Model Verification Using Gaussian Mixture Models
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1 Model Verification Using Gaussian Mixture Models A Parametric, Feature-Based Method Valliappa Lakshmanan 1,2 John Kain 2 1 Cooperative Institute of Mesoscale Meteorological Studies University of Oklahoma 2 Radar Research and Development Division National Severe Storms Laboratory Jan Lakshmanan et. al (OU/NSSL) Verification using GMM AMS Prob. and Stats. Conf. 1 / 20
2 What is a GMM? Introduction Intuitively: find an optimal way to place Gaussian functions at various points in the image such that the sum of these Gaussians mimics the input gridded field. Original 5 Gaussians Lakshmanan et. al (OU/NSSL) Verification using GMM AMS Prob. and Stats. Conf. 2 / 20
3 Number of Gaussians Introduction The accuracy of fit gets better as you increase number of Gaussians. Original 10 Gaussians Lakshmanan et. al (OU/NSSL) Verification using GMM AMS Prob. and Stats. Conf. 3 / 20
4 Introduction Number of Gaussians Gaussians 50 Gaussians Lakshmanan et. al (OU/NSSL) Verification using GMM AMS Prob. and Stats. Conf. 4 / 20
5 Diminishing Returns Introduction Keeps getting more and more accurate: but at some point, the benefits of a parametric model are lost and you might as well just use the pixel values. Lakshmanan et. al (OU/NSSL) Verification using GMM AMS Prob. and Stats. Conf. 5 / 20
6 The GMM GMM The GMM is defined as a weighted sum of K two-dimensional Gaussians: K G(x, y) = π k f k (x, y) (1) Σ xy is: 1 f (x, y) = 2π Σ xy k=1 1 )(y µy ))Σ e ((x µx xy ((x µ x )(y µ y )) T /2 ( ) σ 2 x σ xy σ xy Solved using an iterative approach [Lakshmanan and Kain, 2009]. σ 2 y (2) (3) Lakshmanan et. al (OU/NSSL) Verification using GMM AMS Prob. and Stats. Conf. 6 / 20
7 GMM Intensity? The GMM is defined so as to sum to 1, and the iterative method optimizes the likelihood of the parameters given the positions of the pixels (and not the intensity). Two minor changes: 1 The total intensity associated with all the pixels in the image is used to scale the GMM 2 More intensive locations are repeated several (m) times: m = 1 + γround( CDF(I xy) freq(i mode ) ) I xy < I mode (4) Lakshmanan et. al (OU/NSSL) Verification using GMM AMS Prob. and Stats. Conf. 7 / 20
8 Synthetic case Example Verification: Geometric Geometric dataset from [Ahijevych et al., 2009]. Chose 3 Gaussians without much intensity correction geom000 geom003 Lakshmanan et. al (OU/NSSL) Verification using GMM AMS Prob. and Stats. Conf. 8 / 20
9 Synthetic case Geometric: why is the fit so bad? 1 Abrupt changes in intensity: need more Gaussians 2 Fewer high-intensity pixels: need intensity correction Orig γ = 0 γ = 0.5 γ = 1 γ = 3 γ = 5 GMM fit better on real-world images Lakshmanan et. al (OU/NSSL) Verification using GMM AMS Prob. and Stats. Conf. 9 / 20
10 Synthetic case Example Verification: Geometric µ x µ y σx 2 σ xy σy 2 π k 0 Original pts right pts right pts right too big Lakshmanan et. al (OU/NSSL) Verification using GMM AMS Prob. and Stats. Conf. 10 / 20
11 Synthetic case Example Verification: Geometric µ x µ y σx 2 σ xy σy 2 π k pts right turned pts right huge Lakshmanan et. al (OU/NSSL) Verification using GMM AMS Prob. and Stats. Conf. 11 / 20
12 Perturbed dataset Perturbed cases 2km WRF from CAPS perturbed (See [Ahijevych et al., 2009]). fake000 fake003 fake007 Lakshmanan et. al (OU/NSSL) Verification using GMM AMS Prob. and Stats. Conf. 12 / 20
13 Perturbed cases GMM parameters for perturbed cases µ x µ y σx 2 σ xy σy 2 π k Original pts. right pts. down pts. right pts. down pts. right pts. down Lakshmanan et. al (OU/NSSL) Verification using GMM AMS Prob. and Stats. Conf. 13 / 20
14 Perturbed cases GMM parameters for perturbed cases Description µ x µ y σx 2 σ xy σy 2 π k 24 pts. right pts. down pts. right pts. down pts. right pts. down times pts. right pts. down minus 2 mm Lakshmanan et. al (OU/NSSL) Verification using GMM AMS Prob. and Stats. Conf. 14 / 20
15 Error measures Error measures Translation error: e tr = (µ xf µ xo ) 2 + (µ yf µ yo ) 2 (5) Rotation error: e rot = 180 π cos 1 (v f.v o ) (6) v f and v o are the maximum-variance eigen vectors of the covariance matrices (Σ) Scaling error: e sc = π k f π ko (7) Lakshmanan et. al (OU/NSSL) Verification using GMM AMS Prob. and Stats. Conf. 15 / 20
16 Associating Gaussians Error measures For each Gaussian in the forecast field, compute the overall error with each Gaussian of observed field and pick the one with the lowest error. e = 0.3 min( e tr 100, 1)+ 0.2 min(e rot, 180 e rot )/ (max(e sc, 1/e sc ) 1) This can also be used to rank model forecasts. The weights are subjective. (8) Lakshmanan et. al (OU/NSSL) Verification using GMM AMS Prob. and Stats. Conf. 16 / 20
17 Error measures Synthetic images: Rank Description e tr e rot e sc e 1 50 pts. right pts. right pts. right, too big pts. right, wrong orient pts. right, huge Rotation error on circular objects is undefined. Ranking: geom001, geom002, geom004, geom003 and finally geom005. Lakshmanan et. al (OU/NSSL) Verification using GMM AMS Prob. and Stats. Conf. 17 / 20
18 Error measures June 1, 2005 Model runs Lakshmanan et. al (OU/NSSL) Verification using GMM AMS Prob. and Stats. Conf. 18 / 20
19 Acknowledgements Error measures Funding for this research was provided under NOAA-OU Cooperative Agreement NA17RJ1227. The GMM fitting technique described in this paper has been implemented within the Warning Decision Support System Integrated Information (WDSSII; [Lakshmanan et al., 2007]) as part of the w2smooth process. It is available for download at Lakshmanan et. al (OU/NSSL) Verification using GMM AMS Prob. and Stats. Conf. 19 / 20
20 Error measures References Ahijevych, D., Gilleland, E., Brown, B., and Ebert, E. (2009). Application of spatial verification methods to idealized and NWP gridded precipitation forecasts. Weather and Forecasting, 0(0):InPress. Lakshmanan, V. and Kain, J. (2009). A Gaussian mixture model approach to forecast verification. Weather and Forecasting, 0(0):subm. Lakshmanan, V., Smith, T., Stumpf, G. J., and Hondl, K. (2007). The warning decision support system integrated information. Weather and Forecasting, 22(3): Lakshmanan et. al (OU/NSSL) Verification using GMM AMS Prob. and Stats. Conf. 20 / 20
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