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

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

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.

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

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

Down Sampling

Original raster at 10 meter resolution (DEM)

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

Mean Raster Just for fun!

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

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

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

Point-Spread Function AVHRR

Resampling

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

Original raster at 10 meter resolution (DEM)

Original raster with noise injection Original raster with error Original raster with noise injection

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.)

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

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

We’re Missing Data! Interpolated Raster Canyon Where Are the weather stations? Canyon

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?

Lab This Week Characterizing the uncertainty in Remotely Sensed data