Abstract

Differential absorption lidar (DIAL) is a well-established technology for estimating the concentration and its path integral CL of vapor materials using two closely spaced wavelengths. The recent development of frequency-agile lasers (FAL’s) with as many as 60 wavelengths that can be rapidly scanned motivates the need for detection and estimation algorithms that are optimal for lidar employing these new sources. I derive detection and multimaterial CL estimation algorithms for FAL applications using the likelihood ratio test methodology of multivariate statistical inference theory. Three model sets of assumptions are considered with regard to the spectral properties of the backscatter from either topographic or aerosol targets. The calculations are illustrated through both simulated and actual lidar data.

© 1996 Optical Society of America

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References

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  1. R. E. Warren, “Detection and discrimination using multiple-wavelength differential absorption lidar,” Appl. Opt. 24, 3541–3545 (1985).
    [CrossRef] [PubMed]
  2. R. V. Hogg, A. T. Craig, Introduction to Mathematical Statistics, 4th ed. (Macmillan, New York, 1978).
  3. J. Neyman, E. S. Pearson, “On the use and interpretation of certain test criteria for purposes of statistical inference: parts 1 and 2,” Biometrika A 20, 175–240, 263–294 (1928).
  4. We note that, because the maximization of the numerator in Eq. (4) is over the whole parameter space Ω, whereas that of the denominator is over a proper subset Ω0, L(x) ≥ 1 (neglecting numerical roundoff errors).
  5. L. Carr, L. Fletcher, M. Crittenden, C. Carlisle, S. Gotoff, F. Reyes, F. D’Amico, “Frequency agile CO2 DIAL for environmental monitoring,” in International Symposium on Optical Sensing for Environmental Monitoring (International Society for Optical Engineering, Atlanta, Ga., 1993), Vol. 2112, pp. 282–294.
  6. L. Carr, C. Carlisle, D. Cooper, M. Crittenden, L. Fletcher, S. Gotoff, F. Reyes, “Design and testing of a mobile vapor chamber for multiple material vapor analysis with a frequency-agile CO2 DIAL system,” presented at the Third Workshop on Standoff Detection for Chemical and Biological Defense, Williamsburg, Va. (1994).
  7. R. E. Warren, “Adaptive Kalman–Bucy filter for differential absorption lidar time series data,” Appl. Opt. 26, 4755–4760 (1987).
    [CrossRef] [PubMed]
  8. R. E. Warren, “Concentration estimation from differential absorption lidar using nonstationary Wiener filtering,” Appl. Opt. 28, 5047–5051 (1989).
    [CrossRef] [PubMed]
  9. S. M. Hannon, R. E. Warren, “Sequential detection of multiple materials using multiwavelength lidar time series data,” in Coherent Laser Radar: Technology and Applications, Vol. 12 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991).
  10. A. Wald, “Tests of statistical hypotheses concerning several parameters when the number of observations is large,” Trans. Amer. Math. Soc. 54, 426–482 (1943).
    [CrossRef]
  11. T. W. Anderson, An Introduction to Multivariate Statistical Analysis, 2nd ed. (Wiley, New York, 1984), p. 594.

1989

1987

1985

1943

A. Wald, “Tests of statistical hypotheses concerning several parameters when the number of observations is large,” Trans. Amer. Math. Soc. 54, 426–482 (1943).
[CrossRef]

1928

J. Neyman, E. S. Pearson, “On the use and interpretation of certain test criteria for purposes of statistical inference: parts 1 and 2,” Biometrika A 20, 175–240, 263–294 (1928).

Anderson, T. W.

T. W. Anderson, An Introduction to Multivariate Statistical Analysis, 2nd ed. (Wiley, New York, 1984), p. 594.

Carlisle, C.

L. Carr, C. Carlisle, D. Cooper, M. Crittenden, L. Fletcher, S. Gotoff, F. Reyes, “Design and testing of a mobile vapor chamber for multiple material vapor analysis with a frequency-agile CO2 DIAL system,” presented at the Third Workshop on Standoff Detection for Chemical and Biological Defense, Williamsburg, Va. (1994).

L. Carr, L. Fletcher, M. Crittenden, C. Carlisle, S. Gotoff, F. Reyes, F. D’Amico, “Frequency agile CO2 DIAL for environmental monitoring,” in International Symposium on Optical Sensing for Environmental Monitoring (International Society for Optical Engineering, Atlanta, Ga., 1993), Vol. 2112, pp. 282–294.

Carr, L.

L. Carr, L. Fletcher, M. Crittenden, C. Carlisle, S. Gotoff, F. Reyes, F. D’Amico, “Frequency agile CO2 DIAL for environmental monitoring,” in International Symposium on Optical Sensing for Environmental Monitoring (International Society for Optical Engineering, Atlanta, Ga., 1993), Vol. 2112, pp. 282–294.

L. Carr, C. Carlisle, D. Cooper, M. Crittenden, L. Fletcher, S. Gotoff, F. Reyes, “Design and testing of a mobile vapor chamber for multiple material vapor analysis with a frequency-agile CO2 DIAL system,” presented at the Third Workshop on Standoff Detection for Chemical and Biological Defense, Williamsburg, Va. (1994).

Cooper, D.

L. Carr, C. Carlisle, D. Cooper, M. Crittenden, L. Fletcher, S. Gotoff, F. Reyes, “Design and testing of a mobile vapor chamber for multiple material vapor analysis with a frequency-agile CO2 DIAL system,” presented at the Third Workshop on Standoff Detection for Chemical and Biological Defense, Williamsburg, Va. (1994).

Craig, A. T.

R. V. Hogg, A. T. Craig, Introduction to Mathematical Statistics, 4th ed. (Macmillan, New York, 1978).

Crittenden, M.

L. Carr, L. Fletcher, M. Crittenden, C. Carlisle, S. Gotoff, F. Reyes, F. D’Amico, “Frequency agile CO2 DIAL for environmental monitoring,” in International Symposium on Optical Sensing for Environmental Monitoring (International Society for Optical Engineering, Atlanta, Ga., 1993), Vol. 2112, pp. 282–294.

L. Carr, C. Carlisle, D. Cooper, M. Crittenden, L. Fletcher, S. Gotoff, F. Reyes, “Design and testing of a mobile vapor chamber for multiple material vapor analysis with a frequency-agile CO2 DIAL system,” presented at the Third Workshop on Standoff Detection for Chemical and Biological Defense, Williamsburg, Va. (1994).

D’Amico, F.

L. Carr, L. Fletcher, M. Crittenden, C. Carlisle, S. Gotoff, F. Reyes, F. D’Amico, “Frequency agile CO2 DIAL for environmental monitoring,” in International Symposium on Optical Sensing for Environmental Monitoring (International Society for Optical Engineering, Atlanta, Ga., 1993), Vol. 2112, pp. 282–294.

Fletcher, L.

L. Carr, L. Fletcher, M. Crittenden, C. Carlisle, S. Gotoff, F. Reyes, F. D’Amico, “Frequency agile CO2 DIAL for environmental monitoring,” in International Symposium on Optical Sensing for Environmental Monitoring (International Society for Optical Engineering, Atlanta, Ga., 1993), Vol. 2112, pp. 282–294.

L. Carr, C. Carlisle, D. Cooper, M. Crittenden, L. Fletcher, S. Gotoff, F. Reyes, “Design and testing of a mobile vapor chamber for multiple material vapor analysis with a frequency-agile CO2 DIAL system,” presented at the Third Workshop on Standoff Detection for Chemical and Biological Defense, Williamsburg, Va. (1994).

Gotoff, S.

L. Carr, C. Carlisle, D. Cooper, M. Crittenden, L. Fletcher, S. Gotoff, F. Reyes, “Design and testing of a mobile vapor chamber for multiple material vapor analysis with a frequency-agile CO2 DIAL system,” presented at the Third Workshop on Standoff Detection for Chemical and Biological Defense, Williamsburg, Va. (1994).

L. Carr, L. Fletcher, M. Crittenden, C. Carlisle, S. Gotoff, F. Reyes, F. D’Amico, “Frequency agile CO2 DIAL for environmental monitoring,” in International Symposium on Optical Sensing for Environmental Monitoring (International Society for Optical Engineering, Atlanta, Ga., 1993), Vol. 2112, pp. 282–294.

Hannon, S. M.

S. M. Hannon, R. E. Warren, “Sequential detection of multiple materials using multiwavelength lidar time series data,” in Coherent Laser Radar: Technology and Applications, Vol. 12 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991).

Hogg, R. V.

R. V. Hogg, A. T. Craig, Introduction to Mathematical Statistics, 4th ed. (Macmillan, New York, 1978).

Neyman, J.

J. Neyman, E. S. Pearson, “On the use and interpretation of certain test criteria for purposes of statistical inference: parts 1 and 2,” Biometrika A 20, 175–240, 263–294 (1928).

Pearson, E. S.

J. Neyman, E. S. Pearson, “On the use and interpretation of certain test criteria for purposes of statistical inference: parts 1 and 2,” Biometrika A 20, 175–240, 263–294 (1928).

Reyes, F.

L. Carr, L. Fletcher, M. Crittenden, C. Carlisle, S. Gotoff, F. Reyes, F. D’Amico, “Frequency agile CO2 DIAL for environmental monitoring,” in International Symposium on Optical Sensing for Environmental Monitoring (International Society for Optical Engineering, Atlanta, Ga., 1993), Vol. 2112, pp. 282–294.

L. Carr, C. Carlisle, D. Cooper, M. Crittenden, L. Fletcher, S. Gotoff, F. Reyes, “Design and testing of a mobile vapor chamber for multiple material vapor analysis with a frequency-agile CO2 DIAL system,” presented at the Third Workshop on Standoff Detection for Chemical and Biological Defense, Williamsburg, Va. (1994).

Wald, A.

A. Wald, “Tests of statistical hypotheses concerning several parameters when the number of observations is large,” Trans. Amer. Math. Soc. 54, 426–482 (1943).
[CrossRef]

Warren, R. E.

Appl. Opt.

Biometrika A

J. Neyman, E. S. Pearson, “On the use and interpretation of certain test criteria for purposes of statistical inference: parts 1 and 2,” Biometrika A 20, 175–240, 263–294 (1928).

Trans. Amer. Math. Soc.

A. Wald, “Tests of statistical hypotheses concerning several parameters when the number of observations is large,” Trans. Amer. Math. Soc. 54, 426–482 (1943).
[CrossRef]

Other

T. W. Anderson, An Introduction to Multivariate Statistical Analysis, 2nd ed. (Wiley, New York, 1984), p. 594.

We note that, because the maximization of the numerator in Eq. (4) is over the whole parameter space Ω, whereas that of the denominator is over a proper subset Ω0, L(x) ≥ 1 (neglecting numerical roundoff errors).

L. Carr, L. Fletcher, M. Crittenden, C. Carlisle, S. Gotoff, F. Reyes, F. D’Amico, “Frequency agile CO2 DIAL for environmental monitoring,” in International Symposium on Optical Sensing for Environmental Monitoring (International Society for Optical Engineering, Atlanta, Ga., 1993), Vol. 2112, pp. 282–294.

L. Carr, C. Carlisle, D. Cooper, M. Crittenden, L. Fletcher, S. Gotoff, F. Reyes, “Design and testing of a mobile vapor chamber for multiple material vapor analysis with a frequency-agile CO2 DIAL system,” presented at the Third Workshop on Standoff Detection for Chemical and Biological Defense, Williamsburg, Va. (1994).

S. M. Hannon, R. E. Warren, “Sequential detection of multiple materials using multiwavelength lidar time series data,” in Coherent Laser Radar: Technology and Applications, Vol. 12 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991).

R. V. Hogg, A. T. Craig, Introduction to Mathematical Statistics, 4th ed. (Macmillan, New York, 1978).

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Figures (14)

Fig. 1
Fig. 1

Hypothetical absorptivity versus wavelength channel for two vapor materials.

Fig. 2
Fig. 2

Histogram-based density compared to theoretical f 0(l) for 1000 trials of synthetic data made from a uniform backscatter model with CL 1 = CL 2 = 0.

Fig. 3
Fig. 3

Histogram-based density compared to theoretical f 1(l) for 1000 trials of synthetic data made from a uniform backscatter model with CL 1 = 0.1, CL 2 = 0.2.

Fig. 4
Fig. 4

Histogram of estimated CL from the data in Fig. 3.

Fig. 5
Fig. 5

Synthetic lidar data versus range bin (300-m total range).

Fig. 6
Fig. 6

Log-likelihood ratio contribution versus range.

Fig. 7
Fig. 7

Estimated CL versus range.

Fig. 8
Fig. 8

Histogram-based density compared to theoretical f 0(l) for 1000 trials of synthetic data made from a measured backscatter model with CL 1 = CL 2 = 0.

Fig. 9
Fig. 9

Histogram-based density compared to theoretical f 1(l) for 1000 trials of synthetic data made from a measured backscatter model with CL 1 = 0.1, CL 2 = 0.2.

Fig. 10
Fig. 10

Histogram of the CL estimates shown in Fig. 9.

Fig. 11
Fig. 11

Log-LRT statistic versus test data set index for methyl ethyl ketone vapor.

Fig. 12
Fig. 12

ML estimates of methyl ethyl ketone vapor path-integrated concentration versus test data set index.

Fig. 13
Fig. 13

Log-LRT statistic for three material injections.

Fig. 14
Fig. 14

ML estimates of CL for three material injections.

Tables (1)

Tables Icon

Table 1 SRI International FAL Lidar System Specifications

Equations (106)

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f ( x | θ ) = i = 1 N 1 2 πθ 2 exp [ 1 2 ( x i θ 1 ) 2 θ 2 ] .
L ( x ) = f ( x | Ω 1 ) f ( x | Ω 0 )
L ( x ) = max θ Ω 1 f ( x | θ ) max θ Ω 0 f ( x | θ ) .
L ( x ) = max θ Ω f ( x | θ ) max θ Ω 0 f ( x | θ ) .
P ( i , j ) = G + n ( i , j ) ,
Λ n ( i , j , i , j ) E n ( i , j ) n ( i , j ) = Λ ( j , j ) δ i i ,
f 0 ( P | G , Λ ) = i = 1 N ( 2 π ) M / 2 | Λ | 1 / 2 × exp { 1 2 j , j = 1 M [ P ( i , j ) G ] × Λ 1 ( j , j ) [ P ( i , j ) G ] } ,
P ( i , j ) = G exp ( 2 l = 1 Q ρ j l C L l ) + n ( i , j ) ,
f 1 ( P | G , Λ , C L ) = i = 1 N ( 2 π ) M / 2 | Λ | 1 / 2 × exp { 1 2 j , j = 1 M [ P ( i , j ) G T j ] × Λ 1 ( j , j ) [ P ( i , j ) G T j ] } ,
T j ( C L ) exp ( 2 l = 1 Q ρ j l C L l )
L = max G , Λ , CL f 1 ( P | G , Λ , C L ) max G , Λ f 0 ( P | G , Λ )
l ln L = max G , Λ , C L ln f 1 ( P | G , Λ , C L ) max G , Λ ln f 0 ( P | G , Λ ) .
G ˆ 0 = 1 N M i j P ( i , j ) ,
Λ ˆ 0 ( j , j ) = 1 N i = 1 N [ P ( i , j ) G ˆ 0 ] [ P ( i , j ) G ˆ 0 ] .
Λ ˆ 0 ( j , j ) = 1 N i = 1 N [ P ( i , j ) G ˆ 1 T ˆ j ] [ P ( i , j ) G ˆ 1 T ˆ j ] ,
G ˆ 1 = j , j T ˆ j Λ ˆ 1 1 ( j , j ) P ¯ ( j ) j , j T ˆ j Λ ˆ 1 1 ( j , j ) T ˆ j
F l ( C L ) j , j T ˆ j Λ ˆ 1 1 ( j , j ) P ¯ ( j ) j j ρ j l T ˆ j Λ ˆ 1 1 ( j , j ) T ˆ j j , j ρ j l T ˆ j Λ ˆ 1 1 ( j , j ) P ¯ ( j ) × j , j T ˆ j Λ ˆ 1 1 ( j , j ) T ˆ j = 0 ,
P ¯ ( j ) = 1 N i = 1 N P ( i , j ) ,
T ˆ j exp [ 2 l = 1 Q ρ j l C L l ] .
[ T ˆ 1 P ¯ 2 T ˆ 2 P ¯ 1 ] [ ( ρ 1 T ˆ ) 1 T ˆ 2 ( ρ 1 T ˆ ) 2 T ˆ 1 ] = 0 ,
x j j = 1 M C ( j , j ) x j
0 = T ˆ 1 P ¯ 2 T ˆ 2 P ¯ 1 = | C | ( T ˆ 1 P ¯ 2 T ˆ 2 P ¯ 1 ) ,
C L = 1 2 ( ρ 11 ρ 21 ) ln P ¯ 2 P ¯ 1
C L l ( k + 1 ) = C L l ( k ) l = 1 Q H l l 1 F l [ C L ( k ) ] ,
H l l ( C L ( k ) ) F l [ C L ( k ) ] C L l ( k ) .
δ F l ( C L ) = l H l l δ C L l + j F l P ¯ ( j ) δ P ¯ ( j ) = 0 .
δ C L l = l H l l 1 j F l P ¯ ( j ) δ P ¯ ( j ) ,
Λ C L ( l , l ) = E δ C L l δ C L l = 1 N l 1 , l 2 H l l 1 1 j , j F l 1 P ¯ ( j ) × Λ ˆ 1 ( j , j ) F l 2 P ¯ ( j ) H l l 2 1 ,
1 N Λ ˆ 1 ( j , j ) = E δ P ¯ ( j ) δ P ¯ ( j )
l = ln f 1 ( P | G ˆ 1 , Λ ˆ 1 , C L ) ln f 0 ( P | G ˆ 0 , Λ ˆ 0 ) .
ln f 0 ( P | G ˆ 0 , Λ ˆ 0 ) = M 2 ln ( 2 π ) N 2 ln | Λ ˆ 0 | 1 2 j j = 1 M × Λ ˆ 0 1 ( j , j ) i = 1 N [ P ( i , j ) G ˆ 0 ] [ P ( i , j ) G ˆ 0 ] = M 2 ln ( 2 π ) N 2 ln | Λ ˆ 0 | N M 2 .
ln f 1 ( P | G ˆ 1 , Λ ˆ 1 , C L ) = M 2 ln ( 2 π ) N 2 ln | Λ ˆ 1 | N M 2 ,
l = N 2 ln | Λ ˆ 0 | | Λ ˆ 1 | ,
P fa = τ f 0 ( l ) d l ,
P d = τ f 1 ( l ) d l .
f 0 ( l ) 1 Γ ( Q / 2 ) l Q / 2 1 exp ( l ) θ ( l ) ,
f 1 ( l ) exp [ ( λ 2 / 2 + l ) ] j = 0 λ 2 j 2 j j ! l Q / 2 + j 1 Γ ( Q / 2 + j ) θ ( l ) ,
λ 2 = N G 2 j , j ( T j T ) Λ 1 ( j , j ) ( T j T ) ,
T M 1 j = 1 M T j .
P ( i , j , k ) = G ( k ) + n ( i , j , k ) ,
Λ ( i , j , k ; i , j , k ) = Λ k ( j , j ) δ k k δ i i .
P ( i , j , k ) = G ( k ) exp [ 2 l = 1 Q ρ j l C L l ( k ) ] + n ( i , j , k ) ,
G ˆ 0 ( k ) = 1 N M i = 1 N j = 1 M P ( i , j , k ) ,
Λ ˆ 0 k ( j , j ) = 1 N i = 1 N [ P ( i , j , k ) G ˆ 0 ( k ) ] × [ P ( i , j , k ) G ˆ 0 ( k ) ] .
l = N 2 k = 1 N r ln | Λ ˆ 0 k | | Λ ˆ 1 k | .
f 0 ( l ) = 1 Γ ( N r Q 2 ) l N r Q / 2 1 exp ( l ) θ ( l )
f 1 ( l ) = exp [ ( l + 1 2 k = 1 N r λ k 2 ) ] j = 0 ( k λ k 2 ) j 2 j j ! l N r Q / 2 + j 1 Γ ( N r Q 2 + j ) θ ( l )
λ k 2 = N G 2 j , j [ T j ( k ) T k ] Λ k 1 ( j , j ) [ T j ( k ) T k ] ,
T k M 1 j = 1 M T j ( k ) .
P ( i , j , k ) = A k g ( j , k ) + n ( i , j , k ) ,
f 0 ( P | A , Λ ) = i = 1 N k = 1 N r ( 2 π ) M / 2 | Λ k | 1 / 2 × exp { 1 2 j , j [ P ( i , j , k ) A k g ( j , k ) ] × Λ k 1 ( j , j ) [ P ( i , j , k ) A k g ( j , k ) ] } .
A ˆ 0 k = j , j g ( j , k ) Λ ˆ 0 k 1 ( j , j ) P ¯ ( j , k ) j , j g ( j , k ) Λ ˆ 0 k 1 ( j , j ) g ( j , k ) ,
Λ ˆ 0 k ( j , j ) = 1 N i = 1 N [ P ( i , j , k ) A ˆ 0 k g ( j , k ) ] × [ P ( i , j , k ) A ˆ 0 k g ( j , k ) ] ,
P ¯ ( j , k ) 1 N i = 1 N P ( i , j , k ) .
P ( i , j , k ) = A k g ( j , k ) T j k ( C L ) + n ( i , j , k ) ,
T j k ( C L ) exp [ 2 l = 1 Q ρ j l C L l ( k ) ]
f 1 ( P | A , Λ , C L ) = i = 1 N k = 1 N r ( 2 π ) M / 2 | Λ k | 1 / 2 × exp [ 1 2 j , j [ P ( i , j , k ) A k g ( j , k ) T j k ] × Λ k 1 ( j , j ) ( P ( i , j , k ) A k g ( j , k ) T j k ) ] .
A ˆ 1 k = j , j g ( j , k ) T ˆ j k Λ ˆ 1 1 ( j , j ) P ¯ ( j , k ) j , j g ( j , k ) T ˆ j k Λ ˆ 1 1 ( j , j ) g ( j , k ) T ˆ j k ,
Λ ˆ 1 k ( j , j ) = 1 N i = 1 N [ P ( i , j , k ) A ˆ 1 k g ( j , k ) T ˆ j k ] × [ P ( i , j , k ) A ˆ 1 k g ( j , k ) T ˆ j k ] ,
F l k ( C L ) = j , j g ( j , k ) T ˆ j k Λ ˆ 1 1 ( j , j ) P ¯ ( j , k ) × j , j ρ j l g ( j , k ) T ˆ j k Λ ˆ 1 1 ( j , j ) g ( j , k ) T ˆ j k j , j ρ j l g ( j , l ) T ˆ j k Λ ˆ 1 1 ( j , j ) × P ¯ ( j , k ) j , j g ( j , k ) T ˆ j k Λ ˆ 1 1 ( j , j ) × g ( j , k ) T ˆ j k = 0 .
l = N 2 k = 1 N r ln | Λ ˆ 0 k | | Λ ˆ 1 k | ,
λ k 2 = N A k 2 j , j g ( j , k ) [ T j ( k ) T k ] Λ k 1 ( j , j ) g ( j , k ) × [ T j ( k ) T k ] ,
T k M 1 j = 1 M T j ( k )
P ( i , j , k ) = G ( j , k ) + n ( i , j , k ) ,
Λ ( i , j , k ; i , j , k ) = Λ k ( j , j ) δ i i δ k k .
G ˆ ( j , k ) = 1 N 0 i = 1 N 0 P ( i , j , k ) ,
Λ ˆ k ( j , j ) = 1 N 0 i = 1 N 0 [ P ( i , j , k ) G ˆ ( j , k ) ] × [ P ( i , j , k ) G ˆ ( j , k ) ] .
P ( i , j , k ) G ˆ ( j , k ) + n ( i , j , k )
f 0 ( P ) = i = 1 N k = 1 N r ( 2 π ) M / 2 | Λ ˆ k | 1 / 2 × exp { 1 2 j , j [ P ( i , j , k ) G ˆ ( j , k ) ] × Λ ˆ k 1 ( j , j ) [ P ( i , j , k ) G ˆ ( j , k ) ] } .
P ( i , j , k ) G ˆ ( j , k ) T j k ( CL ) + n ( i , j , k )
T j k ( C L ) exp | 2 l = 1 Q ρ j l C L l ( k ) |
f 0 ( P | C L ) = i = 1 N k = 1 N r ( 2 π ) M / 2 | Λ ˆ k | 1 / 2 × exp { 1 2 j , j [ P ( i , j , k ) G ˆ ( j , k ) T j k ] × Λ ˆ k 1 ( j , j ) [ P ( i , j , k ) G ˆ ( j , k ) T j k ] } .
F k l ( CL ) = j , j ρ j l G ˆ ( j , k ) T ˆ j k Λ ˆ k 1 ( j , j ) P ¯ ( j , k ) j , j ρ j l G ˆ ( j , k ) T ˆ j k Λ ˆ k 1 ( j , j ) G ˆ ( j , k ) T ˆ j k = 0
P ¯ ( j , k ) = 1 N i = 1 N P ( i , j , k ) ,
T ˆ j k = exp [ 2 l = 1 Q ρ j l C L l ( k ) ] .
Λ C L , k ( l , l ) = 1 N l 1 , l 2 H l l 1 1 ( k ) j , j F k l 1 P ¯ ( j , k ) × Λ ˆ k ( j , j ) F k l 2 P ¯ ( j , k ) H l l 2 1 ( k ) ,
l = k = 1 N r l k ,
l k = N 2 j , j [ P ¯ ( j , k ) G ˆ ( j , k ) ] Λ ˆ k 1 ( j , j ) × [ P ¯ ( j , k ) G ˆ ( j , k ) ] N 2 j , j [ P ¯ ( j , k ) G ˆ ( j , k ) T ˆ j k ] × Λ ˆ k 1 ( j , j ) [ P ¯ ( j , k ) G ˆ ( j , k ) T ˆ j k ] ,
f 0 ( l ) = l N r Q / 2 1 Γ ( N r Q / 2 ) exp ( l ) θ ( l ) ,
f 1 ( l ) = exp [ ( l + 1 2 k λ k 2 ) ] j = 0 ( k λ k 2 ) j 2 j j ! × l N r Q / 2 + j 1 Γ ( N r Q / 2 + j ) θ ( l ) ,
λ k 2 = N j , j G ( j , k ) ( T j k 1 ) Λ k 1 ( j , j ) G ( j , k ) ( T j k 1 ) .
l = max θ Ω ln f ( x | θ ) max θ Ω 0 ln f ( x | θ ) .
ln f ( x | θ ˆ ) ln f ( x | θ ˆ 0 ) + ( θ ˆ θ ˆ 0 ) ln f ( x | θ ) θ | θ= θ ˆ 0 + 1 2 ( θ ˆ θ ˆ 0 ) 2 ln f ( x | θ ) θ θ | θ= θ ˆ 0 ( θ ˆ θ ˆ 0 ) ,
l 1 2 ( θ ˆ θ ˆ 0 ) H ( θ ˆ θ ˆ 0 ) ,
H r r E θ 2 ln f ( x | θ ) θ r θ r | θ= θ ˆ 0 .
H r r = n = 1 Q λ n ϕ n ( r ) ϕ n ( r )
H r r = Λ θ 1 ( r , r ) .
β n r = 1 Q ( θ ˆ r θ ˆ 0 r ) ϕ n ( r ) .
l 1 2 r = 1 Q λ n [ r ( θ ˆ r θ ˆ 0 r ) ϕ n ( r ) ] 2 = 1 2 n λ n β n 2 .
E 0 β n β n = r , r ϕ n ( r ) E 0 ( θ ˆ r θ ˆ 0 r ) ( θ ˆ r θ ˆ 0 r ) ϕ n ( r ) = r , r ϕ n ( r ) Λ θ ˆ ( r , r ) ϕ n ( r ) = n 1 λ n r ϕ n ( r ) ϕ n ( r ) r ϕ n ( r ) ϕ n ( r ) = n 1 λ n δ n n δ n n = 1 λ n δ n n ,
z n = λ n β n ,
l 1 2 n = 1 Q z n 2 ,
f 0 ( l ) = 1 Γ ( Q / 2 ) l Q / 2 1 exp ( l ) θ ( l ) ,
γ r E 1 ( θ ˆ r θ ˆ 0 r ) ,
E 1 β n = r γ r ϕ n ( r )
E 1 z n = λ n r γ r ϕ n ( r ) μ n .
λ 2 = n μ n 2 = n λ n r , r γ r γ r ϕ n ( r ) ϕ n ( r ) = r , r γ r Λ θ 1 ( r , r ) γ r ,
E 1 ( G ˆ 0 ) = 1 N M i = 1 N j = 1 M E 1 [ P ( i , j ) ] = G M j = 1 M T j T G ,
Λ ˆ 0 ( j , j ) = G 2 ( T j T ) ( T j T ) + 1 N i = 1 N G ( T j T ) n ( i , j ) + 1 N i = 1 N G ( T j T ) n ( i , j ) + 1 N i = 1 N n ( i , j ) n ( i , j ) .
Λ ˆ 0 ( j , j ) G 2 ( T j T ) ( T j T ) + Λ ( j , j ) .
| Λ ˆ 0 | | Λ | [ 1 + G 2 j j ( T j T ) Λ 1 ( j , j ) ( T j T ) ] .
G ˆ 1 T ˆ j G T j
l N 2 ln [ 1 + G 2 ( T T ) T Λ 1 ( T T ) ] N 2 G 2 ( T T ) T Λ 1 ( T T ) λ 2 2 ,
l = N 2 [ P ¯ T Λ 1 G ( T ˆ 1 ) + G T ( T ˆ 1 ) T Λ 1 P ¯ G T T ˆ T Λ 1 G T ˆ + G T Λ 1 G ] ,
P ¯ ( j ) 1 N i = 1 N P ( i , j ) = G T j + 1 N i = 1 N n ( i , j )
l N 2 G T ( T 1 ) T Λ 1 G ( T 1 ) λ 2 2 ,

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