Spatial Regression Modeling

Transcription

1 Spatial Regression Modeling Paul Voss & Katherine Curtis The Center for Spatially Integrated Social Science Santa Barbara, CA July 12-17, 2009 Day 1

2 Objective Provide a solid introduction and overview of the concepts and techniques of spatial regression analysis with plenty of hands-on experience in the afternoon lab sessions

3 What s the point? Data that are referenced to location bring an extremely important additional amount of valuable information to a data analysis But it also brings some (possibly unfamiliar) pitfalls that require a new awareness as that analysis proceeds

4 Plan for the week (1) Today: Understanding Spatial Data Broad overview of spatial data, spatial data analysis and core spatial concepts. Why spatial is special Classical linear regression model assumptions underlying OLS consequences of violations of assumptions why spatial processes violate OLS assumptions Introduction to EDA & ESDA Lab: Shapefiles & introduction to GeoDa & EDA using R Tuesday: Diagnosing spatial autocorrelation Understanding & measuring global spatial autocorrelation Weights matrices Understanding & measuring local spatial autocorrelation Moran scatterplot LISA statistics Lab: Global & local measures of spatial autocorrelation

5 Plan for the week (2) Wednesday: Common modeling strategies for spatial processes Understanding spatial processes spatial heterogeneity spatial dependence Spatial regression models OLS in GeoDa understanding GeoDa regression diagnostics spatial lag model; spatial error model No Lab. Free time in Santa Barbara! Thursday: Focus on spatial heterogeneity Spatial heterogeneity in relationships Introduction to GWR GWR theory and approach GWR software Lab: GWR hands-on using R

6 Plan for the week (3) Friday: Alternative spatial processes Reminder that there are some spatial data analysis approaches that have not been covered in this class point pattern analysis geostatistical methods; kriging Exciting things on the horizon Resources for pursuing these topics intensively Lab: Research presentations Open forum

7 Plan for today A good motivational example Why spatial is special characteristics of spatial data problems caused by spatial data Spatial analysis vs. spatial data analysis Broad, gentle overview of spatial data and spatial data analysis Classes of problems in spatial data analysis Review OLS assumptions & violations Exploratory Data Analysis Introduction to afternoon lab

8 Any questions as we get started?

9 Some beginning facts Regression is the workhorse of quantitative social science Much social science data is spatially referenced Spatially referenced data bring special problems to an analysis heterogeneity of observational units heteroskedasticity spatial autocorrelation residual dependence A consequence of these special problems is that the assumption of iid errors in a standard OLS regression specification is violated, and statistical inference from such a model is not valid

10 Motivation Omer R. Galle, Walter R. Gove, & J. Miller McPherson Population Density and Pathology: What Are the Relations for Man Science 176(4030):23-30 data: 75 community areas in Chicago for measures of social pathology as function of crowding, controlling for social class & ethnicity the greater the density, the greater the fertility (p. 176) Colin Loftin & Sally K. Ward A Spatial Autocorrelation Model of the Effects of Population Density on Fertility American Sociological Review 48(1): the GGM findings with regard to fertility are an artifact of the failure to recognize the presence of disturbance variables which are spatially autocorrelated (p. 127) Moral: When analyzing spatially referenced data, it s highly useful to know something about the rudiments of spatial data analysis (i.e., some understanding of why spatial is special )

11 Okay So why is spatial special? Scale dependency Robinson (ASR, 1950) Ecological Fallacy The relationship between ecological and individual correlations which is discussed in this paper provides a definite answer as to whether ecological correlations can validly be used as substitutes for individual correlations. They cannot. (p. 357) MAUP (Modifiable Areal Unit Problem) Habitual users of ecological correlations know that the size of the coefficient depends to a marked degree upon the number of sub-areas. [T]he size of the ecological correlation [will increase numerically as consolidation of smaller areas into larger areas takes place]. (Robinson, pp )

12 Why is spatial special? (2) Observational areas are generally of different size Spatial heterogeneity error heteroskedasticity

13 Counties in U.S. South: 2000 Census Brewster Co. TX: Area: 6,193 mi 2 Fairfax City, VA: Area: 6.3 mi 2 n = 1,387

14 Counties in U.S. South: 2000 Census Loving Co. TX: Pop: 67 Harris Co. TX: Pop: 3,400,578 n = 1,387

15 Why is spatial special? (3) Observational areas are generally of different size (geographic size; population size) Heteroskedasticity in regression errors Neighboring areas are similar Tobler s 1 st law of Geography: Everything is related to everything else, but near things are more related than distant things. (1970:236) (positive) spatial autocorrelation

16 Proportion of Children (under age 18) in Poverty: 2000 Census Loudoun Co. VA: Source: SF3 Table P87 Starr Co. TX: 0.595

17 Why is spatial special? (4) Observational areas are generally of different size Heteroskedasticity in regression errors Neighboring areas are similar Tobler s 1 st law of Geography: Everything is related to everything else, but near things are more related than distant things. (1970:236) (positive) spatial autocorrelation Probable stumbling blocks when modeling the data Again the assumption of iid errors in a standard OLS regression specification is violated and statistical inference is not valid

18 Spatial versus Non-Spatial Data Analysis

19 Take these two maps, for example Any traditional data analysis that does not utilize the location & spatial arrangement (topological information) of the data will lead to identical results for the two maps

20 And why is this important? What makes the methods of modern [spatial data analysis] different from many of their predecessors is that they have been developed with the recognition that spatial data have unique properties and that these properties make the use of methods borrowed from aspatial disciplines highly questionable Fotheringham, Brunsdon & Charlton, Quantitative Geography: Perspectives on Spatial Data Analysis Sage, 2000 p. xii

21 Because of these unique properties, if we blithely carry out an OLS regression using aggregated geographic data surely some large subset of the following undesirable horrors almost certainly awaits us (the curse of Tobler s 1 st Law) Our estimated regression coefficients are biased and inconsistent, or Our estimated regression coefficients are inefficient Our R 2 statistic is exaggerated We ve made incorrect inferences We ll never get it published or shouldn t!

22 Given these problems, why would anyone bother to analyze spatial data? There s lots of it! Occasional need for non-disclosure of individual-level data Space is important Space as a means of organizing human activities Location as a means of integrating interesting data Some interesting questions can (only) be examined with spatial data

23 Some interesting questions that might be addressed using modern spatial analysis: Local tax rates ( spillover in y?) Expenditures for police ( spillover in x?) Demographic analysis: Are Quitman & Tallahatchie counties (two contiguous counties in the Mississippi Delta) really two separate observations? ( spillover in ε?)

24 Now, Let s Define Some Terms

25 What exactly are spatial data? data where, in addition to attribute values relating to the primary phenomena of interest, the relative spatial locations of observations are also recorded

26 And what is spatial regression analysis? Regression using spatial data where explicit attention is given to location and arrangement of geographic units Even if we don t really care about spatial processes

27 Spatial Analysis versus Spatial Data Analysis

28 GIS Spatial Analysis P-median problems Maximal covering problem Location set covering problem Traveling salesman problem (TSP)

29 GIS Spatial Analysis Spatial Data Analysis Spatial Statistics

30 Spatial Data Analysis

31 GIS Spatial Analysis Spatial Data Analysis Spatial Statistics Event Data Lattice Data Spatial Econometrics Spatial Regression Analysis Geostatistical Data

32 Types of Spatial Data Event data (point data) Spatially continuous data (geostatistical data) Lattice data (regionalized data) Spatial interaction data (flow data)

33 Our focus of attention in this workshop will primarily be on Lattice Data On Friday we ll briefly touch on the other kinds of spatial data and other spatial data analytic approaches

34 Lattice Data Have one or more variables whose values are measured over a set of areas Interest focuses on the attribute values, not on the locations which are known and unchanging Objectives: Detecting patterns in the spatial arrangement of attribute values examining the relationships among the set of variables taking into account any spatial effects present Approach: Exploratory spatial data analysis Confirmatory spatial data analysis (spatial regression)

35 Some early cautions: Our goal is to correctly model and draw proper inferences about an unobserved, random DGP (random field) spatial process? Spatial heterogeneity? spatial dependence? time? sampling perspective? Spatial autocorrelation Scale issues; scale dependency; aggregation bias; boundary issues The tools are pretty good, but along the way many subjective decisions are made defining neighborhood choosing a weights matrix

36 Before taking a closer look at the meanings of Spatial Autocorrelation, Spatial Heterogeneity and Spatial Dependence, let s take a closer look at the traditional OLS regression model

37 Standard OLS Regression y i = β + β x + β x β x i 2 2i k ki ε i In matrix notation: y = X β + ε (n x 1) (n x k+1) (k+1 x 1) (n x 1) and where: βˆ = ( X T 1 X ) X T y σˆ 2 = 1 ( n k e 1) T e

38 For those of you who glaze over at the sight of that, the next 3 slides are for you

39 How do we get from this y i = β + β x + β x β x i 2 2i k ki ε i to this y = Xβ + ε???

40 ε β β β ε β β β ε β β β ε β β β ε β β β ε β β β = = = = = = x x y x x y x x y x x y x x y x x y Assume n = 6 and we have two independent variables x 1 and x 2 ε Xβ y + = is shorthand for the above

41 = y y y y y y y = x x x x x x x x x x x x X = β β β β = ε ε ε ε ε ε ε We get to the shorthand matrix algebra expression by realizing the following: ε Xβ y + = and

42 Okay but what about the assumptions underlying the OLS regression model? We must place some conditions both on the population and on the data to establish unbiasedness, consistency and efficiency These conditions are embodied in the Gauss-Markov Theorem

43 The Gauss-Markov Theorem asserts that is a Best Linear Unbiased Estimator (BLUE) of, provided the following assumptions are met: Linearity Mean independence E[ε i x i ] = 0 (implies E[ε] = 0) Homoskedasticity and uncorrelated disturbances Cov[ε] = E[ε ε ] = σ 2 I X is of rank k+1 (k = no. of independent vars.) X is non-stochastic (or stochastic with finite second moments, and E[X ε] = 0 for unbiasedness) Normal disturbance β $

44 To discuss these assumptions we need a few words to describe the performance of estimates Bias. An estimation method is unbiased if it produces estimates that have a statistical expectation equal to the true (population) value Consistency. Estimates converge toward the quantity being estimated as the sample size increases Efficiency. Efficient estimates are those that have smaller standard errors than estimates produced by some competing estimator

45 Regarding the OLS assumptions Linearity and mean independence support unbiased estimates Homoskedasticity and uncorrelated disturbances support efficiency Normality of disturbances means we can do statistical inference using t tables Normality also means we can estimate the model by MLE

46 It is partly with these OLS assumptions in mind that we stress the need to know our data EDA / ESDA We engage in EDA (ESDA) to determine how our data might lead to violations of one or more of the assumptions underlying our model regarding the unknown DGP

47 For example Are we starting out by maximizing the probability of obtaining a normal error structure? Why do we care about this? How can we check for this? What can we do about it? Do we have good linear relationships between our dependent variable and independent variables? Why do we care about this? How can we check for this? What can we do about it? Should any of our variables be transformed? Do you know how to proceed? Do we have any outliers? What kind of outliers? What options are available to us? Fortunately almost everything we ll want to do can be done within GeoDa TM And what we do in GeoDa, we ll try to replicate with R

48 GeoDa TM is a trademark of Luc Anselin & the University of Illinois

49 Exploring Spatial Data Goal is to seek good understanding and description of the data, thus suggesting hypotheses to explore Not much emphasis here on p-values, which are so ubiquitous in most of our training & our statistical instincts Look especially for clues to spatial heterogeneity or spatial dependence Few a priori assumptions about the data Robust ( resistant ) methods Analysis may sometimes end here EDA/ESDA: Despite good tools, at heart EDA is a philosophy; an attitude; best practice way of thinking about your data; a way of staying out of trouble

50 EDA / SDA Tools Maps Descriptive statistics Plots and graphics Classification and clustering methods Software with dynamically linked objects Global and local spatial autocorrelation Look for evidence of Tobler s First Law

51 Maps? Sure, but what kind? So some answers are: What are you trying to discover with the map? What are you wishing to show with the map? PPOV Std. PPOV Dev. Map BoxMap (1.5) PPOV Percentile PPOV Map Quantile Map (9)

52 Descriptive Statistics

53 Plots and Graphs

54 Plots and Graphs (cont.) Checks for normality and symmetry

55 Plots and Graphs (cont.) Checks for linearity & outliers

56 Tomorrow we ll take a close look at the concept of spatial autocorrelation But, in the context of assumptions underlying the OLS model, this raises some serious problems

57 Correlated Disturbances Cov[ε] = E[ε ε ] = σ 2 Σ σ 2 I This can arise from a number of violations of the OLS assumptions Often occurs when there is an important independent variable missing from the specification It causes OLS parameter estimates or their associated standard errors to be unreliable It inflates the value of the R 2 statistic

58 If non-zero covariance between X and ε E[X ε] 0 E[b] = E[(X X) -1 X y] = E[(X X) -1 X (Xβ + ε)] = β + (X X) -1 E[X ε] β OLS parameter estimates are biased plim[b] = β + plim[(x X/n) -1 ] x plim[x ε/n] OLS parameter estimates are inconsistent

59 Readings for today Anselin, Luc What is Special About Spatial Data? Alternative Perspectives on Spatial Data Analysis. NCGIA Technical Paper Ward, Michael D., and Kristian Skrede Gleditsch Spatial Regression Models. Quantitative Applications in the Social Sciences, No Thousand Oaks, CA: Sage. Chapter 1. [Book is available on line at: ] Goodchild, Michael F., Luc Anselin, Richard P. Applebaum, and Barbara Herr Harthorn Toward Spatially Integrated Social Science. International Regional Science Review 23: Loftin, Colin and Sally K. Ward A Spatial Autocorrelation Model of the Effects of Population Density on Fertility. American Sociological Review, 48(1): Galle, Omer R., Walter R. Gove, and J. Miller McPherson Population Density and Pathology: What Are the Relations for Man? Science (new series) 176:23-30.

60 Afternoon Lab Shapefiles and introduction to GeoDa TM and EDA using GeoDa & R

61 Questions?