1. Uncertainty “God does not play dice” –Einstein “the end of certainty” –Prigogine, 1977 Nobel Prize What remains is: –Quantifiable probability with uncertainty

2. Situation All data includes some uncertainty The uncertainty is usually not documented Most modeling methods do not provide uncertainty outputs

3. Solution? Estimate the uncertainty in the measured values and the predictor variables Use “Monte Carlo” methods: 1.Inject error into the input data 2.Create the model 3.Repeat 1 and 2 over and over 4.Find the distribution of the model outputs 1.i.e. the parameters and statistical measures 2.E.g. coefficients, R 2, p values, etc.

4. Douglas-Fir sample data Create the Model Model “Parameters” Precip To Points Extract Text File To Raster Prediction Attributes

5. Estimating Uncertainty Field data –Distribution of x,y values –Measurements Predictor layers –Interpolated –Remotely Sensed

6. Down Sampling

9. Mean Raster

11. Additional Error What was the distribution of the contents of each pixel when it was sampled? What’s in a Pixel? Cracknell, 2010

12. Pixel Sampling Each pixel represents an area that is: –Elliptical –Larger than the pixels dimensions

13. Point-Spread Function AVHRR

14. Resampling

15. Approach? Estimate the standard deviation of the original scene that the pixels represent Use this estimate to create predictor rasters that we down sample for the modeling

18. Monte Carlo Error Injection 1.Create the model with the “mean raster” 2.Inject normally distributed random “error” into the predictors 3.Recreate the model 4.Repeat 2 & 3 saving results 5.Create distribution of the parameters and performance measures (R 2, AIC, AUC, etc.)

19. Interpolated Predictors Many predictors are interpolated from point-source data Kriging provides a standard deviation raster as one of it’s output (these are rarely available) By injecting error into the point data and recreating the interpolated surface, we can characterize the error in it. We can also use this to characterize the error’s impact on the model as above

21. "GSENM" by User:Axcordion - http://en.wikipedia.org/wiki/File:GSENM.JP G. Licensed under CC BY-SA 3.0 via Wikimedia Commons - http://commons.wikimedia.org/wiki/File:GS ENM.jpg#mediaviewer/File:GSENM.jpg

22. http://www.planetware.com/tourist-attractions/utah-usut.htm

23. Zion http://www.planetware.com/tourist-attractions/utah-usut.htm

24. Where was the data collected? On flat spots Near roads Often at airports! The data is not representative of our entire landscape

25. We’re Missing Data! Interpolated Raster Canyon

26. Approach If you created the interpolated surface: –Use Monte Carlo methods to repeatedly recreate the interpolated surface to see the effect of missing data Regardless: –Estimate the variability that was missed Maybe from a DEM? –Use this as an uncertainty raster?

27. Lab This Week Characterizing the uncertainty in Remotely Sensed data

28. Extra slides

30. Uncertainty Factors Inherent uncertainty in the world Limitation of human congnition Limitation of measurement Uncertainty in processing and analysis

31. Dimensions of uncertainty Space Time Attribute Scale Relationships