Lecture 19: Spatial Interpolation II Topics: Point Estimation: 2. Evaluation: References: Chapter 8, Isaaks, E. H., and R. M. Srivastava, 1989. Applied Geostatistics, Oxford University Press, New York Chapter 5, Burrough, P.A. and R.A. McDonnell, 1998. Principles of Geographical Information Systems, Oxford University Press, New York, pp. 98-102.

Outlines 2. Evaluation of Interpolation: What is needed for evaluation: 1) The predictions to be available (the Zi’) This is to be evaluated or validated 2) A validation data set (the Zi) Observation of values at selected locations where predictions were made 3) A set of statistical measures To show the success and the nature of the interpolation (Overview of Evaluation) 2.1 The evaluation (validation or test) data set: 2.1.1 The Nature of validation data set: Needs to be representative for the area Two sampling approaches to ensure the representative nature random sampling approach and regular sampling approach

2.1 The evaluation (validation or test) data set: 2.1.2 Creation of validation data set: 1) independently collected before the interpolation is done. 2) a subset of the original sample data set: the original sample collection is divided into two parts: model building set (sample data set) validation date set (test data set) 3) the same as the sample data but with a cross validation (not preferred) (Cross Validation Figure) 2.2 Statistical Measures: 2.2.1 Quantitative descriptors of errors: 2.2.1.1 Mean of error (average error) 1) Calculation:

2.2 Statistical Measures: (continued …) 2.2.1 Quantitative descriptors of errors: 2.2.1.1 Mean of error (average error) 2) Discussion: Unbiased estimation: when me=0 Is it a perfect estimation? Biased estimation: Overestimation when me>0 Underestimation when me<0 2.2.1.2 Variance of errors (spread) 1) Calculation: Where: ei=(zi’-zi)

2.2 Statistical Measures: (continued …) 2.2.1 Quantitative descriptors of errors: (continued…) 2.2.1.2 Variance of errors (spread) (continued …) 2) Discussion: We want to the variance to be as small as possible Would it be a perfect estimation if the variance is 0? 2.2.1.3 Mean absolute error (MAE) 1) Calculation: 2) Discussion: We want MAE to be as close to 0 as possible What does it mean when it is 0?

2.2 Statistical Measures: (continued …) 2.2.1 Quantitative descriptors of errors: (continued…) 2.2.1.4 Root mean squared error (RMSE) 1) Calculation: 2) Discussion: It is more sensitive to large errors What does it mean when it is 0?

2.2 Statistical Measures: (continued …) 2.2.1 Quantitative descriptors of errors: (continued…) 2.2.1.5 Correlation coefficient (ρ) 1) Calculation: 2) Discussion: It cannot show systematic bias (not good measure). (Figure of same ρ value but different biases)

2.2 Statistical Measures: (continued …) 2.2.2 Graphic plots: 2.2.2.1 Scatter plots (Scatter plot diagram) 2.2.2.2 Quantile plots (Quantifle plots) 2.2.3 Spatial Structure of Errors 2.2.3 Spatial autocorrelation of residuals (errors) Geary’c or Moran’s I Semi-variogram 2.2.3.2 Spatial distribution of residuals (errors) Maps of residuals (errors)

Questions 1. What are the needed components for evaluating the performance of a spatial interpolation? 2. What are the ways for creating a validation data set? Why does the validation data set need to be representative? How to ensure that it is representative? 3. Why do people say that cross validation is not preferred? When should one use it? 4. Describe the different quantitative measures and their strengths and weaknesses? 5. What are the respective roles of scatter plots and residual maps in evaluating a spatial interpolation?