LECTURE SET 6 PROBABILISTIC BEHAVIOUR RECOGNITION ECVision Summer - PowerPoint PPT Presentation

School of Informatics, University of Edinburgh LECTURE SET 6 PROBABILISTIC BEHAVIOUR RECOGNITION ECVision Summer School: 6 - Probabilistic Behaviour Understanding Fisher slide 1

School of Informatics, University of Edinburgh PROBABILISTIC BEHAVIOUR RECOGNITION SUPPOSE AT FRAME i : • HAVE A FRAME DESCRIPTION � f i • FRAME IS IN STATE s i OF A SET OF STATES S = { λ 1 , . . . , λ S } • NO DETERMINISTIC MAP OF � f i to s i . IE. STATE s i HIDDEN. • WE KNOW p ( s i | � f i ) < 1 FOR ALL s i ∈ S . s i ∈S p ( s i | � f i ) = 1. � WHAT TO DO NOW? ECVision Summer School: 6 - Probabilistic Behaviour Understanding Fisher slide 2

School of Informatics, University of Edinburgh BROWSE CONTEXT MODEL M1 B M2 HERE: S = { M 1 , M 2 , B } ASSUME WE CAN COMPUTE: p ( M 1 | � f ) , p ( M 1 | � f ) , p ( B | � f ) EG. BAYES RULE: p ( S | � f ) = p ( � f | S ) p ( S ) /p ( � f ) ECVision Summer School: 6 - Probabilistic Behaviour Understanding Fisher slide 3

School of Informatics, University of Edinburgh MARKOV ASSUMPTION FUTURE STATE IDENTITY DEPENDS ONLY ON THE PRESENT STATE (EG. CURRENT FRAME) AND NOT ON THE EXACT DETAILS OF THE PAST IMPLICIT IN GRAPH MODEL M1 B M2 M 2 DEPENDS ON B , NOT M 1 ECVision Summer School: 6 - Probabilistic Behaviour Understanding Fisher slide 4

School of Informatics, University of Edinburgh BENEFITS OF MARKOV MODEL MORE EFFICIENT EVALUATION (INCREMENTAL) REDUCES NUMBER OF MODEL PARAMETERS REDUCES TRAINING DATA NEEDED ECVision Summer School: 6 - Probabilistic Behaviour Understanding Fisher slide 5

School of Informatics, University of Edinburgh EXTENDED STATE DIAGRAM ASSUME STATE CAN REMAIN SAME IN EACH VIDEO FRAME, SO NEED EXTENDED GRAPH: * * M1 B * M2 ECVision Summer School: 6 - Probabilistic Behaviour Understanding Fisher slide 6

School of Informatics, University of Edinburgh KNOWLEDGE OF TRANSITIONS ASSUME PROBABILITY OF TRANSITION FROM STATE λ i TO EACH STATE λ j AT EACH TIME FRAME: p ( λ i → λ j ) ALSO WANT PERSISTENCE IN A STATE FOR A WHILE: p ( λ i → λ i ) ECVision Summer School: 6 - Probabilistic Behaviour Understanding Fisher slide 7

School of Informatics, University of Edinburgh KNOWLEDGE OF TRANSITIONS II THEREFORE THE PRIOR PROBABILITY OF STAYING IN STATE λ i FOR EXACTLY τ FRAMES IS p ( λ i → λ i ) τ (1 − p ( λ i → λ i )) P τ ECVision Summer School: 6 - Probabilistic Behaviour Understanding Fisher slide 8

School of Informatics, University of Edinburgh NEW BROWSE MODEL PARAMETERS * * M1 B * M2 NEED: p ( M 1 → B ) , p ( B → B ) , p ( B → M 2 ) , p ( M 2 → M 2 ) , p ( M 2 → B ) ECVision Summer School: 6 - Probabilistic Behaviour Understanding Fisher slide 9

School of Informatics, University of Edinburgh HIDDEN MARKOV MODEL (HMM) COMBINES ALL OF ABOVE • STATE TRANSITION PROBABILITIES: p ( λ i → λ j ) • STATE PROBABILITIES A FUNCTION OF OBSERVATIONS: p ( s i | � f i ) • CURRENT STATE DEPENDS ONLY ON PREVIOUS STATE ECVision Summer School: 6 - Probabilistic Behaviour Understanding Fisher slide 10

School of Informatics, University of Edinburgh SEQUENCE LABELLING GIVEN A SEQUENCE OF FRAMES 1 : T OBSERVED FEATURES � f 1: T = { � f 1 , . . . � f T } DESIRE STATE LABELLING OF SEQUENCE s 1: T = { s 1 , . . . s T } WITH THE LARGEST PROBABILITY p ( s 1: T | � f 1: T ) MOST LIKELY SET OF LABELS FOR THE FRAMES ECVision Summer School: 6 - Probabilistic Behaviour Understanding Fisher slide 11

School of Informatics, University of Edinburgh FINDING SEQUENCE LABELLING     T T − 1 p ( s 1: T | � p ( s t | � f 1: T ) = � f t ) � p ( s t → s t +1 )     t =1 t =1 REFORMULATE p ( s 1: T | � f 1: T ) AS p ( s 1: T − 1 | � f 1: T − 1 ) p ( s T | � f T ) p ( s T − 1 → s T ) INCREMENTAL FORM ECVision Summer School: 6 - Probabilistic Behaviour Understanding Fisher slide 12

School of Informatics, University of Edinburgh EFFICIENT SOLUTION DYNAMIC PROGRAMMING KEEPS ONLY BEST LABELLINGS AT EACH TIME t REMOVES INFERIOR LABELLINGS COMPLEXITY O ( | S | 2 T ) ECVision Summer School: 6 - Probabilistic Behaviour Understanding Fisher slide 13

School of Informatics, University of Edinburgh TRANSITION DISTRIBUTION PROBLEM IF FRAME TIME NEARLY SAME AS TIME IN EACH LABEL CLASS, THEN HMM MODEL REASONABLE IF FRAME RATE FASTER: P τ EG: GAMMA DISTRIBUTION ECVision Summer School: 6 - Probabilistic Behaviour Understanding Fisher slide 14

School of Informatics, University of Edinburgh POTENTIAL SOLUTION HSMM - HIDDEN SEMI-MARKOV MODEL EXPLICIT DURATION HMM USES DISTRIBUTION OF TIME SPENT IN EACH STATE CLASS: PROBABILITY THAT THE NUMBER OF CONSECUTIVE STATES THE SAME EFFECTIVELY SPLITS SEQUENCE INTO SEGMENTS LABELLED WITH THE STATE ECVision Summer School: 6 - Probabilistic Behaviour Understanding Fisher slide 15

School of Informatics, University of Edinburgh OPEN RESEARCH PROBLEM STANDARD HSMM ALGORITHM HAS COMPLEXITY O ( | S | 2 T 2 ) COMBINATORIAL EXPLOSION AS SEQUENCES GET LONGER WE ARE INVESTIGATING AN INCREMENTAL ALGORITHM WITH COMPLEXITY O ( | S | 2 T ) RESULTS SOON ECVision Summer School: 6 - Probabilistic Behaviour Understanding Fisher slide 16

School of Informatics, University of Edinburgh SUMMARY • PROBABILITIES SHOULD COPE BETTER WITH PROPERTY NOISE AND AMBIGUITY • HMM CAN MODEL STATE TRANSITION STRUCTURE • HSMM CAN MODEL DIFFERENT DURATIONS IN A GIVEN STATE ECVision Summer School: 6 - Probabilistic Behaviour Understanding Fisher slide 17

School of Informatics, University of Edinburgh Lecture Problem HOW WOULD YOU ESTIMATE THE GAMMA DISTRIBUTION FOR THE BROWSE CONTEXT? SKETCH A REASONABLE BROWSE SITUATION DISTRIBUTION. ECVision Summer School: 6 - Probabilistic Behaviour Understanding Fisher slide 18