s I:

1/𝑛

𝑗=1

𝑛

𝑥𝑗 − ҧ𝑥2= 4.8

This will be our denominator in the following calculations


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Region 1

𝐼1 =(𝑥1 − ҧ𝑥)σ𝑗=1

𝑛 𝑤1𝑗 𝑥𝑗 − ҧ𝑥

σ𝑗=1𝑛 𝑥𝑗 − ҧ𝑥

2/𝑛

= 0.625

𝑗=1

𝑛

𝑤1𝑗 𝑥𝑗 − ҧ𝑥 =1

26 − 5 +

1

26 − 5 = 1

𝑥1 − ҧ𝑥 = 8 − 5

So:

𝐼1 =(𝑥1 − ҧ𝑥)σ𝑗=1



2/𝑛

=(3)(1)

4.8= 0.625


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Region 2

𝐼2 =(𝑥2 − ҧ𝑥)σ𝑗=1



2/𝑛

= 0.1389

𝑗=1

𝑛


38 − 5 +

1

36 − 5 +

1

33 − 5 =

2

3

𝑥2 − ҧ𝑥 = 6 − 5 = 1

So:

𝐼2 =(𝑥2 − ҧ𝑥)σ𝑗=1



2/𝑛

=1 (

23)

4.8= 0.1389


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Region 3

𝐼3 =(𝑥3 − ҧ𝑥)σ𝑗=1



2/𝑛

= 0.1389

𝑗=1

𝑛


38 − 5 +

1

36 − 5 +

1

33 − 5 =

2

3

𝑥3 − ҧ𝑥 = 6 − 5 = 1

So:

𝐼3 =(𝑥3 − ҧ𝑥)σ𝑗=1



2/𝑛

=1 (

23)

4.8= 0.1389


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Region 4

𝐼4 =(𝑥4 − ҧ𝑥) σ𝑗=1



2/𝑛

= 0.1389

𝑗=1

𝑛


36 − 5 +

1

36 − 5 +

1

32 − 5 = −

1

3

𝑥4 − ҧ𝑥 = 3 − 5 = −2

So:

𝐼4 =(𝑥4 − ҧ𝑥)σ𝑗=1



2/𝑛

=(−2)(−

13)

4.8= 0.1389


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Region 5

𝐼5 =(𝑥5 − ҧ𝑥)σ𝑗=1



2/𝑛

= 1.25

𝑗=1

𝑛

𝑤5𝑗 𝑥𝑗 − ҧ𝑥 = 1 3 − 5 = −2

𝑥5 − ҧ𝑥 = 2 − 5 = −3

So:

𝐼5 =(𝑥5 − ҧ𝑥)σ𝑗=1



2/𝑛

=(−2)(−3)

4.8= 1.25


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Does the average of local Moran’s I produce the global Moran’s I calculated

before?

Recall that from our previous example with standardized weights we have:

I = 0.4583

𝐼 =1

5

𝑖

5

𝐼𝑖 =1

50.625 + 0.1389 + 0.1389 + 0.1390 + 1.25 = 0.4583

Interpretation of the results in this example:

- A local spatial cluster is identified around region 5 and to somewhat les

around region 1, as both, 𝐼5 and 𝐼1 exceed the global Moran’s I noticeably

- Since all 𝐼𝑖 values exceed their expected value, no outlying region with

respect to the orientation is identififed


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Goals

Estimate Local Moran’s I using unemployment rates in European

regions using the exponential with cut-off at the third quartile W

matrix. Use the function “localMoranI.m”

Check that the average of the Local Moran I’s add up to the global

Moran I (you need to calculate the global Moran for these Ws)

Create a scatterplot of the estimated Local Moran I’s and another of its

estimated p-values. Is there any pattern?

Create one map of the estimated Local Moran’s I and another one of

the estimated p-values

Practice (Part Local Spatial Autocorr)

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Switch to Matlab“Tutorial3_Local_Spat_Autocorr.m”

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You should obtain a global Moran’s I of 0.8797 and that the average over

Local Moran’s I is also 0.8797

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Local Moran’s I Map


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Local cluster of high

unemployment rates

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Some simplified insights in the south-of Spain local cluster of

unemployment:

(1) Low-valued addded specialization in agriculture (olive oil) dependent on

subsidies and strong public-sector employment share (non-market services)

(2) Distribution of land and income, highly unequal (due to historical reasons)

(1) + (2) → dependent population

(3) Socialist Party stronghold → Monopoly of political power

(4) High levels of corruption

(3) + (4) → clientelistic regime

(5) High levels of religiosity → traditional values and few entrepreneurship

(6) High levels of luminosity + nice weather → strong amenity

All together: corrupt regime with dependent population that does not migrate

to other region (there is not regional re-balancing of the labor supply)


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spatial and regional economic analysis mini-course

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