Mixture Density Mercer Kernels - Intelligent Systems Division - NASA

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Channel 6 Data0.60.5Channel Spectral Wavelength(nm)1 620-6702 841-8763 459-4794 545-5655 1230-12506 1628-16527 2105-21550.40.30.20.1Table 1: The bandwidths for the first seven MODISchannels. The spatial resolutions for Channels 1 and 2are 250 m, and 500 m for Channels 3-7.Figure 5: This figure shows the output of Channel 6 forday 167 in the year of 2002 from the MODIS instrument.Clouds are characterized by regions of greater density ofwhite. This data was used to test the Mixture DensityMercer Kernel as well as the Gaussian Mixture Model.Figure 4: This figure shows the output of Channel 6 forday 188 in the year of 2002 from the MODIS instrument.Clouds are characterized by regions of greater densityof white. This data was used for training the MixtureDensity Mercer Kernel as well as the Gaussian MixtureModel.0.60.50.40.30.20.1Figure 5 shows that there is considerable structure tothe cloud that is missed. Furthermore, other regions,such as the large cloud in the lower right hand portion ofthe image are broken into multiple constituents, whereasChannel 6 does not show such structure.Figure 7 shows the results of applying the same datato the Mixture Density Kernel. We built the kernel onthe same training data, and used 10 mixtures in eachmodel in the ensemble. We built 50 ensembles, resultingin a 500 dimensional feature space. The first area tonotice is the structure of the three-pronged cloud overthe ice sheet. This structure indicates that the cloud isnot a homogeneous entity, but has several constituents.The western side of the image also shows a considerableamount of variation, particularly in the clouds in thelower part of the image. We point out these detailsin contrast to the results for the Gaussian MixtureModel for the same region. These results suggest thatthe Mixture Density Kernel can reveal new featuresin the data in some cases better than the GaussianMixture Model. However, such variation can be foundin other regions of the image, thus making an objectiveevaluation difficult. This difficulty is not unique toMixture Density Kernels and applies equally to generalunsupervised learning algorithms.11 Discussion and ConclusionsWe have shown a method to generate a Mercer Kernelfunction from an ensemble of mixture models. The
GMM Result10Kernel Cluster Uncertainty0.25980.2760.1550.1430.052Figure 6: The results of applying a Gaussian MixtureModel to the test data using 10 mixture coefficients.This model does a good job at segmenting the imageacross the seven spectral bands. Notice that the largethree-pronged cloud is characterized as a single entity.1Figure 8: This figure shows the uncertainty in thecluster assignment as shown by the entropy of classdistribution. It is interesting to note that there is verylow uncertainty over most of the ice sheet, even thoughit is divided into two entities. The three-pronged cloudover the ice sheet shows up as a region with higheruncertainty, as one would expect.Kernel Cluster Result10Figure 7: This figure shows the results of the MixtureDensity Mercer Kernel on the test data using 10 mixturesin the ensemble with 50 models in total. In comparisonto the Gaussian Mixture Model, these resultsshow a difference in the characterization of the threeprongedcloud over the ice sheet. This cloud shows additionalstructure using two different elements. Otherareas of comparison are the three clouds at the top ofthe image. The Mixture Density Mercer Kernel methodcorrectly segments these into the same class, whereasthere is considerable variation in the competing model.987654321Mixture Density Mercer Kernel function is the dotproduct of the vectors of class distributions acrossensembles. We have shown that this function canbe interpreted as a new similarity measure betweentwo data points, and has the feature that regions ofhigher uncertainty can be reduced. We exhibited thealgorithm in an unsupervised clustering setting, whereit shows some promise. We have also shown that theMixture Density Mercer Kernel can incorporate domainknowledge through its inclusion as a prior distributionon the ensemble models. While the algorithm has someadvantages, it is expensive to compute and still exhibitssome run-to-run variation. We are currently researchingmethods to further reduce this variation.This paper has shown the behavior of the new kernelon unsupervised clustering problems. We are currentlypreparing the results of using this kernel function forsupervised prediction problems, in particular on thesnow, ice, and cloud classification.12 AcknowledgementsThe author would like to thank Bill Macready, NikunjOza, and Julienne Stroeve for valuable discussions andfeedback. Dr. Stroeve also provided the MODISdata used as examples in this paper. This work wassupported by the NASA Intelligent Data Understandingsegment of the Intelligent Systems Program.
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Mixture Density Mercer Kernels - Intelligent Systems Division - NASA

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