Abstract

The estimation of an inhomogeneous Poisson process (IHPP) rate function from a set of process observations is an important problem arising in optical communications and a variety of other applications. However, because of practical limitations of detector technology, one is often only able to observe a corrupted version of the original process. In this paper, we consider how inference of the rate function is affected by dead time, a period of time after the detection of an event during which a sensor is insensitive to subsequent IHPP events. We propose a flexible nonparametric Bayesian approach to infer an IHPP rate function given dead-time limited process realizations. Simulation results illustrate the effectiveness of our inference approach and suggest its ability to extend the utility of existing sensor technology by permitting more accurate inference on signals whose observations are dead-time limited. We apply our inference algorithm to experimentally collected optical communications data, demonstrating the practical utility of our approach in the context of channel modeling and validation.

© 2017 Optical Society of America

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References

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  1. R. J. Drost, B. M. Sadler, and G. Chen, “Dead time effects in non-line-of-sight ultraviolet communications,” Opt. Express 23, 15748–15761 (2015).
    [Crossref]
  2. S. Lee, J. R. Wilson, and M. M. Crawford, “Modeling and simulation of a nonhomogeneous Poisson process having cyclic behavior,” Commun. Stat. 20, 777–809 (1991).
    [Crossref]
  3. H. P. Galliher and R. C. Wheeler, “Nonstationary queuing probabilities for landing congestion of aircraft,” Oper. Res. 6, 264–275 (1958).
    [Crossref]
  4. P. A. Lewis and G. S. Shedler, “Statistical analysis of non-stationary series of events in a database system,” IBM J. Res. Dev. 20, 465–482 (1976).
    [Crossref]
  5. P. L. Saldanha, E. A. de Simone, and P. Frutuoso e Melo, “An application of non-homogeneous Poisson point processes to the reliability analysis of service water pumps,” Nucl. Eng. Des. 210, 125–133 (2001).
    [Crossref]
  6. C. Bendjaballah, Introduction to Photon Communication (Springer, 1995), Vol. 29.
  7. U. Hassan and R. Bashir, “Electrical cell counting process characterization in a microfluidic impedance cytometer,” Biomed. Microdevices 16, 697–704 (2014).
    [Crossref]
  8. D. L. Snyder, “Utilizing side information in emission tomography,” IEEE Trans. Nucl. Sci. 31, 533–537 (1984).
    [Crossref]
  9. M. L. Larsen and A. B. Kostinski, “Simple dead-time corrections for discrete time series of non-Poisson data,” Meas. Sci. Technol. 20, 095101 (2009).
    [Crossref]
  10. J. Brenguier and L. Amodei, “Coincidence and dead-time corrections for particle counters. Part I: a general mathematical formalism,” J. Atmos. Ocean. Technol. 6, 575–584 (1989).
    [Crossref]
  11. F. Y. Daniel and J. A. Fessler, “Mean and variance of single photon counting with deadtime,” Phys. Med. Biol. 45, 2043–2056 (2000).
    [Crossref]
  12. J. W. Müller, “Dead-time problems,” Nucl. Instrum. Methods 112, 47–57 (1973).
    [Crossref]
  13. S. Harrod and W. D. Kelton, “Numerical methods for realizing nonstationary Poisson processes with piecewise-constant instantaneous-rate functions,” Simulation 82, 147–157 (2006).
    [Crossref]
  14. L. M. Leemis, “Nonparametric estimation of the cumulative intensity function for a nonhomogeneous Poisson process,” Manag. Sci. 37, 886–900 (1991).
    [Crossref]
  15. G. Vannucci and M. C. Teich, “Dead-time-modified photocount mean and variance for chaotic radiation,” J. Opt. Soc. Am. 71, 164–170 (1981).
    [Crossref]
  16. J. Heikkinen and E. Arjas, “Non-parametric Bayesian estimation of a spatial Poisson intensity,” Scand. J. Stat. 25, 435–450 (1998).
    [Crossref]
  17. K. J. Ryan, “Some flexible families of intensities for non-homogeneous Poisson process models and their Bayes inference,” Qual. Reliab. Eng. Int. 19, 171–181 (2003).
    [Crossref]
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  23. S. Chib and E. Greenberg, “Understanding the Metropolis–Hastings algorithm,” Am. Statist. 49, 327–335 (1995).
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  28. L. Liao, R. J. Drost, Z. Li, T. Lang, B. M. Sadler, and G. Chen, “Long-distance non-line-of-sight ultraviolet communication channel analysis: experimentation and modelling,” IET Optoelectron. 9, 223–231 (2015).
    [Crossref]
  29. R. J. Drost, T. J. Moore, and B. M. Sadler, “UV communications channel modeling incorporating multiple scattering interactions,” J. Opt. Soc. Am. A 28, 686–695 (2011).
    [Crossref]
  30. A. M. Siegel, G. A. Shaw, and J. Model, “Short-range communication with ultraviolet LEDs,” Proc. SPIE 5530, 182–193 (2004).
    [Crossref]

2015 (2)

L. Liao, R. J. Drost, Z. Li, T. Lang, B. M. Sadler, and G. Chen, “Long-distance non-line-of-sight ultraviolet communication channel analysis: experimentation and modelling,” IET Optoelectron. 9, 223–231 (2015).
[Crossref]

R. J. Drost, B. M. Sadler, and G. Chen, “Dead time effects in non-line-of-sight ultraviolet communications,” Opt. Express 23, 15748–15761 (2015).
[Crossref]

2014 (1)

U. Hassan and R. Bashir, “Electrical cell counting process characterization in a microfluidic impedance cytometer,” Biomed. Microdevices 16, 697–704 (2014).
[Crossref]

2011 (1)

2009 (1)

M. L. Larsen and A. B. Kostinski, “Simple dead-time corrections for discrete time series of non-Poisson data,” Meas. Sci. Technol. 20, 095101 (2009).
[Crossref]

2007 (1)

J. Weinberg, L. D. Brown, and J. R. Stroud, “Bayesian forecasting of an inhomogeneous Poisson process with applications to call center data,” J. Am. Stat. Assoc. 102, 1185–1198 (2007).

2006 (1)

S. Harrod and W. D. Kelton, “Numerical methods for realizing nonstationary Poisson processes with piecewise-constant instantaneous-rate functions,” Simulation 82, 147–157 (2006).
[Crossref]

2004 (1)

A. M. Siegel, G. A. Shaw, and J. Model, “Short-range communication with ultraviolet LEDs,” Proc. SPIE 5530, 182–193 (2004).
[Crossref]

2003 (1)

K. J. Ryan, “Some flexible families of intensities for non-homogeneous Poisson process models and their Bayes inference,” Qual. Reliab. Eng. Int. 19, 171–181 (2003).
[Crossref]

2001 (1)

P. L. Saldanha, E. A. de Simone, and P. Frutuoso e Melo, “An application of non-homogeneous Poisson point processes to the reliability analysis of service water pumps,” Nucl. Eng. Des. 210, 125–133 (2001).
[Crossref]

2000 (1)

F. Y. Daniel and J. A. Fessler, “Mean and variance of single photon counting with deadtime,” Phys. Med. Biol. 45, 2043–2056 (2000).
[Crossref]

1998 (1)

J. Heikkinen and E. Arjas, “Non-parametric Bayesian estimation of a spatial Poisson intensity,” Scand. J. Stat. 25, 435–450 (1998).
[Crossref]

1995 (1)

S. Chib and E. Greenberg, “Understanding the Metropolis–Hastings algorithm,” Am. Statist. 49, 327–335 (1995).

1992 (2)

A. Gelman and D. B. Rubin, “Inference from iterative simulation using multiple sequences,” Stat. Sci. 7, 457–472 (1992).
[Crossref]

G. Casella and E. I. George, “Explaining the Gibbs sampler,” Am. Statist. 46, 167–174 (1992).

1991 (2)

L. M. Leemis, “Nonparametric estimation of the cumulative intensity function for a nonhomogeneous Poisson process,” Manag. Sci. 37, 886–900 (1991).
[Crossref]

S. Lee, J. R. Wilson, and M. M. Crawford, “Modeling and simulation of a nonhomogeneous Poisson process having cyclic behavior,” Commun. Stat. 20, 777–809 (1991).
[Crossref]

1989 (1)

J. Brenguier and L. Amodei, “Coincidence and dead-time corrections for particle counters. Part I: a general mathematical formalism,” J. Atmos. Ocean. Technol. 6, 575–584 (1989).
[Crossref]

1987 (1)

S. Duane, A. D. Kennedy, B. J. Pendleton, and D. Roweth, “Hybrid Monte Carlo,” Phys. Lett. B 195, 216–222 (1987).
[Crossref]

1984 (1)

D. L. Snyder, “Utilizing side information in emission tomography,” IEEE Trans. Nucl. Sci. 31, 533–537 (1984).
[Crossref]

1981 (1)

1976 (1)

P. A. Lewis and G. S. Shedler, “Statistical analysis of non-stationary series of events in a database system,” IBM J. Res. Dev. 20, 465–482 (1976).
[Crossref]

1973 (1)

J. W. Müller, “Dead-time problems,” Nucl. Instrum. Methods 112, 47–57 (1973).
[Crossref]

1958 (1)

H. P. Galliher and R. C. Wheeler, “Nonstationary queuing probabilities for landing congestion of aircraft,” Oper. Res. 6, 264–275 (1958).
[Crossref]

Adams, R. P.

R. P. Adams, I. Murray, and D. J. MacKay, “Tractable nonparametric Bayesian inference in Poisson processes with Gaussian process intensities,” in Proceedings of the 26th Annual International Conference on Machine Learning, 2009, pp. 9–16.

Amodei, L.

J. Brenguier and L. Amodei, “Coincidence and dead-time corrections for particle counters. Part I: a general mathematical formalism,” J. Atmos. Ocean. Technol. 6, 575–584 (1989).
[Crossref]

Arjas, E.

J. Heikkinen and E. Arjas, “Non-parametric Bayesian estimation of a spatial Poisson intensity,” Scand. J. Stat. 25, 435–450 (1998).
[Crossref]

Bashir, R.

U. Hassan and R. Bashir, “Electrical cell counting process characterization in a microfluidic impedance cytometer,” Biomed. Microdevices 16, 697–704 (2014).
[Crossref]

Bendjaballah, C.

C. Bendjaballah, Introduction to Photon Communication (Springer, 1995), Vol. 29.

Brenguier, J.

J. Brenguier and L. Amodei, “Coincidence and dead-time corrections for particle counters. Part I: a general mathematical formalism,” J. Atmos. Ocean. Technol. 6, 575–584 (1989).
[Crossref]

Brown, L. D.

J. Weinberg, L. D. Brown, and J. R. Stroud, “Bayesian forecasting of an inhomogeneous Poisson process with applications to call center data,” J. Am. Stat. Assoc. 102, 1185–1198 (2007).

Casella, G.

G. Casella and E. I. George, “Explaining the Gibbs sampler,” Am. Statist. 46, 167–174 (1992).

C. Robert and G. Casella, Monte Carlo Statistical Methods (Springer, 2013).

Chen, G.

L. Liao, R. J. Drost, Z. Li, T. Lang, B. M. Sadler, and G. Chen, “Long-distance non-line-of-sight ultraviolet communication channel analysis: experimentation and modelling,” IET Optoelectron. 9, 223–231 (2015).
[Crossref]

R. J. Drost, B. M. Sadler, and G. Chen, “Dead time effects in non-line-of-sight ultraviolet communications,” Opt. Express 23, 15748–15761 (2015).
[Crossref]

G. Chen, L. Liao, Z. Li, R. J. Drost, and B. M. Sadler, “Experimental and simulated evaluation of long distance NLOS UV communication,” in Proceedings of the 19th International Symposium Communication Systems, Networks & Digital Signal Processing (2014), pp. 904–909.

Chib, S.

S. Chib and E. Greenberg, “Understanding the Metropolis–Hastings algorithm,” Am. Statist. 49, 327–335 (1995).

Crawford, M. M.

S. Lee, J. R. Wilson, and M. M. Crawford, “Modeling and simulation of a nonhomogeneous Poisson process having cyclic behavior,” Commun. Stat. 20, 777–809 (1991).
[Crossref]

Daniel, F. Y.

F. Y. Daniel and J. A. Fessler, “Mean and variance of single photon counting with deadtime,” Phys. Med. Biol. 45, 2043–2056 (2000).
[Crossref]

de Simone, E. A.

P. L. Saldanha, E. A. de Simone, and P. Frutuoso e Melo, “An application of non-homogeneous Poisson point processes to the reliability analysis of service water pumps,” Nucl. Eng. Des. 210, 125–133 (2001).
[Crossref]

Drost, R. J.

R. J. Drost, B. M. Sadler, and G. Chen, “Dead time effects in non-line-of-sight ultraviolet communications,” Opt. Express 23, 15748–15761 (2015).
[Crossref]

L. Liao, R. J. Drost, Z. Li, T. Lang, B. M. Sadler, and G. Chen, “Long-distance non-line-of-sight ultraviolet communication channel analysis: experimentation and modelling,” IET Optoelectron. 9, 223–231 (2015).
[Crossref]

R. J. Drost, T. J. Moore, and B. M. Sadler, “UV communications channel modeling incorporating multiple scattering interactions,” J. Opt. Soc. Am. A 28, 686–695 (2011).
[Crossref]

G. Chen, L. Liao, Z. Li, R. J. Drost, and B. M. Sadler, “Experimental and simulated evaluation of long distance NLOS UV communication,” in Proceedings of the 19th International Symposium Communication Systems, Networks & Digital Signal Processing (2014), pp. 904–909.

Duane, S.

S. Duane, A. D. Kennedy, B. J. Pendleton, and D. Roweth, “Hybrid Monte Carlo,” Phys. Lett. B 195, 216–222 (1987).
[Crossref]

Fessler, J. A.

F. Y. Daniel and J. A. Fessler, “Mean and variance of single photon counting with deadtime,” Phys. Med. Biol. 45, 2043–2056 (2000).
[Crossref]

Frutuoso e Melo, P.

P. L. Saldanha, E. A. de Simone, and P. Frutuoso e Melo, “An application of non-homogeneous Poisson point processes to the reliability analysis of service water pumps,” Nucl. Eng. Des. 210, 125–133 (2001).
[Crossref]

Galliher, H. P.

H. P. Galliher and R. C. Wheeler, “Nonstationary queuing probabilities for landing congestion of aircraft,” Oper. Res. 6, 264–275 (1958).
[Crossref]

Gelman, A.

A. Gelman and D. B. Rubin, “Inference from iterative simulation using multiple sequences,” Stat. Sci. 7, 457–472 (1992).
[Crossref]

George, E. I.

G. Casella and E. I. George, “Explaining the Gibbs sampler,” Am. Statist. 46, 167–174 (1992).

Greenberg, E.

S. Chib and E. Greenberg, “Understanding the Metropolis–Hastings algorithm,” Am. Statist. 49, 327–335 (1995).

Harrod, S.

S. Harrod and W. D. Kelton, “Numerical methods for realizing nonstationary Poisson processes with piecewise-constant instantaneous-rate functions,” Simulation 82, 147–157 (2006).
[Crossref]

Hassan, U.

U. Hassan and R. Bashir, “Electrical cell counting process characterization in a microfluidic impedance cytometer,” Biomed. Microdevices 16, 697–704 (2014).
[Crossref]

Heikkinen, J.

J. Heikkinen and E. Arjas, “Non-parametric Bayesian estimation of a spatial Poisson intensity,” Scand. J. Stat. 25, 435–450 (1998).
[Crossref]

Kelton, W. D.

S. Harrod and W. D. Kelton, “Numerical methods for realizing nonstationary Poisson processes with piecewise-constant instantaneous-rate functions,” Simulation 82, 147–157 (2006).
[Crossref]

Kennedy, A. D.

S. Duane, A. D. Kennedy, B. J. Pendleton, and D. Roweth, “Hybrid Monte Carlo,” Phys. Lett. B 195, 216–222 (1987).
[Crossref]

Kostinski, A. B.

M. L. Larsen and A. B. Kostinski, “Simple dead-time corrections for discrete time series of non-Poisson data,” Meas. Sci. Technol. 20, 095101 (2009).
[Crossref]

Lang, T.

L. Liao, R. J. Drost, Z. Li, T. Lang, B. M. Sadler, and G. Chen, “Long-distance non-line-of-sight ultraviolet communication channel analysis: experimentation and modelling,” IET Optoelectron. 9, 223–231 (2015).
[Crossref]

Larsen, M. L.

M. L. Larsen and A. B. Kostinski, “Simple dead-time corrections for discrete time series of non-Poisson data,” Meas. Sci. Technol. 20, 095101 (2009).
[Crossref]

Lee, S.

S. Lee, J. R. Wilson, and M. M. Crawford, “Modeling and simulation of a nonhomogeneous Poisson process having cyclic behavior,” Commun. Stat. 20, 777–809 (1991).
[Crossref]

Leemis, L. M.

L. M. Leemis, “Nonparametric estimation of the cumulative intensity function for a nonhomogeneous Poisson process,” Manag. Sci. 37, 886–900 (1991).
[Crossref]

Lewis, P. A.

P. A. Lewis and G. S. Shedler, “Statistical analysis of non-stationary series of events in a database system,” IBM J. Res. Dev. 20, 465–482 (1976).
[Crossref]

Li, Z.

L. Liao, R. J. Drost, Z. Li, T. Lang, B. M. Sadler, and G. Chen, “Long-distance non-line-of-sight ultraviolet communication channel analysis: experimentation and modelling,” IET Optoelectron. 9, 223–231 (2015).
[Crossref]

G. Chen, L. Liao, Z. Li, R. J. Drost, and B. M. Sadler, “Experimental and simulated evaluation of long distance NLOS UV communication,” in Proceedings of the 19th International Symposium Communication Systems, Networks & Digital Signal Processing (2014), pp. 904–909.

Liao, L.

L. Liao, R. J. Drost, Z. Li, T. Lang, B. M. Sadler, and G. Chen, “Long-distance non-line-of-sight ultraviolet communication channel analysis: experimentation and modelling,” IET Optoelectron. 9, 223–231 (2015).
[Crossref]

G. Chen, L. Liao, Z. Li, R. J. Drost, and B. M. Sadler, “Experimental and simulated evaluation of long distance NLOS UV communication,” in Proceedings of the 19th International Symposium Communication Systems, Networks & Digital Signal Processing (2014), pp. 904–909.

MacKay, D. J.

R. P. Adams, I. Murray, and D. J. MacKay, “Tractable nonparametric Bayesian inference in Poisson processes with Gaussian process intensities,” in Proceedings of the 26th Annual International Conference on Machine Learning, 2009, pp. 9–16.

Model, J.

A. M. Siegel, G. A. Shaw, and J. Model, “Short-range communication with ultraviolet LEDs,” Proc. SPIE 5530, 182–193 (2004).
[Crossref]

Moore, T. J.

Müller, J. W.

J. W. Müller, “Dead-time problems,” Nucl. Instrum. Methods 112, 47–57 (1973).
[Crossref]

Murray, I.

R. P. Adams, I. Murray, and D. J. MacKay, “Tractable nonparametric Bayesian inference in Poisson processes with Gaussian process intensities,” in Proceedings of the 26th Annual International Conference on Machine Learning, 2009, pp. 9–16.

Pendleton, B. J.

S. Duane, A. D. Kennedy, B. J. Pendleton, and D. Roweth, “Hybrid Monte Carlo,” Phys. Lett. B 195, 216–222 (1987).
[Crossref]

Rasmussen, C. E.

C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning (MIT, 2006).

Robert, C.

C. Robert and G. Casella, Monte Carlo Statistical Methods (Springer, 2013).

Roweth, D.

S. Duane, A. D. Kennedy, B. J. Pendleton, and D. Roweth, “Hybrid Monte Carlo,” Phys. Lett. B 195, 216–222 (1987).
[Crossref]

Rubin, D. B.

A. Gelman and D. B. Rubin, “Inference from iterative simulation using multiple sequences,” Stat. Sci. 7, 457–472 (1992).
[Crossref]

Ryan, K. J.

K. J. Ryan, “Some flexible families of intensities for non-homogeneous Poisson process models and their Bayes inference,” Qual. Reliab. Eng. Int. 19, 171–181 (2003).
[Crossref]

Sadler, B. M.

L. Liao, R. J. Drost, Z. Li, T. Lang, B. M. Sadler, and G. Chen, “Long-distance non-line-of-sight ultraviolet communication channel analysis: experimentation and modelling,” IET Optoelectron. 9, 223–231 (2015).
[Crossref]

R. J. Drost, B. M. Sadler, and G. Chen, “Dead time effects in non-line-of-sight ultraviolet communications,” Opt. Express 23, 15748–15761 (2015).
[Crossref]

R. J. Drost, T. J. Moore, and B. M. Sadler, “UV communications channel modeling incorporating multiple scattering interactions,” J. Opt. Soc. Am. A 28, 686–695 (2011).
[Crossref]

G. Chen, L. Liao, Z. Li, R. J. Drost, and B. M. Sadler, “Experimental and simulated evaluation of long distance NLOS UV communication,” in Proceedings of the 19th International Symposium Communication Systems, Networks & Digital Signal Processing (2014), pp. 904–909.

Saldanha, P. L.

P. L. Saldanha, E. A. de Simone, and P. Frutuoso e Melo, “An application of non-homogeneous Poisson point processes to the reliability analysis of service water pumps,” Nucl. Eng. Des. 210, 125–133 (2001).
[Crossref]

Shaw, G. A.

A. M. Siegel, G. A. Shaw, and J. Model, “Short-range communication with ultraviolet LEDs,” Proc. SPIE 5530, 182–193 (2004).
[Crossref]

Shedler, G. S.

P. A. Lewis and G. S. Shedler, “Statistical analysis of non-stationary series of events in a database system,” IBM J. Res. Dev. 20, 465–482 (1976).
[Crossref]

Siegel, A. M.

A. M. Siegel, G. A. Shaw, and J. Model, “Short-range communication with ultraviolet LEDs,” Proc. SPIE 5530, 182–193 (2004).
[Crossref]

Snyder, D. L.

D. L. Snyder, “Utilizing side information in emission tomography,” IEEE Trans. Nucl. Sci. 31, 533–537 (1984).
[Crossref]

Stroud, J. R.

J. Weinberg, L. D. Brown, and J. R. Stroud, “Bayesian forecasting of an inhomogeneous Poisson process with applications to call center data,” J. Am. Stat. Assoc. 102, 1185–1198 (2007).

Teich, M. C.

Vannucci, G.

Weinberg, J.

J. Weinberg, L. D. Brown, and J. R. Stroud, “Bayesian forecasting of an inhomogeneous Poisson process with applications to call center data,” J. Am. Stat. Assoc. 102, 1185–1198 (2007).

Wheeler, R. C.

H. P. Galliher and R. C. Wheeler, “Nonstationary queuing probabilities for landing congestion of aircraft,” Oper. Res. 6, 264–275 (1958).
[Crossref]

Williams, C. K. I.

C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning (MIT, 2006).

Wilson, J. R.

S. Lee, J. R. Wilson, and M. M. Crawford, “Modeling and simulation of a nonhomogeneous Poisson process having cyclic behavior,” Commun. Stat. 20, 777–809 (1991).
[Crossref]

Am. Statist. (2)

G. Casella and E. I. George, “Explaining the Gibbs sampler,” Am. Statist. 46, 167–174 (1992).

S. Chib and E. Greenberg, “Understanding the Metropolis–Hastings algorithm,” Am. Statist. 49, 327–335 (1995).

Biomed. Microdevices (1)

U. Hassan and R. Bashir, “Electrical cell counting process characterization in a microfluidic impedance cytometer,” Biomed. Microdevices 16, 697–704 (2014).
[Crossref]

Commun. Stat. (1)

S. Lee, J. R. Wilson, and M. M. Crawford, “Modeling and simulation of a nonhomogeneous Poisson process having cyclic behavior,” Commun. Stat. 20, 777–809 (1991).
[Crossref]

IBM J. Res. Dev. (1)

P. A. Lewis and G. S. Shedler, “Statistical analysis of non-stationary series of events in a database system,” IBM J. Res. Dev. 20, 465–482 (1976).
[Crossref]

IEEE Trans. Nucl. Sci. (1)

D. L. Snyder, “Utilizing side information in emission tomography,” IEEE Trans. Nucl. Sci. 31, 533–537 (1984).
[Crossref]

IET Optoelectron. (1)

L. Liao, R. J. Drost, Z. Li, T. Lang, B. M. Sadler, and G. Chen, “Long-distance non-line-of-sight ultraviolet communication channel analysis: experimentation and modelling,” IET Optoelectron. 9, 223–231 (2015).
[Crossref]

J. Am. Stat. Assoc. (1)

J. Weinberg, L. D. Brown, and J. R. Stroud, “Bayesian forecasting of an inhomogeneous Poisson process with applications to call center data,” J. Am. Stat. Assoc. 102, 1185–1198 (2007).

J. Atmos. Ocean. Technol. (1)

J. Brenguier and L. Amodei, “Coincidence and dead-time corrections for particle counters. Part I: a general mathematical formalism,” J. Atmos. Ocean. Technol. 6, 575–584 (1989).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Manag. Sci. (1)

L. M. Leemis, “Nonparametric estimation of the cumulative intensity function for a nonhomogeneous Poisson process,” Manag. Sci. 37, 886–900 (1991).
[Crossref]

Meas. Sci. Technol. (1)

M. L. Larsen and A. B. Kostinski, “Simple dead-time corrections for discrete time series of non-Poisson data,” Meas. Sci. Technol. 20, 095101 (2009).
[Crossref]

Nucl. Eng. Des. (1)

P. L. Saldanha, E. A. de Simone, and P. Frutuoso e Melo, “An application of non-homogeneous Poisson point processes to the reliability analysis of service water pumps,” Nucl. Eng. Des. 210, 125–133 (2001).
[Crossref]

Nucl. Instrum. Methods (1)

J. W. Müller, “Dead-time problems,” Nucl. Instrum. Methods 112, 47–57 (1973).
[Crossref]

Oper. Res. (1)

H. P. Galliher and R. C. Wheeler, “Nonstationary queuing probabilities for landing congestion of aircraft,” Oper. Res. 6, 264–275 (1958).
[Crossref]

Opt. Express (1)

Phys. Lett. B (1)

S. Duane, A. D. Kennedy, B. J. Pendleton, and D. Roweth, “Hybrid Monte Carlo,” Phys. Lett. B 195, 216–222 (1987).
[Crossref]

Phys. Med. Biol. (1)

F. Y. Daniel and J. A. Fessler, “Mean and variance of single photon counting with deadtime,” Phys. Med. Biol. 45, 2043–2056 (2000).
[Crossref]

Proc. SPIE (1)

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Figures (9)

Fig. 1.
Fig. 1. IHPP rate function estimation using histograms of dead-time corrupted observations of 100 simulated independent IHPP realizations, where the actual rate function (which could model an impulse response, for example) is depicted with a solid curve. Each subfigure considers a different distribution of the dead-time duration τd: (a) τd=0, (b) τdf0.01,0.0001N[0,), (c) τdf0.1,0.0001N[0,), and (d) τdf0.5,0.0001N[0,).
Fig. 2.
Fig. 2. Algorithmic description of the generative observation model for a dead-time limited IHPP with a random rate function.
Fig. 3.
Fig. 3. Example generation of dead-time limited events based on the rate function given by the black unimodal curve. The top row of circles depict a realization of a homogeneous Poisson process with rate λ*=100. Thinning the realization of the homogeneous process produces the IHPP realization—corresponding to the rate function—depicted with the second row of circles. The last two rows depict two realizations of the dead-time limited IHPP based on the IHPP realization, where two different dead-time distributions are considered: fτ=f0.1,0.0001N[0,) (top) and fτ=f0.5,0.0001N[0,) (bottom).
Fig. 4.
Fig. 4. Rate functions used in simulations. We label the functions as (a) IF1, (b) IF2, and (c) IF3.
Fig. 5.
Fig. 5. Inference results for IF1, where (a)–(c) consider inference ignoring dead time and (d)–(f) consider dead time in the inference procedure. The dead-time duration associated with each plot is (a, d) short; (b, e) moderate; and (c, f) long. The true rate function, mean function, and quantile functions are depicted, respectively, with a solid gray line, solid black line, and dashed lines.
Fig. 6.
Fig. 6. Inference results for IF2, where (a)–(c) consider inference ignoring dead time and (d)–(f) consider dead time in the inference procedure. The dead-time duration associated with each plot is (a, d) short; (b, e) moderate; and (c, f) long. The true rate function, mean function, and quantile functions are depicted, respectively, with a solid gray line, solid black line, and dashed lines.
Fig. 7.
Fig. 7. Inference results for IF3, where (a)–(c) consider inference ignoring dead time and (d)–(f) consider dead time in the inference procedure. The dead-time duration associated with each plot is (a, d) short; (b, e) moderate; and (c, f) long. The true rate function, mean function, and quantile functions are depicted, respectively, with a solid gray line, solid black line, and dashed lines.
Fig. 8.
Fig. 8. Inference results for IF1 with a moderate dead-time duration, where the inference procedure includes dead-time effects and a value of λ*=200 is assumed. The true rate function, mean function, and quantile functions are depicted, respectively, with a solid gray line, solid black line, and dashed lines.
Fig. 9.
Fig. 9. Inference results for UVC rate functions based on (a) simulated data and (b) experimentally collected data. In each plot, a model prediction for the rate function is depicted by a solid blue line; mean and quantile functions for inference ignoring dead-time effects are depicted with, respectively, solid and dashed red lines; and mean and quantile functions for inference accounting for dead-time effects are depicted with, respectively, solid and dashed green lines. In addition, the cyan curve in (b) depicts a scaled version of the model-based rate function. The scaling is chosen to minimize the mean squared error between the model-based rate function and the mean inferred rate function estimate. The cyan curve enables direct comparison of the shape of the estimated rate function curve with the model prediction.

Tables (3)

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Table 1. Area under Rate Functions and Dead-Time Losses in Simulated Data

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Table 2. Error in Inference Ignoring and Including Dead-Time Effects

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Table 3. UVC Experiment Parameters

Equations (29)

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k(t1,t2)=σv2exp{|t1t2|2/(2σl2)},
λ(t)=λ*σ(g(t)),
Ii{Ii[1],Ii[2],,Ii[|Oi|1],Ii[|Oi|]}={(Oi[1],Oi[2]),,(Oi[|Oi|1],Oi[|Oi|]),(Oi[|Oi|],)}.
τimin(k)[max{Ii[k]Di}]IiL[k]
f(ψ|λ*,θ,Fτ)f(O,D,R,g|E||λ*,θ,Fτ)=f(O,D|R,E,Fτ)f(R|g|E|,E)fθGP(g|E||E)×f(E||E1|,,|EN|,λ*)f(|E1|,,|EN||λ*),
f(|E1|,,|EN||λ*)=i=1Nfλ*TPoisson(|Ei|)=eNλ*T(λ*T)|E|/i=1N|Ei|!.
f(E||E1|,,|EN|,λ*)=i=1N(1/T)|Ei||Ei|!=(1/T)|E|i=1N|Ei|!.
f(R|g|E|,E)=i=1Nk=1|Ei|[1σ(g|E|(Ei[k]))]1Ri(Ei[k])×[σ(g|E|(Ei[k]))]11Ri(Ei[k]),
f(O,D|R,E,Fτ)=i=1Nf(Oi,Di|Ri,Ei,Fτ),
f(Oi,Di|Ri,Ei,Fτ)=f(Oi[1]|Ri,Ei,Fτ)×[k=1|Oi|1f(Oi[k+1]|Oi[k],Ri,Ei,Fτ)]×f(DiIi[|Oi|]|Oi[|Oi|],Ri,Ei,Fτ),
f(Oi[k+1]|Oi[k],Ri,Ei,Fτ)=f(Oi[k+1],DiIi[k]|Oi[k],Ri,Ei,Fτ).
f(Oi[1]|Ri,Ei,Fτ)=1{minEiRi}(Oi[1]).
f(Oi[k+1]|Oi[k],Ri,Ei,Fτ)=Fτ(τimax(k))Fτ(τimin(k)).
f(DiIi[|Oi|]|Oi[|Oi|],Ri,Ei,Fτ)=1Fτ(τimin(|Oi|))=Fτ(τimax(|Oi|))Fτ(τimin(|Oi|)).
α=f(ψ|λ*,θ,Fτ)f(ψ|λ*,θ,Fτ)×q(ψ|ψ)q(ψ|ψ),
β|Ei|fλ*TPoisson(|Ei|+1)fλ*TPoisson(|Ei|1)+fλ*TPoisson(|Ei|+1)
q(ψ{s,g(s)}R,i|ψ)=β|Ei|(1/T)fθGP(g(s)|g|E|,E,s).
αR,iins=λ*T(1β|Ei|+1)[1σ(g(s))]fθGP(g|E|+1|E,s)β|Ei|(|Ri|+1)fθGP(g|E||E)fθGP(g(s)|g|E|,E,s)=λ*T(1β|Ei|+1)[1σ(g(s))]β|Ei|(|Ri|+1),
q(ψ{s,g(s)}R,i|ψ)=(1β|Ei|)/|Ri|.
αR,idel=β|Ei|1|Ri|λ*T(1β|Ei|)[1σ(g(s))].
q(ψ{s,g(s)}D,i|ψ)=β|Ei|fθGP(g(s)|g|E|,E,s)TOi[1].
αD,iins=λ*(TOi[1])(1β|Ei|+1)σ(g(s))β|Ei|(|Di|+1)×Fτ(τimax(k))Fτ(τ˜imin(k))Fτ(τimax(k))Fτ(τimin(k)).
q(ψ{s,g(s)}D,i|ψ)=(1β|Ei|)/|Di|.
αD,idel=β|Ei|1|Di|λ*(TOi[1])(1β|Ei|)σ(g(s))×Fτ(τimax(k))Fτ(τ˜imin(k))Fτ(τimax(k))Fτ(τimin(k)).
q(ψ|ψ)=(1/|Ri|)fs,θl2N[0,T)(s)fθGP(g(s)|g|E|,E,s).
αD,imove=fs,θl2N[0,T)(s)(1σ(g(s)))fs,θl2N[0,T)(s)(1σ(g(s))).
q(ψ|ψ)=(1/|Di|)fOi[k],(μD/3)2NIi[k][0,T]fθGP(g(s)|g|E|,E,s).
αD,imove=fOi[k],(μD/3)2NIi[k][0,T](s)σ(g(s))fOi[k],(μD/3)2NIi[k][0,T](s)σ(g(s))×Fτ(τimax(k))Fτ(τ˜imin(k))Fτ(τimax(k))Fτ(τimin(k)).
e(λ^)0T[λ(t)λ^(t)]2dt0Tλ(t)dt,

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