Abstract

We continue the theme of previous papers [J. Opt. Soc. Am. A 7, 1266 (1990); J. Opt. Soc. Am. A 12, 834 (1995)] on objective (task-based) assessment of image quality. We concentrate on signal-detection tasks and figures of merit related to the ROC (receiver operating characteristic) curve. Many different expressions for the area under an ROC curve (AUC) are derived for an arbitrary discriminant function, with different assumptions on what information about the discriminant function is available. In particular, it is shown that AUC can be expressed by a principal-value integral that involves the characteristic functions of the discriminant. Then the discussion is specialized to the ideal observer, defined as one who uses the likelihood ratio (or some monotonic transformation of it, such as its logarithm) as the discriminant function. The properties of the ideal observer are examined from first principles. Several strong constraints on the moments of the likelihood ratio or the log likelihood are derived, and it is shown that the probability density functions for these test statistics are intimately related. In particular, some surprising results are presented for the case in which the log likelihood is normally distributed under one hypothesis. To unify these considerations, a new quantity called the likelihood-generating function is defined. It is shown that all moments of both the likelihood and the log likelihood under both hypotheses can be derived from this one function. Moreover, the AUC can be expressed, to an excellent approximation, in terms of the likelihood-generating function evaluated at the origin. This expression is the leading term in an asymptotic expansion of the AUC; it is exact whenever the likelihood-generating function behaves linearly near the origin. It is also shown that the likelihood-generating function at the origin sets a lower bound on the AUC in all cases.

© 1998 Optical Society of America

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References

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  1. H. H. Barrett, “Objective assessment of image quality: effects of quantum noise and object variability,” J. Opt. Soc. Am. A 7, 1266–1278 (1990).
    [CrossRef] [PubMed]
  2. H. H. Barrett, J. L. Denny, R. F. Wagner, K. J. Myers, “Objective assessment of image quality. II. Fisher information, Fourier crosstalk, and figures of merit for task performance,” J. Opt. Soc. Am. A 12, 834–852 (1995).
    [CrossRef]
  3. H. H. Barrett, D. W. Wilson, B. M. W. Tsui, “Noise properties of the EM algorithm I: theory,” Phys. Med. Biol. 39, 833–846 (1994).
    [CrossRef] [PubMed]
  4. D. W. Wilson, B. M. W. Tsui, H. H. Barrett, “Noise properties of the EM algorithm II: Monte Carlo simulations,” Phys. Med. Biol. 39, 847–872 (1994).
    [CrossRef] [PubMed]
  5. H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968), Vol. I.
  6. J. O. Berger, Statistical Decision Theory and Bayesian Analysis (Springer-Verlag, New York, 1985).
  7. J. L. Melsa, D. L. Cohn, Decision and Estimation Theory (McGraw-Hill, New York, 1978).
  8. K. Fukunaga, Introduction to Statistical Pattern Recognition, 2nd ed. (Academic, New York, 1990).
  9. H. H. Barrett, T. A. Gooley, K. A. Girodias, J. P. Rolland, T. A. White, J. Yao, “Linear discriminants and image quality,” Image Vision Comput. 10, 451–460 (1992).
    [CrossRef]
  10. H. H. Barrett, J. Yao, J. Rolland, K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. USA 90, 9758–9765 (1993).
    [CrossRef] [PubMed]
  11. C. E. Metz, Department of Radiology, University of Chicago, Chicago, Ill. (personal communication, 1998).
  12. R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).
  13. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products, 4th ed., Formula 3.954 (Academic, New York, 1965).
  14. H. H. Barrett, C. K. Abbey, “Bayesian detection of random signals on random backgrounds,” in Information Processing in Medical Imaging, Proceedings of the 15th International Conference, IPMI 97, Poultney, Vt., June 9–13, 1997, Lecture Notes in Computer Science, J. Duncan, G. Gindi, eds. (Springer-Verlag, Berlin, 1997).
  15. D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).
  16. R. M. Swensson, D. M. Green, “On the relations between random walk models for two-choice response times,” J. Math. Psychol. 15, 282–291 (1977).
    [CrossRef]
  17. A. N. Shiryayev, Probability (Springer-Verlag, New York, 1984).
  18. D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1966).
  19. L. Rade, B. Westergren, Beta Mathematics Handbook, 2nd ed. (CRC Press, Boca Raton, Fla., 1990).
  20. R. V. Churchill, J. W. Brown, R. F. Verkey, Complex Variables and Applications, 3rd ed. (McGraw-Hill, New York, 1974).
  21. S. Lang, Real and Functional Analysis, 3rd ed. (Springer-Verlag, New York, 1993).
  22. R. E. A. C. Paley, N. Wiener, Fourier Transforms in the Complex Domain (American Mathematical Society, New York, 1934), Vol. 19.
  23. R. Strichartz, A Guide to Distribution Theory and Fourier Transforms (CRC Press, Boca Raton, Fla., 1994).

1995 (1)

1994 (2)

H. H. Barrett, D. W. Wilson, B. M. W. Tsui, “Noise properties of the EM algorithm I: theory,” Phys. Med. Biol. 39, 833–846 (1994).
[CrossRef] [PubMed]

D. W. Wilson, B. M. W. Tsui, H. H. Barrett, “Noise properties of the EM algorithm II: Monte Carlo simulations,” Phys. Med. Biol. 39, 847–872 (1994).
[CrossRef] [PubMed]

1993 (1)

H. H. Barrett, J. Yao, J. Rolland, K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. USA 90, 9758–9765 (1993).
[CrossRef] [PubMed]

1992 (1)

H. H. Barrett, T. A. Gooley, K. A. Girodias, J. P. Rolland, T. A. White, J. Yao, “Linear discriminants and image quality,” Image Vision Comput. 10, 451–460 (1992).
[CrossRef]

1990 (1)

1977 (1)

R. M. Swensson, D. M. Green, “On the relations between random walk models for two-choice response times,” J. Math. Psychol. 15, 282–291 (1977).
[CrossRef]

Abbey, C. K.

H. H. Barrett, C. K. Abbey, “Bayesian detection of random signals on random backgrounds,” in Information Processing in Medical Imaging, Proceedings of the 15th International Conference, IPMI 97, Poultney, Vt., June 9–13, 1997, Lecture Notes in Computer Science, J. Duncan, G. Gindi, eds. (Springer-Verlag, Berlin, 1997).

Barrett, H. H.

H. H. Barrett, J. L. Denny, R. F. Wagner, K. J. Myers, “Objective assessment of image quality. II. Fisher information, Fourier crosstalk, and figures of merit for task performance,” J. Opt. Soc. Am. A 12, 834–852 (1995).
[CrossRef]

D. W. Wilson, B. M. W. Tsui, H. H. Barrett, “Noise properties of the EM algorithm II: Monte Carlo simulations,” Phys. Med. Biol. 39, 847–872 (1994).
[CrossRef] [PubMed]

H. H. Barrett, D. W. Wilson, B. M. W. Tsui, “Noise properties of the EM algorithm I: theory,” Phys. Med. Biol. 39, 833–846 (1994).
[CrossRef] [PubMed]

H. H. Barrett, J. Yao, J. Rolland, K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. USA 90, 9758–9765 (1993).
[CrossRef] [PubMed]

H. H. Barrett, T. A. Gooley, K. A. Girodias, J. P. Rolland, T. A. White, J. Yao, “Linear discriminants and image quality,” Image Vision Comput. 10, 451–460 (1992).
[CrossRef]

H. H. Barrett, “Objective assessment of image quality: effects of quantum noise and object variability,” J. Opt. Soc. Am. A 7, 1266–1278 (1990).
[CrossRef] [PubMed]

H. H. Barrett, C. K. Abbey, “Bayesian detection of random signals on random backgrounds,” in Information Processing in Medical Imaging, Proceedings of the 15th International Conference, IPMI 97, Poultney, Vt., June 9–13, 1997, Lecture Notes in Computer Science, J. Duncan, G. Gindi, eds. (Springer-Verlag, Berlin, 1997).

Berger, J. O.

J. O. Berger, Statistical Decision Theory and Bayesian Analysis (Springer-Verlag, New York, 1985).

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

Brown, J. W.

R. V. Churchill, J. W. Brown, R. F. Verkey, Complex Variables and Applications, 3rd ed. (McGraw-Hill, New York, 1974).

Churchill, R. V.

R. V. Churchill, J. W. Brown, R. F. Verkey, Complex Variables and Applications, 3rd ed. (McGraw-Hill, New York, 1974).

Cohn, D. L.

J. L. Melsa, D. L. Cohn, Decision and Estimation Theory (McGraw-Hill, New York, 1978).

Denny, J. L.

Fukunaga, K.

K. Fukunaga, Introduction to Statistical Pattern Recognition, 2nd ed. (Academic, New York, 1990).

Girodias, K. A.

H. H. Barrett, T. A. Gooley, K. A. Girodias, J. P. Rolland, T. A. White, J. Yao, “Linear discriminants and image quality,” Image Vision Comput. 10, 451–460 (1992).
[CrossRef]

Gooley, T. A.

H. H. Barrett, T. A. Gooley, K. A. Girodias, J. P. Rolland, T. A. White, J. Yao, “Linear discriminants and image quality,” Image Vision Comput. 10, 451–460 (1992).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products, 4th ed., Formula 3.954 (Academic, New York, 1965).

Green, D. M.

R. M. Swensson, D. M. Green, “On the relations between random walk models for two-choice response times,” J. Math. Psychol. 15, 282–291 (1977).
[CrossRef]

D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1966).

Lang, S.

S. Lang, Real and Functional Analysis, 3rd ed. (Springer-Verlag, New York, 1993).

Melsa, J. L.

J. L. Melsa, D. L. Cohn, Decision and Estimation Theory (McGraw-Hill, New York, 1978).

Metz, C. E.

C. E. Metz, Department of Radiology, University of Chicago, Chicago, Ill. (personal communication, 1998).

Middleton, D.

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

Myers, K. J.

Paley, R. E. A. C.

R. E. A. C. Paley, N. Wiener, Fourier Transforms in the Complex Domain (American Mathematical Society, New York, 1934), Vol. 19.

Rade, L.

L. Rade, B. Westergren, Beta Mathematics Handbook, 2nd ed. (CRC Press, Boca Raton, Fla., 1990).

Rolland, J.

H. H. Barrett, J. Yao, J. Rolland, K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. USA 90, 9758–9765 (1993).
[CrossRef] [PubMed]

Rolland, J. P.

H. H. Barrett, T. A. Gooley, K. A. Girodias, J. P. Rolland, T. A. White, J. Yao, “Linear discriminants and image quality,” Image Vision Comput. 10, 451–460 (1992).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products, 4th ed., Formula 3.954 (Academic, New York, 1965).

Shiryayev, A. N.

A. N. Shiryayev, Probability (Springer-Verlag, New York, 1984).

Strichartz, R.

R. Strichartz, A Guide to Distribution Theory and Fourier Transforms (CRC Press, Boca Raton, Fla., 1994).

Swensson, R. M.

R. M. Swensson, D. M. Green, “On the relations between random walk models for two-choice response times,” J. Math. Psychol. 15, 282–291 (1977).
[CrossRef]

Swets, J. A.

D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1966).

Tsui, B. M. W.

H. H. Barrett, D. W. Wilson, B. M. W. Tsui, “Noise properties of the EM algorithm I: theory,” Phys. Med. Biol. 39, 833–846 (1994).
[CrossRef] [PubMed]

D. W. Wilson, B. M. W. Tsui, H. H. Barrett, “Noise properties of the EM algorithm II: Monte Carlo simulations,” Phys. Med. Biol. 39, 847–872 (1994).
[CrossRef] [PubMed]

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968), Vol. I.

Verkey, R. F.

R. V. Churchill, J. W. Brown, R. F. Verkey, Complex Variables and Applications, 3rd ed. (McGraw-Hill, New York, 1974).

Wagner, R. F.

Westergren, B.

L. Rade, B. Westergren, Beta Mathematics Handbook, 2nd ed. (CRC Press, Boca Raton, Fla., 1990).

White, T. A.

H. H. Barrett, T. A. Gooley, K. A. Girodias, J. P. Rolland, T. A. White, J. Yao, “Linear discriminants and image quality,” Image Vision Comput. 10, 451–460 (1992).
[CrossRef]

Wiener, N.

R. E. A. C. Paley, N. Wiener, Fourier Transforms in the Complex Domain (American Mathematical Society, New York, 1934), Vol. 19.

Wilson, D. W.

H. H. Barrett, D. W. Wilson, B. M. W. Tsui, “Noise properties of the EM algorithm I: theory,” Phys. Med. Biol. 39, 833–846 (1994).
[CrossRef] [PubMed]

D. W. Wilson, B. M. W. Tsui, H. H. Barrett, “Noise properties of the EM algorithm II: Monte Carlo simulations,” Phys. Med. Biol. 39, 847–872 (1994).
[CrossRef] [PubMed]

Yao, J.

H. H. Barrett, J. Yao, J. Rolland, K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. USA 90, 9758–9765 (1993).
[CrossRef] [PubMed]

H. H. Barrett, T. A. Gooley, K. A. Girodias, J. P. Rolland, T. A. White, J. Yao, “Linear discriminants and image quality,” Image Vision Comput. 10, 451–460 (1992).
[CrossRef]

Image Vision Comput. (1)

H. H. Barrett, T. A. Gooley, K. A. Girodias, J. P. Rolland, T. A. White, J. Yao, “Linear discriminants and image quality,” Image Vision Comput. 10, 451–460 (1992).
[CrossRef]

J. Math. Psychol. (1)

R. M. Swensson, D. M. Green, “On the relations between random walk models for two-choice response times,” J. Math. Psychol. 15, 282–291 (1977).
[CrossRef]

J. Opt. Soc. Am. A (2)

Phys. Med. Biol. (2)

H. H. Barrett, D. W. Wilson, B. M. W. Tsui, “Noise properties of the EM algorithm I: theory,” Phys. Med. Biol. 39, 833–846 (1994).
[CrossRef] [PubMed]

D. W. Wilson, B. M. W. Tsui, H. H. Barrett, “Noise properties of the EM algorithm II: Monte Carlo simulations,” Phys. Med. Biol. 39, 847–872 (1994).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. USA (1)

H. H. Barrett, J. Yao, J. Rolland, K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. USA 90, 9758–9765 (1993).
[CrossRef] [PubMed]

Other (16)

C. E. Metz, Department of Radiology, University of Chicago, Chicago, Ill. (personal communication, 1998).

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products, 4th ed., Formula 3.954 (Academic, New York, 1965).

H. H. Barrett, C. K. Abbey, “Bayesian detection of random signals on random backgrounds,” in Information Processing in Medical Imaging, Proceedings of the 15th International Conference, IPMI 97, Poultney, Vt., June 9–13, 1997, Lecture Notes in Computer Science, J. Duncan, G. Gindi, eds. (Springer-Verlag, Berlin, 1997).

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968), Vol. I.

J. O. Berger, Statistical Decision Theory and Bayesian Analysis (Springer-Verlag, New York, 1985).

J. L. Melsa, D. L. Cohn, Decision and Estimation Theory (McGraw-Hill, New York, 1978).

K. Fukunaga, Introduction to Statistical Pattern Recognition, 2nd ed. (Academic, New York, 1990).

A. N. Shiryayev, Probability (Springer-Verlag, New York, 1984).

D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1966).

L. Rade, B. Westergren, Beta Mathematics Handbook, 2nd ed. (CRC Press, Boca Raton, Fla., 1990).

R. V. Churchill, J. W. Brown, R. F. Verkey, Complex Variables and Applications, 3rd ed. (McGraw-Hill, New York, 1974).

S. Lang, Real and Functional Analysis, 3rd ed. (Springer-Verlag, New York, 1993).

R. E. A. C. Paley, N. Wiener, Fourier Transforms in the Complex Domain (American Mathematical Society, New York, 1934), Vol. 19.

R. Strichartz, A Guide to Distribution Theory and Fourier Transforms (CRC Press, Boca Raton, Fla., 1994).

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Figures (1)

Fig. 1
Fig. 1

Illustration of the behavior of the function M0(β).

Equations (182)

Equations on this page are rendered with MathJax. Learn more.

TPF(x)=Pr(tx|H1)=xdtpr(t|H1),
FPF(x)=Pr(tx|H0)=xdtpr(t|H0).
R=i=01j=01CijPr(Di, Hj)
=i=01j=01CijPr(Di|Hj)Pr(Hj).
R=i=01j=01CijPr(Hj)ΓidMgpr(g|Hj).
Γ0dMgpr(g|Hj)+Γ1dMgpr(g|Hj)=1.
R=C01Pr(H1)+C00Pr(H0)+Γ1dMg  C11Pr(H1)pr(g|H1)+C10Pr(H0)pr(g|H0)-C01Pr(H1)pr(g|H1)-C00Pr(H0)pr(g|H0).
(C00-C10)Pr(H0)pr(g|H0)
>(C11-C01)pr(g|H1)Pr(H1).
pr(g|H1)pr(g|H0)>(C10-C00)Pr(H0)(C01-C11)Pr(H1).
Λ(g)=pr(g|H1)pr(g|H0).
x=(C10-C00)Pr(H0)(C01-C11)Pr(H1).
SNRt2[t1-t0]212var1(t)+12var0(t),
AUC=12+12erfSNRt2,
SNR(AUC)2 erf-1(2 AUC-1),
pr(t|Hj)pj(t).
AUC=01TPFd(FPF),
AUC=--dx TPF(x) ddxFPF(x)
ddxFPF(x)=-p0(x),
AUC=-dxp0(x)xdtp1(t).
P1(x)Pr(t<x|H1)=-xdtp1(t)=1-xdtp1(t).
AUC=1--dxp0(x)P1(x).
AUC=-dx-dtp0(x)p1(t)step(t-x).
AUC=-dx-dyp0(x)p1(y+x)step(y)
=0dy[p0p1](y),
step(x)=12+12sgn(x),
AUC=12+12-dx-dtp0(x)p1(t)sgn(t-x).
F{step(x)}=12δ(ξ)+P12πiξ,
step(x)=12+12πiP- dξξexp(2πiξx),
AUC=12+12πiP- dξξ-dx-dtp0(x)×p1(t)exp[2πiξ(t-x)]
=12+12πiP- dξξψ0(ξ)ψ1*(ξ),
ψj(ξ)exp(-2πiξt)j=-dtpj(t)exp(-2πiξt)=F{pj(t)}.
Mj(β)=exp(βt)j=-dtpj(t)exp(βt)=ψjiβ2π.
Lj(β)=log[Mj(β)].
Lj(β)=n=0 1n!Lj(n)(0)βn,
tnj=M(n)(0).
Lj(0)(0)=log[Mj(0)]=0,
Lj(1)(0)=Mj(1)(0)Mj(0)=tjt¯j,
Lj(2)(0)=(t-t¯j)2jσj2,
Lj(3)(0)=(t-t¯j)3jσj3Sj,
Lj(4)(0)=[(t-t¯j)4j-3(t-t¯j)2j2]σj4Kj.
ψj(ξ)=Mj(-2πiξ)=exp-2πit¯jξ-2π2σj2ξ2+i 4π33σj3Sjξ3+2π43σj4Kjξ4+.
AUC=12+12πP- dξξ×sin2π(t¯1-t¯0)ξ-4π33(σ13S1-σ03S0)ξ3+exp[-2π2(σ02+σ12)ξ2+(2π4/3)(σ14K1+σ04K0)ξ4+].
Mj(β)=exp(tj¯β+12σj2β2),
ψj(ξ)=exp(-2πitj¯ξ-2π2σj2ξ2).
AUC=12+12πP- dξξsin[2π(t1¯-t0¯)ξ]
×exp[-2π2(σ02+σ12)ξ2].
AUC=12+12πlim0-dξ ξξ2+sin[2π(t1¯-t0¯)ξ]
exp[-2π2(σ02+σ12)ξ2].
pr(t|g)=δ[t-θ(g)],
pr(t|Hj)=dMgpr(t|g)pr(g|Hj)
=dMgpr(g|Hj)δ[t-θ(g)].
qj(g)pr(g|Hj).
pj(t)=dMgqj(g)δ[t-θ(g)].
AUC=-dx-dtdMgq0(g)δ[x-θ(g)]×dMgq1(g)δ[t-θ(g)]step(t-x).
AUC=dMgdMgq0(g)q1(g)step[θ(g)-θ(g)].
AUC=12+12πiP- dξξdMgdMgq0(g)×q1(g)exp{2πiξ[θ(g)-θ(g)]}
=12+12πiP- dξξexp[-2πiξθ(g)]0×exp[2πiξθ(g)]1
=12+12πiP- dξξψ0(ξ)ψ1*(ξ).
Pr(correct)=Pr[θ(g)>θ(g)]
=dMgdMgq0(g)q1(g)step[θ(g)-θ(g)],
θ(g)=wtg,
AUClin=dMgdMgq0(g)q1(g)step[wt(g-g)].
AUClin=dMgdMgq0(g)q1(g+g)step(wtg)=dMg[q0q1](g)step(wtg),
Ψj(ρ)=dMgp(g|Hj)exp(-2πiρtg),
ψj(ξ)=dMgp(g|Hj)exp(-2πiξwtg)=Ψj(wξ),
AUClin=12+12πiP- dξξΨ0(wξ)Ψ1*(wξ).
Λk+10=dMgq0(g)q1(g)q0(g)k+1
=dMgq1(g)q1(g)q0(g)k=Λk1.
Λ0=Λ01=dMgq0(g) q1(g)q0(g)=dMgq1(g)=1,
var0(Λ)=Λ20-Λ02=Λ1-1.
exp[(k+1)λ]0=exp(kλ)1.
M0(β+1)=M1(β),
ψ0ξ+i2π=ψ1(ξ).
λk0=M0(k)(0),
λk1=M1(k)(0)=M0(k)(1).
Λk0=exp(kλ)0=M0(k)
Λk1=exp[(k+1)λ]0=M0(k+1).
p1(λ)=F-1{ψ1(ξ)}=-dξψ0ξ+i2πexp(2πiξλ)
=eλ-+i/2π+i/2πdzψ0(z)exp(2πizλ),
p1(λ)=eλ-dzψ0(z)exp(2πizλ).
p1(λ)=eλp0(λ).
-dλ eλp0(λ)=1.
pj(λ)=pr(Λ|Hj)|dλ/dΛ|.
pr(Λ|H1)=eλpr(Λ|H0)=Λpr(Λ|H0).
pr(Λ|H1)pr(Λ|H0)=Λ.
pr(g|Hj)=k=1Kpr(gk|Hj).
λ=k=1K[log pr(gk|H1)-log pr(gk|H0)].
M0(1)=exp[λ¯0+12var0(λ)]=1.
λ¯0=-12var0(λ),
M0(β)=exp[12(β2-β)var0(λ)]
λ¯1=M0(1)(1)=12var0(λ)=-λ¯0,
var1(λ)=M0(2)(1)-[M0(1)(1)]2=var0(λ).
M1(β)=M0(β+1)=exp[12(β2+β)var0(λ)].
p0(λ)=1[2π var0(λ)]1/2exp-[λ+12var0(λ)]22 var0(λ),
p1(λ)=1[2π var0(λ)]1/2exp-[λ-12var0(λ)]22 var0(λ).
var0(λ)=logΛ1.
SNRλ2=[λ1-λ0]212var1(λ)+12var0(λ)=var0(λ)=logΛ1.
AUC=12+12erf[12logΛ1].
p0(λ)=exp(-12λ)f(λ),p1(λ)=exp(12λ)f(λ).
ψ0(ξ)=Fξ-i4π,ψ1(ξ)=Fξ+i4π,
M0(β)=FL(β-12),M1(β)=FL(β+12),
FL(β)=- dλf(λ)exp(βλ).
F(ξ)=expξ+i4πξ-i4πT(ξ),
FL(β)=exp[(β+12)(β-12)G(β)].
T(ξ)=-4π2G(-2πiξ).
ψ0(ξ)=expξξ-i2πTξ-i4π
=exp-4π2ξξ-i2πG-2πiξ-12,
ψ1(ξ)=expξξ+i2πTξ+i4π
=exp-4π2ξξ+i2πG-2πiξ+12,
M0(β)=exp[β(β-1)G(β-12)],
M1(β)=exp[β(β+1)G(β+12)].
G(β)=log M0(β+12)(β-12)(β+12)=log M1(β-12)(β-12)(β+12).
logΛk1=logΛk+10=k(k+1)G(k+12).
λ¯0=-G(-12),λ¯1=G(12);
var0(λ)=2[G(-12)-G(-12)],
var1(λ)=2[G(12)+G(12)],
SNRλ2=[G(12)+G(-12)]2G(12)+G(-12)+G(12)-G(-12).
SNRλ2λ¯1-λ¯0=G(12)+G(-12)2G(0).
G(2πiξ)=G*(-2πiξ),(ξreal).
β(β+1)G(β+12)=logexp(βλ)1βλ¯1=βG(12).
(β+12)G(β)G(12),βreal,β12,
logΛαjαλj,j=0,1.
-G(-12)=λ¯00,
λ¯1log Λ¯1=2G(32).
AUC=12+12πiP- dξξexp-4π2ξ2-iξ2π×G-2πiξ-12+G2πiξ+12
=12+12πiP- dξξexp-4π2ξ2-iξ2πH(ξ),
H(ξ)G(-2πiξ-12)+G(2πiξ+12).
H(ξ)=Hr(ξ)+iHi(ξ),
Hr(-ξ)=Hr(ξ),Hi(-ξ)=-Hi(ξ).
ξ2Hr(ξ)+ξ2πHi(ξ)0.
AUC=12-12πP- dξξ×exp-4π2ξ2Hr(ξ)+ξ2πHi(ξ)×sin4π2ξ2Hi(ξ)-ξ2πHr(ξ).
G(β)=n=0 1n!G(n)(0)βn.
H(ξ)=2k=0 1(2k)!G(2k)(0)2πiξ+122k.
AUC=12+12erf[122G(0)].
AUC=1-12π0 dαa2+14exp[-2α2+14Re G(iα)].
exp[-2(α2+14)G(iα)]=M0(12+iα)=Λ1/2Λiα0.
exp[-2(α2+14)Re G(iα)]=|M0(12+iα)|=|Λ1/2Λiα0|,(αreal).
|Λ1/2Λiα0||Λ1/2Λiα|0=Λ1/20=exp[-14G(0)],
exp[-2(α2+14)Re G(iα)]exp[-14G(0)].
AUC1-12exp[-12G(0)].
G(0)=-4 log[FL(0)]=-4 log- dλf(λ).
G(0)=-4 log- dλp1(λ)exp(-12λ)=-4 logΛ-1/21=-4 log- dλp0(λ)exp(12λ)=-4 logΛ1/20.
G(0)=-4 log dMgq0(g)q1(g).
M0(12)=Λ1/20=exp[-14G(0)]=- dλf(λ).
ψj(z)uj(ξ,η)+ivj(ξ,η)=- dλpj(λ)exp(-2πizλ)=- dλpj(λ)exp(-2πiξλ)exp(2πηλ),
uj(ξ,η)ξ=vj(ξ,η)η,vj(ξ,η)ξ=-uj(ξ,η)η.
uj(ξ,η)=- dλpj(λ)cos(2πξλ)exp(2πηλ),
vj(ξ,η)=-- dλpj(λ)sin(2πξλ)exp(2πηλ).
- dλpj(λ)exp(2πηλ)<,
- dλpj(λ)|λ|exp(2πηλ)<.
Λ2πηj<.
λ2jΛ4πηj<.
-dλp0(λ)exp(2πηλ)
-dλp0(λ)exp(λ)=exp(λ)0.
Gβ-12=log M0(β)β(β-1).
AUC=12+12πiP - dξξψ0(ξ) ψ0-ξ+i2π.
AUC=12+12πiC dzzψ0(z)×ψ0-z+i2π+12πiπi.
AUC=1+12πi- dξξ+i4πψ0ξ+i4π×ψ0-ξ+i4π.
AUC=1+12πi- dξξ+i/(4π)F(ξ)F(-ξ).
AUC=1+12πi- dξξ+i/(4π)×exp2ξ2+116π2Re T(ξ).
1ξ+i/(4π)=ξξ2+1/(16π2)-14πiξ2+1/(16π2).
AUC=1-14π20 dξξ2+1/(16π2)×exp2ξ2+116π2Re T(ξ).
AUC=1-14π20 dξξ2+1/(16π2)×exp-8π2ξ2+116π2Re G(2πiξ).
AUC=1-I0+I,
I0=14π20 dξξ2+1/16π2×exp-8π2ξ2+116π2G(0),
I=14π20dξ exp-8π2ξ2+116π2G(0)b(ξ),
b(ξ)=1-exp{-8π2[ξ2+1/(16π2)]Re G˜(2πiξ)}ξ2+1/(16π2).
h(ξ)=exp(-8π2ξ2),
I=exp-G(0)20dξh[G(0)ξ]b(ξ).
AUC=12+12erf[122G(0)]+I
b(ξ)=n=1bnξ2n,
Mh(2n+1)=0 dξξh(ξ)ξ2n+1.
I14π2exp-G(0)2 n=1bnMh(2n+1) [G(0)]-2n-1.
4π2I expG(0)2-n=1N-1bnMh(2n+1)[G(0)]-2n-1
<EN[G(0)]-2N-1.
AUC-12+12erf122G(0)
E14π2[G(0)]3exp-12G(0)
Mh(2n+1)=1×2××(2n-1)4(4π)2n12π,
Re G˜(2πiξ)=-2π2G(0)ξ2+2π43G(4)(0)ξ4+.
AUC12+12erf122G(0)-G(0)8exp-G(0)2[G(0)]-3.
AUC12+12 erf[12 2G(0)]

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