(MIRU2010) 2010 7 ROC 567-0047 8-1 E-mail: {makihara,hossain,yagi}@am.sanken.osaka-u.ac.jp ROC 1 1 2 ( ) ROC 2 ROC Adaptive Acceptance Threshold Control using Matching s with Abstract Confidence Values for ROC Curve Optimization Yasushi MAKIHARA, Md. ALTAB HOSSAIN, and Yasushi YAGI Osaka university 8-1 Mihogaoka, Ibaraki, Osaka, 567-0047 E-mail: {makihara,hossain,yagi}@am.sanken.osaka-u.ac.jp In two-class classification problems such as one-to-one verification and object detection, the performance is usually evaluated by a so-called Receiver Operating Characteristics (ROC) curve expressing a tradeoff between False Rejection Rate (FRR) and False Acceptance Rate (FAR). On the other hand, it is also well known that the performance is significantly affected by the situation differences between enrollment and test phases. This paper describes a method to adaptively control an acceptance threshold with confidence values derived from situation differences so as to optimize the ROC curve. We show that the optimal evolution of the adaptive threshold in the domain of the distance and confidence value is equivalent to a constant evolution in the domain of the error gradient defined as a ratio of a total error rate to a total acceptance rate. An experiment with simulation and real data demonstrates the effectiveness of the proposed method, particularly under a lower FAR or FRR tolerance condition. Key words rejection rate ROC curve, Acceptance threshold, Confidence values, Error gradient, False acceptance rate, False 1. 2 1 1 2 [1] 2 [2] ID 1 1 2 ( ) ( ) 2
( False Acceptance Rate: FAR ) ( False Rejection Rate: FRR) (ROC) [3] 2 [4] [5], [6] SN (Quality measure) [7] [8] [9] [10] Hossain [11] [12] [13] [14] [15] [16] Kryszczuk [17] evidence space [17] ROC ROC ROC ROC ROC 1 1 2 2. 2. 1 ROC ROC 1 1 ( ) ( ) ( ) 1(a) (Probability Distribution Function: PDF) T A F R (False Rejection Rate: FRR) A F A T T (< T ) FRR FAR ( 1(a)) T L (> T ) FRR FAR ( 1(a)) FRR 100% FAR 0% +. FRR 0% FAR 100% ( ) ( + ) 1(b) FRR FAR ROC [3] ROC 1 1 FRR FAR FRR FAR (Equal Error Rate: EER) ROC 1 1 PDF FRR FAR 3( ) PDF
Probability 1 A FA A FR T T T T L (a) PDF Positive Negative False Rejection Rate P FR 1.0 T T Tight Loose 0.0 T T L 0.0 1.0 False Acceptance Rate P FA (b) ROC PDF ROC Low confidence 2 High confidence Positive Negative Confidence ROC 3( ) PDF ROC ROC ROC PDF ROC 2. 2 2 2 ( 2 ) ( 2 ) PDF ROC 3 FAR 0% T L NoF A T H NoF A (> T L NoF A ) FRR 0% T L NoF R and T H NoF R (< T L NoF R ) ROC T L NoF A T L NoF A T H NoF A ( 3(a) ) T H NoF R T H NoF R T L NoF R T L NoF A T H NoF A T H NoF R T L NoF R 2. 3 t PDF p P (t) p N (t) T FAR R F A (T ) FRR R F R (T )
Probability Probability 3 T L NoFA T H NoFA False acceptance (a) PDF Positive Negative Positive Negative False Rejection Rate P FR False Rejection Rate P FR 1.0 T L NoFA T H NoFA 0.0 0.0 1.0 False Acceptance Rate P FA 1.0 T L NoFA T H NoFA 0.0 0.0 1.0 False Acceptance Rate P FA (b) ROC ( ) ( ) PDF ROC R F A (T ) = T R F R (T ) = 1 T p N (t)dt (1) p P (t)dt. (2) R E (T ) R A (T ) R E (T ) = R F A (T ) + R F R (T ) (3) R A (T ) = R F A (T ) + (1 R F R (T )). (4) R A (T ) R E (T ) g(t ) = dr E(T ) dr A (T ) = pn (T ) p P (T ) p N (T ) + p P (T ). (5) ( g(t ) 1.0) ( g(t ) 1.0) ( g(t ) 0.0) FAR FRR g(t ) g(t ) 2 t c g(t) g(t ) 2. 4 N P N N i (t P i, cp i ) (tn i, cn i ) t c t j =t min + js t, j Z, 0< = j < = (t max t min )/s t (6) c k =c min + ks c, k Z, 0< = k < = (c max c min )/s c (7) i k w P i,k w P i,k = max(1.0 c P i c k /s c, 0) (8) j k PDF p j,k p P j,k = 1 N P wi,k P exp ( (tp i t j ) 2 ) Z k 2σ 2, (9) i Z k PDF j pp j,k s t = 1 σ PDFp N j,k (t j, c k ) g j,k (5). c k g = {g j,k } g =arg min S(ĝ) (10) ĝ S(ĝ)= k,j {(ĝ j,k g j,k ) 2 +α(ĝ j,k ĝ j 1,k ) 2 } (11) subject to ĝ j 1,k > = ĝ j,k, (12) α 2
3. 3. 1 PDF c PDF 0 < = c < = 1 PDF N (µ P (c), σ P (c)) N (µ N (c), σ N (c)) PDF c µ P (c) = 5.0 3.0c, µ N (c) = 7.0 2.0c, σ P (c) = 1.0 0.5c, σ N (c) = 1.0 0.5c, PDF 10,000 ( 4(a)) ( c = 1.0) ( c = 0.0) 10,000 PDF σ = 0.3, s t = 0.01, s c = 0.1 t min = 0.0, t max = 10.0 α = 1.0 z [3] z 1 1 µ(c) σ(c) 4(b) z 0 1 3. 2 5 4 8.0 7.0 6.0 5.0 4.0 (a) (b) z 3.0 0.0 0.2 0.4 0.6 0.8 1.0 Confidence value 5 ( 0.1) 6 ( g = 0.0), ( g = 1.0) ( g = 0.0) ( g = 1.0) ROC 7(a) (AATC) (const) z (Z-norm)
False Rejection Rate 0.5 0.4 0.3 0.2 0.1 Constant Z- norm AATC 6 FAR ( 7(b)) FRR ( 7(c)) z FAR FRR 3. 3 68 32 2,120 8 20 48 1 1 1 Hossain [6] [6] [18] 9 10-1 +1 1 1 11 ROC z False Rejection Rate False Rejection Rate 0.0 0.0 0.1 0.2 0.3 0.4 0.5 False Acceptance Rate 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 (a) Z-Norm AATC 1.E-04 1.E-03 1.E-02 1.E-01 False Acceptance Rate 1.0E-01 1.0E-02 1.0E-03 (b) FAR Z-Norm AATC 1.0E-04 0.0 0.2 0.4 0.6 0.8 1.0 False Acceptance Rate 8 (c) FRR 7 ROC 4. z
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