Abstract
The three technologies that are surveyed here are (1) wavelet approximations, (2) hidden Markov models, and (3) the Markov chain Renaissance. The intention of the article is to provide an introduction to the benefits these technologies offer and to explain as far as possible the sources of their effectiveness. We also hope to suggest some useful relationships between these technologies and issues of importance on the agenda of biological and medical research.
Research partially supported by NSF DMS92-11634
Research partially supported by ARO Grant DAAL03-91-G-0110 and NSF DMS92-11634
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Chui, C.K. (1992a), An Introduction to Wavelets. Academic Press, New York.
Chui, C.K. (1992b), ed., Wavelets: A Tutorial in Theory and Applications. Academic Press, New York.
Coifman, R.R. and Meyer, Y. (1991), “Remarques sur l’analyse de Fourier â fenêtre,” C.R. Acad. Sci. Paris Sér. I, 259–261.
Daubechies, I. (1988), “Orthonormal bases of compactly supported wavelets,” Comm. Pure Appl. Math., 41, 909–996.
Daubechies, I., (1992) Ten Lectures on Wavelets, SIAM Publications, Philadelphia PA.
Duffin, R.J. and Schaeffer, A.C. (1952), “A class of nonharmonic Fourier series,” Trans. Amer. Math. Soc., 72, 341–366.
Gröchenig, G.K. (1991), “Describing functions: atomic decompositions versus frames,” Monatsh. Math., 112, 1–42.
Gröchenig, K. and Madych, W.R. (1992), “Multiresolution analysis, Haar bases and self-similar filings of V,” IEEE Trans. Inform. Theory, 38, 556–568.
Kovaevie, J. and Vetterli, M. (1991), “Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for Nn, ” IEEE Trans. Inform. Theory, 38, 533–535.
Mallat, S. (1989), “Multiresolution approximation and wavelets,” Trans. Amer. Math. Soc., 315, 69–88.
Meyer, Y. (1990), Ondelettes ét opéateurs, I: Ondelettes, II: Opéatcurs dc CalderónZygrnund, III: Opéatcurs multilinéaires. Hermann, Paris.
Meyer, Y. (1993), Wavelets: Algorithms and Applications. SIAM Publications, Philadelphia, PA.
Strang, G. (1993), “ Wavelet Transforms versus Fourier Transforms”, Bulletin of the American Mathematical Society 28,2, 288–305.
Ruskai, M.B., 13eylkin, G., Coffman, R.R., Daubechies, 1., Mallat, S. Meyer, Y. and Raphael, L. (1992), eds., Wavelets and their Applications. Jones and Bartlett, Boston.
HMM
Aggoun, L. and Elliot, R.J. (1992), “Finite dimensional predictors for bidders Markov chains,” System Control Lett., 19, 335–340.
Baum, L.E. and Eagon, J.A. (1967). An inequality with applications to statistical estimation for probabilistic functions of Markov processes and to a model of Ecology,” Bull. Arner. Math. Soc., 73, 360–363.
Baum, L.E. and Petrie, T. (1966), “Statistical inference for probabilistic functions of finite state Markov chains,” Ann. Math. Statist., 37, 1554–1563.
Baum, L.E., Petrie, ’l’., Smiles, G. and Weiss, N. (1970), “A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains,” Ann. Math. Statist., 41. I61–171.
Bickel, P.J. and Ritov, Y. (1993), “Inference in hidden Markov models 1 Local asymptotic normality in the stationary case,” Technical Report, Department of Statistics, University of California, Berkeley.
Blackwell, D. and Koopmans, L. (1957), “On the identifiability problem for functions of finite Markov chains,” Ann. Math. Statist., 28, 1011–1015.
Chung, S.H., Moore, J.B., Xia, L., Premkumar, L.S. and Gages, P.W. (1990), “Characterization of single channel currents using digital signal processing techniques based on hidden Markov models,” Phil. Trans. Roy. Soc. Land. Ser. B, 329, 265–285.
Churchill, G.A. (1989), “Stochastic models for heterogeneous DNA sequences,” Bull. Math. Biol., 51, 79–94.
Coast, l).A., Cano, G.G. and Briller, S.A. (1991), “Use of hidden Markov models for Electrocardiographic signal analysis,” J. of Electrocardiology, 23, 184–191.
Forney, G.D. (1973), “The Viterbi algorithm,” Proc. IEEE, 61, 268–278.
Fredkin, D.R. and Rice, J.A. (1992), “Bayesian restoration of single channel patch clamp recordings,” Biometrics,48, 427–448.
Fridman, M. (1993), “Hidden Markov model regression,” Unpublished Thesis, University of Pennsylvania.
Gilbert, E.J. (1959), “On the identifiability problem for functions of finite Markov chains,” Ann. Math. Statist.,30, 688–697.
Guttorp, P., Newton, M.A. and Abkowitz, J.L. (1990), “A stochastic model for haematopoiesi in cats,” IMA J. Math. Appl. Med. Biol.,7, 125–143.
Juang, B.H. (1985), “Maximum Likelihood estimation for mixture multivariate stochastic observations of Markov chains,” AT T Tech. J.,64, 1235–1249.
Juang, B.H., Levinson, S.E. and Sondhi, M.M. (1986), “Maximum likelihood estimation for multivariate mixture observations of Markov chains,” IEEE Trans. Inform. Theory, IT-32, 307–309.
Juang, B.H. and Rabiner, L.R. (1990), “The Segmental K-Means algorithm for estimating parameters of hidden Markov models,” IEEE Trans. Acoust., Speech, Signal Process, ASSP-38, 1639–1641.
Juang, B.H. and Rabiner, L.R. (1991), “Hidden Markov models for speech recognition,” Technometrics, 33, 251–272.
Khasminskii, R.Z., Lazareva, B.V. and Stapleton, J. (1993), “Some procedures for state estimation of a hidden Markov chain with two states,” Technical Report, Department of Statistics and Probability, Michigan State University.
Kogan, J.A. (1991), “Optional reconstruction of Markov sequences through indirect observations,” 6th USSR-Japan Symp. on Probab. and Math. Statist., Kiev.
Kullback, S. and Leibler, R.A. (1951), “On information and sufficiency,” Ann. Math. Statist., 22, 79–86.
Leroux, B.G. (1992), “Maximum-likelihood estimation for hidden Markov models,” Stochastic Process. Appl., 40, 127–143.
Levinson, S.E. (1986), “Continuously variable duration hidden Markov models for automatic speech recognition,” Computer, Speech and Language, 1, 29–45.
Liporace, L.A. (1982), “Maximum Likelihood estimation for multivariate observations of Markov sources,” IEEE Trans. Inform. Theory,IT-28, 729–734.
Paul, D.B. (1985), “Training of HMM recognizers by simulated annealing,” Proc. ICASSP, New York: IEEE, 13–16.
Petrie, T. (1969), “Probabilistic functions of finite state Markov chains,” Ann. Math. Statist.,40, 97–115.
Quandt, R.E. (1972), “A new approach to estimating switching regressions,” J. Amer. Statist. Assoc., 67, 306–310.
Quandt, R.E. and Ramsey, J.B. (1978), “Estimating mixtures of Normal distributions and switching regressions,” J. Amer. Statist. Assoc., 73, 730–738.
Rabiner, L.R. (1989), “A tutorial on hidden Markov models and selected applications in speech recognition,” Proc. IEEE, 77, 257–285.
Viterbi, A.D. (1967), “Error bounds for convolutional codes and an asymptotically optimal decoding algorithm,” IEEE Trans. Inform.. Theory, IT-13, 260–269.
Markov Renaissance
Albert, 3.H. and Chib, S. (1993), “Bayes inference via Gibbs sampling of autoregressive time series subject to Markov mean and variance shifts,” J. Bus. h’ Econ. Statist., 11, 1–16.
Besag, J. (1974), “Spatial interaction and the statistical analysis of lattice systems (with discussion),” J. Roy. Statist. Soc. Ser. 13, 36, 192–236.
Besag, J. and Green, P.J. (1993), “Spatial statistics and Bayesian computations,” J. Roy. Statist. Soc. Ser. B, 55, 25–37.
Cerny, V. (1985), “A thermodynamic approach to the traveling salesman problem: An efficient simulation”, J. Optirn.. Theory Appl.. 45, 41–51.
Gelfand, A.E. and Smith, A.F.M. (1990), “Sampling-based approaches to calculating marginal densities,” J. Amer. Statist. Assoc., 85, 398–409.
Geman, S. and Geman, D. (1984), “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattn Anal. Mae’i. Intcll., 6, 72 1741
Geyer, C.J. (1991), “Markov chain Monte Carlo maximum likelihood,” Computer Science and Statistics: Proc. 2.3rd Swap. Interface, Fairfax Station: Interface Foundation, 156–163.
Gilks, W.R., Clayton, D.G., Spiegelhalter, D.J., Best, N.G., McNeil, A.J., Sharpies, L.D. and Kirby, A.J. (1993), “Modelling complexity: Applications of Gibbs sampling in medicine,” J. Roy. Statist. Soc. Ser. B, 55, 39–52.
Johnson, D.S., Aragon, C., McGeoch, L. and Schevon, C. (1990), “Optimization by simulated annealing: An experimental evaluation, Part I: Graph partitioning,” Oper. Res., 37, 865–892.
Johnson, D.S., Aragon, C., McGeoch, L. and Schevon, C. (1991), “Optimization by simulated annealing: An experimental evaluation, Part II: Graph coloring and number partitioning,” Oper. Res., 39, 378–406.
Johnson, D.S., Aragon, C., McGeoch, L. and Schevon, C. (1992), “Optimization by simulated annealing: An experimental evaluation, Part III: The traveling salesman problem,” in preparation.
IIajek, B. (1988), “Cooling schedules for optimal annealing,” Math. Oper. Res., 13, 311–329.
Hastings, W.K. (1970), “Monte Carlo sampling methods using Markov chains and their applications,” Biometrika, 57, 97–109.
Kirkpatrick, S., Gelett, C.D. and Vecchi, M.P (1983), “Optimization by simulated annealing,” Science, 220, 621–630.
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H. and Teller, E. (1953), “Equation of state calculation by fast computing machines,” J. Chem. Phys., 21, 1087–1092.
Smith, A.F.M. and Roberts, G.O. (1993), “Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion),” J. Roy. Statist. Soc. Ser. B, 55, 3–23.
Tanner, M.A and Wong, W. (1987), “The calculation of posterior distributions by data augmentation (with discussion),” J. Amer. Statist. Assoc., 82, 528–550.
Tierney, L. (1991), “Exploring posterior distributions using Markov chains,” Computer Science and Statistics: Proc. 23rd Symp. Interface, Fairfax Station: Interface Foundation, 563–570.
Zeger, S. and Rizaul Karim, M. (1991), Generalized linear models with random effects: A Gibbs sampling approach,” J. Amer. Statist. Assoc., 86, 79–86.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer Science+Business Media New York
About this chapter
Cite this chapter
Fridman, M., Steele, J.M. (1994). Three Statistical Technologies with High Potential in Biological Imaging and Modeling. In: Varma, M.N., Chatterjee, A. (eds) Computational Approaches in Molecular Radiation Biology. Basic Life Sciences, vol 63. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-9788-6_15
Download citation
DOI: https://doi.org/10.1007/978-1-4757-9788-6_15
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4757-9790-9
Online ISBN: 978-1-4757-9788-6
eBook Packages: Springer Book Archive