Umeå University Sara Sjöstedt-de Luna Time series analysis and spatial statistics

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1 Umeå University Sara Sjöstedt-de Luna Time series analysis and spatial statistics Laboration in ArcGIS Geostatistical Analyst These exercises are aiming at helping you understand ArcGIS Geostatistical Analyst, with respect to spatial interpolation methods (Geostatistical Wizard) and exploratory tools (Explore Data). Exercise A studies the Inverse distance weighted method. Exercise B focus on Ordinary Kriging. As a complement you are encouraged to read the Quick-Start Tutorial of hapter in the manual Using ArcGIS Geostatistical Analyst. Start ArcMap. Activate Geostatistical Analyst and its toolbar by choosing Customize -> Extensions -> Geostatistical Analyst and then Customize ->Toolbars-> Geostatistical Analyst. Exercise A In this exercise you will work with the spatial interpolation method Inverse distance weighted method (IDW). ArcGIS commands will be in italics and underlined. Open the ozone data set from California by loading the files ca_ozone_pts.shp and ca_outline.shp (File -> Add Data ). When you choose Geostatistical Wizard from the toolbar Geostatistical Analyst, all available spatial interpolation methods available can be seen. To choose the IDW method for the ozone data do as follows and then choose Next>: Press the arrow to get the data set of interest. Press the arrow to get the variable of interest

2 Press here to optimize the power (the value of p). Press + to see the weights a) Construct predicted surfaces for ozone with 8 neighbours using Power p = 1, and p =7. Register the weights for p=1 and p=7 and the Root-Mean-Square error/rmse (go to Next > step) of the different models. Compare the two predicted surfaces. How do they differ? Can you see a connection with how the weights change? Try to give an explanation. b) Try IDW with different number of neighbours, 1,, 3, 4, 5, 10, 0 and 40 and let for each number of neighbours ArcGIS choose the optimal value of p. Which variant yields the smallest RMSE? Try to explain why. c) Suppose there is a stronger dependence in the northwest southeast direction (anisotropy). Use an elliptic form in this direction to see whether you get a lower RMSE for 8 neighbours (Angle, Major semiaxis Minor semiaxis). What is your conclusion for this example? Exercise B In this exercise you will work with the spatial interpolation method Ordinary Kriging. ArcGIS commands will be in italics and underlined. Download the data from the file lab.txt via File->Add Data->Add XY Data. To view the locations, click (with the button to the right at your mouse) the name lab.txt Events in the left marginal and choose Zoom To Layer. There are two variables available in the file. The first, zisot, consists of 100 observations simulated from a process with a constant (unknown) trend and with a dependence structure according to the following isotropic exponential semivariogram:

3 γ ( h) = σ (1 exp( 3h / r)), (1) where the sill is σ = 1 and the range is r = 15. The second variable, zanisot, is simulated from an anisotropic process with constant trend and with a dependence structure according to the following semivariogram: γ h, h ) = σ (1 exp( 3 h / r 3 h / )) () ( r where σ = 1, r 1 = 15 and r = 6. Note that h in equation (1) corresponds to h = h 1 + h. The variable zanisot thus have different strengths of dependence in different directions. Choose the variable zisot. In order to do Ordinary kriging interpolation on the variable zisot, first choose Kriging and the variable zisot within the Geostatistical Wizard and go to Next >: Now choose Kriging Type: Ordinary, Output Type: Prediction and Order of trend removal: Constant. These commands are the starting point for Ordinary Kriging.

4 a) What is the estimated level of the constant trend (go to Next > )? Estimate an exponential variogram (go to Next > and choose Exponential). What are the estimated values of the sill and the range? Now estimate the covariogram instead and register the estimates of the sill and the range. Compare all estimates to the corresponding true values of σ and r. Which method seems to work better? b) Change the lag size (0.5, 1,, 3) and the number of lags (7, 15, 30, 40). For each combination, register the lag size multiplied by the number of lags as well as the estimates of the sill σ and the range r. How do the estimates of the sill and the range change depending on the lag size, the number of lags and (lag size)x(number of lags)? Compare them with the true values of σ and r.

5 c) Continue to step 4 and choose to use 10 neighbours (include at least 10, circular window without sectors) and study how the RMSE changes depending on lag size and the number of lags (e.g.. those that you used in b) above). Which combination seems to work best (in terms of RMSE)? d) Try different numbers of neighbours (circular window without sectors) and different ways of estimating the variogram (number of lags and lag size). Which combination seems to give the smallest RMSE? e) Study the RMSE when you predict zisot via ordinary kriging and exponential variogram, and compare when the variogram is isotropically estimated versus anisotropically (everything else kept the same). Do the anisotropic or the isotropic predictions perform better? Is it reasonable? f) Use your best ordinary kriging model to predict the variable at the coordinate (100,100) and estimate its prediction error. Note that prediction error maps are constructed by choosing Geostatistical Analyst -> Geostatistical Wizard->Kriging->Next and then Kriging Type: Ordinary, Output Type: Prediction Standard Error and then continue to specify your best kriging model (trend, variogram etc.). An alternative is given in the below picture.

6 Click the right button on your mouse and choose Change output to Prediction Standard Error to generate the prediction error map. Exercise C Load the ozone data that you worked with in Exercise A. a) Study the trend surface for ozone via Geostatistical Analyst -> Explore Data -> Trend Analysis.Which type of (deterministic) trend would you suggest is appropriate by visual inspection? b) Use Universal kriging to predict ozone as well as possible (minimizing the RMSE). Think about that the following can be varied: the order of the trend polynomial, the variogram (type, isotropy/anisotropy, lag size, no. of lags), neighbourhood (number of neighbours, sectors). Present your best model and the corresponding RMSE and bias. Explain how you came up with your best model. In order to perform Universal kriging, please go to Geostatistical Analyst -> Geostatistical Wizard, choose Kriging/Cocriging and variable, then use Ordinary kriging with option order of trend remval: first, second or third. There seems to be something wrong (a bug) with the universal kriging command in ArcGIS.