Clustering Analysis of Univariate Probability Distribution Model
Article Sidebar
Main Article Content
Abstract
Accurately modeling chlorophyll distribution is essential for understanding plankton dynamics and assessing marine ecosystem health. This study evaluates the suitability of various univariate probability distribution models for chlorophyll data, including the exponential, gamma, log-normal, logistic, log-logistic, normal, Rayleigh, Generalized Extreme Value (GEV), and Weibull distributions. The model selection process incorporates Goodness-of-Fit (GoF) tests (Kolmogorov-Smirnov and Anderson-Darling) and information criteria (AIC, BIC, AICc, CAIC, and HQC) to assess both statistical fit and model complexity. Additionally, the k-means clustering algorithm is applied to classify distributions based on their GoF test performance, with the Calinski-Harabasz Index (CHI) used to determine the optimal number of clusters. The results indicate that the gamma, log-normal, logistic, log-logistic, normal, and GEV distributions consistently demonstrate superior fit across multiple sample sizes, making them the most appropriate models for chlorophyll data. Conversely, the Exponential and Rayleigh distributions exhibit poor performance, particularly for larger datasets, due to their inability to capture data heterogeneity. The clustering analysis reveals that the GEV distribution tends to group with the exponential distribution in small samples ( ) but aligns with the gamma and log-normal distributions as the sample size increases ( ). This finding suggests that sample size significantly influences distribution stability and clustering behavior. This study contributes to probability distribution modeling by integrating GoF-based clustering approaches, reducing ambiguity in model selection for heterogeneous marine datasets. The findings emphasize the importance of selecting flexible distributions, particularly in ecological and oceanographic applications, where data variability is high. Future research should explore alternative GoF tests (e.g., Cramer-von Mises, Kuiper) and advanced clustering methods (e.g., DBSCAN, Gaussian Mixture Models) to enhance distribution selection for complex marine environments, supporting Marine Life-related SDGs.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Copyright Transfer Statement
The copyright of this article is transferred to Current Applied Science and Technology journal with effect if and when the article is accepted for publication. The copyright transfer covers the exclusive right to reproduce and distribute the article, including reprints, translations, photographic reproductions, electronic form (offline, online) or any other reproductions of similar nature.
The author warrants that this contribution is original and that he/she has full power to make this grant. The author signs for and accepts responsibility for releasing this material on behalf of any and all co-authors.
Here is the link for download: Copyright transfer form.pdf
References
Ashari, I. F., Dwi Nugroho, E., Baraku, R., Novri Yanda, I., & Liwardana, R. (2023). Analysis of elbow, silhouette, Davies-Bouldin, Calinski-Harabasz, and Rand-index evaluation on K-means algorithm for classifying flood-affected areas in Jakarta. Journal of Applied Informatics and Computing, 7(1), 95-103. https://doi.org/10.30871/jaic.v7i1.4947
Badr, M. M. (2019). Goodness-of-fit tests for the Compound Rayleigh distribution with application to real data. Heliyon, 5(8), Article e02225. https://doi.org/10.1016/j.heliyon.2019.e02225
Balakrishnan, N., Chimitova, E., & Vedernikova, M. (2015). An empirical analysis of some nonparametric goodness-of-fit tests for censored data. Communications in Statistics: Simulation and Computation, 44(4), 1101-1115. https://doi.org/10.1080/03610918.2013.796982
Basheer, A. M. (2019). Alpha power inverse Weibull distribution with reliability application. Journal of Taibah University for Science, 13(1), 423-432. https://doi.org/10.1080/16583655.2019.1588488
Behrenfeld, M. J., & Boss, E. S. (2014). Resurrecting the ecological underpinnings of ocean plankton blooms. Annual Review of Marine Science, 6(1), 167-194. https://doi.org/10.1146/annurev-marine-052913-021325
Bishop, C. H. (2016). The GIGG-EnKF: Ensemble Kalman filtering for highly skewed non-negative uncertainty distributions. Quarterly Journal of the Royal Meteorological Society, 142, 1395-1412. https://doi.org/10.1002/qj.2742
Burnham, K. P., & Anderson, D. R. (2002). Model selection and multimodel inference: A practical information-theoretic approach (2nd ed.). Springer.
Campbell, J. W. (1995). The lognormal distribution as a model for bio-optical variability in the sea. Journal of Geophysical Research, 100(C7), 13237-13254. https://doi.org/10.1029/95jc00458
de Souza, A., Aristone, F., Fernandes, W. A., Oliveira, A. P. G., Olaofe, Z., Abreu, M. C., Junior, J. F. de O., Cavazzana, G., dos Santos, C. M., & Pobocikova, I. (2020). Analysis of ozone concentrations using probability distributions. Ozone: Science and Engineering, 42(6), 539-550. https://doi.org/10.1080/01919512.2020.1736987
Devred, E., Sathyendranath, S., Stuart, V., & Platt, T. (2011). A three-component classification of phytoplankton absorption spectra: Application to ocean-color data. Remote Sensing of Environment, 115(9), 2255-2266. https://doi.org/10.1016/j.rse.2011.04.025
Ding, J., Tarokh, V., & Yang, Y. (2018). Model selection techniques: An overview. IEEE Signal Processing Magazine, 35(6), 16-34. https://doi.org/10.1109/MSP.2018.2867638
Dutkiewicz, S., Hickman, A. E., Jahn, O., Gregg, W. W., Mouw, C. B., & Follows, M. J. (2015). Capturing optically important constituents and properties in a marine biogeochemical and ecosystem model. Biogeosciences, 12(14), 4447-4481. https://doi.org/10.5194/bg-12-4447-2015
Goual, H., & Yousof, H. M. (2020). Validation of Burr XII inverse Rayleigh model via a modified chi-squared goodness-of-fit test. Journal of Applied Statistics, 47(3), 393-423. https://doi.org/10.1080/02664763.2019.1639642
Guedes, K. S., de Andrade, C. F., Rocha, P. A. C., Mangueira, R. dos S., & de Moura, E. P. (2020). Performance analysis of metaheuristic optimization algorithms in estimating the parameters of several wind speed distributions. Applied Energy, 268, Article 114952. https://doi.org/10.1016/j.apenergy.2020.114952
Gulati, S., & Shapiro, S. (2009). A new goodness of fit test for the logistic distribution. Journal of Statistical Theory and Practice, 3(3), 567-576. https://doi.org/10.1080/15598608.2009.10411947
Haggag, M. M. M. (2014). New criteria of model selection and model averaging in linear regression models. American Journal of Theoretical and Applied Statistics, 3(5), 148-166. https://doi.org/10.11648/j.ajtas.20140305.15
Indrayani, E., Dimara, L., Paiki, K., & Reba, F. (2018). The analysis of phytoplankton abundance using Weibull distribution (a case study in the coastal area of East Yapen in the regency of Yapen Islands, Papua). Journal of Education and Learning, 7(3), 251-258. https://doi.org/10.5539/jel.v7n3p251
Irwin, A. J., Finkel, Z. V., Schofield, O. M. E., & Falkowski, P. G. (2006). Scaling-up from nutrient physiology to the size-structure of phytoplankton communities. Journal of Plankton Research, 28(5), 459-471. https://doi.org/10.1093/plankt/fbi148
Jain, A. K. (2010). Data clustering: 50 years beyond K-means. Pattern Recognition Letters, 31(8), 651-666. https://doi.org/10.1016/j.patrec.2009.09.011
Jiang, H., Wang, J., Wu, J., & Geng, W. (2017). Comparison of numerical methods and metaheuristic optimization algorithms for estimating parameters for wind energy potential assessment in low wind regions. Renewable and Sustainable Energy Reviews, 69, 1199-1217. https://doi.org/10.1016/j.rser.2016.11.241
Jung, C., & Schindler, D. (2017). Global comparison of the goodness-of-fit of wind speed distributions. Energy Conversion and Management, 133, 216-234. https://doi.org/10.1016/j.enconman.2016.12.006
Kumar, V., Shanu, S., & Jahangeer, J. (2017). Statistical distribution of rainfall in Uttarakhand, India. Applied Water Science, 7(8), 4765-4776. https://doi.org/10.1007/s13201-017-0586-5
Lima, A. O., Lyra, G. B., Abreu, M. C., Oliveira-Júnior, J. F., Zeri, M., & Cunha-Zeri, G. (2021). Extreme rainfall events over Rio de Janeiro State, Brazil: Characterization using probability distribution functions and clustering analysis. Atmospheric Research, 247, Article 105221. https://doi.org/10.1016/j.atmosres.2020.105221
Lombard, F., Boss, E., Waite, A. M., Vogt, M., Uitz, J., Stemmann, L., Sosik, H. M., Schulz, J., Romagnan, J.-B., Picheral, M., Pearlman, J., Ohman, M. D., Niehoff, B., Möller, K. O., Miloslavich, P., Lara-Lopez, A., Kudela, R. M., Lopes, R. M., Kiko, R., Karp-Boss, L., Jaffe, J. S.,... Appeltans, W. (2019). Globally consistent quantitative observations of planktonic ecosystems. Frontiers in Marine Science, 6, Article 196. https://doi.org/10.3389/fmars.2019.00196
Mansoor, M., Tahir, M. H., Cordeiro, G. M., Provost, S. B., & Alzaatreh, A. (2019). The Marshall-Olkin logistic-exponential distribution. Communications in Statistics - Theory and Methods, 48(2), 220-234. https://doi.org/10.1080/03610926.2017.1414254
Mbah, A. K., & Paothong, A. (2015). Shapiro–Francia test compared to other normality test using expected p-value. Journal of Statistical Computation and Simulation, 85(15), 3002-3016. https://doi.org/10.1080/00949655.2014.947986
Mohammadi, K., Alavi, O., & McGowan, J. G. (2017). Use of Birnbaum-Saunders distribution for estimating wind speed and wind power probability distributions: A review. Energy Conversion and Management, 143, 109-122. https://doi.org/10.1016/j.enconman.2017.03.083
Ouarda, T. B. M. J., Charron, C., & Chebana, F. (2016). Review of criteria for the selection of probability distributions for wind speed data and introduction of the moment and L-moment ratio diagram methods, with a case study. Energy Conversion and Management, 124, 247-265. https://doi.org/10.1016/j.enconman.2016.07.012
Ozay, C., & Celiktas, M. S. (2016). Statistical analysis of wind speed using two-parameter Weibull distribution in Alaçati region. Energy Conversion and Management, 121, 49-54. https://doi.org/10.1016/j.enconman.2016.05.026
Pobočíková, I., Sedliačková, Z., & Michalková, M. (2017). Application of Four Probability Distributions for Wind Speed Modeling. Procedia Engineering, 192, 713-718. https://doi.org/10.1016/j.proeng.2017.06.123
Rajakaruna, H., VandenByllaardt, J., Kydd, J., & Bailey, S. (2018). Modeling the distribution of colonial species to improve estimation of plankton concentration in ballast water. Journal of Sea Research, 133, 166-176. https://doi.org/10.1016/j.seares.2017.08.005
Siegel, D. A., Doney, S. C., & Yoder, J. A. (2002). The North Atlantic spring phytoplankton bloom and Sverdrup’s critical depth hypothesis. Science, 296(5568), 730-733. https://doi.org/10.1126/science.1069174
ul Haq, M. A., Rao, G. S., Albassam, M., & Aslam, M. (2020). Marshall–Olkin Power Lomax distribution for modeling of wind speed data. Energy Reports, 6, 1118-1123. https://doi.org/10.1016/j.egyr.2020.04.033
Van, T. V., & Pham-Gia, T. (2010). Clustering probability distributions. Journal of Applied Statistics, 37(11), 1891-1910. https://doi.org/10.1080/02664760903186049
Wang, J., Hu, J., & Ma, K. (2016). Wind speed probability distribution estimation and wind energy assessment. Renewable and Sustainable Energy Reviews, 60, 881-899. https://doi.org/10.1016/j.rser.2016.01.057
Wang, X., & Xu, Y. (2019). An improved index for clustering validation based on silhouette index and Calinski-Harabasz index. IOP Conference Series: Materials Science and Engineering, 569(5), Article 052024. https://doi.org/10.1088/1757-899X/569/5/052024
Wasserman, L. (2000). Bayesian model selection and model averaging. Journal of Mathematical Psychology, 44(1), 92-107. https://doi.org/10.1006/jmps.1999.1278
Yazici, B., & Yolacan, S. (2007). A comparison of various tests of normality. Journal of Statistical Computation and Simulation, 77(2), 175-183. https://doi.org/10.1080/10629360600678310
Zheng, S., Fan, K., Hou, Y., Feng, J., & Fu, Y. (2023). Clustering by the probability distributions from extreme value theory. IEEE Transactions on Artificial Intelligence, 4(2), 292-303. https://doi.org/10.1109/TAI.2022.3153592
