Download PDF

Discretely Observed Brownian Motion Governed by Telegraph Process: Estimation

Methodology and Computing in Applied Probability volume 21, pages907–920(2019)Cite this article

Abstract

A Brownian motion whose infinitesimal variance alternates according to a telegraph process is considered. This stochastic process can be employed to model a variety of real-word situations, such as animal movement in ecology and stochastic volatility in mathematical finance. The main goal is to develop an estimation procedure for the underlying model parameters when the process is observed at discrete, possibly irregularly spaced time points. The sequence of observations is not Markov, but the sequence of the state of the telegraph process, if observed, is Markov. The observed sequence is therefore from a hidden Markov model. Likelihood inference is developed via dynamic programming, and is demonstrated to have much higher efficiency than the composite likelihood approach that was applied in an earlier work. The model is applied to model the movement of a mountain lion.

References

  1. Beier P, Choate D, Barrett R H (1995) Movement patterns of mountain lions during different behaviors. J Mammal 76:1056–1070

    Article  Google Scholar 

  2. Calenge C (1035) The package adehabitat for the R software: Tool for the analysis of space and habitat use by animals, vol 197

  3. Cane V R (1959) Behavior sequences as semi-Markov chains. J R Stat Soc Ser B 21:36–58

    MathSciNet  MATH  Google Scholar 

  4. Cappé O, Moulines E, Rydén T (2005) Inference in Hidden Markov Models, vol 6, Springer

  5. Di Crescenzo A, Di Nardo E, Ricciardi L M (2005) Simulation of first-passage times for alternating Brownian motions. Methodol Comput Appl Probab 7:161–181

    MathSciNet  Article  MATH  Google Scholar 

  6. Di Crescenzo A, Martinucci B, Zacks S (2014) Mathematical and statistical methods for actuarial sciences and finance. In: On the geometric brownian motion with alternating trend Perna C, Sibillo M (eds)

  7. Di Crescenzo A, Pellerey F (2002) On prices’ evolutions based on geometric telegrapher’s process. Appl Stoch Models Bus Ind 18:171–184

    MathSciNet  Article  MATH  Google Scholar 

  8. Di Crescenzo A, Zacks S (2015) Probability law and flow function of Brownian motion driven by a generalized telegraph process. Methodol Comput Appl Probab 17:761–780

    MathSciNet  Article  MATH  Google Scholar 

  9. Farmer C, Safi K, Barber D, Martel M, Bildstein K (2010) Efficacy of migration counts for monitoring continental populations of raptors: an example using the osprey (pandion haliaetus). The Auk 127:863–970

    Article  Google Scholar 

  10. Fouque J-P, Papanicolaou G, Sircar R, Sølna k (2011) Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives. Cambridge University Press, Cambridge

    Google Scholar 

  11. Horne J S, Garton E O, Krone S M, Lewis S J (2007) Analyzing animal movements using Brownian bridges. Ecology 88:2354–2363

    Article  Google Scholar 

  12. Hornocker M G (1970) An analysis of mountain lion predation upon mule deer and elk in the idaho primitive area, Wildlife Society. Technical Report 21

  13. Jeschke J M (2007) When carnivores are “full and lazy”. Oecologia 152:357–364

    Article  Google Scholar 

  14. Jonsen I D, Flemming J M, Myers R A (2005) Robust state-space modeling of animal movement data. Ecology 86:2874–2880

    Article  Google Scholar 

  15. Kolesnik A D, Ratanov N (2013) Springer Briefs in Statistics Telegraph processes and option pricing. Springer, Heidelberg

  16. Kranstauber B, Kays R, LaPoint S D, Wikelski M, Safi K (2012) A dynamic brownian bridge movement model to estimate utilization distributions for heterogeneous animal movement. J Anim Ecol 81:738–746

    Article  Google Scholar 

  17. Kranstauber B, Smolla M (2013) move: Visualizing and analyzing animal track data. R package version 1.1.360/r365

  18. Lonergan M, Fedak M, McConnell B (2009) The effects of interpolation error and location quality on animal track reconstruction. Marine Mammal Science 25:275–282

    Article  Google Scholar 

  19. Nielson R M, Sawyer H, McDonald T L (2012) BBMM: Brownian Bridge Movement Model. R package version 2.3

  20. Page E S (1960) Theoretical considerations of routine maintenance. The Computer Journal 2:199–204

    Article  MATH  Google Scholar 

  21. Patterson T, Thomas L, Wilcox C, Ovaskainen O, Matthiopoulos J (2008) State-space models of individual animal movement. Trends in Ecology and Evolution 23:87–94

    Article  Google Scholar 

  22. Perry D, Stadje W, Zacks S (1999) First-exit times for increasing compound processes. Communications in Statistics: Stochastic Models 15:977–992

    MathSciNet  MATH  Google Scholar 

  23. Pierce B M, Bleich V C (2003) Wild Mammals of North America, Biology, Management and Conservation. In: Feldhamer G A, Thompson B C, Chapman J A (eds) Mountain lion, 2edn. Johns Hopkins University Press, Baltimore, pp 744–757

    Google Scholar 

  24. Pierce B M, Bleich V C, Chetkiewicz C-L B, Wehausen J D (1998) Timing of feeding bouts of mountain lions. J Mammal 79:222–226

    Article  Google Scholar 

  25. Pozdnyakov V, Meyer T H, Wang Y-B, Yan J (2014) On modeling animal movements using Brownian motion with measurement error. Ecology 95:247–253

    Article  Google Scholar 

  26. Preisler H K, Ager A A, Johnson B K, Kie J G (2004) Modeling animal movements using stochastic differential equations. Environmetrics 15:643–657

    Article  Google Scholar 

  27. Schaller G B, Crawshaw Jr P G (1980) Movement patterns of jaguar. Biotropica 12:161–168

  28. Stadje W, Zacks S (2004) Telegraph processes with random velocities. J Appl Probab 41:665–678

    MathSciNet  Article  MATH  Google Scholar 

  29. Takekawa J, Newman S, Xiao X, Prosser D, Spragens K, Palm E, Yan B, Li T, Lei F, Zhao D, Douglas D, Muzaffar S, Ji W (2010) Migration of waterfowl in the east asian flyway and spatial relationship to hpai h5n1 outbreaks. Avian Dis 54(s1):466–476

    Article  Google Scholar 

  30. Willems E, Hill R (2009) Predator-specific landscapes of fear and resource distribution: effects on spatial range use. Ecology 90:546–555

    Article  Google Scholar 

  31. Yan J, Chen Y-W, Lawrence-Apfel K, Ortega I, Pozdnyakov V, Williams S, Meyer T (2014) A moving–resting process with an embedded brownian motion for animal movements. Popul Ecol 56:401–415

    Article  Google Scholar 

  32. Yan J, Pozdnyakov V (2016) smam: Statistical Modeling of Animal Movements. R package version 0.3-0

  33. Zacks S (2004) Generalized integrated telegraph processes and the distribution of related stopping times. J Appl Probab 41:497–507

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

  1. Department of Statistics, University of Connecticut, 215 Glenbrook Rd., Storrs, CT, 06269, USA

    Vladimir Pozdnyakov, Anthony Labarga & Jun Yan

  2. Panthera, 8 W 40th St, 18th Floor, New York, NY, 10018, USA

    L. Mark Elbroch

  3. Center for Environmental Sciences and Engineering, University of Connecticut, 3107 Horsebarn Hill Rd, Storrs, CT, 06269, USA

    Thomas Meyer & Jun Yan

  4. Department of Natural Resources and the Environment, University of Connecticut, 1376 Storrs Road, Unit 4087, Storrs, CT, 06268, USA

    Thomas Meyer

Authors
  1. Vladimir Pozdnyakov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. Mark Elbroch
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Anthony Labarga
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Thomas Meyer
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Jun Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vladimir Pozdnyakov.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Pozdnyakov, V., Elbroch, L.M., Labarga, A. et al. Discretely Observed Brownian Motion Governed by Telegraph Process: Estimation. Methodol Comput Appl Probab 21, 907–920 (2019). https://doi.org/10.1007/s11009-017-9547-6

Download citation

Keywords

  • Markov process
  • Dynamic programming
  • Likelihood estimation

Mathematics Subject Classification (2010)

  • 62M05
  • 62P10