## Probability density function estimation of laser light scintillation via Bayesian mixtures |

JOSA A, Vol. 31, Issue 3, pp. 580-590 (2014)

http://dx.doi.org/10.1364/JOSAA.31.000580

Enhanced HTML Acrobat PDF (508 KB)

### Abstract

A method for probability density function (PDF) estimation using Bayesian mixtures of weighted gamma distributions, called the Dirichlet process gamma mixture model (DP-GaMM), is presented and applied to the analysis of a laser beam in turbulence. The problem is cast in a Bayesian setting, with the mixture model itself treated as random process. A stick-breaking interpretation of the Dirichlet process is employed as the prior distribution over the random mixture model. The number and underlying parameters of the gamma distribution mixture components as well as the associated mixture weights are learned directly from the data during model inference. A hybrid Metropolis–Hastings and Gibbs sampling parameter inference algorithm is developed and presented in its entirety. Results on several sets of controlled data are shown, and comparisons of PDF estimation fidelity are conducted with favorable results.

© 2014 Optical Society of America

**OCIS Codes**

(000.5490) General : Probability theory, stochastic processes, and statistics

(010.1300) Atmospheric and oceanic optics : Atmospheric propagation

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(010.7060) Atmospheric and oceanic optics : Turbulence

(290.5930) Scattering : Scintillation

(150.1135) Machine vision : Algorithms

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: August 29, 2013

Manuscript Accepted: November 18, 2013

Published: February 14, 2014

**Citation**

Eric X. Wang, Svetlana Avramov-Zamurovic, Richard J. Watkins, Charles Nelson, and Reza Malek-Madani, "Probability density function estimation of laser light scintillation via Bayesian mixtures," J. Opt. Soc. Am. A **31**, 580-590 (2014)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-31-3-580

Sort: Year | Journal | Reset

### References

- E. Jakeman and P. Pusey, “Significance of the k-distribution in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978). [CrossRef]
- V. Gudimetla and J. Holmes, “Probability density function of the intensity for a laser-generated speckle field after propagation through the turbulent atmosphere,” J. Opt. Soc. Am. 72, 1213–1218 (1982). [CrossRef]
- R. Phillips and L. Andrews, “Universal statistical model for irradiance fluctuations in a turbulent medium,” J. Opt. Soc. Am. 72, 864–870 (1982). [CrossRef]
- L. Bissonette and P. Wizniowich, “Probability distribution of turbulent irradiance in a saturation regime,” Appl. Opt. 18, 1590–1599 (1979). [CrossRef]
- W. Strohbein, T. Wang, and J. Speck, “On the probability distribution of line-of-sight fluctuations for optical signals,” Radio Sci. 10, 59–70 (1975).
- J. Churnside and R. Hill, “Probability density of irradiance scintillations for strong path-integrated refractive turbulence,” J. Opt. Soc. Am. 4, 727–733 (1987). [CrossRef]
- D. Mudge, A. Wedd, J. Craig, and J. Thomas, “Statistical measurements of irradiance fluctuations produced by a reflective membrane optical scintillator,” J. Opt. Laser Technol. 28, 381–387 (1996). [CrossRef]
- P. Orbanz and Y. Teh, “Bayesian nonparametric models,” in Encyclopedia of Machine Learning (Springer, 2010), pp. 81–89.
- J. McLaren, J. Thomas, J. Mackintosh, K. Mudge, K. Grant, B. Clare, and W. Cowley, “Comparison of probability density functions for analyzing irradiance statistics due to atmospheric turbulence,” J. Appl. Opt. 51, 5996–6002 (2012). [CrossRef]
- Y. Jiang, J. Ma, L. Tan, S. Yu, and W. Du, “Measurement of optical intensity fluctuation over an 11.8 km turbulent path,” Opt. Express 16, 6963–6973 (2008). [CrossRef]
- M. Al-Habash, L. Andrews, and R. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” J. Opt. Eng. 40, 1554–1562 (2001). [CrossRef]
- R. Barakat, “Second-order statistics of integrated intensities and detected photons, the exact analysis,” J. Mod. Opt. 43, 1237–1252 (1996). [CrossRef]
- R. Barakat, “First-order intensity and log-intensity probability density functions of light scattered by the turbulent atmosphere in terms of lower-order moments,” J. Opt. Soc. Am. A 16, 2269–2274 (1999). [CrossRef]
- O. Korotkova, S. Avramov-Zamurovic, R. Malek-Madani, and C. Nelson, “Probability density function of the intensity of a laser beam propagating the maritime environment,” Opt. Express 19, 20322–20331 (2011). [CrossRef]
- M. Welling, “Robust series expansions for probability density estimation,” Technical Note (Department of Electrical and Computer Engineering, California Institute of Technology, 2001).
- B. Silverman, “Survey of existing methods,” in Density Estimates for Statistics and Data Analysis, Monographs on Statistics and Applied Probability (Chapman & Hall, 1986), pp. 1–22.
- C. Nelson, S. Avramov-Zamurovic, R. Malek-Madani, O. Korotkova, R. Sova, and F. Davidson, “Measurements and comparison of the probability density and covariance functions of laser beam intensity fluctuations in a hot-air turbulence emulator with the maritime atmospheric environment,” Proc. SPIE 8517, 851707 (2012).
- M. Escobar and M. West, “Bayesian density estimation and inference using mixtures,” J. Am. Stat. Assoc. 90, 577–588 (1995). [CrossRef]
- M. Rosenblatt, “Remarks on some nonparametric estimates of density function,” Ann. Math. Sci. 27, 832–837 (1956). [CrossRef]
- E. Parzen, “On estimation of a probability density function and mode,” Ann. Math. Sci. 33, 1065–1076 (1962). [CrossRef]
- T. Ferguson, “Bayesian density estimation by mixtures of normal distributions,” Recent Advances in Statistics (Academic, 1983), pp. 287–302.
- P. Orbanz and J. Buhmann, “Smooth image segmentation by nonparametric Bayesian inference,” in European Conference on Computer Vision (Springer, 2006), Vol. 1, pp. 444–457.
- Y. Teh, M. Jordan, M. Beal, and M. Jordan, “Hierarchical Dirichlet processes,” J. Am. Stat. Assoc. 101, 1566–1581 (2005). [CrossRef]
- E. Wang, D. Liu, J. Silva, and L. Carin, “Joint analysis of time-evolving binary matrices and associated documents,” in Advances in Neural Information Processing Systems (Curran Associates, 2010), pp. 2370–2378.
- J. Sethuraman, “A constructive definition of Dirichlet priors,” Statistica Sinica 4, 639–650 (1994).
- A. Webb, “Gamma mixture models for target recognition,” Pattern Recogn. 33, 2045–2054 (2000). [CrossRef]
- K. Corsey and A. Webb, “Bayesian gamma mixture model approach to radar target recognition,” IEEE Trans. Aerosp. Electron. Syst. 39, 1201–1217 (2003).
- M. Wiper, D. Insua, and F. Ruggeri, “Mixtures of gamma distributions with applications,” J. Comput. Graph. Stat. 10, 440–454 (2001). [CrossRef]
- R. Hill and J. Churnside, “Observational challenges of strong scintillations of irradiance,” J. Opt. Soc. Am. A 5, 445–447 (1988). [CrossRef]
- D. Blei, A. Ng, and M. Jordan, “Latent Dirichlet allocation,” J. Mach. Learn. Res. 3, 993–1022 (2003).
- J. Neyman and E. Pearson, “On the problem of the most efficient tests of statistical hypothesis,” Philos. Trans. R. Soc. London 231, 289–337 (1933). [CrossRef]
- M. J. Beal, “Variational algorithms for approximate Bayesian inference,” Ph.D. Thesis (University College London, 2003).
- A. Gelfand and A. Smith, “Sampling-based approaches to calculating marginal densities,” J. Am. Stat. Assoc. 85, 398–409 (1990). [CrossRef]
- A. Dempster, N. Laird, and D. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. 39, 1–38 (1977).
- M. Aitkin and D. Rubin, “Estimation and hypothesis testing in finite mixture models,” J. R. Stat. Soc. 47, 67–75 (1985).
- J. Marin, K. Mengersen, and C. Roberts, Handbook of Statistics: Bayesian Thinking - Modeling and Computation (Elsevier, 2011), Chap. 25.
- H. Ishwaran and M. Zarepour, “Exact and approximate sum representations for the Dirichlet process,” Can. J. Stat. 30, 269–283 (2002). [CrossRef]
- C. Andrieu, N. de Freitas, A. Doucet, and M. Jordan, “An introduction to MCMC for machine learning,” Mach. Learn. 50, 5–43 (2003). [CrossRef]
- N. Metropolis, A. Rosenbluth, M. Rosenbluth, and A. Teller, “Equations of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953). [CrossRef]
- W. Hastings, “Monte Carlo sampling methods using Markov chains and their applications,” Biometrika 57, 97–109 (1970). [CrossRef]
- Q. An, C. Wang, I. Shterev, E. Wang, L. Carin, and D. Dunson, “Hierarchical kernel stick-breaking process for multi-task image analysis,” in International Conference on Machine Learning (ICML) (Omnipress, 2008), pp. 17–24.
- D. Dunson and N. Pillai, “Bayesian density regression,” J. R. Stat. Soc. 69, 163–183 (2007). [CrossRef]
- I. Pruteanu-Malcini, L. Ren, J. Paisley, E. Wang, and L. Carin, “Hierarchical Bayesian modeling of topics in time-stamped documents,” IEEE Trans. Pattern Anal. Mach. Intell. 32, 996–1011 (2010). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.