Continuum Fusion: A New Approach to Composite Hypothesis Testing

Continuum Fusion:A New Approach to

Composite Hypothesis Testing

A. SchaumNaval Research Laboratory

Washington, D.Cschaum@nrl.navy.mil

Quantitative Methods in Defense and National Security 2010

George Mason University May 25-26, 2010

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

OUTLINE

• CONTEXT/BACKGROUND

• MOTIVATING EXAMPLE: ANOMALY DETECTION

• CREATING A SYSTEMATIC METHODOLOGY

• RESULTS

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

NAVY CONTEXTMANNED & UNMANNED LONG STANDOFF RANGE

RECCE/SURVEILLANCE

ANALYST DISPLAY STATION

APPROACH HSI autonomous detection system cues image analyst to region of interest on high resolution imager

NRL CONOPS is theINDUSTRY STANDARD

BACKGROUND:REPRESENTATIONS OF HYPERSPECTRAL DATA

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.

Each pixel is an N-dim. vector.

Hyperspectral Imagery

Hyperspectral Scatter Plot

Target of interest

A point represents a single HSI pixel.

Decision boundary

Likelihood ratio decision boundary

“Linear matched filter”

ADDITIVE TARGET MODELS

pC x 1

2 Ndet M

2x C t

M 1 x C

TARGET DISTRIBUTION

pT x 1

2 Ndet M

2x T t

M 1 x T

Target mean T can depend on parameters with unknown values.

Clutter mean and covariance can usually be estimated from field data.

CLUTTER DISTRIBUTION

Whitening

EUCLIDEAN SPACE

OUTLINE

• CONTEXT/BACKGROUND

• MOTIVATING EXAMPLE: ANOMALY DETECTION

• RESULTS

INITIAL MOTIVATION: SUPPRESSING ONE MECHANISM

OF FALSE ALARMS IN ANOMALY DETECTION

WHITENED SPACE

GLR SOLUTION DOES NOT

KNOW THE PHENOMENOLOGY

THE CFAR FUSION METHOD GIVES THE INTUITIVE ANSWER

PRIMARY CLUTTER

CLUTTER IN SHADOW GLR SURFACES

(CFAR) FLR SURFACES

PRIMARY

CLUTTER

REDUCED-SCALE VERSIONS OF

PRIMARY DETECTOR,

MATCHED TO CLUTTER SCALE (CFAR)

UNION OF ALL “CLUTTER”

DECISION REGIONS

STANDARD ANOMALY DETECTOR

DECISION BOUNDARY

pC x 1

2 N exp 1

2x C t

OUTLINE

• CONTEXT

• MOTIVATING EXAMPLE

• RESULTS

CREATING A SYSTEMATIC METHODOLOGY

1. FUSION LOGIC

2. GETTING THE RIGHT RESULT IN THE “CLASSICAL LIMIT”

3. HANDLING THE GENERAL CASE

1. 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

2. GETTING THE RIGHT RESULT IN THE “CLASSICAL LIMIT”– Generating the optimal answer, when it exists

3. HANDLING THE GENERAL CASE– Deriving the “Fusion Relations”

– A surprise: Unification

– Fusion flavors

3. HANDLING THE GENERAL CASE– Deriving the “Fusion Relations”

Summary of “Getting the Right Results”

• 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

Some “unknown parameter” problems have optimal solutions (UMP: “uniformly most powerful”) that do no depend on those parameters.

– Fusion flavors

3. HANDLING THE GENERAL CASE– Introductory example

– The “Fusion Relations”

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

UNKNOWN PARAMETER VALUES IN TARGET SIGNATURE PREDICTION

Mean target spectrum

Covariancematrix

VRC: Virtual Relative Calibration

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

Application: use of laboratory reflectance signature to detect material in remote sensing system

Issue: Sensor measures radiance, not reflectance

Clutter

DIMENSION 1

AN AFFINE TARGET SUBSPACE MODEL FOLLOWS FROM VRC

Target

TARGET DISTRIBUTIONHAS UNKNOWN MEAN

LAB REFLECTANCESIGNATURE

Clutter mean

Target

Clutter

DIMENSION 1

AN AFFINE SUBSPACE TARGET MODEL

TARGET HAS KNOWN VARIANCE,KNOWN MEAN DIRECTION,

BUT UNKNOWN MEAN AMPLITUDE

GENERALIZED LIKELIHOODRATIO TEST

Target

Clutter

AMF DecisionBoundaries

DIMENSION 1

THE AFFINE MATCHED FILTERSOLVES A GLR PROBLEM

DIMENSION 1

CFAR FUSION SOLUTION TO THE AFFINE TARGET SUBSPACE MODEL

TARGET HAS KNOWN VARIANCEBUT UNKNOWN MEAN

“DECLARE TARGET” REGION

Target

DIMENSION 1

FUSED CFAR DECISION SURFACE FOR THE

AFFINE SUBSPACE PROBLEM

LR FUSIONDecision Boundary(Comet shape) is a

combination ofasymptotes and envelopes of the constituent curves

Target

DIMENSION 1

FLR DECISION SURFACES FOR THE

CFAR FUSIONDecision

Boundaries

Target

GLR DecisionBoundaries

DIMENSION 1

GLR vs FLR DECISION SURFACES FOR THE

CFAR FUSIONDecision

Boundaries

DIMENSION 1

CPD FUSION SOLUTION TO THE AFFINE TARGET SUBSPACE MODEL

TARGET HAS KNOWN VARIANCEBUT UNKNOWN MEAN

“DECLARE TARGET” REGION

ENVELOPE CANNOT BE DEDUCED FROM GEOMETRICAL ARGUMENTS

OPTIMAL DETECTORS BASED ON THE LIKELIHOOD RATIO TEST

The FUNDAMENTAL THEOREM of statistical binary testing.

LR pT x : t pC x : c threshold x represents target

d x : t, c pT x : t pC x : c discriminant function

IF the values of all parameters t, c are knownIF the values of all parameters t, c are known

DECISION BOUNDARY IS DEFINED BY NULLS IN THE DISCRIMINANT FUNCTION: d(x:t,c) = 0

(CLR) FUSION IN ACTION

CLUTTER AT 1

1-D TARGET SUBSPACE

FUSED DECISION SURFACE

The FUNDAMENTAL THEOREM OFCONTINUUM FUSION

d x; t, c pT (x; t)

pC x; c t, c 0

d x; t, c ti

0;d x; t, c

2d x; t, c 2ti

0;2d x; t, c

LAPLACIAN ECD ADDITIVE TARGET MODEL

CLUTTER

BALLTARGET DIRECTION

Each curve is characterized by a number:

the likelihood ratio

d x : t pT x : t 8 pC x 0

Clutter (& target) modeled as Laplacian-distributed-more realistic-matched filter alarms falsely on half of all outliers

MEAN TARGET = 8

“Seed” algorithm for fusion

CLR FUSION SOLUTION TO THE LAPLACIAN ECD ADDITIVE TARGET MODEL

CLR FUSION SOLUTION

CONSTRAINT ON CONSTITUENT DETECTORSALL MUST HAVE THE SAME LR VALUES FOR THE

CORRESPONDING TARGET DISTRIBUTION MODELS

MEAN TARGET VALUES: 9 8 7 6 5 4.5

MEAN TARGET = 8EXAMPLE

MEAN TARGET = 9EXAMPLE

CLUTTER

BALL TARGET DIRECTION

Each curve is characterized by a number:

the likelihood ratio

d x : t pT x : t pC x 0

DETECTORS BASED ON THE GENERALIZED LIKELIHOOD RATIO TEST

GLR x t

Max pT x; t

Max pC x; c threshold x represents target

t = target parameters, c = clutter parameters

Standard recipe for composite hypothesis problem:

GLR test

RELATION OF GLR TO FUSION

GLR recipe:

GLR x t

Max pT x; t

Max pC x; c

pT x; t ti

0;pC x; c

2 pT x; t ti

2 0;2 pC x; c

General Fusion Equations:

d x; t, c pT (x; t)

pC x; c t, c 0

d x; t, c ti

0;d x; t, c

2d x; t, c 2ti

0;2d x; t, c

1. GLR is equivalent to a fusion method!2. independent of t,c means GLR can be derived from a“Constant Likelihood Ratio” (CLR) Fusion Method.Therefore:3. The fusion formalism always includes the GLR as a

special case!

CLR FUSION SOLUTION TO THE LAPLACIAN ECD ADDITIVE TARGET MODEL

IS THE GLR TEST

CLR FUSION/GLR TEST

MEAN TARGET VALUES: 9 8 7 6 5 4.5

CLUTTER

Each curve produces the same likelihood ratio for its own target distribution

d x : t pT x : t pC x 0

that is, is independent of t

GLR (= CLR FUSION FLAVOR) FOR THE LAPLACIAN ECD ADDITIVE TARGET MODEL

CLUTTER

CLR FUSION SOLUTION

1. LR was constant in the fusion process.2. Constituents were hyperboloids.3. Asymptotes grow linearly with target

mean because LR kept constant.

Surface is a paraboloidResembles matched filter asymptoticallyOutlier rejection is lost in the fusion process

Can prevent growth of asymptotic slopes by allowing log(LR) to vary (linearly) with target meanLLLR FUSION (Log Linear Likelihood Ratio)

• Recoups outlier rejection• Captures bulk statistical rejection

CONTINUUM OF LLLR FUSION FLAVORS FOR THE LAPLACIAN ECD

ADDITIVE TARGET MODEL

CLUTTER

CLR FUSION SOLUTION

MINIMUM TARGET CONTRAST SET TO 4 STANDARD DEVIATIONS FOR ALL 4 DETECTORS

THREE OTHER DETECTOR FLAVORS(LINEAR LOG LIKELIHOOD RATIO)

FOR THE ADDITIVE TARGET MODEL

DESIGNED TO MINIMIZE OUTLIER DETECTIONS

DECISION BOUNDARIES FOR THREE VERSIONS OF THE AFFINE MATCHED FILTER

3 FUSION FLAVORSfor

GAUSSIAN DISTRIBUTIONS

OUTLIERS PRESENT?

GLR FOR THE AFFINE LAPLACIAN MODEL

CLUTTER AT 1

1-D TARGET SUBSPACE

FUSED DECISION SURFACE

OUTLIERS?

LLLR FUSION FLAVOR FORAFFINE LAPLACIAN MODEL

OUTLIERS?

FUSION FLAVOR TAILORED TO1. REJECT BULK CLUTTER2. REJECT OUTLIERS

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

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

MULTIMODAL FUSION APPLICATION OF CONTINUUM FUSION

PROTOTYPE PROBLEMTWO SENSORSFUSE SIGNALS FROM BOTH

SENSOR 1

SENSOR 2 PROCESSING

1-D SIGNALS

FUSION

DETECTIONS

MODE 1

MODE 2

TARGET DISTRIBUTION

DIMENSION 1

DIMENSION 2

BIVARIATETARGET DISTRIBUTION

DIMENSION 2

DIMENSION 1CLUTTER

TARGET

1. Clutter signals are uncorrelated, due to whitening transformation.

2. Target and clutter distributions have different means, but identical variances.

3. Target signals have unknown level of correlation .

= 0 = -3/4

The correlation is a target parameter with unknown value, which defines a composite hypothesis testing problem.

GLR method is nearly unsolvable!

BIMODAL FUSION

Detectors corresponding to3 different threshold values

BIMODAL FUSION

= -1/2

f = .8

Picking a seed algorithm

Detectors corresponding to3 different threshold values

BIMODAL FUSION

f = .8

FI Fusion Flavor

Expectation for the sensor fusion problem:1. FI should be approximately CFAR and approximately CPD2. Therefore it should also be approximately CLR (i.e. GLR)

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

FI Fusion is solvable in closed formGLR method is virtually unsolvable

BIMODAL FUSION f = .8

Selected values of > 0

BIMODAL FUSION

f = .8Selected values of < 0

BIMODAL FUSION f = .8All

d x; pT (x;)

pC x 0

Bounding surfaces correspondsto extreme allowed values of parameters

Removing the spurious boundaries

FI fuseesvs

GLR fusees

= -.95

= -.95 = 1/2