Continuum Fusion: A New Approach to Composite Hypothesis Testing

1. Continuum Fusion:A New Approach to Composite Hypothesis Testing A. Schaum Naval Research Laboratory Washington, D.C schaum@nrl.navy.mil Quantitative Methods in Defense and National Security 2010 George Mason University May 25-26, 2010

2. CONTINUUM FUSION:A NEW THEORY of INFERENCE A FRAMEWORK FOR GENERATING DETECTION ALGORITHMS WHEN USING AMBIGUOUS MODELS PROBLEM CLASS • FOR MAKING DECISIONS BASED ON MODELS CONTAINING PARAMETERS WHOSE VALUES ARE FIXED BUT UNKNOWN (CALLED THE “COMPOSITE HYPOTHESIS” TESTING PROBLEM) • CAN BE SUBSTITUTED FOR ANY GENERALIZED LIKELIHOOD RATIO (GLR) TEST SYNOPSIS • FUSES A CONTINUUM OF OPTIMAL METHODS WHEN YOU DON’T KNOW WHICH ONE IS REALLY OPTIMAL ADVANTAGES • GROWS THE GLR RECIPE INTO A FULL MENU OF DETECTION ALGORITHM “FLAVORS” • CAN PRODUCE DETECTORS FOR MODELS WHERE GLR IS UNSOLVABLE • FLEXIBILITY ALLOWS SIMULTANEOUS TREATMENT OF STATISTICAL AND NON-STATISTICAL MODELS • ALLOWS OPTIMIZATION OF NEW DESIGN METRICS

3. OUTLINE • CONTEXT/BACKGROUND • MOTIVATING EXAMPLE: ANOMALY DETECTION • CREATING A SYSTEMATIC METHODOLOGY • RESULTS

4. CONTEXT:DETECTION & DISCRIMINATION ALGORITHMS • DATA DRIVEN/AGNOSTIC (“MACHINE LEARNING”) • ARTIFICIAL NEURAL NETWORKS • GENETIC • SVMs ... • MODEL-BASED • UBIQUITOUS IN MANY SENSING MODALITIES • KNOWN PHYSICS, UNKNOWN PARAMETERS • COMMONEST: SIGNAL AMPLITUDE • VARIABLE RANGE • UNKNOWN SIGNAL DRIVER • ALLOW GENERALIZATION TO UNTRAINED SITUATIONS

5. NAVY CONTEXTMANNED & UNMANNED LONG STANDOFF RANGE RECCE/SURVEILLANCE NRL CONOPS is the INDUSTRY STANDARD APPROACH HSI autonomous detection system cues image analyst to region of interest on high resolution imager ANALYST DISPLAY STATION

6. Hyperspectral Scatter Plot A point represents a single HSI pixel. Target of interest Green Each pixel is an N-dim. vector. Red BACKGROUND:REPRESENTATIONS OF HYPERSPECTRAL DATA Hyperspectral Imagery • Algorithms operate in an N-dimensional spectral space (N=64 for WAR HORSE) • Similar objects in HSI imagery occupy similar regions in the spectral space. • Multivariate detection algorithms generate a “decision surface” that identifies where targets lie in the vector space. • Number of dimensions should exceed number of different constituents.

7. Target mean T can depend on parameters with unknown values. Likelihood ratio decision boundary “Linear matched filter” TARGET DISTRIBUTION Decision boundary Whitening EUCLIDEAN SPACE CLUTTER DISTRIBUTION ADDITIVE TARGET MODELS Clutter mean and covariance can usually be estimated from field data.

8. OUTLINE • CONTEXT/BACKGROUND • MOTIVATING EXAMPLE: ANOMALY DETECTION • CREATING A SYSTEMATIC METHODOLOGY • RESULTS

9. UNION OF ALL “CLUTTER” DECISION REGIONS (CFAR) FLR SURFACES STANDARD ANOMALY DETECTOR DECISION BOUNDARY PRIMARY CLUTTER CLUTTER IN SHADOW GLR SURFACES REDUCED-SCALE VERSIONS OF PRIMARY DETECTOR, MATCHED TO CLUTTER SCALE (CFAR) INITIAL MOTIVATION: SUPPRESSING ONE MECHANISM OF FALSE ALARMS IN ANOMALY DETECTION WHITENED SPACE PRIMARY CLUTTER GLR SOLUTION DOES NOT KNOW THE PHENOMENOLOGY THE CFAR FUSION METHOD GIVES THE INTUITIVE ANSWER

10. OUTLINE • CONTEXT • MOTIVATING EXAMPLE • CREATING A SYSTEMATIC METHODOLOGY • RESULTS

11. CREATING A SYSTEMATIC METHODOLOGY • FUSION LOGIC • GETTING THE RIGHT RESULT IN THE “CLASSICAL LIMIT” • HANDLING THE GENERAL CASE

12. CREATING A SYSTEMATIC METHODOLOGY • FUSION LOGIC • Form UNION of “decide clutter” regions (if fusing over clutter parameters) • Form UNION of “decide target” regions (if fusing over target parameters) • Fusion flavors • GETTING THE RIGHT RESULT IN THE “CLASSICAL LIMIT” • Generating the optimal answer, when it exists • HANDLING THE GENERAL CASE • Deriving the “Fusion Relations” • A surprise: Unification

13. CREATING A SYSTEMATIC METHODOLOGY • FUSION LOGIC • Form UNION of “decide clutter” regions (if fusing over clutter parameters) • Form UNION of “decide target” regions (if fusing over target parameters) • Fusion flavors • GETTING THE RIGHT RESULT IN THE “CLASSICAL LIMIT” • Generating the optimal answer, when it exists • HANDLING THE GENERAL CASE • Deriving the “Fusion Relations” • A surprise: Unification

14. Summary of “Getting the Right Results” Some “unknown parameter” problems have optimal solutions (UMP: “uniformly most powerful”) that do no depend on those parameters. • CFAR and CPD flavors both give the matched filter answer to the Gaussian additive target problem (unknown target amplitude) • GLR solution does not always give the right answer to the Gaussian additive target problem • CFAR, CPD, and GLR flavors all give the correct Gaussian anomaly detector

15. CREATING A SYSTEMATIC METHODOLOGY • FUSION LOGIC • Form UNION of “decide clutter” regions (if fusing over clutter parameters) • Form UNION of “decide target” regions (if fusing over target parameters) • Fusion flavors • GETTING THE RIGHT RESULT IN THE “CLASSICAL LIMIT” • Generating the optimal answer, when it exists • HANDLING THE GENERAL CASE • Introductory example • The “Fusion Relations” • A surprise: Unification

16. Mean target spectrum 1 3 2 Covariance matrix UNKNOWN PARAMETER VALUES IN TARGET SIGNATURE PREDICTION In-scene target radiance prediction • 1st order: reflectance to radiance • (Solar spectrum) x (reflectivity) • 2nd order: column densities • aerosols • water vapor • CO2 • BRDF effects • contamination • Other unknowns • Downwelling radiances • solar • sky • Background interactions • reflections • adjacency effects • Upwelling effects

17. VRC: Virtual Relative Calibration Application: use of laboratory reflectance signature to detect material in remote sensing system Issue: Sensor measures radiance, not reflectance • Most Short Wave IR mineral reflectance spectra are flat (“graybody”) • Model the mean reflectance of an image as gray body • A non-flat mean background radiance spectrum seen by a remote systems reflects the spectral content of illumination/attenuation effects • Mean spectrum can serve as relative calibration source

18. Target Subspace Clutter mean . X AN AFFINE TARGET SUBSPACE MODEL FOLLOWS FROM VRC TARGET DISTRIBUTION HAS UNKNOWN MEAN Clutter DIMENSION 2 LAB REFLECTANCE SIGNATURE DIMENSION 1

19. TARGET HAS KNOWN VARIANCE, KNOWN MEAN DIRECTION, BUT UNKNOWN MEAN AMPLITUDE GENERALIZED LIKELIHOOD RATIO TEST AN AFFINE SUBSPACE TARGET MODEL Clutter Target Subspace DIMENSION 2 DIMENSION 1

20. THE AFFINE MATCHED FILTER SOLVES A GLR PROBLEM AMF Decision Boundaries Clutter Target Subspace DIMENSION 2 DIMENSION 1

21. CFAR FUSION SOLUTION TO THE AFFINE TARGET SUBSPACE MODEL “DECLARE TARGET” REGION DIMENSION 2 TARGET HAS KNOWN VARIANCE BUT UNKNOWN MEAN DIMENSION 1

22. FUSED CFAR DECISION SURFACE FOR THE AFFINE SUBSPACE PROBLEM LR FUSION Decision Boundary (Comet shape) is a combination of asymptotes and envelopes of the constituent curves Target Subspace DIMENSION 2 DIMENSION 1

23. FLR DECISION SURFACES FOR THE AFFINE SUBSPACE PROBLEM CFAR FUSION Decision Boundaries Target Subspace DIMENSION 2 DIMENSION 1

24. GLR vs FLR DECISION SURFACES FOR THE AFFINE SUBSPACE PROBLEM CFAR FUSION Decision Boundaries GLR Decision Boundaries Target Subspace DIMENSION 2 DIMENSION 1

25. CPD FUSION SOLUTION TO THE AFFINE TARGET SUBSPACE MODEL “DECLARE TARGET” REGION ENVELOPE CANNOT BE DEDUCED FROM GEOMETRICAL ARGUMENTS DIMENSION 2 TARGET HAS KNOWN VARIANCE BUT UNKNOWN MEAN DIMENSION 1

26. OPTIMAL DETECTORS BASED ON THE LIKELIHOOD RATIO TEST The FUNDAMENTAL THEOREM of statistical binary testing. IF the values of all parameters t, care known DECISION BOUNDARY IS DEFINED BY NULLS IN THE DISCRIMINANT FUNCTION: d(x:t,c) = 0

27. (CLR) FUSION IN ACTION CLUTTER AT 1  1-D TARGET SUBSPACE FUSED DECISION SURFACE

29. The FUNDAMENTAL THEOREM OFCONTINUUM FUSION

30. “Seed” algorithm for fusion MEAN TARGET = 8 TARGET DIRECTION LAPLACIAN ECD ADDITIVE TARGET MODEL CLUTTER BALL • Clutter (& target) modeled as Laplacian-distributed • more realistic • matched filter alarms falsely on half of all outliers

31. CLR FUSION SOLUTION MEAN TARGET = 8 EXAMPLE MEAN TARGET = 9 EXAMPLE TARGET DIRECTION CLR FUSION SOLUTION TO THE LAPLACIAN ECD ADDITIVE TARGET MODEL MEAN TARGET VALUES: 9 8 7 6 5 4.5 CLUTTER BALL CONSTRAINT ON CONSTITUENT DETECTORS ALL MUST HAVE THE SAME LR VALUES FOR THE CORRESPONDING TARGET DISTRIBUTION MODELS

32. Standard recipe for composite hypothesis problem: GLR test DETECTORS BASED ON THE GENERALIZED LIKELIHOOD RATIO TEST

33. RELATION OF GLR TO FUSION GLR is equivalent to a fusion method!  independent of t,c means GLR can be derived from a “Constant Likelihood Ratio” (CLR) Fusion Method. Therefore: The fusion formalism always includes the GLR as a special case!

34. CLR FUSION/GLR TEST TARGET DIRECTION CLR FUSION SOLUTION TO THE LAPLACIAN ECD ADDITIVE TARGET MODELIS THE GLR TEST MEAN TARGET VALUES: 9 8 7 6 5 4.5 CLUTTER BALL

35. GLR (= CLR FUSION FLAVOR) FOR THE LAPLACIAN ECD ADDITIVE TARGET MODEL CLR FUSION SOLUTION CLUTTER BALL Surface is a paraboloid Resembles matched filter asymptotically Outlier rejection is lost in the fusion process TARGET DIRECTION LR was constant in the fusion process. Constituents were hyperboloids. Asymptotes grow linearly with target mean because LR kept constant. • Can prevent growth of asymptotic slopes by allowing log(LR) to vary (linearly) with target mean LLLR FUSION (Log Linear Likelihood Ratio) • Recoups outlier rejection • Captures bulk statistical rejection

36. DESIGNED TO MINIMIZE OUTLIER DETECTIONS CONTINUUM OF LLLR FUSION FLAVORS FOR THE LAPLACIAN ECD ADDITIVE TARGET MODEL CLR FUSION SOLUTION CLUTTER BALL TARGET DIRECTION THREE OTHER DETECTOR FLAVORS (LINEAR LOG LIKELIHOOD RATIO) FOR THE ADDITIVE TARGET MODEL MINIMUM TARGET CONTRAST SET TO 4 STANDARD DEVIATIONS FOR ALL 4 DETECTORS

37. * * * * OUTLIERS PRESENT? * DECISION BOUNDARIES FOR THREE VERSIONS OF THE AFFINE MATCHED FILTER GLR CFAR CPD 3 FUSION FLAVORS for GAUSSIAN DISTRIBUTIONS

38. * * * * OUTLIERS? GLR FOR THE AFFINE LAPLACIAN MODEL CLUTTER AT 1  FUSED DECISION SURFACE 1-D TARGET SUBSPACE

39. * * * * OUTLIERS? LLLR FUSION FLAVOR FORAFFINE LAPLACIAN MODEL FUSION FLAVOR TAILORED TO REJECT BULK CLUTTER REJECT OUTLIERS

40. SUMMARY: CONTINUUM FUSION • Provides a new framework for designing detection algorithms in model-based problem sets • Reduces to the desired results in the appropriate limits • Matched filter, RX • Does so more naturally and generally than GLR • Comes in many flavors • Includes “vanilla,” the GLR,the only prior solution to the general CH problem • New metrics of performance can be optimized (min/max) • Any CF method can be customized by manipulating the fusion process • (including GLR) • Non-statistical criteria can be accommodated • Constitutes a new branch of Statistical Detection Theory • Future • Any model that has used a GLR can be revisited • Thousands of published results • Some CH problems unsolvable with GLR can be solved with other CF methods • Theoretical issues • Relationship to specialized apps (“parameter testing holy trinity”) • UMPs and Invariance • Studying “fusion characteristics” • References • -A. Schaum, Continuum Fusion, a theory of inference, with applications to hyperspectral detection, 12 April 2010 / Vol. 18, No. 8 / OPTICS EXPRESS 8171-8188. • -A. Schaum, Continuum Fusion Detectors for Affine and Oblique Spectral Subspace Models, Special Issue of IEEE Transactions on Geoscience and Remote Sensing on Hyperspectral Image and Signal Processing, in review

41. Matrix of CF Problems (System model) x (Statistical model) x (Flavor) • System Model • Physical: Sensing mode/environment • Structural • Statistical models • Gaussian, Laplacian, t-score, ... • Homoskedastic, Ampliskedastic, Heteroskedastic • Some flavors • CFAR: constant false alarm rate • CPD: constant probability of detection • CRL: constant likelihood ratio (= GLR) • FIF: Fixed Intercept Fraction • FI: Fixed Intercept • LLLR: Linear log likelihood ratio • Geometrical

42. MULTIMODAL FUSION APPLICATION OF CONTINUUM FUSION PROTOTYPE PROBLEM TWO SENSORS FUSE SIGNALS FROM BOTH SENSOR 1 FUSION SENSOR 2 PROCESSING DETECTIONS 1-D SIGNALS

43. TARGET DISTRIBUTION TARGET DISTRIBUTION MULTIMODAL FUSION APPLICATION OF CONTINUUM FUSION MODE 2 MODE 1 CLUTTER DISTRIBUTION

44. BIVARIATE TARGET DISTRIBUTION MULTIMODAL FUSION APPLICATION OF CONTINUUM FUSION DIMENSION 2 DIMENSION 1 CLUTTER DISTRIBUTION

45. = 3/4 = 0 = -3/4 MULTIMODAL FUSION APPLICATION OF CONTINUUM FUSION DIMENSION 2 Clutter signals are uncorrelated, due to whitening transformation. Target and clutter distributions have different means, but identical variances. Target signals have unknown level of correlation . TARGET DIMENSION 1 CLUTTER The correlation  is a target parameter with unknown value, which defines a composite hypothesis testing problem. GLR method is nearly unsolvable!

46. BIMODAL FUSION Detectors corresponding to 3 different threshold values = 1/2

47. f = .8 BIMODAL FUSION Detectors corresponding to 3 different threshold values = -1/2 Picking a seed algorithm

48. BIMODAL FUSION = 1/2 f = .8

49. FI Fusion Flavor Definition of Fixed Intercept fusion: For different parameter values, fuse optimal algorithms whose decision boundaries have the same intercept with line from target-to-clutter means Expectation for the sensor fusion problem: • FI should be approximately CFAR and approximately CPD • Therefore it should also be approximately CLR (i.e. GLR) FI Fusion is solvable in closed form GLR method is virtually unsolvable

50. f = .8 BIMODAL FUSION Selected values of > 0