A Remark on L2 Distance Function And Non-Identifiability Problem of Finite Mixture Distribution Models in Model-Based Classification

Yıl 2019, Cilt: 14 Sayı: 4, 139 - 146, 26.10.2019

Yüksel Öner Fikriye Kabakcı Burçin Öner Mehmet Gürcan

Öz

          Finite
mixture models provide flexible method of modeling data obtained from
population consisting of finite number of homogeneous subpopulations. One of
the main areas in which the finite mixture model structures is practically used
in statistics is model based classification. However, the result of non
identifiability problem arising from the structure of the finite mixture models
may cause unreliable results on classification. In this paper we compare the
probability density functions () of
the finite mixture distribution models for two different populations by L2
distance. We propose the componentwise L2 distance function to compare the  of finite mixture distribution models for two
different populations in the presence of non identifiability problem. Besides,
a condition is proposed to control whether the L2 distance function gives
similar results with the componentwise L2 distance function to compare the  of finite mixture distribution models for two
different populations. 

Anahtar Kelimeler

Finite Mixture Distribution, L2 Distance Function, Model Based Classification, Mixture Model, Non-identifiability

Kaynakça

• [1] McNicholas, P.D., (2016). Mixture Model-Based Classification: Chapman and Hall/CRC.
• [2] McLachlan, G. and Peel, D., (2000). Finite Mixture Models, Willey Series in Probability and Statistics. In: John Wiley & Sons, New York.
• [3] Erol, H., (2004). A Note on Non-identifibiality Problem of Finite Mixture Distribution Models in Model-based Classification. Selcuk Journal of Applied Mathematics, 5(1):3-10.
• [4] Beran, R., (1977). Minimum Hellinger Distance Estimates for Parametric Models. The annals of Statistics, 5(3):445-463.
• [5] Titterington, D.M., Smith, A.F., and Makov, U.E., (1985). Statistical Analysis of Finite Mixture Distributions: Wiley.
• [6] Dempster, A.P., Laird, N.M., and Rubin, D.B., (1977). Maximum Likelihood from Incomplete Data Via the EM Algorithm. Journal of the Royal Statistical Society. Series B (methodological), pp:1-38.
• [7] Thayasivam, U., Kuruwita, C., and Ramachandran, R.P., (2015). Robust L_2 E Parameter Estimation of Gaussian Mixture Models: Comparison with Expectation Maximization. 22nd International Conference, ICONIP 2015, Istanbul, Proceedings Books, pp:281-288.
• [8] Scott, D.W., (2001). Parametric Statistical Modeling by Minimum Integrated Square Error. Technometrics, 43(3):274-285. doi: 10.1198/004017001316975880.
• [9] Thayasivam, U., (2009). L_2 E Estimation of Mixture Complexity. UGA.
• [10] Aitkin, M. and Rubin, D.B., (1985). Estimation and Hypothesis-Testing in Finite Mixture-Models. Journal of the Royal Statistical Society Series B-Methodological, 47(1):67-75.
• [11] Teicher, H., (1963). Identifiability of Finite Mixtures. The Annals of Mathematical statistics, 34(4):1265-1269.
• [12] Yakowitz, S.J. and Spragins, J.D., (1968). On the Identifiability of Finite Mixtures. The annals of Mathematical statistics, 39(1):209-214.
• [13] Kadane, J.B., (1975). The Role of Identification in Bayesian Theory. Studies in Bayesian econometrics and statistics.
• [14] Akdağ, S.A., (2018). Rüzgar Enerjisi Potansiyel Analizinde Karışım Dağılımları Temelli Tekniklerin Kullanılması, Doktora Tezi, İstanbul Teknik Üniversitesi, Enerji Enstitüsü.
• [15] Toher, D., Downey, G., and Murphy, T., (2005). A Comparison of Model-based and Regression Classification Techniques Applied to Near-infrared Spectroscopic data in Food Authentication Studies. Technical Report 5/10, Department of Statistics, Trinity College Dublin.
• [16] Mclachlan, G.J. and Basford, K.E., (1988). Mixture Models: Inference and Applications to Clustering. Marcel Dekker, New York.
• [17] Fraley, C., Raftery, A.E., and Wehrens, R., (2005). Incremental Model Based Clustering for Large Data Sets with Small Clusters. Journal of Computational and Graphical Statistics, Vol:14, pp:1–18.
• [18] Erol, H. and Akdeniz, F., (2005). A Per-field Classification Method Based on Mixture Distribution Models and an Application to Landsat Thematic Mapper data. International Journal of Remote Sensing Vol:26, No:6, pp:1229–1244.
• [19] Dean, N., Murphy, T.B., and Downey, G., (2006). Updating Classification Rules with Unlabeled Data with Applications in Food Authenticity Studies. Journal of the Royal Statistical Society, Series C (Applied Statistics), Vol:55, pp:1–14.
• [20] Wehrens, R., Buydens, L., Fraley, C., and Raftery, A.E., (2004). Model Based Clustering for Image Segmentation and Large Data Sets Via Sampling. Journal of Classification, Vol:21, pp:231–253.
• [21] Yeung, K.Y., Fraley, C., Murua, A., Raftery, A.E. and Ruzzo, W.L., (2001). Model Based Clustering and Data Transformations for Gene Expression Data. Bioinformatics, 17(10):977-987.
• [22] Servi, T., (2009). Çok Değişkenli Karma Dağılım Modeline Dayalı Kümeleme Analizi, Doktora Tezi, Çukurova Üniversitesi Fen Bilimleri Enstitüsü.
• [23] Everitt, B.S. and David, J.H., (1981). Finite Mixture Distributions. Monographs on Applied Probability and Statistics. Chapman and Hall, London, New York.