Kriging by Example: Regression of oceanographic data. Paris Perdikaris. Brown University, Division of Applied Mathematics

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1 Kriging by Example: Regression of oceanographic data Paris Perdikaris Brown University, Division of Applied Mathematics! January, 0 Sea Grant College Program Massachusetts Institute of Technology Cambridge, MA

2 Outline Overview of Kriging An academic example in D Pirate s Cove data-set: Bathymetry Plant height /9

3 Kriging Model observed data Y (x) as a realization of a Gaussian process Z(x) up to measurement error E(x): Construction steps: Y (x) =Z(x)+E(x). Assume covariance models ( ), E ( E ) parametrized by (, E ).. Explore spatial correlations in the observed data y and estimate the optimal kriging hyper-parameters (, E ) through optimization.. The conditional expectation E[Z Y ] provides a predictive scheme for estimating y at new locations x.. The conditional covariance Z Y quanties the uncertainty of the kriging predictor. /9

4 Kriging Some remarks: Choosing a covariance model should reect our prior belief of the nature of the data (smoothness, stationarity, anisotropy). Error models can capture the statistical behavior of the measurement noise. If E(x) is neglected then the kriging predictor interpolates the data. The computational cost is dominated by the optimization for learning the kriging hyper-parameters (typically scales os O(N )). Matlab code is available. /9

5 D Example f(x) =(x ) sin (x ) 0 y x /9

6 D Example f(x) =(x ) sin (x ) 0 y x /9

7 D Example f(x) =(x ) sin (x ) 0 y x 7/9

8 D Example f(x) =(x ) sin (x ) y x 8/9

9 D Example f(x) =(x ) sin (x ) 0 y x 9/9

10 D Example f(x) =sin(x)+ϵ f(x) Observations Kriging mean f(x) x 0/9

11 Pirate s Cove Mike Sacarny:, depth measurements, plant height measurements /9

12 Bathymetry Predictor Depth(m) 7 Uncertainty Depth(m) Longitude Longitude Latitude Matérn / covariance observations Latitude 0. /9

13 Bathymetry 0% measurement noise Predictor Depth(m) 7 Uncertainty Depth(m) Longitude Longitude Latitude Latitude Matérn / covariance observations /9

14 Bathymetry 0% measurement noise Predictor Uncertainty Depth(m) Depth(m) Longitude Longitude Latitude Matérn / covariance observations Latitude /9

15 Bathymetry Predictor accuracy 7 0% noise 0% noise 0% noise 7 7 ŷ ŷ ŷ 7 y 7 y 7 y /9

16 Bathymetry Kriging prediction based on observations /9

17 ... Plant Height 0% measurement noise Predictor Uncertainty Depth(m) Depth(m) Longitude Longitude Latitude Matérn / covariance observations Latitude 7/9

18 Plant Height Kriging prediction based on observations (overlay with predicted bathymetry) 8/9

19 Summary Kriging as a statistical regression tool for oceanographic data: Next steps: Is built on exploring spatial correlations in the data. Provides a scheme for making predictions at new spatial locations. Provides a measure of uncertainty quantication for the predicted values. Co-kriging for exploring cross-correlations between dierent variables. Choose new sampling locations using the maximum expected improvement criterion. Extend formulation to spatio-temporal data-sets. Scale algorithms to very large data-sets. 9/9

20 References [] N. Cressie. Statistics for Spatial Data. Wiley-Interscience, 99. [] A. Forrester, A. Sóbester, and A. Keane. Engineering Design via Surrogate Modelling: A Practical Guide. John Wiley & Sons, 008. [] A. I. J. Forrester, A. Sóbester, and A. J. Keane. Multi-fidelity optimization via surrogate modelling. P. Roy. Soc. Lond. A Mat., (088): 9, 007. [] T. Hastie, R. Tibshirani, J. Friedman, T. Hastie, J. Friedman, and R. Tibshirani. The elements of statistical learning. Springer, 009. [] M. C. Kennedy and A. O Hagan. Predicting the output from a complex computer code when fast approximations are available. Biometrika, 87():, 000. [] D. G. Krige. A Statistical Approach to Some Mine Valuation and Allied Problems on the Witwatersrand: By DG Krige. PhD thesis, University of the Witwatersrand, 9. [7] G. Matheron. Principles of geostatistics. Econ. Geol., 8(8):, 9. [8] C. E. Rasmussen and C. K. I. Williams. Gaussian Processes for Machine Learning. The MIT Press, 00. [9] J. Sacks, W. J. Welch, T. J. Mitchell, and H. P. Wynn. Design and analysis of computer experiments. Stat. Sci., ():09, 989.