Spatial autocorrelation analysis of residuals and geographically weighted regression

Transcription

1 Spatial autocorrelation analysis of residuals and geographically weighted regression Materials: Use your project from the tutorial Temporally dynamic aspatial regression in SpaceStat Objective: You will undertake a LISA analysis to determine whether regression residuals are spatially autocorrelated. You will then conduct a geographically weighted regression (GWR) to: (1) Improve local predictive power of the regression; (2) Reduce autocorrelation in the residuals; (3) Relax the assumption of stationary regression coefficients. Why GWR? The aspatial regressions you ran in the tutorial Temporally dynamic aspatial regression in SpaceStat applied over the entire study area and assumed the regression coefficients were the same in all locations on the map (were stationary). GWR relaxes this assumption, and may be an appropriate method in those instances when you think the regression coefficients may indeed differ from one location to another. Alternative regression models such as spatial lag regression, spatial error regression, and spatial multi-level regression are also available in SpaceStat. You also may use geostatistical models for spatial prediction. Here we ll be using GWR. The analysis of residuals: We analyze residuals to determine whether the assumptions of regression have been met, primarily that the residuals are IID independent and identically distributed. This means: (1) The errors are independent of one another, and their values do not depend on the value of residuals at neighboring locations (i.e. no spatial autocorrelation in the residuals) (2) The errors have a constant variance (homoskedastic). We therefore inspect (i) timeplots of the residuals to see whether their dispersion changes through time; and (ii) scatterplots of the residuals vs. the predictors to see whether their dispersion depends on the value of the predictors. (3) The errors are normally distributed. We verify this using histograms of the residuals. The relative importance of these assumptions is (1) constant variance, (2) independence, and (3) normality (Griffith and Layne 2000). What happens if these assumptions aren t met? Heteroskedasticity in the residuals causes estimates of the regression coefficients to be less precise. Spatial autocorrelation in the residuals results in an underestimation of the standard error of the estimates of the regression coefficients and a bias towards rejecting the null hypothesis that the value of the coefficient is zero. Non-normality of the residuals compromises interpretability of significance tests of the regression coefficients. Finally, multicollinearity results in over-estimates of the variances of the regression coefficients. For the model to be correct we also assume (1) The relationships between dependent and independent variables are linear (since we used a linear model); and (2) all independent variables are included (the model is properly specified). There are statistical tests for evaluating these assumptions. The approach taken in SpaceStat is primarily through visualization.

2 In the tutorial Temporally dynamic aspatial regression in SpaceStat you checked the assumptions of homoskedasticity and normality. We now assess spatial independence of the regression residuals. Step 1: Spatial autocorrelation analysis of residuals Spatial autocorrelation in the residuals is often interpreted to mean that (1) an important independent variable (predictor) is missing from the regression, or (2) an underlying spatial process that induces spatial autocorrelation in some of the variables is missing from the model (e.g. groundwater flow inducing spatial autocorrelation in heavy metals). Load the project you created in Temporally dynamic aspatial regression in SpaceStat. Complete the analysis of residuals by determining whether they are spatially autocorrelated. Click on Methods.. Clustering.. Local Moran.. Univariate Local Moran. Change the dataset to be Residuals. Look at the advanced and output tabs, using help to explore them (you don t need to make any changes). Run the method. When the method finishes the local Moran map and Moran scatterplot will appear, and the global Moran s I will be written to the log view. If they are not already shown, turn on the graph statistics for the Moran scatterplot. The local Moran map and the Moran scatterplot are already time-synchronized. Animate the Moran scatterplot and local Moran map through time (you will only be able to do this when at least one of the variables in your regression changes through time, otherwise your residuals will be static). Recall the slope of the line on the Moran scatterplot is the global Moran s I coefficient. Is global spatial autocorrelation increasing, decreasing or not changing through time? Open the log view and inspect the table of local Moran coefficients. Is there significant global spatial autocorrelation in the residuals? Is the significance of the global Moran s I changing through time? Inspect the local Moran map, recalling that residuals are calculated as: (observed value of the dependent variable Estimated mean). Red clusters are areas of high regression residuals where the observed value is under predicted, and blue clusters are areas of low residuals where the observed value is over predicted. Does it make sense to interpret the local Moran map when global autocorrelation in the residuals is absent? Why or why not? Identification of localities where your aspatial model is over predicting or under predicting can lead to insights regarding mis-specification of your regression model. Further inspect the local Moran map. Where is your model over predicting? Where is it under predicting? Does this give you any additional insights into some other variable you might choose to incorporate into your model?

3 Step 2: Geographically Weighted Regression Geographically Weighted Regression (GWR) may be used when there is spatial autocorrelation in the residuals from the aspatial regression, or when you have reason to believe the regression coefficients might change from one location to another (e.g. the regression coefficients are not stationary). In SpaceStat GWR uses point geographies. If you are analyzing a polygon geography SpaceStat will automatically calculate polygon centroids, and then apply GWR to those centroids. Should you wish to calculate a centroid geography outside of GWR you would use Tools..create centroid geography, and click ok on the create centroid geography dialog. You can check the data view to verify the centroid geography was created. Now select Methods.. Regression.. Geographically weighted regression. The model you created for aspatial regression should automatically be available for GWR. If not, you will need to recreate your regression model, specifying the dependent and independent variables as you did earlier for aspatial regression. Be sure to create the same regression model you used in aspatial regression, as you will compare results from the two approaches later in this exercise. Click ok on the define regression tab. Open the other tabs ( regression settings, bandwidth settings, and more settings ), and use help to explore how they can be used. For now you don t need to change anything on these other tabs. Now run the GWR method. SpaceStat will then calculate a local regression for every local area in the dataset, using the data from 20 nearest neighbors as the input to each regression. Open the dataview if it isn t already open. Look under your regression s dependent variable for the folder GWR. New datasets created by GWR will be highlighted. Reported are the local weighted means of your dependent and independent variables, standard deviations, and local correlations between the dependent variable and each of your independent variables. You also will see residuals, std error of the mean and R- square. Map the predicted value: Create a map of the estimated mean of [name of your dependent variable] under the GWR model and color it in a fashion similar to what you used for the estimated mean from your aspatial model. How do these compare? Do either of the models appear to do a better job of estimating the observed values of the dependent variable? Map uncertainty in the predicted value: When creating models of spatial data it is essential to always display maps of the uncertainty in the predicted value. Do this by mapping the Std err of mean. This value is the standard error of the estimated mean of [name of your dependent variable], which of course is the value predicted by GWR. Now create a scatterplot with the estimated mean on the x-axis and std err of the mean on the y-axis. Ideally, uncertainty in the estimated mean would be independent of the value of the estimated mean. Is this true for your model? Why or why not?

4 Map the residuals from GWR: The GWR procedure should have created a map of the residuals, using a continuous color scheme with color breaks such that 0 is a neutral color, negative values are a cool color (green), and positive values are a hot color (Purple). Undertake a LISA analysis of the residuals from GWR, and inspect how the clusters (should there be any) vary through time. Inspect the table of global Moran values. Compare these to the global Moran values you obtained for your Moran analysis of the residuals from the aspatial regression. Are the residuals from GWR more autocorrelated or less autocorrelated than those from the aspatial regression? Why might this be? Evaluate local correlations: GWR also calculates local correlations between the dependent and independent variables e.g. correlation b/t ARSENIC and WPROSTATE. Map these correlations and inspect them for pattern (Use a color ramp centered on 0, with red indicating high positive values and blue indicating negative values). This tells you where the underlying correlations between the variables are the same, and where they are different. Strong underlying spatial variability in the correlations among the variables can support the use of a GWR model since the strength of the dependencies varies from one place to another. Should these correlations be stationary you may wish to employ a modeling approach other than GWR. Compare predicted to observed values: As for the aspatial model, one measure of the predictive ability of the model is how strongly the predicted value of the dependent variable corresponds to the observed value of the dependent variable. Assess this by creating a scatterplot of the estimated mean of the dependent variable from GWR to its observed value. Scale the x and y axes to the same range. If your GWR model was perfect the points would fall on the 45 degree line and the correlation shown in the graph statistics would be 1.0. How large is this correlation for your GWR model? Does it change through time? What does this tell you about the predictive ability of your GWR model? Step 3: Compare the aspatial and GWR models Compare the predictive powers of the aspatial and GWR models: Place the scatterplots of the predicted and observed values for your aspatial and GWR models side by side. If you have time-dynamic models, synchronize the two scatterplots and play them. How do the predictive abilities of the aspatial and GWR models compare? Evaluate goodness of fit using R-square: Now explore the local goodness of fit using the R-square from GWR. Right click on R-square in the GWR folder and create a new map. Also create a histogram of the GWR R-square and time synchronize the new map and histogram. Use these to answer the following questions. Do the locations with the largest R-square change through time? What is the maximum of the local R-square from GWR? Is the average R-square from GWR larger or smaller than the R-square from the aspatial linear regression?

5 Recall that GWR relaxes the assumption of stationary regression coefficients. Why might your regression using GWR fit better in some locations than others? Step 4: Think about your results Based on these analyses, how might you change your original question/hypothesis?