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 Apr Lakshmanan et. al (OU/NSSL) Verification using GMM TAMU-CC 1 / 26
2 Model Verification Are these good forecasts? 3 hr Forecast 15-hr Forecast Observed = Lakshmanan et. al (OU/NSSL) Verification using GMM TAMU-CC 2 / 26
3 Model Verification Traditional verification methods Can simply compare the forecast and observed images pixel by pixel, but: 1 Small-scale structure differences lead to large errors 2 Position errors lead to double-penalty 3 Difficult to diagnose errors from MSE, POD/FAR/CSI, Brier score, etc. Lakshmanan et. al (OU/NSSL) Verification using GMM TAMU-CC 3 / 26
4 Model Verification 1. Neighborhood methods Instead of comparing pixel-by-pixel, evaluate forecasts within a local neighborhood: compute fractional coverage and Brier score for different box sizes (Roberts and Lean, 2008) Addresses only problem of small-scale structure differences. Lakshmanan et. al (OU/NSSL) Verification using GMM TAMU-CC 4 / 26
5 Model Verification 2. Scale decomposition Instead of computing errors for different box sizes, decompose the images into images using wavelets and parcel out the error at different scales and intensity thresholds (Casati et al. 2004) Addresses problems of small-scale structure differences and double penalty. Lakshmanan et. al (OU/NSSL) Verification using GMM TAMU-CC 5 / 26
6 3. Find deformations Model Verification Find deformations needed to make forecast field resemble observed field and compute error measures from deformation matrix (Gilleland 2009, Marzban 2009) Addresses only double penalty issue. Lakshmanan et. al (OU/NSSL) Verification using GMM TAMU-CC 6 / 26
7 Model Verification 4. Verify based on objects Identify objects based on thresholding the spatial fields and then associate objects to find object attributes (Davis et. al, 2009) Addresses all three issues, but object identification and assocation can be problematic. Lakshmanan et. al (OU/NSSL) Verification using GMM TAMU-CC 7 / 26
8 What is a GMM? Parametric 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 TAMU-CC 8 / 26
9 Number of Gaussians Parametric The accuracy of fit gets better as you increase number of Gaussians. Original 10 Gaussians Lakshmanan et. al (OU/NSSL) Verification using GMM TAMU-CC 9 / 26
10 Parametric Number of Gaussians Gaussians 50 Gaussians Lakshmanan et. al (OU/NSSL) Verification using GMM TAMU-CC 10 / 26
11 Diminishing Returns Parametric 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 TAMU-CC 11 / 26
12 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, 2010]. σ 2 y (2) (3) Lakshmanan et. al (OU/NSSL) Verification using GMM TAMU-CC 12 / 26
13 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 TAMU-CC 13 / 26
14 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 TAMU-CC 14 / 26
15 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 TAMU-CC 15 / 26
16 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 TAMU-CC 16 / 26
17 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 TAMU-CC 17 / 26
18 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 TAMU-CC 18 / 26
19 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 TAMU-CC 19 / 26
20 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 TAMU-CC 20 / 26
21 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 TAMU-CC 21 / 26
22 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 TAMU-CC 22 / 26
23 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 TAMU-CC 23 / 26
24 Error measures June 1, 2005 Model runs Lakshmanan et. al (OU/NSSL) Verification using GMM TAMU-CC 24 / 26
25 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 TAMU-CC 25 / 26
26 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. Wea. Forecasting, 24: Lakshmanan, V. and Kain, J. (2010). A Gaussian mixture model approach to forecast verification. Wea. Forecasting, 25(3): Lakshmanan, V., Smith, T., Stumpf, G. J., and Hondl, K. (2007). The warning decision support system integrated information. Wea. Forecasting, 22(3): Lakshmanan et. al (OU/NSSL) Verification using GMM TAMU-CC 26 / 26
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