analyzing spatial nonstationarity in multivariate relationships uvm math/stat department lecture,...

Analyzing spatial Analyzing spatial nonstationarity in multivariate nonstationarity in multivariate

relationships relationships UVM math/stat department lecture, march 31, 2004UVM math/stat department lecture, march 31, 2004

By Austin TroyBy Austin Troy11

1.1. University of Vermont, Rubenstein School of Environment University of Vermont, Rubenstein School of Environment and Natural Resourcesand Natural Resources

Spatial Nonstationarity

• The notion that relationships change across space.– That is, the relationship between y and (x1,x2....xn ) is non

constant from one location to the next.

• Can a variable be included in a model that proxies for space?

• Sometimes this is possible, but very often the factors that make one location different from another are non-quantifiable, or involve extremely complex interactions that cannot be parsimoniously modeled. Especially in social research

Approaches to nonstationarity

• There are several approaches to dealing with this problem. Among them are:

1. Create zones of homogeneity and stratify

2. Allow parameters to vary constantly

First Approach

1. Parameterize a global model and look at the residuals to detect patterns

First Approach

k khkh xxf )(

k kiki xxf )(

k kjkj xxf )(

•Use patterns in residuals to define patches

•Specify a separate equation for each patch, or stratum

Second Approach

k kiikii xvuvuxf ),(),()(

’s vary continuously as a function of location (u,v) at each point i (Fotheringham 2002)

2. Continuous variation: Create measures of statistical relationships that are continuously varying across space, such as GWR

GWR• Weighted moving window regression method developed

by Foterhingham and Brundson (2000, 2002), building on works of Hastie and Tibshirani (1990) and Loader (1999)

• Expanded form of simple multiple regression equation

• Coefficients are deterministic functions of location in space

• Uses weighted least squares approach• Fully unbiased estimate of local coefficient is impossible,

but estimates with only slight bias are possible

ik kiikii exvuvuxY ),(),()(

GWR estimation• Separate regression is run for each

observation, using a spatial kernel that centers on a given point and weights observations subject to a distance decay function.

• Can used fixed size kernel or adaptive kernel to determine number of local points that will be included in each local regression

• Adaptive kernels used when data is not evenly distributed

GWR kernel

From Fotheringham, Brundson and Charlton. 2002. Geographically Weighted Regression

GWR with fixed kernel GWR with adaptive kernel

Points are weighted based on distance from center of kernele.g. Gaussian kernel where weighting is given by:

wi(g) = exp[-1/2(dij/b)2 where b is bandwidth

GWR kernel Adaptive kernel width is determined through

minimization of the Akaike Information Criterion

)tr(2

)tr()2(log)ˆ(log2AIC

S

S

n

nnnn eec

where tr(S) is the trace of the hat matrix and n is the number of observations

Bias and variance tradeoff

• Tradeoff between bias and standard error• The smaller the bandwidth, the more variance but

the lower the bias, the larger the bandwidth, the more bias but the more variance is reduced

• This is because we assume there are many betas over space and the more it is like a global regression, the more biased it is.

• AIC minimization provides a way of choosing bandwidth that makes optimal tradeoff between bias and variance.

GWR outputs

• The result is a statistical output showing global summary stats and parameters estimates, local model summary stats and non-stationarity stats for each parameter

• Also produces a map output of points with parameter estimates, standard errors and “pseudo t statistics” for each variable for each point

A simple (?) question we can address with this approach

• How is proximity to trees and other “green assets” reflected in property values?

We can ask this with hedonic analysis

• Econometric method for disaggregating observed housing prices into a series of unobserved “implicit” prices, reflective of WTP for a given marginal change in attribute

• Price= fn(structure, lot, location)

• Has been used extensively for valuing environmental amenities and disamenities

eXXXP LLNNSS )ln(

Hedonics and space• Hedonic analysis is generally taken to be

spatially stationary; that is, it is assumed that marginal WTP for an attribute is fixed within a housing market.

• Usually these markets are assumed to be quite big and tools for determining these market boundaries are poorly developed (see approach 1, earlier)

Case Study:Two Scales of analysis

1. Block group level analysis of median housing price as fn of tree cover, controlling for many other factors

2. Property level analysis of housing prices as a function of tree cover and parks, controlling for structural, locational and environmental attributes (hedonics)

Research is part of the Baltimore Ecosystem Study, an NSF-LTER

Block group analysis

Initial pattern of housing values tells us what we know intuitively: suburbs are more valuable than central city, but value goes down again as you get too far from city center

Block group analysis

Average canopy cover by block group, as derived from 2000 USGS 30 m canopy cover analysis

Running GWR at the BG level

• Given the spatial dependence in the data, it is likely that the coefficients of a multivariate relationship will be related to the spatial processes underlying the spatial dependence. Hence, we choose to run GWR and compare it to global regression.

Equation

Predictor variables=• Median age• Percent with income greater than $100k• Median household income• Percent owner occupied housing• Percent vacant buildings• Percent single family detached homes• Mean tree canopy percentage• Median number of rooms per house• Median age of housing• Percent with mortgage• Percent high school educated• Percent African American• Percent Protected Land

iiiiiiiiii exvuxvuxvuvuP 13132211 ),(....),(),(),(

Data

• This was run using census block group data from 2000 for the five counties in the Baltimore Metro Area

• Observational problem: while population of BGs is relatively constant, size is not, hence may be some form of Modifiable Areal Unit Problem; may be varying levels of heterogeneity within block groups

Results: Global Model

• Global model: highly significant

• Canopy is significant at α = .05, while percent protected land is not

• All other control variables are

Comparison of Local and Global• The ANOVA tests the null hypothesis that the GWR model

represents no improvement over a global model and rejects the null

• Notice also that Coefficient of Determination increases significantly and AIC decreases

Local Parameter variabilty

• We also conduct a Monte Carlo Significance test which finds that almost all variables are spatially non-stationary, although protected land is not

How does this testing work?• We expect all parameters to have slight spatial variations; is

that variation sufficient to reject the null hypothesis that it is globally fixed?

• If so, then any permutation of regression variable against locations is equally likely, allowing us to model a null distribution of the variance

• A Monte Carlo approach is adopted to create this distribution in which the geographical coordinates of the observations are randomly permuted aginst the variables n times; results in n values of the variance of the coefficient of interest which we use as an experimental distribution

• We can then compare the actual value of the variance against these values to obtain the experimental significance level

Local R squared shows that model fit varies by locations

Notice also how under GWR, there is no pattern to the residuals because it accounts for spatial effects

Results showthat the city center is where tree canopy cover is valued highest on the margin; this might be because trees are scarcest there

However, the pseudo t statistic reveals that not all areas are significant

When we blank out the non-significant observations, we see that trees are only significantly reflected in property values in some areas, and it’s negative in Howard County

Interpretation

• In some areas, canopy cover is valued more highly than in others. The degree to which canopy is valued may relate to the scarcity or spatial distribution of trees at multiple scales. It may be negative in Howard county because trees are associated with some other factor, like “ruralness,” which is not properly quantified by the census and which exerts downward influence on housing prices

Other variables

• We’ll ignore protected land, since it’s not non-stationary. However, some of the other control variables are telling. Let’s look at median house age.

Pattern makes sense: older homes yield a positive premium in the more affluent suburb, but a negative premium in the poorer central city

When we do a cluster analysis (PAM) based on 3 parameter values (canopy, house age and number of rooms), we get something looking like markets

Here’s what it looked like with 6

Note that the silhouette score for the PAM was .55 for the 4 class and .5 for the 6 class—strong structuring

Problems with this approach

• Using block group level data is not ideal for a number of reasons– Modifiable Areal Unit Problem– BG obscures significant within unit

heterogeneity– Level of heterogeneity (variance) varies between

observations– There may be spatial clustering within BGs– Data is not very accurate at this level– Attribution not great either

Property Level Analysis• Property level analysis is way of dealing

with the local spatial heterogeneity and of getting much better attribution

• Used Maryland Property View data set for half of Baltimore City

• Regressed log price against 22 variables, including structural, neighborhood and locational variables, including several environmental variables

•Simple plot of log price shows that while there are slight patterns, there is still considerable heterogeneity over space and no clear patches are emergent

•Moreover, price is only one factor and social patches are defined on a number of factors

•Plotting out log price normalized by square footage gives a similarly heterogeneous result

•Other plots of other variables also show similar heterogeneity

Overall results• Local model was a considerable

improvement over the global model

Non-stationarity of parameters

• Monte Carlo significance test used to determine whether parameters are significantly non-stationary

• All but 4 were at the = .05 significance level

Plotting out Parameter Estimates and T values

Parameter on dummy variable for trees within 20 meters

T value on tree dummy variable

The only real significant relationship is on east side of study area

Note that 1458 out of 2350 observations had a “1” for this value. A better approach for future would be tree density index

Parameter value on distance to nearest park

T value on Distance to nearest park

Areas where parks are highly valued in the real estate market

Number of Baths

Parameter values show clear patchiness

Here an additional bathroom adds 6-9% to home value

Here it adds 1.5 to 3%

T value of parameter on number of bathrooms shows that number of bathrooms only significantly related to price in northern neighborhoods

(small dots are observations where parameter estimate is insignificant)

Parameter value on dummy for single family home

T value on SFH dummy

Note that even though there are SFHs in the south, SFH status only appears to add value in the North

Parameter values on Age of Structure

Shows clear patchiness

T values on Age of Structure

Shows age is most significantly related to price near center of town. Only in northeast is it positively related, suggest that older homes have “historic” value there but not elsewhere

Generating fuzzy Surfaces

• With this data, surfaces can be interpolated showing the change in parameter values over space

• Unfortunately, this only works well where most points have significant parameter estimates

Surface Example:Parameter on

Distance to park• Interpolation

allows us to much more clearly see “patches” because it smoothes out some of the within-group heterogeneity

Surface Example:Parameter on structure age

T Value on structure age

• Note that interpolation of t value looks almost the same

Combining interpolated layers• In a preliminary attempt we

took four interpolated layers for dummy variables (hence all on the same scale), including structure age>80, trees w/in 20 meter, SFH and brick and added them together to get a test composite layer; the more layers we add, the more we begin to see clear differentiations between neighborhoods

Patch Classification

• This can then be used to classify areas into distinct patches or zones

• These zone boundaries are based on a vector of parameters from the multivariate equation

• This is extremely preliminary—an example for display

Conclusion• GWR offers an excellent way to define meaningful

socio-economic boundaries• It allows us to look at the spatial arrangement of the

relationship between y and z, controlling for all other variation

• In particular, it offers an excellent way to look at the spatial variability of social phenomena, which are often mediated by spatial processes that cannot be quantified, like trying to stay friendly with your neighbors and neighborhood association

• Suggests that preferences do vary over space and that future analyses must take this into account