Geographically Weighted Regression and Kriging: Alternative Approaches to Interpolation A Stewart Fotheringham

Transcription

1 Geographically Weighted Regression and Kriging: Alternative Approaches to Interpolation A Stewart Fotheringham National Centre for Geocomputation National University of Ireland, Maynooth

2 Outline 1. The prediction problem 2. Predictions based on nearby values only 1. IDW 2. Ordinary Kriging 3. Predictions based on nearby values and location 1. Global Regression 2. Universal Kriging 3. GWR 4. Predictions based on nearby values and external covariates 1. Global regression 2. Global spatial lag model 3. Universal riging with external drift 4. GWR 5. Comparison of Results 6. Combining GWR and Kriging

3 The General Prediction Problem We are interested in predicting the value of Y at location (Y ) given we have values of Y i where i represents a location of a point in the set 1.n which does not include One of the following circumstances will also occur: 1. We have no further information 2. We have information on a set of external variables X for the points in the set 1.n and at 3. We have information on a set of external variables X for the points in the set 1 n but not for

4 Our Specific Problem We have a set of housing mortgage records from a Building Society for a sample of properties they provide: the final sale price (our Y variable) some attributes of the property (floor space etc.) - our set of X variables the spatial coordinates of the houses We have the same set of attributes for another sample taen in the same year these too have been geocoded Can we reliably predict the sale price of the properties in the second sample?

5 Data for the experiments The data are a sample of Building Society mortgage records for Greater London in Among the attributes are Sale price Postcode of the property (used for geocoding) Type of the property (house, flat, &c) Number of bedrooms, bathrooms Date of construction Floor area Provision of garage Some neighbourhood variables (social class, unemployment) have been extracted from the 2001 Census of Population

6 General Approach Using the first sample as a training set we can estimate the relationship between the sale price at location i and various X variables measured at i and the prices at nearby locations We can then use the estimated relationship to predict the sale price for the properties in the second validation set. In this case we now the selling prices for the validation set, so we can compare our predictions with what happened in reality.

7 Data locations in the training and validation sets observations Training Set Validation Set Subset 1 (random) 1500 overlapping coords. = 1442 obs. Subset 2 (random) 1500 overlapping coords. = 1445 obs.

8 Software There are plenty of alternatives: IDW Most GIS pacages GWR Fotheringham/Brunsdon/Charlton s GWR3.0 Bivand s spgwr pacage for R Kriging Pebesma s standalone DOS gstat program Pebesma s gstat pacage for R GeoR in R Pacages in various GIS

9 Modelling house price variation A class of models for modelling the relationship between the price of a house and its attributes exists. They are nown as hedonic models Categorical attributes are modelled as dummy variables. The coefficient for each of these variables represents the additional value which accrues from the presence of that attribute Usually formulated in an OLS regression framewor but houses in the same neighbourhood tend to have similar attributes and prices and therefore problems with non-independent observation Also, house prices are notoriously sewed we therefore generally wor with log of prices

10 Price transformation Histogram of Price Histogram of logprice Frequency Frequency e+00 4 e+05 8 e+05 Price logprice Logging the price variable removes the sew

11 Predictions based on nearby values only 1. IDW 2. Ordinary Kriging

12 Inverse distance weighting (IDW) Y = n i= 1 Y d i β i n i= 1 d β i Weighted mean of Y variables. β < 0

13 Ordinary Kriging n Y = Y i i= 1 λ where λ i = 1 and the weights are generated from an empirical semivariogram i 0.3 semivariance Training Nug(0) Sph(2500) distance

14 Predictions based on nearby values and location 1. Global Regression 2. Universal Kriging 3. GWR

15 Global Regression Y = β + β x + β y where β 0 is the value of Y at the origin and x,y represent the cartesian coordinates of a location.

16 Universal Kriging = + = = + + = + + = + = n i i i i i i i i i m Y m Y y x m y x m m Y λ μ μ β β β β β β μ Value is mean plus error Model mean in terms of location from training data Obtain estimates of errors for training data and rige

17 Geographically Weighted Regression A model of spatial heterogeneity u is either the location of a sample point or any location in the study area (so these can be the validation point locations, or the prediction points Weights W(u) are generated from a ernel function which uses a bandwidth found by optimising a goodness-of-fit criterion W Y X X W X x x x Y T T m m ) (... = = β β β β β

18 Spatially Adaptive Weighting Scheme

19 Using GWR for prediction Three stage process 1. Determine optimal bandwidth using the training set and fit the model at these locations (obtain the local parameter estimates for these locations etc) 2. Using this bandwidth, estimate the local parameters at location using data from locations in the training set 3. Using the X variables at location along with estimated local parameters for location, predict Y

20 GWR outputs Predicted dependent variable values at all locations Local parameter estimates for all locations Local standard errors for these parameter estimates Local T values for these parameter estimates In addition, for the locations where Y is nown (e.g. in the training set), we also get Local influence measures Residuals (and standardised residuals) Local goodness of fit measure (R 2 ) Corrected Aaie Information Criterion (Hurvich/Tsai)

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23 Predictions based on nearby values and external covariates 1. Global regression 2. Global spatial lag model 3. Universal riging with external drift 4. GWR

24 Global Spatial Lag Model = = n i i i m m y w x x x Y ρ β β β β where ρ is a spatial autocorrelation parameter and w i is a externally derived weight for the observation at i on the regression at

25 Comparison of Results ,392 1, , , , ,243 1, ,217 1, , ,243 1, IDW_U OK_U LM_U UK_U GWR_U LM_M SAR_M UK_M GWR_M

26 Ordinary Kriging

27 Linear Model - locations

28 Universal Kriging linear drift

29 Universal Kriging - hedonic

30 Linear Model - hedonic

31 GWR - hedonic

32 Combining GWR and Kriging To this point, we have assumed that the X variables for location are nown but the Y variable is unnown. What about the more common situation where both the Y and the X variables at location are unnown? Here is the potential to combine the power of Kriging and GWR. Step 1: use riging to estimate the X variables at location from the nown X variables at the locations in the training set. Step 2: use GWR to estimate the local parameter estimates at location from data in training set. Step 3: use GWR model at location to predict Y

33 Summary Predicting unnown quantities at given locations is a generic spatial problem It is neither new nor surprising that adding covariates to the prediction process is much better than simply relying on spatially weighted functions of Y or on modelling trends through spatial coordinates alone. What is perhaps surprising is the extent to which the latter two processes are still used. GWR can be used to predict values accurately (at least as well as most if not all riging techniques) and is arguably easier to use and more intuitive in some circumstances. Combining riging (to predict the X variables) and GWR (to obtain local parameter estimates) to predict the Y variables is the next logical step.

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35 GWR Kernels There are several ernels which are used with GWR the shape is not as important as the bandwidth h Bandwidth is chosen to minimise either a cross-validation score or an Aaie Information Criterion Models with different bandwidths have different hat matrices and therefore different degrees of freedom Bisquare ernel is frequently used W i (u) = [1 - d(u i - u) 2 ] 2 if d(u i - u) < h W i (u) = 0 otherwise d(u i - u) is distance from location u i to regression point u

36 Results Method SSE e 12 Does it use the HP model? Method SSE e 12 Does it use the HP model? Global Model 1.88 Y IDW 5.56 N GWR 1.41 Y Ordinary Kriging 5.46 N Spatial Lag Model 1.56 Y Simple Kriging 5.45 N Ordinary Kriging 5.65 N Universal Kriging 5.95 N Simple Kriging 2.76 Y Universal Kriging 1.32 Y Ordinary Coriging 5.52 N Simple Coriging 5.27 N Universal Coriging 1.30 Y

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