Fitting a Gaussian mixture model through the Gini index. (English) Zbl 1504.62082
Summary: A linear combination of Gaussian components is known as a Gaussian mixture model. It is widely used in data mining and pattern recognition. In this paper, we propose a method to estimate the parameters of the density function given by a Gaussian mixture model. Our proposal is based on the Gini index, a methodology to measure the inequality degree between two probability distributions, and consists in minimizing the Gini index between an empirical distribution for the data and a Gaussian mixture model. We will show several simulated examples and real data examples, observing some of the properties of the proposed method.
MSC:
| 62H30 | Classification and discrimination; cluster analysis (statistical aspects) |
| 62G07 | Density estimation |
References:
| [1] | Bassetti, F., Bodini, A. and Regazzini, E. (2006). On minimum Kantorovich distance estimators, Statistics and Probability Letters76(12): 1298-1302. · Zbl 1090.62030 |
| [2] | Bishop, C.M. (2006). Pattern Recognition and Machine Learning, Springer, New York. · Zbl 1107.68072 |
| [3] | 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)39(1): 1-22. · Zbl 0364.62022 |
| [4] | Elkan, C. (1997). Boosting and naive Bayesian learning, Proceedings of the International Conference on Knowledge Discovery and Data Mining, Newport Beach, USA. |
| [5] | Flach, P.A. and Lachiche, N. (2004). Naive Bayesian classification of structured data, Machine Learning57(3): 233-269. · Zbl 1079.68085 |
| [6] | Giorgi, G.M. and Gigliarano, C. (2017). The Gini concentration index: A review of the inference literature, Journal of Economic Surveys31(4): 1130-1148. |
| [7] | Greenspan, H., Ruf, A. and Goldberger, J. (2006). Constrained Gaussian mixture model framework for automatic segmentation of MR brain images, IEEE Transactions on Medical Imaging25(9): 1233-1245. |
| [8] | Kłopotek, R., Kłopotek, M. and Wierzchoń, S. (2020). A feasible k-means kernel trick under non-Euclidean feature space, International Journal of Applied Mathematics and Computer Science30(4): 703-715, DOI: 10.34768/amcs-2020-0052. · Zbl 1467.62089 |
| [9] | Kulczycki, P. (2018). Kernel estimators for data analysis, in M. Ram and J.P. Davim (Eds), Advanced Mathematical Techniques in Engineering Sciences, CRC/Taylor & Francis, Boca Raton, pp. 177-202. |
| [10] | López-Lobato, A.L. and Avendaño-Garrido, M.L. (2020). Using the Gini index for a Gaussian mixture model, in L. Martínez-Villaseñor et al. (Eds), Advances in Computational Intelligence. MICAI 2020, Lecture Notes in Computer Science, Vol. 12469, Springer, Cham, pp. 403-418. |
| [11] | Mao, C., Lu, L. and Hu, B. (2020). Local probabilistic model for Bayesian classification: A generalized local classification model, Applied Soft Computing93: 106379. |
| [12] | Meng, X.-L. and Rubin, D.B. (1994). On the global and componentwise rates of convergence of the EM algorithm, Linear Algebra and its Applications199(Supp. 1): 413-425. · Zbl 0818.65153 |
| [13] | Povey, D., Burget, L., Agarwal, M., Akyazi, P., Kai, F., Ghoshal, A., Glembek, O., Goel, N., Karafiát, M., Rastrow, A., Rose, R., Schwarz, P. and Thomas, S. (2011). The subspace Gaussian mixture model: A structured model for speech recognition, Computer Speech & Language25(2): 404-439. |
| [14] | Rachev, S., Klebanov, L., Stoyanov, S. and Fabozzi, F. (2013). The Methods of Distances in the Theory of Probability and Statistics, Springer, New York, pp. 659-663. · Zbl 1280.60005 |
| [15] | Reynolds, D.A. (2009). Gaussian mixture models, in S.Z. Li (Ed.), Encyclopedia of Biometrics, Springer, New York, pp. 659-663. |
| [16] | Rubner, Y., Tomasi, C. and Guibas, L.J. (2000). The Earth mover’s distance as a metric for image retrieval, International Journal of Computer Vision40(2): 99-121. · Zbl 1012.68705 |
| [17] | Singh, R., Pal, B.C. and Jabr, R.A. (2009). Statistical representation of distribution system loads using Gaussian mixture model, IEEE Transactions on Power Systems25(1): 29-37. |
| [18] | Torres-Carrasquillo, P.A., Reynolds, D.A. and Deller, J.R. (2002). Language identification using Gaussian mixture model tokenization, 2002 IEEE International Conference on Acoustics, Speech, and Signal Processing, Orlando, USA, pp. I-757. |
| [19] | Ultsch, A. and Lötsch, J. (2017). A data science based standardized Gini index as a Lorenz dominance preserving measure of the inequality of distributions, PloS One12(8): e0181572. |
| [20] | Vaida, F. (2005). Parameter convergence for EM and MM algorithms, Statistica Sinica15(2005): 831-840. · Zbl 1087.62035 |
| [21] | Villani, C. (2003). Topics in Optimal Transportation, American Mathematical Society, Providence. · Zbl 1106.90001 |
| [22] | Xu, L. and Jordan, M.I. (1996). On convergence properties of the EM algorithm for Gaussian mixtures, Neural Computation8(1): 129-151. |
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