Kriging: an Introduction to Concepts and Applications Nicholas M

Kriging: An Introduction to Concepts and Applications Nicholas M. Giner – Esri Agenda

• What is interpolation? • Interpolation applications • Spatial autocorrelation • Deterministic vs. Geostatistical interpolators • Building up interpolation • Kriging theory • Empirical Bayesian Kriging (EBK) • EBK Regression • EBK 3D • Areal Interpolation What is interpolation?

• Process of predicting values at unknown locations using values at known location • Transforms measurements of a continuous phenomenon into a continuous surface • Interpolation predicts within region; Extrapolation predicts outside region What is interpolation? Interpolation applications

• Many continuous phenomena (z) - Elevation - Soil (pH, nutrient levels, porosity) - Precipitation / Snowfall - Temperature - Windspeed - Air pollution / Air quality - Ozone - Water quality - Mining - Heavy metal concentrations - Environmental contaminants - Noise - Disease occurrence Spatial autocorrelation

• Tobler’s First Law of Geography - “…everything is related to everything else, but near things are more related than distant things” • O’Sullivan and Unwin, 2003 - “If geography is worth studying at all, it must be because phenomena do not vary randomly across space” Deterministic vs. Geostatistical interpolators

• Deterministic interpolators - Based on mathematical functions, not statistical theory - Model parameters are determined by the user - Does not include randomness - No estimates of prediction error (uncertainty/accuracy/confidence) - Examples: Inverse Distance Weighting (IDW), Spline, Global Polynomial Interpolation

• Geostatistical interpolators - Based on mathematical functions, AND statistical theory - Model parameters are estimated based on the data (spatial autocorrelation) - Includes randomness to approximate the variation present in geographic data - Produces estimates of prediction error (uncertainty/accuracy/confidence) - Example: Kriging Two components of all interpolators

• Neighborhood definition – distance or number of points

• Estimation function – mathematics used to make the estimation (e.g. determine the weights) Building up interpolation

• Average of all data points: 49

Source: Geographic Information Analysis – O’Sullivan and Unwin Building up interpolation

• Local spatial average: 40.75 - All points in the local neighborhood are weighted equally Building up interpolation

• Inverse Distance Weighted (IDW): 41.01 - Closer points have higher weights and more influence

Source: Geographic Information Analysis – O’Sullivan and Unwin Building up interpolation

• Inverse Distance Weighted (IDW): 49.8 - More influence from points below simply because they are within the neighborhood and closer in distance Building up interpolation

• Kriging: 56.2 - Prediction is based on how correlated points are based on distance - There can be negative weights Geostatistics and Kriging

• Geostatistics - statistics of spatially correlated data • Quantify spatial autocorrelation and incorporate it into the interpolation • Kriging – “optimal” interpolator given that data meets certain conditions (assumptions) - Based on the foundational work by Daniel Krige and George Matheronin the 1950s-1960s predicting gold ores in South Africa - Main idea is that spatial data can be decomposed into two main components

1) Deterministic variation (global trend) • Can be constant mean or mathematical function

2) Spatially correlated, random variation (local autocorrelation)

Z (s) = µ + ε(s)

Prediction = mean + error What makes it “optimal”?

• Estimates true value, on average (unbiased) • Lowest expected prediction error • Can use information about covariates • Can be generalized to different geometries • Estimates a prediction distribution at each location (not just one value)

• Kriging assumptions - Normally distributed - No trends - Spatially autocorrelated - Stationary Kriging assumption: Normal distribution

• If your input data is normally distributed, you can guarantee that your predicted distribution will be normally distributed • Many transformation options if not

Histogram

QQ Plot Kriging assumption: No trends

• Systematic patterns and trends in an area might impact the interpolation • Trade-off with spatial autocorrelation Kriging assumption: Spatial autocorrelation

• How correlated are points based on how far apart they are from one another • Once you know expected correlation in known values given distance, you can predict the value at unknown locations Kriging assumption: Stationarity

• The correlation between points is defined only by the distance between them, not their location - Mean stationarity - Local stationarity Kriging workflow

1) Map your data 2) Exploratory 3) Variography – 4) Fit model – 5) Use model to Spatial Data Describe spatial Summarize spatial determine weights Analysis (ESDA) variation in the variation with a in search Configure options data math. function neighborhood

8) Repeat Steps 2-7 7) Evaluate 6) Interpolate

(Cross-validation) Demo #1 Map the data, Geostatistical Wizard, ESDA, Configure options Variography (Modeling)

• Examining and modeling spatial autocorrelation Variography (Modeling)

1) Calculate empirical semivariogram - Calculate distance and difference between each pair of points

Semivariogram (distance h) = 0.5 * average (location i – location j)2 2) Bin the semivariogram - Group the pairs of locations into a specified range of distances (lags)

3) Average the semivariogram - Calculate the average distance and difference (semivariance) for each lag

4) Fit a model - Find the best fit line for the average semivariances Semivariogram

• Represents the expected difference in data value for pairs of points that are a given distance apart, regardless of their spatial location

Nugget – semivariance at 0 distance (measurement error)

Range – distance at which autocorrelation falls off, where semivariance is constant, where there is no more spatial structure in the data. Points are uncorrelated after the range. (data correlation)

Sill – constant semivariance value beyond the range (data variance) Demo #2 Simple kriging Validation

• Full validation - Split data into ~80% training, ~20% testing • Cross-validation (“Leave-one out”) - Remove a single known point, use all remaining points to interpolate at that location, then compare measured value to predicted value • Diagnostics - Predictions should be unibiased (e.g. over- and under-predictions should cancel each other out) - Mean Error should be near zero (unbiased) - Mean Standardized Error should be near zero - Predictions should be closed to known values - Root Mean Square Error (RMSE) should be as small as possible - Assessment of model stability and accuracy of standard errors - Root Mean Square Standardized should be close to 1 - Average Standard Error close to RMSE Empirical Bayesian Kriging (EBK)

• Automates the most difficult aspects of building a valid kriging model • Not as many parameters • Relaxes the stationarity assumption of kriging • More accurate estimates of prediction standard errors • Handles uncertainty associated with one semivariogram (true) How EBK works

1. Divide data into local subsets of a given size (can overlap) 2. For each subset, estimate the semivariogram 3. Use this semivariogram to simulate a new set of values for the points (sim #1) 4. Produce a semivariogram from the simulated points (semiv #1) 5. Repeat step 3 many times, resulting in a distribution of semivariograms 6. Mix the local prediction surfaces together to get the final surface Demo #3 EBK EBK Regression Prediction

• Combines regression with kriging • Allows covariates (explanatory variables to improve predictions) • Both regression models and kriging models are estimated locally • Uses Principal Components Analysis (PCA)

Kriging Regression Kriging

Prediction = mean + error Prediction (DV) = intercept + (v1 * coef1) + (v2 * coef2) +… (vk * coefk) + error • Mean is constant and error term is estimated • Regression equation estimates the mean for kriging from surrounding points • Error is modeled with the semivariogram, and kriging is performed • Estimation focuses on the error terms, and does little with the mean • If semivariogram is flat, you essentially have OLS • If there are no explanatory variables, you essentially have simple kriging Regression (OLS)

Prediction (DV) = intercept + (v1 * coef1) + (v2 * coef2) +… (vk * coefk) + error

• Error term is assumed to be random noise (unmodellable) • Estimation focuses on the mean, and does little with the error terms Demo #4 EBK Regression EBK 3D

• Applies the EBK model to 3D - Distances are calculated using 3D Euclidean Distance - Subsets are created in 3D - Search neighborhoods are 3D - Vertical trend can be removed • Elevation Inflation Factor - Vertical variation happens at ta different rate than horizontal variation Demo #5 EBK 3D Areal Interpolation

• Applies kriging theory to polygon data • Two main use cases - Fill missing data - Downscale from larger polygons to smaller polygons • Three data inputs - Average (Gaussian) - Rate (Binomial) - Count (Poisson) Demo #5 Areal Interpolation Print Your Certificate of Attendance

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