Creating and Managing a W Matrix

Transcription

1 Creating and Managing a W Matrix Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign hrtdmrt2@illinois.edu Junel 22th, 2016 C. Hurtado (UIUC - Economics) Spatial Econometrics

2 On the Agenda 1 Main Feature of Spatial Data: Spatial Dependence 2 Creating and Managing a W Matrix 3 Measuring Spatial Correlation: A simple example 4 Creation of a W Matrix for Irregular Data C. Hurtado (UIUC - Economics) Spatial Econometrics

3 Main Feature of Spatial Data: Spatial Dependence On the Agenda 1 Main Feature of Spatial Data: Spatial Dependence 2 Creating and Managing a W Matrix 3 Measuring Spatial Correlation: A simple example 4 Creation of a W Matrix for Irregular Data C. Hurtado (UIUC - Economics) Spatial Econometrics

4 Main Feature of Spatial Data: Spatial Dependence Main Feature of Spatial Data: Spatial Dependence The observation of a phenomenon in one place provides statistical information also about the spatial contex. Tobler s first law of geography: Everything is related to everything else, but close things are more related than things that are far apart Positive dependence: - Objects that are close in space tend to assume similar values - Point tend to cluster in space Negative dependence: - Objects that are close in space tend to assume dissimilar values - Point tend to overdisperse C. Hurtado (UIUC - Economics) Spatial Econometrics 1 / 11

5 Main Feature of Spatial Data: Spatial Dependence Main Feature of Spatial Data: Spatial Dependence The observation of a phenomenon in one place provides statistical information also about the spatial contex. Tobler s first law of geography: Everything is related to everything else, but close things are more related than things that are far apart Positive dependence: - Objects that are close in space tend to assume similar values - Point tend to cluster in space Negative dependence: - Objects that are close in space tend to assume dissimilar values - Point tend to overdisperse C. Hurtado (UIUC - Economics) Spatial Econometrics 1 / 11

6 Main Feature of Spatial Data: Spatial Dependence Spatial autocorrelation Is the correlation of a variable with itself corr(x i, X j) If correlation is positive, objects that are close in the plane tend to have similar values of X We need to define the concept of close or neighbourhood. There are various possible definitions. - Critical cut-off distance neighbourhood: Two sites are neighbours if 0 d d, for a given distance. - Nearest neighbour: d i,j = min d i,k k - Contiguity-based neighbourhood: based on the adjacency between two polygons, i.e., if they share a common boundary. C. Hurtado (UIUC - Economics) Spatial Econometrics 2 / 11

7 Main Feature of Spatial Data: Spatial Dependence Spatial autocorrelation Is the correlation of a variable with itself corr(x i, X j) If correlation is positive, objects that are close in the plane tend to have similar values of X We need to define the concept of close or neighbourhood. There are various possible definitions. - Critical cut-off distance neighbourhood: Two sites are neighbours if 0 d d, for a given distance. - Nearest neighbour: d i,j = min d i,k k - Contiguity-based neighbourhood: based on the adjacency between two polygons, i.e., if they share a common boundary. C. Hurtado (UIUC - Economics) Spatial Econometrics 2 / 11

8 Creating and Managing a W Matrix On the Agenda 1 Main Feature of Spatial Data: Spatial Dependence 2 Creating and Managing a W Matrix 3 Measuring Spatial Correlation: A simple example 4 Creation of a W Matrix for Irregular Data C. Hurtado (UIUC - Economics) Spatial Econometrics

9 Creating and Managing a W Matrix Creating and Managing a W Matrix Define N i as the set of all neighbours of site S i. We consider that S i / N i. Detonte by η i the cardinality of N i. The W matrix can be definde by: { 1 if si N i w ij = 0 otherwise Notice that η i = j w ij Most of the time, for the practical applications, the W matrix is row-standardized, that is w ij = wij j wij = wij η i The virtue of the row-standardized matrix is that wij = 1 j C. Hurtado (UIUC - Economics) Spatial Econometrics 3 / 11

10 Creating and Managing a W Matrix Creating and Managing a W Matrix Define N i as the set of all neighbours of site S i. We consider that S i / N i. Detonte by η i the cardinality of N i. The W matrix can be definde by: { 1 if si N i w ij = 0 otherwise Notice that η i = j w ij Most of the time, for the practical applications, the W matrix is row-standardized, that is w ij = wij j wij = wij η i The virtue of the row-standardized matrix is that wij = 1 j C. Hurtado (UIUC - Economics) Spatial Econometrics 3 / 11

11 Creating and Managing a W Matrix Creating and Managing a W Matrix Define N i as the set of all neighbours of site S i. We consider that S i / N i. Detonte by η i the cardinality of N i. The W matrix can be definde by: { 1 if si N i w ij = 0 otherwise Notice that η i = j w ij Most of the time, for the practical applications, the W matrix is row-standardized, that is w ij = wij j wij = wij η i The virtue of the row-standardized matrix is that wij = 1 j C. Hurtado (UIUC - Economics) Spatial Econometrics 3 / 11

12 Creating and Managing a W Matrix Spatial Lag In spatial context the concept of lag is difficult to extend due to the multilaterality of proximity in space. A lagged value can be any of the neighbours according to the neighbourhood definition. Let us define the spatial lag operator as L[X i] = 1 η i = 1 η i = X j j N i n i w ijx j j=1 n i j=1 = WX w ij X j C. Hurtado (UIUC - Economics) Spatial Econometrics 4 / 11

13 Creating and Managing a W Matrix Spatial Lag In spatial context the concept of lag is difficult to extend due to the multilaterality of proximity in space. A lagged value can be any of the neighbours according to the neighbourhood definition. Let us define the spatial lag operator as L[X i] = 1 η i = 1 η i = X j j N i n i w ijx j j=1 n i j=1 = WX w ij X j C. Hurtado (UIUC - Economics) Spatial Econometrics 4 / 11

14 Creating and Managing a W Matrix Measures of Spatial Dependence If L is the lag operator, in time series we have: ρ k = cov(xt, LK X t) var(x) In space we can use the same expresion using the concept of spatial lag. Let us deffine Moran s I as I = i j w ij (X i X)(X j X) i (Xi X) 2 = Y WY Y Y where Y = X X C. Hurtado (UIUC - Economics) Spatial Econometrics 5 / 11

15 Creating and Managing a W Matrix Measures of Spatial Dependence If L is the lag operator, in time series we have: ρ k = cov(xt, LK X t) var(x) In space we can use the same expresion using the concept of spatial lag. Let us deffine Moran s I as I = i j w ij (X i X)(X j X) i (Xi X) 2 = Y WY Y Y where Y = X X C. Hurtado (UIUC - Economics) Spatial Econometrics 5 / 11

16 Creating and Managing a W Matrix Measures of Spatial Dependence Moran s statistic can be used to test the spatial correlation using the regression residual. It plays the role of the Durbin Watson test in time series. It takes the form of a correlation between the regression residuals and their spatially laged values I = ɛ W ɛ ɛ ɛ I Moran test suffers from the limitation of not being based on an explicit alternative hypothesis, but it is equivalent to the LM test. C. Hurtado (UIUC - Economics) Spatial Econometrics 6 / 11

17 Measuring Spatial Correlation: A simple example On the Agenda 1 Main Feature of Spatial Data: Spatial Dependence 2 Creating and Managing a W Matrix 3 Measuring Spatial Correlation: A simple example 4 Creation of a W Matrix for Irregular Data C. Hurtado (UIUC - Economics) Spatial Econometrics

18 Measuring Spatial Correlation: A simple example Creating and Managing a W Matrix The creation and the management of a W matrix is the trickiest part in running a spatial regression in any software. It is also what distinguishes software with spatial capabilities from standard econometric software. For this reason we will devote a significant part of the present chapter to discuss some of the most important steps needed for its creation. All the R procedures that will be illustrated in the present section are contained in the package spdep. To install the package: install.packages( spdep ) To call the package: library(spdep) C. Hurtado (UIUC - Economics) Spatial Econometrics 7 / 11

19 Measuring Spatial Correlation: A simple example Creation of a W Matrix for Regular Grid Data Consider the case of a regular square lattice grid of dimension 3 by 3. R generates the list of neighbors automatically with the command: Wnb< cell2nb(3,3,type= rook ) C. Hurtado (UIUC - Economics) Spatial Econometrics 8 / 11

20 Measuring Spatial Correlation: A simple example Creation of a W Matrix for Regular Grid Data The command indicates that we want to change our data from a cell system (cell) to (2) a list of neighbors (nb). The object Wnb is just a list of neighbors. If we type Wnb we obtain a summary of the information contained in it Once this object is created, we have to transform it into an actual matrix, say W. W< nb2listw(wnb) The command indicates that we want to change our data from a list of neighbors (nb) to (2) a weight matrix (listw). In order to visualize the actual neighbors type: W$weights Once the weight matrix W is created, the spatially lagged variable of a variable X can be easily obtained through the command WX< lag.listw(w,x) C. Hurtado (UIUC - Economics) Spatial Econometrics 9 / 11

21 Creation of a W Matrix for Irregular Data On the Agenda 1 Main Feature of Spatial Data: Spatial Dependence 2 Creating and Managing a W Matrix 3 Measuring Spatial Correlation: A simple example 4 Creation of a W Matrix for Irregular Data C. Hurtado (UIUC - Economics) Spatial Econometrics

22 Creation of a W Matrix for Irregular Data Creation of a W Matrix for Irregular Data Consider now the case of an irregular set of regions and let us as an example the 20 Italian regions. Use: library( maptools ) Create an object with the polygons of Itally: Ita< readshapepoly( Reg2014 ED50 ) To calculate the contiguity-based neighbors use: contnb< poly2nb(ita,queen=t) The Queen criterion specified ensures that two regions are considered neighbors is they have a common boundary. The command indicates that we want to change our data from a list of polygons (poly) to (2) a list of neighbors (nb). However, this command does not tolerate the presence of isolated areas (i.e. iland) C. Hurtado (UIUC - Economics) Spatial Econometrics 10 / 11

23 Creation of a W Matrix for Irregular Data Creation of a W Matrix for Irregular Data We can force the command to create the W matrix to include these areas. In this case the W matrixwill be generated with one or more areas with all zero in the corresponding line. The problem may be eliminated by generating a list of neighbors base on minimum threshold distance use the comand: nbit< dnearneigh(coords,0,380000,longlat=f) C. Hurtado (UIUC - Economics) Spatial Econometrics 11 / 11