1. Uncertainty “God does not play dice”
Einstein
“The more we know about our universe, the more difficult it becomes to believe in determinism.”
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:
Inject “noise” into the input data
Create the model
Repeat 1 and 2 over and over
Find the distribution of the model outputs
i.e. the parameters and statistical measures
E.g. coefficients, R2, p values, etc.

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

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

6. Down Sampling

7. Original raster at 10 meter resolution (DEM)

8. Down sampled to 90 meters (each pixel is the mean of the pixels it overlaps with)

9. Mean Raster
Just for fun!

10. Standard deviation of the pixels that each pixel was derived from.

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

16. Original raster at 10 meter resolution (DEM)

17. Original raster with noise injection
Original raster with error
Original raster with noise injection

18. Monte Carlo Error Injection
Create the model with the “mean raster”
Inject normally distributed random “error” into the predictors
Recreate the model
Repeat 2 & 3 saving results
Create distribution of the parameters and performance measures (R2, 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
"GSENM" by User:Axcordion - Licensed under CC BY-SA 3.0 via Wikimedia Commons -

23. Zion

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
Where Are the weather stations?
Canyon

26. Approach If you created the interpolated surface: Regardless:
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