Applied Intelligence

, Volume 49, Issue 7, pp 2699–2722 | Cite as

On solving the SPL problem using the concept of probability flux

  • Asieh Abolpour MofradEmail author
  • Anis Yazidi
  • Hugo Lewi Hammer


The Stochastic Point Location (SPL) problem Oommen is a fundamental learning problem that has recently found a lot of research attention. SPL can be summarized as searching for an unknown point in an interval under faulty feedback. The search is performed via a Learning Mechanism (LM) (algorithm) that interacts with a stochastic Environment which in turn informs it about the direction of the search. Since the Environment is stochastic, the guidance for directions could be faulty. The first solution to the SPL problem, which was pioneered two decades ago by Oommen, relies on discretizing the search interval and performing a controlled random walk on it. The state of the random walk at each step is considered to be the estimation of the point location. The convergence of the latter simplistic estimation strategy is proved for an infinite resolution, i.e., infinite memory. However, this strategy yields rather poor accuracy for low discretization resolutions. In this paper, we present two major contributions to the SPL problem. First, we demonstrate that the estimation of the point location can significantly be improved by resorting to the concept of mutual probability flux between neighboring states along the line. Second, we are able to accurately track the position of the optimal point and simultaneously show a method by which we can estimate the error probability characterizing the Environment. Interestingly, learning this error probability of the Environment takes place in tandem with the unknown location estimation. We present and analyze several experiments discussing the weaknesses and strengths of the different methods.


Stochastic Point Location (SPL) Mutual probability flux Flux-based Estimation Solution (FES) Last Transition-based Estimation Solution (LTES) Stochastic Learning Weak Estimation (SLWE) Estimating environment effectiveness 



  1. 1.
    Arntzen E, Steingrimsdottir HS (2014) On the use of variations in a delayed matching-to-sample procedure in a patient with neurocognitive disorder. In: Braunstein SM, Swahn MH, Palmier JB (eds) Mental disorder. iConcept PressGoogle Scholar
  2. 2.
    Camp CJ, Foss JW, O’Hanlon AM, Stevens AB (1996) Memory interventions for persons with dementia. Appl Cogn Psychol 10(3):193–210CrossRefGoogle Scholar
  3. 3.
    De Santo M, Percannella G, Sansone C, Vento M (2004) A multi-expert approach for shot classification in news videos. In: International conference image analysis and recognition, pp 564–571. SpringerGoogle Scholar
  4. 4.
    Friedman J, Hastie T, Tibshirani R (2010) Regularization paths for generalized linear models via coordinate descent. J Stat Softw 33(1):1CrossRefGoogle Scholar
  5. 5.
    Granmo O-C, John Oommen B (2010) Optimal sampling for estimation with constrained resources using a learning automaton-based solution for the nonlinear fractional knapsack problem. Appl Intell 33(1):3–20CrossRefGoogle Scholar
  6. 6.
    Granmo O-C, John Oommen B (2010) Solving stochastic nonlinear resource allocation problems using a hierarchy of twofold resource allocation automata. IEEE Trans Comput 59(4):545–560MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Guo Y, Ge H, Huang J, Li S (2016) A general strategy for solving the stochastic point location problem by utilizing the correlation of three adjacent nodes. In: IEEE international conference on data science in cyberspace (DSC). IEEE, pp 215–221Google Scholar
  8. 8.
    Hammer HL, Yazidi A (2018) Parameter estimation in abruptly changing dynamic environments using stochastic learning weak estimator. Appl Intell 48(11):4096–4112CrossRefGoogle Scholar
  9. 9.
    Havelock J, Oommen BJ, Granmo O-C (2018) Novel distance estimation methods using ?stochastic learning on the line? strategies. IEEE Access 6:48438–48454CrossRefGoogle Scholar
  10. 10.
    Hossain MA, Parra J, Atrey PK, El Saddik A (2009) A framework for human-centered provisioning of ambient media services. Multimed Tools Appl 44(3):407–431CrossRefGoogle Scholar
  11. 11.
    Huang D-S, Jiang W (2012) A general cpl-ads methodology for fixing dynamic parameters in dual environments. IEEE Trans Syst Man Cybern B (Cybernetics) 42(5):1489–1500CrossRefGoogle Scholar
  12. 12.
    Jang YM (2000) Estimation and prediction-based connection admission control in broadband satellite systems. ETRI J 22(4):40–50CrossRefGoogle Scholar
  13. 13.
    Jiang W, Huang D-S, Li S (2016) Random walk-based solution to triple level stochastic point location problem. IEEE Trans Cybern 46(6):1438–1451CrossRefGoogle Scholar
  14. 14.
    Kelly FP (2011) Reversibility and stochastic networks. Cambridge University Press, CambridgezbMATHGoogle Scholar
  15. 15.
    Kpamegan EE, Flournoy N (2008) Up-and-down designs for selecting the dose with maximum success probability. Seq Anal 27(1):78–96MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Landauer TK, Bjork RA (1978) Optimum rehearsal patterns and name learning. In: Gruneberg MM, Morris PE, Sykes RN (eds) Practical aspects of memory. Academic Press, London, pp 625–632Google Scholar
  17. 17.
    Manning CD, Schütze H (1999) Foundations of statistical natural language processing. MIT Press, CambridgezbMATHGoogle Scholar
  18. 18.
    Mofrad AA, Yazidi A, Hammer HL (2017) Solving stochastic point location problem in a dynamic environment with weak estimationGoogle Scholar
  19. 19.
    Randolph N (2013) Probability, stochastic processes, and queueing theory: the mathematics of computer performance modeling. Springer Science & Business Media, BerlinGoogle Scholar
  20. 20.
    Oommen BJ (1997) Stochastic searching on the line and its applications to parameter learning in nonlinear optimization. IEEE Trans Syst Man Cybern B (Cybernetics) 27(4):733–739MathSciNetCrossRefGoogle Scholar
  21. 21.
    Oommen BJ, Calitoiu D (2008) Modeling and Simulating a disease outbreak by learning a contagion parameter-based model. In: Proceedings of the spring simulation multiconference, pp 547–555. Society for computer simulation internationalGoogle Scholar
  22. 22.
    Oommen BJ, Kim S-W, Samuel MT, Granmo O-C (2008) A solution to the stochastic point location problem in metalevel nonstationary environments. IEEE Trans Syst Man Cybern B (Cybernetics) 38(2):466–476CrossRefGoogle Scholar
  23. 23.
    Oommen BJ, Raghunath G (1998) Automata learning and intelligent tertiary searching for stochastic point location. IEEE Trans Syst Man Cybern B (Cybernetics) 28(6):947–954CrossRefGoogle Scholar
  24. 24.
    Oommen BJ, Raghunath G, Kuipers B (2006) Parameter learning from stochastic teachers and stochastic compulsive liars. IEEE Trans Syst Man Cybern B (Cybernetics) 36(4):820–834CrossRefGoogle Scholar
  25. 25.
    Oommen BJ, Rueda L (2006) Stochastic learning-based weak estimation of multinomial random variables and its applications to pattern recognition in non-stationary environments. Pattern Recogn 39(3):328–341CrossRefzbMATHGoogle Scholar
  26. 26.
    Tao T, Tao H, Cai G, Li S (2013) ALidaptive step searching for solving stochastic point location problem. In: International conference on intelligent computing. Springer, pp 192–198Google Scholar
  27. 27.
    R Core Team (2017) R: A language and environment for statistical computingGoogle Scholar
  28. 28.
    Yazidi A, Granmo O-C, Oommen BJ, Goodwin M (2014) A novel strategy for solving the stochastic point location problem using a hierarchical searching scheme. IEEE Trans Cybern 44(11):2202–2220CrossRefGoogle Scholar
  29. 29.
    Yazidi A, Hammer H, Oommen BJ (2018) Higher-fidelity frugal and accurate quantile estimation using a novel incremental discretized paradigm. IEEE Access 6:24362–24374CrossRefGoogle Scholar
  30. 30.
    Yazidi A, Oommen BJ (2017) A novel technique for stochastic root-finding: enhancing the search with adaptive d-ary search. Inf Sci 393:108–129CrossRefGoogle Scholar
  31. 31.
    Yazidi A, Oommen BJ (2017) The theory and applications of the stochastic point location problem. In: 2017 international conference on new trends in computing sciences (ICTCS). IEEE, pp 333–341Google Scholar
  32. 32.
    Yazidi A, Oommen BJ (2016) Novel discretized weak estimators based on the principles of the stochastic search on the line problem. IEEE Trans Cybern 46(12):2732–2744CrossRefGoogle Scholar
  33. 33.
    Zhang J, Lu S, Zang D, Zhou M (2016) Integrating particle swarm optimization with stochastic point location method in noisy environment. In: 2016 international conference on systems, man, and cybernetics (SMC). IEEE, pp 002067–002072Google Scholar
  34. 34.
    Zhang J, Wang Y, Wang C, Zhou M (2017) Symmetrical hierarchical stochastic searching on the line in informative and deceptive environments. IEEE Trans Cybern 47(3):626–635CrossRefGoogle Scholar
  35. 35.
    Zhang J, Zhang L, Zhou M (2015) Solving stationary and stochastic point location problem with optimal computing budget allocation. In: 2015 IEEE international conference on systems, man, and cybernetics (SMC). IEEE, pp 145–150Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Computer ScienceOsloMet - Oslo Metropolitan UniversityOsloNorway

Personalised recommendations