Abstract

An estimation problem for statistical reconstruction of heterogeneous three-dimensional objects from two-dimensional tomographic data (single-particle cryoelectron microscope images) is posed as the problem of estimating class probabilities, means, and covariances for a Gaussian mixture where both the mean and covariance are stochastically structured. Both discrete (i.e., classes) and continuous heterogeneity is included. A maximum likelihood solution computed by a generalized expectation-maximization algorithm is presented and demonstrated on experimental images of Flock House Virus.

© 2012 Optical Society of America

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  2. S. H. W. Scheres, H. Gao, M. Valle, G. T. Herman, P. P. B. Eggermont, J. Frank, and J.-M. Carazo, “Disentangling conformational states of macromolecules in 3D-EM through likelihood optimization,” Nat. Methods 4, 27–29(2007).
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    [CrossRef]
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    [CrossRef]
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  22. B. Efron and D. V. Hinkley, “Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher information,” Biometrika 65, 457–482 (1978).
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    [CrossRef]
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    [CrossRef]
  25. J. Lanman, J. Crum, T. J. Deerinck, G. M. Gaietta, A. Schneemann, G. E. Sosinsky, M. H. Ellisman, and J. E. Johnson, “Visualizing flock house virus infection in Drosophila cells with correlated fluorescence and electron microscopy,” J. Struct. Biol. 161, 439–446 (2008).
    [CrossRef]
  26. URL, “Flock House Virus (FHV) web page,” http://viperdb.scripps.edu/info_page.php?VDB=2q25.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  30. J. M. Hogle, “Poliovirus cell entry: common structural themes in viral cell entry pathways,” Annu. Rev. Microbiol. 56, 677–702 (2002).
    [CrossRef]
  31. M. Tihova, K. A. Dryden, T. L. Le, S. C. Harvey, J. E. Johnson, M. Yeager, and A. Schneemann, “Nodavirus coat protein imposes dodecahedral RNA structure independent of nucleotide sequence and length,” J. Virol. 78, 2897–2905(2004).
    [CrossRef]
  32. E. F. Pettersen, T. D. Goddard, C. C. Huang, G. S. Couch, D. M. Greenblatt, E. C. Meng, and T. E. Ferrin, “UCSF Chimera–A visualization system for exploratory research and analysis,” J. Comput. Chem. 25, 1605–1612 (2004).
    [CrossRef]
  33. URL http://www.mathworks.com/ .

2009 (1)

C. J. Prust, P. C. Doerschuk, G. C. Lander, and J. E. Johnson, “Ab initio maximum likelihood reconstruction from cryo electron microscopy images of an infectious virion of the tailed bacteriophage P22 and maximum likelihood versions of Fourier Shell Correlation appropriate for measuring resolution of spherical or cylindrical objects,” J. Struct. Biol. 167, 185–199 (2009).
[CrossRef]

2008 (1)

J. Lanman, J. Crum, T. J. Deerinck, G. M. Gaietta, A. Schneemann, G. E. Sosinsky, M. H. Ellisman, and J. E. Johnson, “Visualizing flock house virus infection in Drosophila cells with correlated fluorescence and electron microscopy,” J. Struct. Biol. 161, 439–446 (2008).
[CrossRef]

2007 (3)

H. Snoussi and A. Mohammad-Djafari, “Estimation of structured Gaussian mixtures: the inverse EM algorithm,” IEEE Trans. Signal Process. 55, 3185–3191 (2007).
[CrossRef]

S. H. W. Scheres, H. Gao, M. Valle, G. T. Herman, P. P. B. Eggermont, J. Frank, and J.-M. Carazo, “Disentangling conformational states of macromolecules in 3D-EM through likelihood optimization,” Nat. Methods 4, 27–29(2007).
[CrossRef]

J. Lee, P. C. Doerschuk, and J. E. Johnson, “Exact reduced-complexity maximum likelihood reconstruction of multiple 3-D objects from unlabeled unoriented 2-D projections and electron microscopy of viruses,” IEEE Trans. Image Process 16, 2865–2878 (2007).
[CrossRef]

2006 (5)

J. Tang, J. M. Johnson, K. A. Dryden, M. J. Young, A. Zlotnick, and J. E. Johnson, “The role of subunit hinges and molecular “switches’’ in the control of viral capsid polymorphism,” J. Struct. Biol. 154, 59–67 (2006).
[CrossRef]

A. Berge and A. H. Schistad Solberg, “Structured Gaussian components for hyperspectral image classification,” IEEE Trans. Geosci. Remote Sens. 44, 3386–3396 (2006).
[CrossRef]

K. C. Sim and M. J. F. Gales, “Minimum phone error training of precision matrix models,” IEEE Trans. Audio Speech Lang. Process. 14, 882–889 (2006).
[CrossRef]

Y. Tian, J.-L. Zhou, H. Lin, and H. Jiang, “Tree-based covariance modeling of hidden Markov models,” IEEE Trans. Audio Speech Lang. Process. 14, 2134–2146 (2006).
[CrossRef]

S. Dharanipragada and K. Visweswariah, “Gaussian mixture models with covariances or precisions in shared multiple subspaces,” IEEE Trans. Audio Speech Lang. Process. 14, 1255–1266 (2006).
[CrossRef]

2005 (1)

D. Bubeck, D. J. Filman, and J. M. Hogle, “Cryo-electron microscopy reconstruction of a poliovirus-receptor-membrane complex,” Nat. Struct. Mol. Biol. 12, 615–618 (2005).
[CrossRef]

2004 (2)

M. Tihova, K. A. Dryden, T. L. Le, S. C. Harvey, J. E. Johnson, M. Yeager, and A. Schneemann, “Nodavirus coat protein imposes dodecahedral RNA structure independent of nucleotide sequence and length,” J. Virol. 78, 2897–2905(2004).
[CrossRef]

E. F. Pettersen, T. D. Goddard, C. C. Huang, G. S. Couch, D. M. Greenblatt, E. C. Meng, and T. E. Ferrin, “UCSF Chimera–A visualization system for exploratory research and analysis,” J. Comput. Chem. 25, 1605–1612 (2004).
[CrossRef]

2003 (1)

Z. Yin, Y. Zheng, P. C. Doerschuk, P. Natarajan, and J. E. Johnson, “A statistical approach to computer processing of cryo electron microscope images: Virion classification and 3-D reconstruction,” J. Struct. Biol. 144, 24–50(2003).
[CrossRef]

2002 (1)

J. M. Hogle, “Poliovirus cell entry: common structural themes in viral cell entry pathways,” Annu. Rev. Microbiol. 56, 677–702 (2002).
[CrossRef]

2000 (2)

Y. Zheng and P. C. Doerschuk, “Explicit computation of orthonormal symmetrized harmonics with application to the identity representation of the icosahedral group,” SIAM J. Math. Anal. 32, 538–554 (2000).
[CrossRef]

P. C. Doerschuk and J. E. Johnson, “Ab initio reconstruction and experimental design for cryo electron microscopy,” IEEE Trans. Inf. Theory 46, 1714–1729 (2000).
[CrossRef]

1999 (1)

T. S. Baker, N. H. Olson, and S. D. Fuller, “Adding the third dimension to virus life cycles: three-dimensional reconstruction of icosahedral viruses from cryo-electron micrographs,” Microbiol. Molec. Biol. Rev. 63, 862–922 (1999).

1994 (1)

R. H. Cheng, V. S. Reddy, N. H. Olson, A. J. Fisher, T. S. Baker, and J. E. Johnson, “Functional implications of quasi-equivalence in a T=3 icosahedral animal virus established by cryo-electron microscopy and x-ray crystallography,” Structure 2, 271–282 (1994).
[CrossRef]

1993 (1)

A. J. Fisher and J. E. Johnson, “Ordered duplex RNA controls capsid architecture in an icosahedral animal virus,” Nature 361, 176–179 (1993).
[CrossRef]

1987 (1)

M. van Heel, “Similarity measures between images,” Ultramicroscopy 21, 95–100 (1987).
[CrossRef]

1986 (1)

G. Harauz and M. van Heel, “Exact filters for general geometry three dimensional reconstruction,” Optik 73, 146–156(1986).

1984 (1)

R. A. Redner, and H. F. Walker, “Mixture densities, maximum likelihood and the EM algorithm,” SIAM Rev. 26, 195–239 (1984).
[CrossRef]

1978 (1)

B. Efron and D. V. Hinkley, “Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher information,” Biometrika 65, 457–482 (1978).
[CrossRef]

Anderson, B. D. O.

B. D. O. Anderson and J. B. Moore, Optimal Filtering (Prentice-Hall, 1979).

Baker, T. S.

T. S. Baker, N. H. Olson, and S. D. Fuller, “Adding the third dimension to virus life cycles: three-dimensional reconstruction of icosahedral viruses from cryo-electron micrographs,” Microbiol. Molec. Biol. Rev. 63, 862–922 (1999).

R. H. Cheng, V. S. Reddy, N. H. Olson, A. J. Fisher, T. S. Baker, and J. E. Johnson, “Functional implications of quasi-equivalence in a T=3 icosahedral animal virus established by cryo-electron microscopy and x-ray crystallography,” Structure 2, 271–282 (1994).
[CrossRef]

Berge, A.

A. Berge and A. H. Schistad Solberg, “Structured Gaussian components for hyperspectral image classification,” IEEE Trans. Geosci. Remote Sens. 44, 3386–3396 (2006).
[CrossRef]

Berger, R. L.

G. Casella and R. L. Berger, Statistical Inference, 2nd ed.(Duxbury2002).

Bilmes, J. A.

J. A. Bilmes, “A gentle tutorial of the EM algorithm and its application to parameter estimation for Gaussian mixture and hidden Markov models,” Tech. Rep. TR-97-021 (Department of Electrical Engineering and Computer Science, University of California at Berkeley, 1998).

Bubeck, D.

D. Bubeck, D. J. Filman, and J. M. Hogle, “Cryo-electron microscopy reconstruction of a poliovirus-receptor-membrane complex,” Nat. Struct. Mol. Biol. 12, 615–618 (2005).
[CrossRef]

Carazo, J.-M.

S. H. W. Scheres, H. Gao, M. Valle, G. T. Herman, P. P. B. Eggermont, J. Frank, and J.-M. Carazo, “Disentangling conformational states of macromolecules in 3D-EM through likelihood optimization,” Nat. Methods 4, 27–29(2007).
[CrossRef]

Casella, G.

G. Casella and R. L. Berger, Statistical Inference, 2nd ed.(Duxbury2002).

Cheng, R. H.

R. H. Cheng, V. S. Reddy, N. H. Olson, A. J. Fisher, T. S. Baker, and J. E. Johnson, “Functional implications of quasi-equivalence in a T=3 icosahedral animal virus established by cryo-electron microscopy and x-ray crystallography,” Structure 2, 271–282 (1994).
[CrossRef]

Couch, G. S.

E. F. Pettersen, T. D. Goddard, C. C. Huang, G. S. Couch, D. M. Greenblatt, E. C. Meng, and T. E. Ferrin, “UCSF Chimera–A visualization system for exploratory research and analysis,” J. Comput. Chem. 25, 1605–1612 (2004).
[CrossRef]

Crum, J.

J. Lanman, J. Crum, T. J. Deerinck, G. M. Gaietta, A. Schneemann, G. E. Sosinsky, M. H. Ellisman, and J. E. Johnson, “Visualizing flock house virus infection in Drosophila cells with correlated fluorescence and electron microscopy,” J. Struct. Biol. 161, 439–446 (2008).
[CrossRef]

Deerinck, T. J.

J. Lanman, J. Crum, T. J. Deerinck, G. M. Gaietta, A. Schneemann, G. E. Sosinsky, M. H. Ellisman, and J. E. Johnson, “Visualizing flock house virus infection in Drosophila cells with correlated fluorescence and electron microscopy,” J. Struct. Biol. 161, 439–446 (2008).
[CrossRef]

Dharanipragada, S.

S. Dharanipragada and K. Visweswariah, “Gaussian mixture models with covariances or precisions in shared multiple subspaces,” IEEE Trans. Audio Speech Lang. Process. 14, 1255–1266 (2006).
[CrossRef]

Doerschuk, P. C.

C. J. Prust, P. C. Doerschuk, G. C. Lander, and J. E. Johnson, “Ab initio maximum likelihood reconstruction from cryo electron microscopy images of an infectious virion of the tailed bacteriophage P22 and maximum likelihood versions of Fourier Shell Correlation appropriate for measuring resolution of spherical or cylindrical objects,” J. Struct. Biol. 167, 185–199 (2009).
[CrossRef]

J. Lee, P. C. Doerschuk, and J. E. Johnson, “Exact reduced-complexity maximum likelihood reconstruction of multiple 3-D objects from unlabeled unoriented 2-D projections and electron microscopy of viruses,” IEEE Trans. Image Process 16, 2865–2878 (2007).
[CrossRef]

Z. Yin, Y. Zheng, P. C. Doerschuk, P. Natarajan, and J. E. Johnson, “A statistical approach to computer processing of cryo electron microscope images: Virion classification and 3-D reconstruction,” J. Struct. Biol. 144, 24–50(2003).
[CrossRef]

P. C. Doerschuk and J. E. Johnson, “Ab initio reconstruction and experimental design for cryo electron microscopy,” IEEE Trans. Inf. Theory 46, 1714–1729 (2000).
[CrossRef]

Y. Zheng and P. C. Doerschuk, “Explicit computation of orthonormal symmetrized harmonics with application to the identity representation of the icosahedral group,” SIAM J. Math. Anal. 32, 538–554 (2000).
[CrossRef]

Dryden, K. A.

J. Tang, J. M. Johnson, K. A. Dryden, M. J. Young, A. Zlotnick, and J. E. Johnson, “The role of subunit hinges and molecular “switches’’ in the control of viral capsid polymorphism,” J. Struct. Biol. 154, 59–67 (2006).
[CrossRef]

M. Tihova, K. A. Dryden, T. L. Le, S. C. Harvey, J. E. Johnson, M. Yeager, and A. Schneemann, “Nodavirus coat protein imposes dodecahedral RNA structure independent of nucleotide sequence and length,” J. Virol. 78, 2897–2905(2004).
[CrossRef]

Efron, B.

B. Efron and D. V. Hinkley, “Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher information,” Biometrika 65, 457–482 (1978).
[CrossRef]

Eggermont, P. P. B.

S. H. W. Scheres, H. Gao, M. Valle, G. T. Herman, P. P. B. Eggermont, J. Frank, and J.-M. Carazo, “Disentangling conformational states of macromolecules in 3D-EM through likelihood optimization,” Nat. Methods 4, 27–29(2007).
[CrossRef]

Ellisman, M. H.

J. Lanman, J. Crum, T. J. Deerinck, G. M. Gaietta, A. Schneemann, G. E. Sosinsky, M. H. Ellisman, and J. E. Johnson, “Visualizing flock house virus infection in Drosophila cells with correlated fluorescence and electron microscopy,” J. Struct. Biol. 161, 439–446 (2008).
[CrossRef]

Ferrin, T. E.

E. F. Pettersen, T. D. Goddard, C. C. Huang, G. S. Couch, D. M. Greenblatt, E. C. Meng, and T. E. Ferrin, “UCSF Chimera–A visualization system for exploratory research and analysis,” J. Comput. Chem. 25, 1605–1612 (2004).
[CrossRef]

Filman, D. J.

D. Bubeck, D. J. Filman, and J. M. Hogle, “Cryo-electron microscopy reconstruction of a poliovirus-receptor-membrane complex,” Nat. Struct. Mol. Biol. 12, 615–618 (2005).
[CrossRef]

Fisher, A. J.

R. H. Cheng, V. S. Reddy, N. H. Olson, A. J. Fisher, T. S. Baker, and J. E. Johnson, “Functional implications of quasi-equivalence in a T=3 icosahedral animal virus established by cryo-electron microscopy and x-ray crystallography,” Structure 2, 271–282 (1994).
[CrossRef]

A. J. Fisher and J. E. Johnson, “Ordered duplex RNA controls capsid architecture in an icosahedral animal virus,” Nature 361, 176–179 (1993).
[CrossRef]

Frank, J.

S. H. W. Scheres, H. Gao, M. Valle, G. T. Herman, P. P. B. Eggermont, J. Frank, and J.-M. Carazo, “Disentangling conformational states of macromolecules in 3D-EM through likelihood optimization,” Nat. Methods 4, 27–29(2007).
[CrossRef]

J. Frank, Three-Dimensional Electron Microscopy of Macromolecular Assemblies (Academic, 1996).

Fuller, S. D.

T. S. Baker, N. H. Olson, and S. D. Fuller, “Adding the third dimension to virus life cycles: three-dimensional reconstruction of icosahedral viruses from cryo-electron micrographs,” Microbiol. Molec. Biol. Rev. 63, 862–922 (1999).

Gaietta, G. M.

J. Lanman, J. Crum, T. J. Deerinck, G. M. Gaietta, A. Schneemann, G. E. Sosinsky, M. H. Ellisman, and J. E. Johnson, “Visualizing flock house virus infection in Drosophila cells with correlated fluorescence and electron microscopy,” J. Struct. Biol. 161, 439–446 (2008).
[CrossRef]

Gales, M. J. F.

K. C. Sim and M. J. F. Gales, “Minimum phone error training of precision matrix models,” IEEE Trans. Audio Speech Lang. Process. 14, 882–889 (2006).
[CrossRef]

Gao, H.

S. H. W. Scheres, H. Gao, M. Valle, G. T. Herman, P. P. B. Eggermont, J. Frank, and J.-M. Carazo, “Disentangling conformational states of macromolecules in 3D-EM through likelihood optimization,” Nat. Methods 4, 27–29(2007).
[CrossRef]

Goddard, T. D.

E. F. Pettersen, T. D. Goddard, C. C. Huang, G. S. Couch, D. M. Greenblatt, E. C. Meng, and T. E. Ferrin, “UCSF Chimera–A visualization system for exploratory research and analysis,” J. Comput. Chem. 25, 1605–1612 (2004).
[CrossRef]

Greenblatt, D. M.

E. F. Pettersen, T. D. Goddard, C. C. Huang, G. S. Couch, D. M. Greenblatt, E. C. Meng, and T. E. Ferrin, “UCSF Chimera–A visualization system for exploratory research and analysis,” J. Comput. Chem. 25, 1605–1612 (2004).
[CrossRef]

Harauz, G.

G. Harauz and M. van Heel, “Exact filters for general geometry three dimensional reconstruction,” Optik 73, 146–156(1986).

Harvey, S. C.

M. Tihova, K. A. Dryden, T. L. Le, S. C. Harvey, J. E. Johnson, M. Yeager, and A. Schneemann, “Nodavirus coat protein imposes dodecahedral RNA structure independent of nucleotide sequence and length,” J. Virol. 78, 2897–2905(2004).
[CrossRef]

Herman, G. T.

S. H. W. Scheres, H. Gao, M. Valle, G. T. Herman, P. P. B. Eggermont, J. Frank, and J.-M. Carazo, “Disentangling conformational states of macromolecules in 3D-EM through likelihood optimization,” Nat. Methods 4, 27–29(2007).
[CrossRef]

Hinkley, D. V.

B. Efron and D. V. Hinkley, “Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher information,” Biometrika 65, 457–482 (1978).
[CrossRef]

Hogle, J. M.

D. Bubeck, D. J. Filman, and J. M. Hogle, “Cryo-electron microscopy reconstruction of a poliovirus-receptor-membrane complex,” Nat. Struct. Mol. Biol. 12, 615–618 (2005).
[CrossRef]

J. M. Hogle, “Poliovirus cell entry: common structural themes in viral cell entry pathways,” Annu. Rev. Microbiol. 56, 677–702 (2002).
[CrossRef]

Huang, C. C.

E. F. Pettersen, T. D. Goddard, C. C. Huang, G. S. Couch, D. M. Greenblatt, E. C. Meng, and T. E. Ferrin, “UCSF Chimera–A visualization system for exploratory research and analysis,” J. Comput. Chem. 25, 1605–1612 (2004).
[CrossRef]

Jiang, H.

Y. Tian, J.-L. Zhou, H. Lin, and H. Jiang, “Tree-based covariance modeling of hidden Markov models,” IEEE Trans. Audio Speech Lang. Process. 14, 2134–2146 (2006).
[CrossRef]

Johnson, J. E.

C. J. Prust, P. C. Doerschuk, G. C. Lander, and J. E. Johnson, “Ab initio maximum likelihood reconstruction from cryo electron microscopy images of an infectious virion of the tailed bacteriophage P22 and maximum likelihood versions of Fourier Shell Correlation appropriate for measuring resolution of spherical or cylindrical objects,” J. Struct. Biol. 167, 185–199 (2009).
[CrossRef]

J. Lanman, J. Crum, T. J. Deerinck, G. M. Gaietta, A. Schneemann, G. E. Sosinsky, M. H. Ellisman, and J. E. Johnson, “Visualizing flock house virus infection in Drosophila cells with correlated fluorescence and electron microscopy,” J. Struct. Biol. 161, 439–446 (2008).
[CrossRef]

J. Lee, P. C. Doerschuk, and J. E. Johnson, “Exact reduced-complexity maximum likelihood reconstruction of multiple 3-D objects from unlabeled unoriented 2-D projections and electron microscopy of viruses,” IEEE Trans. Image Process 16, 2865–2878 (2007).
[CrossRef]

J. Tang, J. M. Johnson, K. A. Dryden, M. J. Young, A. Zlotnick, and J. E. Johnson, “The role of subunit hinges and molecular “switches’’ in the control of viral capsid polymorphism,” J. Struct. Biol. 154, 59–67 (2006).
[CrossRef]

M. Tihova, K. A. Dryden, T. L. Le, S. C. Harvey, J. E. Johnson, M. Yeager, and A. Schneemann, “Nodavirus coat protein imposes dodecahedral RNA structure independent of nucleotide sequence and length,” J. Virol. 78, 2897–2905(2004).
[CrossRef]

Z. Yin, Y. Zheng, P. C. Doerschuk, P. Natarajan, and J. E. Johnson, “A statistical approach to computer processing of cryo electron microscope images: Virion classification and 3-D reconstruction,” J. Struct. Biol. 144, 24–50(2003).
[CrossRef]

P. C. Doerschuk and J. E. Johnson, “Ab initio reconstruction and experimental design for cryo electron microscopy,” IEEE Trans. Inf. Theory 46, 1714–1729 (2000).
[CrossRef]

R. H. Cheng, V. S. Reddy, N. H. Olson, A. J. Fisher, T. S. Baker, and J. E. Johnson, “Functional implications of quasi-equivalence in a T=3 icosahedral animal virus established by cryo-electron microscopy and x-ray crystallography,” Structure 2, 271–282 (1994).
[CrossRef]

A. J. Fisher and J. E. Johnson, “Ordered duplex RNA controls capsid architecture in an icosahedral animal virus,” Nature 361, 176–179 (1993).
[CrossRef]

Johnson, J. M.

J. Tang, J. M. Johnson, K. A. Dryden, M. J. Young, A. Zlotnick, and J. E. Johnson, “The role of subunit hinges and molecular “switches’’ in the control of viral capsid polymorphism,” J. Struct. Biol. 154, 59–67 (2006).
[CrossRef]

Krishnan, T.

G. J. McLachlan and T. Krishnan, The EM Algorithm and Extensions (Wiley-Interscience, 1997).

Lander, G. C.

C. J. Prust, P. C. Doerschuk, G. C. Lander, and J. E. Johnson, “Ab initio maximum likelihood reconstruction from cryo electron microscopy images of an infectious virion of the tailed bacteriophage P22 and maximum likelihood versions of Fourier Shell Correlation appropriate for measuring resolution of spherical or cylindrical objects,” J. Struct. Biol. 167, 185–199 (2009).
[CrossRef]

Lanman, J.

J. Lanman, J. Crum, T. J. Deerinck, G. M. Gaietta, A. Schneemann, G. E. Sosinsky, M. H. Ellisman, and J. E. Johnson, “Visualizing flock house virus infection in Drosophila cells with correlated fluorescence and electron microscopy,” J. Struct. Biol. 161, 439–446 (2008).
[CrossRef]

Le, T. L.

M. Tihova, K. A. Dryden, T. L. Le, S. C. Harvey, J. E. Johnson, M. Yeager, and A. Schneemann, “Nodavirus coat protein imposes dodecahedral RNA structure independent of nucleotide sequence and length,” J. Virol. 78, 2897–2905(2004).
[CrossRef]

Lee, J.

J. Lee, P. C. Doerschuk, and J. E. Johnson, “Exact reduced-complexity maximum likelihood reconstruction of multiple 3-D objects from unlabeled unoriented 2-D projections and electron microscopy of viruses,” IEEE Trans. Image Process 16, 2865–2878 (2007).
[CrossRef]

Lin, H.

Y. Tian, J.-L. Zhou, H. Lin, and H. Jiang, “Tree-based covariance modeling of hidden Markov models,” IEEE Trans. Audio Speech Lang. Process. 14, 2134–2146 (2006).
[CrossRef]

McLachlan, G. J.

G. J. McLachlan and T. Krishnan, The EM Algorithm and Extensions (Wiley-Interscience, 1997).

Meng, E. C.

E. F. Pettersen, T. D. Goddard, C. C. Huang, G. S. Couch, D. M. Greenblatt, E. C. Meng, and T. E. Ferrin, “UCSF Chimera–A visualization system for exploratory research and analysis,” J. Comput. Chem. 25, 1605–1612 (2004).
[CrossRef]

Mohammad-Djafari, A.

H. Snoussi and A. Mohammad-Djafari, “Estimation of structured Gaussian mixtures: the inverse EM algorithm,” IEEE Trans. Signal Process. 55, 3185–3191 (2007).
[CrossRef]

Moore, J. B.

B. D. O. Anderson and J. B. Moore, Optimal Filtering (Prentice-Hall, 1979).

Natarajan, P.

Z. Yin, Y. Zheng, P. C. Doerschuk, P. Natarajan, and J. E. Johnson, “A statistical approach to computer processing of cryo electron microscope images: Virion classification and 3-D reconstruction,” J. Struct. Biol. 144, 24–50(2003).
[CrossRef]

Olson, N. H.

T. S. Baker, N. H. Olson, and S. D. Fuller, “Adding the third dimension to virus life cycles: three-dimensional reconstruction of icosahedral viruses from cryo-electron micrographs,” Microbiol. Molec. Biol. Rev. 63, 862–922 (1999).

R. H. Cheng, V. S. Reddy, N. H. Olson, A. J. Fisher, T. S. Baker, and J. E. Johnson, “Functional implications of quasi-equivalence in a T=3 icosahedral animal virus established by cryo-electron microscopy and x-ray crystallography,” Structure 2, 271–282 (1994).
[CrossRef]

Pettersen, E. F.

E. F. Pettersen, T. D. Goddard, C. C. Huang, G. S. Couch, D. M. Greenblatt, E. C. Meng, and T. E. Ferrin, “UCSF Chimera–A visualization system for exploratory research and analysis,” J. Comput. Chem. 25, 1605–1612 (2004).
[CrossRef]

Prust, C. J.

C. J. Prust, P. C. Doerschuk, G. C. Lander, and J. E. Johnson, “Ab initio maximum likelihood reconstruction from cryo electron microscopy images of an infectious virion of the tailed bacteriophage P22 and maximum likelihood versions of Fourier Shell Correlation appropriate for measuring resolution of spherical or cylindrical objects,” J. Struct. Biol. 167, 185–199 (2009).
[CrossRef]

Reddy, V. S.

R. H. Cheng, V. S. Reddy, N. H. Olson, A. J. Fisher, T. S. Baker, and J. E. Johnson, “Functional implications of quasi-equivalence in a T=3 icosahedral animal virus established by cryo-electron microscopy and x-ray crystallography,” Structure 2, 271–282 (1994).
[CrossRef]

Redner, R. A.

R. A. Redner, and H. F. Walker, “Mixture densities, maximum likelihood and the EM algorithm,” SIAM Rev. 26, 195–239 (1984).
[CrossRef]

Scheres, S. H. W.

S. H. W. Scheres, H. Gao, M. Valle, G. T. Herman, P. P. B. Eggermont, J. Frank, and J.-M. Carazo, “Disentangling conformational states of macromolecules in 3D-EM through likelihood optimization,” Nat. Methods 4, 27–29(2007).
[CrossRef]

Schistad Solberg, A. H.

A. Berge and A. H. Schistad Solberg, “Structured Gaussian components for hyperspectral image classification,” IEEE Trans. Geosci. Remote Sens. 44, 3386–3396 (2006).
[CrossRef]

Schneemann, A.

J. Lanman, J. Crum, T. J. Deerinck, G. M. Gaietta, A. Schneemann, G. E. Sosinsky, M. H. Ellisman, and J. E. Johnson, “Visualizing flock house virus infection in Drosophila cells with correlated fluorescence and electron microscopy,” J. Struct. Biol. 161, 439–446 (2008).
[CrossRef]

M. Tihova, K. A. Dryden, T. L. Le, S. C. Harvey, J. E. Johnson, M. Yeager, and A. Schneemann, “Nodavirus coat protein imposes dodecahedral RNA structure independent of nucleotide sequence and length,” J. Virol. 78, 2897–2905(2004).
[CrossRef]

Sim, K. C.

K. C. Sim and M. J. F. Gales, “Minimum phone error training of precision matrix models,” IEEE Trans. Audio Speech Lang. Process. 14, 882–889 (2006).
[CrossRef]

Snoussi, H.

H. Snoussi and A. Mohammad-Djafari, “Estimation of structured Gaussian mixtures: the inverse EM algorithm,” IEEE Trans. Signal Process. 55, 3185–3191 (2007).
[CrossRef]

Sosinsky, G. E.

J. Lanman, J. Crum, T. J. Deerinck, G. M. Gaietta, A. Schneemann, G. E. Sosinsky, M. H. Ellisman, and J. E. Johnson, “Visualizing flock house virus infection in Drosophila cells with correlated fluorescence and electron microscopy,” J. Struct. Biol. 161, 439–446 (2008).
[CrossRef]

Tang, J.

J. Tang, J. M. Johnson, K. A. Dryden, M. J. Young, A. Zlotnick, and J. E. Johnson, “The role of subunit hinges and molecular “switches’’ in the control of viral capsid polymorphism,” J. Struct. Biol. 154, 59–67 (2006).
[CrossRef]

Tian, Y.

Y. Tian, J.-L. Zhou, H. Lin, and H. Jiang, “Tree-based covariance modeling of hidden Markov models,” IEEE Trans. Audio Speech Lang. Process. 14, 2134–2146 (2006).
[CrossRef]

Tihova, M.

M. Tihova, K. A. Dryden, T. L. Le, S. C. Harvey, J. E. Johnson, M. Yeager, and A. Schneemann, “Nodavirus coat protein imposes dodecahedral RNA structure independent of nucleotide sequence and length,” J. Virol. 78, 2897–2905(2004).
[CrossRef]

Valle, M.

S. H. W. Scheres, H. Gao, M. Valle, G. T. Herman, P. P. B. Eggermont, J. Frank, and J.-M. Carazo, “Disentangling conformational states of macromolecules in 3D-EM through likelihood optimization,” Nat. Methods 4, 27–29(2007).
[CrossRef]

van Heel, M.

M. van Heel, “Similarity measures between images,” Ultramicroscopy 21, 95–100 (1987).
[CrossRef]

G. Harauz and M. van Heel, “Exact filters for general geometry three dimensional reconstruction,” Optik 73, 146–156(1986).

Visweswariah, K.

S. Dharanipragada and K. Visweswariah, “Gaussian mixture models with covariances or precisions in shared multiple subspaces,” IEEE Trans. Audio Speech Lang. Process. 14, 1255–1266 (2006).
[CrossRef]

Walker, H. F.

R. A. Redner, and H. F. Walker, “Mixture densities, maximum likelihood and the EM algorithm,” SIAM Rev. 26, 195–239 (1984).
[CrossRef]

Wang, Q.

Q. Wang, is preparing a Ph.D. dissertation called “Maximum likelihood reconstruction of heterogeneous 3-D objects from 2-D projections of unknown orientation and application to electron microscope images of viruses” (School of Electrical and Computer Engineering, Cornell University, Ithaca, N.Y., USA; expected completion date 2013).

Yeager, M.

M. Tihova, K. A. Dryden, T. L. Le, S. C. Harvey, J. E. Johnson, M. Yeager, and A. Schneemann, “Nodavirus coat protein imposes dodecahedral RNA structure independent of nucleotide sequence and length,” J. Virol. 78, 2897–2905(2004).
[CrossRef]

Yin, Z.

Z. Yin, Y. Zheng, P. C. Doerschuk, P. Natarajan, and J. E. Johnson, “A statistical approach to computer processing of cryo electron microscope images: Virion classification and 3-D reconstruction,” J. Struct. Biol. 144, 24–50(2003).
[CrossRef]

Young, M. J.

J. Tang, J. M. Johnson, K. A. Dryden, M. J. Young, A. Zlotnick, and J. E. Johnson, “The role of subunit hinges and molecular “switches’’ in the control of viral capsid polymorphism,” J. Struct. Biol. 154, 59–67 (2006).
[CrossRef]

Zheng, Y.

Z. Yin, Y. Zheng, P. C. Doerschuk, P. Natarajan, and J. E. Johnson, “A statistical approach to computer processing of cryo electron microscope images: Virion classification and 3-D reconstruction,” J. Struct. Biol. 144, 24–50(2003).
[CrossRef]

Y. Zheng and P. C. Doerschuk, “Explicit computation of orthonormal symmetrized harmonics with application to the identity representation of the icosahedral group,” SIAM J. Math. Anal. 32, 538–554 (2000).
[CrossRef]

Y. Zheng, “Novel statistical models and a high-performance computing toolkit for the solution of cryo electron microscopy inverse problems in viral structural biology,” Ph.D. dissertation (School of Electrical and Computer Engineering, Purdue University, West Lafayette, Ind., USA, 2008).

Zhou, J.-L.

Y. Tian, J.-L. Zhou, H. Lin, and H. Jiang, “Tree-based covariance modeling of hidden Markov models,” IEEE Trans. Audio Speech Lang. Process. 14, 2134–2146 (2006).
[CrossRef]

Zlotnick, A.

J. Tang, J. M. Johnson, K. A. Dryden, M. J. Young, A. Zlotnick, and J. E. Johnson, “The role of subunit hinges and molecular “switches’’ in the control of viral capsid polymorphism,” J. Struct. Biol. 154, 59–67 (2006).
[CrossRef]

Annu. Rev. Microbiol. (1)

J. M. Hogle, “Poliovirus cell entry: common structural themes in viral cell entry pathways,” Annu. Rev. Microbiol. 56, 677–702 (2002).
[CrossRef]

Biometrika (1)

B. Efron and D. V. Hinkley, “Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher information,” Biometrika 65, 457–482 (1978).
[CrossRef]

IEEE Trans. Audio Speech Lang. Process. (3)

K. C. Sim and M. J. F. Gales, “Minimum phone error training of precision matrix models,” IEEE Trans. Audio Speech Lang. Process. 14, 882–889 (2006).
[CrossRef]

Y. Tian, J.-L. Zhou, H. Lin, and H. Jiang, “Tree-based covariance modeling of hidden Markov models,” IEEE Trans. Audio Speech Lang. Process. 14, 2134–2146 (2006).
[CrossRef]

S. Dharanipragada and K. Visweswariah, “Gaussian mixture models with covariances or precisions in shared multiple subspaces,” IEEE Trans. Audio Speech Lang. Process. 14, 1255–1266 (2006).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

A. Berge and A. H. Schistad Solberg, “Structured Gaussian components for hyperspectral image classification,” IEEE Trans. Geosci. Remote Sens. 44, 3386–3396 (2006).
[CrossRef]

IEEE Trans. Image Process (1)

J. Lee, P. C. Doerschuk, and J. E. Johnson, “Exact reduced-complexity maximum likelihood reconstruction of multiple 3-D objects from unlabeled unoriented 2-D projections and electron microscopy of viruses,” IEEE Trans. Image Process 16, 2865–2878 (2007).
[CrossRef]

IEEE Trans. Inf. Theory (1)

P. C. Doerschuk and J. E. Johnson, “Ab initio reconstruction and experimental design for cryo electron microscopy,” IEEE Trans. Inf. Theory 46, 1714–1729 (2000).
[CrossRef]

IEEE Trans. Signal Process. (1)

H. Snoussi and A. Mohammad-Djafari, “Estimation of structured Gaussian mixtures: the inverse EM algorithm,” IEEE Trans. Signal Process. 55, 3185–3191 (2007).
[CrossRef]

J. Comput. Chem. (1)

E. F. Pettersen, T. D. Goddard, C. C. Huang, G. S. Couch, D. M. Greenblatt, E. C. Meng, and T. E. Ferrin, “UCSF Chimera–A visualization system for exploratory research and analysis,” J. Comput. Chem. 25, 1605–1612 (2004).
[CrossRef]

J. Struct. Biol. (4)

C. J. Prust, P. C. Doerschuk, G. C. Lander, and J. E. Johnson, “Ab initio maximum likelihood reconstruction from cryo electron microscopy images of an infectious virion of the tailed bacteriophage P22 and maximum likelihood versions of Fourier Shell Correlation appropriate for measuring resolution of spherical or cylindrical objects,” J. Struct. Biol. 167, 185–199 (2009).
[CrossRef]

J. Lanman, J. Crum, T. J. Deerinck, G. M. Gaietta, A. Schneemann, G. E. Sosinsky, M. H. Ellisman, and J. E. Johnson, “Visualizing flock house virus infection in Drosophila cells with correlated fluorescence and electron microscopy,” J. Struct. Biol. 161, 439–446 (2008).
[CrossRef]

J. Tang, J. M. Johnson, K. A. Dryden, M. J. Young, A. Zlotnick, and J. E. Johnson, “The role of subunit hinges and molecular “switches’’ in the control of viral capsid polymorphism,” J. Struct. Biol. 154, 59–67 (2006).
[CrossRef]

Z. Yin, Y. Zheng, P. C. Doerschuk, P. Natarajan, and J. E. Johnson, “A statistical approach to computer processing of cryo electron microscope images: Virion classification and 3-D reconstruction,” J. Struct. Biol. 144, 24–50(2003).
[CrossRef]

J. Virol. (1)

M. Tihova, K. A. Dryden, T. L. Le, S. C. Harvey, J. E. Johnson, M. Yeager, and A. Schneemann, “Nodavirus coat protein imposes dodecahedral RNA structure independent of nucleotide sequence and length,” J. Virol. 78, 2897–2905(2004).
[CrossRef]

Microbiol. Molec. Biol. Rev. (1)

T. S. Baker, N. H. Olson, and S. D. Fuller, “Adding the third dimension to virus life cycles: three-dimensional reconstruction of icosahedral viruses from cryo-electron micrographs,” Microbiol. Molec. Biol. Rev. 63, 862–922 (1999).

Nat. Methods (1)

S. H. W. Scheres, H. Gao, M. Valle, G. T. Herman, P. P. B. Eggermont, J. Frank, and J.-M. Carazo, “Disentangling conformational states of macromolecules in 3D-EM through likelihood optimization,” Nat. Methods 4, 27–29(2007).
[CrossRef]

Nat. Struct. Mol. Biol. (1)

D. Bubeck, D. J. Filman, and J. M. Hogle, “Cryo-electron microscopy reconstruction of a poliovirus-receptor-membrane complex,” Nat. Struct. Mol. Biol. 12, 615–618 (2005).
[CrossRef]

Nature (1)

A. J. Fisher and J. E. Johnson, “Ordered duplex RNA controls capsid architecture in an icosahedral animal virus,” Nature 361, 176–179 (1993).
[CrossRef]

Optik (1)

G. Harauz and M. van Heel, “Exact filters for general geometry three dimensional reconstruction,” Optik 73, 146–156(1986).

SIAM J. Math. Anal. (1)

Y. Zheng and P. C. Doerschuk, “Explicit computation of orthonormal symmetrized harmonics with application to the identity representation of the icosahedral group,” SIAM J. Math. Anal. 32, 538–554 (2000).
[CrossRef]

SIAM Rev. (1)

R. A. Redner, and H. F. Walker, “Mixture densities, maximum likelihood and the EM algorithm,” SIAM Rev. 26, 195–239 (1984).
[CrossRef]

Structure (1)

R. H. Cheng, V. S. Reddy, N. H. Olson, A. J. Fisher, T. S. Baker, and J. E. Johnson, “Functional implications of quasi-equivalence in a T=3 icosahedral animal virus established by cryo-electron microscopy and x-ray crystallography,” Structure 2, 271–282 (1994).
[CrossRef]

Ultramicroscopy (1)

M. van Heel, “Similarity measures between images,” Ultramicroscopy 21, 95–100 (1987).
[CrossRef]

Other (9)

URL, “Flock House Virus (FHV) web page,” http://viperdb.scripps.edu/info_page.php?VDB=2q25.

URL http://www.mathworks.com/ .

G. J. McLachlan and T. Krishnan, The EM Algorithm and Extensions (Wiley-Interscience, 1997).

J. A. Bilmes, “A gentle tutorial of the EM algorithm and its application to parameter estimation for Gaussian mixture and hidden Markov models,” Tech. Rep. TR-97-021 (Department of Electrical Engineering and Computer Science, University of California at Berkeley, 1998).

B. D. O. Anderson and J. B. Moore, Optimal Filtering (Prentice-Hall, 1979).

G. Casella and R. L. Berger, Statistical Inference, 2nd ed.(Duxbury2002).

J. Frank, Three-Dimensional Electron Microscopy of Macromolecular Assemblies (Academic, 1996).

Y. Zheng, “Novel statistical models and a high-performance computing toolkit for the solution of cryo electron microscopy inverse problems in viral structural biology,” Ph.D. dissertation (School of Electrical and Computer Engineering, Purdue University, West Lafayette, Ind., USA, 2008).

Q. Wang, is preparing a Ph.D. dissertation called “Maximum likelihood reconstruction of heterogeneous 3-D objects from 2-D projections of unknown orientation and application to electron microscope images of viruses” (School of Electrical and Computer Engineering, Cornell University, Ithaca, N.Y., USA; expected completion date 2013).

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Figures (3)

Fig. 1.
Fig. 1.

Reconstruction of FHV from experimental images. (a) Example boxed experimental images (same color map). (b) Surface plot of the mean ρ¯^η=1(x) pseudocolored by the variance r^η=1(x,x). The variance is highest near the fivefold symmetry axes, which is consistent with the idea that the binding of the particle to a new host cell and possibly later events concerning RNA translocation occur around a fivefold axis [2830]. (c) Cross section of the mean though the center of the particle perpendicular to a fivefold symmetry axis pseudocolored by the variance. (d) RNA core after removal of the protein capsid. Surface plot of the mean pseudocolored by the variance. The ordered dodecahedral RNA cage [24,31] is detected and, as expected, the variance of the cage is low (blue), while that of the surrounding less well-ordered material is high (red). Visualizations in panels (b)–(d) by UCSF Chimera [32].

Fig. 2.
Fig. 2.

Cross sections through the origin normal to fivefold, threefold, and twofold symmetry axes of the estimated 3-D mean (ρ¯^η=1(x)) function. The symmetries can clearly be seen in the cross sections, although the fivefold symmetry is approximately a tenfold symmetry and the threefold symmetry is approximately a sixfold symmetry. The same color map is used in all images.

Fig. 3.
Fig. 3.

FSC curve between ρ¯^η=1(x) computed with V constrained to V=0 versus the optimal diagonal V. The independent variable k is the magnitude of the spatial frequency vector measured in Å. The pixel size of 4.7 Å implies a Nyquist frequency of 1/(2×4.7)=0.106Å1, which determines the upper limit of k=0.1Å1. Since the curve remains above 1/2 for the entire range k[0,0.1]Å1, from the biological point of view, the two mean structures are equivalent.

Equations (61)

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

ρi(x)=τ=1Nc(ηi)ci,τϕτ(ηi)(x),
yi,j=Li,j(θi,ηi)ci+wi,j,
yi=Li(θi,ηi)ci+wi,
μi(θi,ηi,c¯ηi)E[yi|θi,ηi,c¯ηi,Vηi,Qi]
=Li(θi,ηi)c¯ηi,
Σi(θi,ηi,Vηi,Qi)Cov[yi|θi,ηi,c¯ηi,Vηi,Qi]
=Li(θi,ηi)VηiLiT(θi,ηi)+Qi.
p(yi|θi,ηi,c¯ηi,Vηi,Qi)=N(μi(θi,ηi,c¯ηi),Σi(θi,ηi,Vηi,Qi))(yi).
ρi(x)=ρ0(η)(x)+si(η)(x)IS(η)(x),
ci,τS(η)[ρ0(η)(x)+si(η)(x)IS(η)(x)]ϕτ(η)(x)d3x,
c¯τηE[ci,τ|η]
=S(η)ρ0(η)(x)ϕτ(η)(x)d3x,
(Vη)τ,τE[(ci,τE[ci,τ|η])(ci,τE[ci,τ|η])|η]
=νηδτ,τ.
Vη=νηINc(η).
ρ¯η(x)E[ρ(x)|η=η]=τ=1Nc(η)c¯τηϕτ(η)(x),
rη(x,x)E[[ρ(x)ρ¯η(x)][ρ(x)ρ¯η(x)]|η=η]
=τ=1Nc(η)τ=1Nc(η)(Vη)τ,τϕτ(η)(x)ϕτ(η)(x).
lnp(y|c¯,V,q,Q)=i=1Nvln[ηi=1Nηθip(yi|θi,ηi,c¯ηi,Vηi,Qi)qηip(θi)dθi],
Q(c¯,V,q,Q|c¯0,V0,q0,Q0,y)=θη[lnp(y,θ,η|c¯,V,q,Q)]p(θ,η|c¯0,V0,q0,Q0,y)dθ
=i=1Nνθiηi=1Nη[lnp(yi|θi,ηi,c¯ηi,Vηi,Qi)+lnp(θi)+lnqηi]p(θi,ηi|yi,c¯ηi0,Vηi0,q0,Qi0)dθi,
p(θi,ηi|yi,c¯,V,q,Q)=p(yi|θi,ηi,c¯ηi,Vηi,Qi)p(θi)qηiη=1Nηθqηp(θ)p(yi|η,θ,c¯η,Vη,Qi)dθ.
η=1Nηqη=1,
qη0η{1,,Nη},
Q3(q|c¯0,V0,q0,Q0,y)i=1Nvθiηi=1Nη[lnqηi]p(θi,ηi|yi,c¯ηi0,Vηi0,q0,Qi0)dθi,
qη=1Nνi=1Nvθip(θi,η|yi,c¯η0,Vη0,q0,Qi0,)dθiη{1,,Nη},
Vη=VηT,
Vη>0,
Q1(c¯,V,Q|c¯0,V0,q0,Q0,y)i=1Nvθiηi=1Nη[lnp(yi|θi,ηi,c¯ηi,Vηi,Qi)]p(θi,ηi|yi,c¯ηi0,Vηi0,q0,Qi0)dθi
=Ny2ln(2π)Nv12i=1Nvθiηi=1Nηlndet(Σi(θi,ηi,Vηi,Qi))p(θi,ηi|yi,c¯ηi0,Vηi0,q0,Qi0)dθi12i=1Nνθiηi=1Nη(yiμi(θi,ηi,c¯ηi))TΣi1(θi,ηi,Vηi,Qi)(yiμi(θi,ηi,c¯ηi))p(θi,ηi|yi,c¯ηi0,Vηi0,q0,Qi0)dθi
c¯ηQ1(c¯,V,Q|c¯0,V0,q0,Q0,y)=i=1NvθiLiT(θi,η)Σi1(θi,η,Vη,Qi)(Li(θi,η)c¯ηyi)p(θi,η|yi,c¯η0,Vη0,q0,Qi0)dθi.
0=c¯ηQ1(c¯,V,q,Q|c¯0,V0,q0,Q0,y)
[i=1NvθiLiT(θi,η)Σi1(θi,η,Vη,Qi)Li(θi,η)p(θi,η|yi,c¯η0,Vη0,q0,Qi0)dθi]c¯η=[i=1NvθiLiT(θi,η)Σi1(θi,η,Vη,Qi)yip(θi,η|yi,c¯η0,Vη0,q0,Qi0)dθi],
Ni(yi,θi,ηi,c¯ηi)(yiμi(θi,ηi,c¯ηi))(yiμi(θi,ηi,c¯ηi))T,
Q1(c¯,V,q,Q|c¯0,V0,q0,Q0,y)=Ny2ln(2π)Nv+12i=1Nvθiηi=1Nηlndet(Σi1(θi,ηi,Vηi,Qi))p(θi,ηi|yi,c¯ηi0,Vηi0,q0,Qi0)dθi12i=1Nvθiηi=1Nηtr[Σi1(θi,ηi,Vηi,Qi)Ni(yi,θi,ηi,c¯ηi)]p(θi,ηi|yi,c¯ηi0,Vηi0,q0,Qi0)dθi.
Q1(c¯,V,q,Q|c¯0V0q0Q0,y)Vη=12i=1Nvθilndet(Σi1(θi,η,Vη,Qi))Vηp(θi,η|yi,c¯η0Vη0q0Qi0)dθi12i=1Nvθitr[Σi1(θi,η,Vη,Qi)Ni(yi,θi,η,c¯η)]Vηp(θi,η|yi,,c¯η0Vη0q0Qi0)dθi.
lndet(R(Σ(V)))V=[LT(Σ1+ΣT)Ldiag(LTΣ1L)],
tr(R(Σ(V))N)V=-[LT(Σ1NΣ1+ΣTNΣT)Ldiag(LTΣ1NΣ1L)],
Q1(c¯,V,q,Q|c¯0,V0,q0,Q0,y)Vη=2S(y,η,c¯η,Vη,c¯η0,Vη0,q0)diag(S(y,η,c¯η,Vη,c¯η0,Vη0,q0)),
S(y,η,c¯η,Vη,c¯η0,Vη0,q0)12i=1NvθiMi(yi,θi,η,c¯η,Vη)p(θi,η|yi,c¯η0,Vη0,q0,Qi0)dθi,
Mi(yi,θi,η,c¯η,Vη)LiT(θi,η)Σi1(θi,η,Vη,Qi)Ni(yi,θi,η,c¯η)Σi1(θi,η,Vη,Qi)Li(θi,η)LiT(θi,η)Σi1(θi,η,Vη,Qi)Li(θi,η),
0=S(y,η,c¯η,Vη,c¯η0,Vη0,q0),
vηQ1(c¯,V,q,Q|c¯0,V0,q0,Q0,y)=12i=1Nvθidiag(Mi(yi,θi,η,c¯η,Vη))p(θi,η|yi,c¯0,V0,q0,Q0)dθi,
2Q1(c¯,V,q,Q|c¯0,V0,q0,Q0,y)vη2=12i=1Nvθi(2DiE+EE)dθi,
D=LiT(θi,η)Σi1(θi,η,Vη)Ni(yi,θi,η,c¯η)Σi1(θi,η,Vη)Li(θi,η),
E=LiT(θi,η)Σi1(θi,η,Vη)Li(θi,η).
Σi1(θi,η,Vη,Qi)=Qi1Qi1Li(θi,η)(LiT(θi,η)Qi1Li(θi,η)+Vη1)1LiT(θ,η)Qi1.
Di(θi,η)LiT(θi,η)Qi1Li(θi,η),
Δη(y,c¯η0,Vη0,q0)i=1NvθiDi(θi,η)p(θi,η|yi,c¯η0,Vη0,q0,Qi0)dθi,
bi(θi,η,yi)LiT(θi,η)Qi1yi,
βη(y,c¯η0,Vη0,q0)i=1Nνθibi(θi,η,yi)p(θi,η|yi,c¯η0,Vη0,q0,Qi0)dθi
Ti(θi,η)(Di(θi,η)+Vη1)1.
[Δη(y,c¯η0,Vη0,q0)i=1NνθiDi(θi,η)(Di(θi,η)+Vη1)1Di(θi,η)p(θi,η|yi,c¯η0,Vη0,q0,Qi0)dθi]c¯η=βη(y,0c¯η,0Vη,0q)i=1NνθiDi(θi,η)(Di(θi,η)+Vη1)1bi(θi,η,yi)p(θi,η|yi,c¯η0,Vη0,q0,Qi0)dθi.
M=[(IDT)(bDc)][(IDT)(bDc)]T(IDT)D.
ρ¯η(Rβ1x)=ρ¯η(x),
rη(Rβ1x,Rβ1x)=rη(x,x),
p(θi,Hi|yi,c¯,V,q,Q)=p(yi|θi,ηi,c¯ηi,Vηi,Qi)p(θi|Hi)qHiH=1NHθp(θ|H)qHp(yi|η,θ,c¯η,Vη)dθ,
p(θi,ηi|yi,c¯,V,q,Q)={Hi:η(Hi)=ηi}p(θi,Hi|yi,c¯,V,q,Q).
qH=1Nvi=1Nvθip(θi,H|yi,c¯η0Vη0q0)dθiH{1,,NH},
p(θi,Hi=(ηi,ξi)|yi,c¯,V,q,Q)=p(yi|θξi,ηi,c¯ηi,Vηi,Qi)δθi,θξiqHi=(ηi,ξi)η=1Nηξ=1NξqH=(η,ξ)p(yi|η,θξ,c¯η,Vη),
p(θi,ηi|yi,c¯,V,q,Q)=p(yi|θi,ηi,c¯ηi,Vηi,Qi)qHi=(ηi,ξ(θi))η=1Nηξ=1NξqH=(η,ξ)p(yi|η,θξ,c¯η,Vη).

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