Abstract
The Maximum Likelihood (ML) and Bayesian estimation paradigms work within the model that the data, from which the parameters are to be estimated, is treated as a set rather than as a sequence. The pioneering paper that dealt with the field of sequence-based estimation [2] involved utilizing both the information in the observations and in their sequence of appearance. The results of [2] introduced the concepts of Sequence Based Estimation (SBE) for the Binomial distribution, where the authors derived the corresponding MLE results when the samples are taken two-at-a-time, and then extended these for the cases when they are processed three-at-a-time, four-at-a-time etc. These results were generalized for the multinomial “two-at-a-time” scenario in [3]. This paper (This paper is dedicated to the memory of Dr. Mohamed Kamel, who was a close friend of the first author.) now further generalizes the results found in [3] for the multinomial case and for subsequences of length 3. The strategy used in [3] (and also here) involves a novel phenomenon called “Occlusion” that has not been reported in the field of estimation. The phenomenon can be described as follows: By occluding (hiding or concealing) certain observations, we map the estimation problem onto a lower-dimensional space, i.e., onto a binomial space. Once these occluded SBEs have been computed, the overall Multinomial SBE (MSBE) can be obtained by combining these lower-dimensional estimates. In each case, we formally prove and experimentally demonstrate the convergence of the corresponding estimates.
B. John Oommen is a Fellow: IEEE and Fellow: IAPR. The work was done while he was visiting at Myongji University, Yongin, Korea. He also holds an Adjunct Professorship with the Department of Information and Communication Technology, University of Agder, Grimstad, Norway. The work was partially supported by NSERC, the Natural Sciences and Engineering Research Council of Canada and a grant from the National Research Foundation of Korea. This work was also generously supported by the National Research Foundation of Korea funded by the Korean Government (NRF-2012R1A1A2041661).
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- 1.
This information is, of course, traditionally used when we want to consider dependence information, as in the case of Markov models and n-gram statistics.
- 2.
- 3.
The contents of this section is quite identical to the corresponding section in [3]. This is unavoidable because the notation is quite cumbersome. Besides, the fundamental theory of using “occlusion” is identical in both the papers. Unfortunately, it is futile to omit these concepts and to refer the reader to [3] - it will render the present paper to be quite incomprehensible.
- 4.
We apologize for this cumbersome notation, but this is unavoidable considering the complexity of the problem and the ensuing analysis.
- 5.
For the present, we consider non-overlapping subsequences. We shall later extend this to overlapping sequences when we report the experimental results.
- 6.
The reader must take pains to differentiate between the q’s and the s’s, because the former refer to the BSBEs and the latter to the MSBEs.
- 7.
The issue of how BSBEs are obtained for specific instantiations of \(\pi (a,b)\) is discussed in the subsequent sections.
- 8.
The fact that c is a dummy variable will not be repeated in the future invocations of this result.
- 9.
This, of course, makes sense only if \(\forall c, \widehat{q}_{a}\Big |_ {{\pi (a,c)}} ^{ac} \ne 0\). This condition will not be explicitly stated in the future.
- 10.
In practice, this is augmented by the fact that the SBEs sometimes lead to complex solutions or to unrealistic solutions when the number of samples processed is too small.
References
Fukunaga, K.: Introduction to Statistical Pattern Recognition. Academic Press, San Diego (1990)
Oommen, B.J., Kim, S.-W., Horn, G.: On the estimation of independent binomial random variables using occurrence and sequential information. Pattern Recogn. 40, 3263–3276 (2007)
Oommen, B.J., Kim, S.-W.: Multinomial Sequence Based Estimation: The Case of Pairs of Contiguous Occurrences. In: To appear in the Proceedings of AI 2016, The 2016 Canadian Artificial Intelligence Conference, Victoria, Canada, This talk will be a Plenary/Keynote Talk at the Conference, May 2016
Oommen, B.J., Kim, S.-W.: Occlusion-based Estimation of Independent Multinomial Random Variables Using Occurrence and Sequential Information (To be submitted for Publication)
Ross, S.: Introduction to Probability Models, 2nd edn. Academic Press, San Diego (2002)
Shao, J.: Mathematical Statistics, 2nd edn. Springer, New York (2003)
Sprinthall, R.: Basic Statistical Analysis, 2nd edn. Allyn and Bacon, Boston (2002)
van der Heijden, F., Duin, R.P.W., de Ridder, D., Tax, D.M.J.: Classification, Parameter Estimation and State Estimation: An Engineering Approach using MATLAB. Wiley, England (2004)
Webb, A.: Statistical Pattern Recognition, 2nd edn. Wiley, New York (2002)
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Oommen, B.J., Kim, SW. (2016). Multinomial Sequence Based Estimation Using Contiguous Subsequences of Length Three . In: Campilho, A., Karray, F. (eds) Image Analysis and Recognition. ICIAR 2016. Lecture Notes in Computer Science(), vol 9730. Springer, Cham. https://doi.org/10.1007/978-3-319-41501-7_28
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