Abstract
Approximate normalizing transformations are derived for Poisson counting systems affected by nonparalyzable and paralyzable dead time. In the nonparalyzable case the transformation takes the form of a simple inverse hyperbolic function whereas in the paralyzable case it is an inverse trigonometric function. The results are expected to find use in neural counting, photon counting, and nuclear counting, as well as in queuing theory.
Similar content being viewed by others
References
Albert GE, Nelson L (1953) Contributions to the statistical theory of counter data. Ann Math Stat 24:9–22
Anscombe FJ (1948) The transformation of Poisson, binomial and negative binomial data. Biometrika 35:246–254
Bartlett MS (1936) The square-root transformation in the analysis of variance. J R Stat Soc Suppl 3:68–78
Bédard G (1967) Dead-time corrections to the statistical distribution of photoelectrons. Proc Phys Soc (Lond) 90:131–141
Cantor BI, Teich MC (1975) Dead-time-corrected photocounting distributions for laser radiation. J Opt Soc Am 65:786–791
Cochran WG (1940) The analysis of variance when experimental errors follow the Poisson or binomial laws. Ann Math Stat 9:335–347
Cox DR (1962) Renewal theory. Methuen, London
Curtiss JH (1943) On transformations used in the analysis of variance. Ann Math Stat 14:107–122
DeLotto I, Manfredi PF, Principi P (1964) Counting statistics and dead-time losses, Part 1. Energ Nucl (Milan) 11:557–564
Dwight HB (1961) Tables of integrals and other mathematical data. 4th ed. Macmillan, New York
Feller W (1948) On probability problems in the theory of counters. In: Studies and essays: a volume for the anniversary of courant. Wiley, New York, pp 105–115
Freeman MF, Tukey JW (1950) Transformations related to the angular and square root. Ann Math Stat 21:607–611
Kendall MG, Stuart A (1966) The advanced theory of statistics, vol. 3. Hafner, New York, pp 88–94
Lachs G, Al-Shaikh R, Bi Q, Saia RA, Teich MC (1984) A neuralcounting model based on physiological characteristics of the peripheral auditory system. V. Application to loudness estimation and intensity discrimination. IEEE Trans SMC-14:819–836
Libert J (1976) Comparaison des distributions statistiques de comptage des systèmes radioactifs. Nucl Instrum Methods 136:563–568
Mattick ATR, McClemont J, Irwin JO (1935) The plate count of milk. Dairy Res 6:130–147
Mueller, CG (1954) A quantitative theory of visual excitation for the single photoreceptor. Proc Natl Acad Sci USA 40:853–863
Müller JW (1973) Dead-time problems. Nucl Instrum Methods 112:47–57
Müller JW (1974) Some formulae for a dead-time-distorted Poisson process. Nucl Instrum Methods 117:401–404
Müller JW (ed) (1981) Bibliography on dead time effects. Report BIP M-81/11. Bureau International des Poids et Mesures, Sèvres, France
Parzen E (1962) Stochastic processes. Holden-Day, San Francisco, pp 117–186
Prucnal PR (1980) Receiver performance evaluation using photocounting cumulants with application to atmospheric turbulence. Appl Opt 19:3611–3616
Prucnal PR, Saleh BEA (1981) Transformation of imagesignal-dependent noise into image-signal-independent noise. Opt Lett 6:316–318
Prucnal PR, Teich MC (1980) An increment threshold law for stimuli of arbitrary statistics. J Math Psychol 21:168–177
Prucnal PR, Teich MC (1982) Multiplication noise in the human visual system at threshold. 2. Probit estimation of parameters. Biol Cybern 43:87–96
Prucnal PR, Teich MC (1983) Refractory effects in neural counting processes with exponentially decaying rates. IEEE Trans SMC-13:1028–1033
Ricciardi LM, Esposito F (1966) On some distribution functions for non-linear switching elements with finite dead time. Kybernetik (Biol Cybern) 3:148–152
Saleh BEA (1978) Photoelectron statistics. Springer, Berlin Heidelberg New York, p 19
Teich MC, Cantor BI (1978) Information, error, and imaging in deadtime-perturbed doubly-stochastic Poisson counting systems. IEEE J QE-14:993–1003
Teich MC, Diament P (1980) Relative refractoriness in visual information processing. Biol Cybern 38:187–191
Teich MC, Lachs G (1979) A neural counting model incorporating refractoriness and spread of excitation: I. Application to intensity discrimination. J Acoust Soc Am 66:1738–1749
Teich MC, Lachs G (1983) A neural counting model incorporating refractoriness and spread of excitation: III. Application to intensity discrimination and loudness estimation for variable-bandwidth noise stimuli. Acustica 53:225–236
Teich MC, Vannucci G (1978) Observation of dead-time-modified photocounting distributions for modulated laser radiation. J Opt Soc Am 68:1338–1342
Teich MC, Matin L, Cantor BI (1978) Refractoriness in the maintained discharge of the cat's retinal ganglion cell. J Opt Soc Am 68:386–402
Tukey JW (1957) On the comparative anatomy of transformations. Ann Math Stat 28:602–632
Vannucci G, Teich MC (1979) Equivalence of threshold detection with and without dead time. Appl Opt 18:3886–3887
Young ED, Barta PE (1985) Rate responses of auditory nerve fibers to tones in noise near masked threshold. J Acoust Soc Am (to be published)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Teich, M.C. Normalizing transformations for dead-time-modified Poisson counting distributions. Biol. Cybern. 53, 121–124 (1985). https://doi.org/10.1007/BF00337028
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00337028