Kriging: An Introduction to Concepts and
ApplicationsEric Krause
Prahlad Jat
Sessions of note…Tuesday
• Interpolating Surfaces in ArcGIS (1:00-2:00 SDCC Rm33C)
• Kriging: An Introduction to Concepts and Applications (2:30-3:30 SDCC Rm33C)
• Geostatistical Analyst: An Introduction (4:00-5:00 SDCC Rm30C)
Wednesday
Thursday
• Surface Interpolation in ArcGIS (11:15-12:00 SDCC Demo Theater 10)
• Empirical Bayesian Kriging and EBK Regression Prediction in ArcGIS (2:30-3:15 SDCC Demo Theater 10)
• Geostatistics in Practice: Learning Interpolation Through Examples (8:30-9:30 SDCC Rm30A)
• Polygon-to-Polygon Predictions Using Areal Interpolation (11:15-12:00 SDCC Demo Theater 10)
• Geostatistical Analyst: An Introduction (1:00-2:00 SDCC Rm30A)
• Using Living Atlas Data and ArcGIS Pro for 3D Interpolation (2:30-3:30 SDCC Rm 30C)
• Interpolating Surfaces in ArcGIS (4:00-5:00 SDCC Rm15A)
• Kriging: An Introduction to Concepts and Applications (4:00-5:00 SDCC Rm15B)
2
Geostatistical Analyst Resourceshttp://esriurl.com/GeostatGetStarted
• GeoNet – community.esri.com
- Blogs
- Free textbook and datasets
- Spatial Statistical Analysis For GIS Users
- Lots of discussions and Q&A
• Learn GIS – learn.arcgis.com
- Model Water Quality Using Interpolation
- Analyze Urban Heat Using Kriging
- Interpolate 3D Oxygen Measurements in Monterey Bay
Outline
• Introduction to interpolation
• Introduction to kriging
• Validating interpolation results
• Empirical Bayesian Kriging and EBK Regression Prediction
• Empirical Bayesian Kriging 3D
• Areal Interpolation
• Questions
What is interpolation?
• Predict values at unknown locations using values at measured locations
• Many interpolation methods: kriging, IDW, LPI, etc
What is autocorrelation?
Tobler’s first law of geography:
"Everything is related to everything else, but near things are more related than distant things."
Building up interpolation
• Given the data on the right,
how would you predict the
value of the red point?
• Simplest answer: guess the
average of all the points: 49
- Ignores spatial information
Building up interpolation
• Next best guess: average of a
local neighborhood
• Average of points within 12km
buffer: 40.75
- Better, but points that are
closer should have more
influence
Building up interpolation
• Next best guess: Weight
points by inverse distance
• Weighted average of points
within 12km buffer: 41.01
- This seems like a reasonable
guess with no obvious
problems
Building up interpolation
• What about this point?
• More points below than above
Building up interpolation
• What about this point?
• More points below than above
• Inverse-distance weighted
average in 40km buffer: 49.8
Building up interpolation
• What about this point?
• More points below than above
• Kriging prediction: 56.2
- Some weights are even
negative
Semivariogram
What is kriging?
• Kriging is the optimal interpolation method if the data meets certain
conditions.
• What are these conditions?
- Normally distributed
- Stationary
- No trends
• How do I check these conditions?
- Exploratory Analysis and charting
What is an “optimal” interpolator?
• Estimates the true value, on average
• Lowest expected prediction error
• Able to use extra information, such as covariates
• Can be generalized to polygons (Areal interpolation, Geostatistical
simulations)
• Estimates a prediction distribution, not just a single prediction
Prediction Error of Predictions Probability Quantile
Kriging output surface types
Kriging workflow
1. Explore the data – check kriging assumptions
2. Configure options: trend removal, transformations, etc
3. Estimate a semivariogram model
4. Validate the results
5. Repeat steps 2-4 as necessary
6. Map the data for decision-making
Does my data follow a normal distribution?
• Histogram chart
- Symmetric and bell-shaped
- Look for outliers
- Mean ≈ Median
• What can I do if my data is not
normally distributed?
- Apply a transformation
- Log, Box Cox, Arcsin, Normal Score
Transformation
Why the normal distribution is important
• Kriging predicts mean and standard error at every location
Why the normal distribution is important
• Kriging predicts mean and standard error at every location
Skewed distributions and outliers
Skewed Distribution Outlier
Assessing models using cross validation
• Used to determine the reliability of the model
- Iteratively discard each sample
- Use remaining points to estimate value at measured location
- Compare predicted versus measured value
• Calculates various statistics
- Root-mean-square : root of average squared deviation from true value
- Smaller is better
- Mean : the average of the deviations
- Should be close to zero
- Root-mean-square standardized : measures whether standard errors are
estimated correctly
- Should be close to one
- Average standard error : should be small and close to the root-mean-square
Demo
Kriging in the
Geostatistical
Wizard
Empirical Bayesian Kriging
• Advantages
- Requires minimal interactive modeling, spatial relationships are modeled
automatically
- Usually more accurate, especially for small or nonstationary datasets
- Uses local models to capture small scale effects
- Doesn’t assume one model fits the entire data
- Standard errors of prediction are more accurate than other kriging methods
• Disadvantages
- Processing is slower than other kriging methods
- Limited customization
How does EBK work?
1. Divide the data into subsets of a given size
- Controlled by “Subset Size” parameter
- Subsets can overlap, controlled by “Overlap Factor”
2. For each subset, estimate the semivariogram
3. Simulate data at input point locations and estimate new
semivariogram
4. Repeat step 3 many times. This results in a distribution of
semivariograms
5. Mix the local surfaces together to get the final surface.
EBK Regression Prediction
• Allows you to use explanatory variable rasters to improve predictions
• Automatically extracts useful information from explanatory variables
• Uses Principle Components to handle multicollinearity
Demo
EBK and EBK
Regression
Prediction
Empirical Bayesian Kriging 3D
• New in ArcGIS Pro 2.3
• Interpolate points measured in 3D using EBK
• Visualize 2D transects using Range Slider
• Export to rasters, contours, and 3D points
Demo
EBK 3D
Areal Interpolation
• Predict data in a different geometry
- School zones to census tracts
• Estimate values for missing data
Obesity by school zone Obesity surface and
error surface
Obesity by census block
Areal Interpolation Workflow