Volume-9 , Issue-4 , Aug 2022, ISSN 2348-4519 (Online) Go Back

• Open Access   Article

  Modified Ratio Type Estimator for the Estimation of Population Mean under Adaptive Cluster Sampling

  A.A. Bhat, M. Sharma, M.I.J. Bhat, S.H. Bhat

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.9 , Issue.4 , pp.1-4, Aug-2022

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  Abstract
  Adaptive Cluster Sampling (ACS) design is a leading sampling design developed for estimating the population parameters of rare, clustered, endangered, hard to reach and challenging population. In this article a modified ratio type estimator for the estimation of population mean has been developed under ACS design. The mathematical equation for approximate mean square error (MSE) has been derived upto first order of approximation and the optimum values (p and q) of the developed estimators have also been obtained. The effectiveness of the developed estimator has been empirically evaluated using the simulated x- and y-values. The results on the basis of percentage relative efficiency indicated that the developed estimator is superior as compared to other known conventional and ACS estimators when there is a positive correlation between study and the auxiliary variable.

  Key-Words / Index Term
  ACS, Ratio Estimator, Bias, Mean Square Error, Efficiency

  References
  [1] A.L. Dryver & C.T. Chao, “Ratio Estimators in Adaptive Cluster Sampling,” Environmetrics, Vol.18, Issue.6, pp. 607–620, 2007.
  [2] B. Kadilar & H. Cingi, “A Study on the Chain Ratio-Type Estimator, Hacettepe Journal of Mathematics and Statistics,” Vol.32, pp.105-108, 2003.
  [3] B. Kumar, M. Sharma, S.E.H. Rizvi, & M. I. Jeelani, “Estimation of Population Mean Through Improved Class Of Ratio Type Estimators using Auxiliary Information,” Journal of Reliability and Statistical Studies, Vol.11, Issue.1, pp. 01-07, 2018.
  [4] D.G. Horvitz & D.J. Thompson, “A Generalization of Sampling without Replacement from a Finite Universe,” Journal of American Statistical Association, Vol.47, Issue.260, pp. 663–685, 1952.
  [5] D.S. Robson, “Applications of Multivariate Polykays to the Theory of Unbiased Ratio-Type Estimation,” Journal of American Statistical Association, Vol.52, Issue.280, pp. 511-522, 1957.
  [6] H.P. Singh & R. Tailor, Use of Known Correlation Coefficient in Estimating the Final Population Mean, Statistics in Transition, Vol.6, pp.555-560, 2003.
  [7] M. I. Jeelani, S.E.H. Rizvi, M. Sharma, S. A. Mir, S.A. Raja, S.A. Maqbool, N. Nazir, & F. Jeelani, Modified Ratio Estimator under Rank Set Sampling,” Revista Investigacion Operacional, Vol.38, Issue.1, pp.103-106, 2017.
  [8] M. Sharma, & S. Bhatnagar, “A General Class of Estimator for the Estimation of Population Mean using Auxiliary Information,” Journal of the Indian Statistical Association, Vol.46, Issue.2, pp. 155-161, 2008.
  [9] M.M. Hansen & W.N. Hurwitz, “On the Theory of Sampling from Finite Population,” Ann Math Stat, Vol.14, Issue.4, pp. 333–362, 1943.
  [10] N. Chutiman, “Adaptive Cluster Sampling using Auxiliary Variable,” J. Math. & Stat., Vol.9, Issue.3, pp. 249-255, 2013.
  [11] N. Upadhyaya & H.P. Singh, “Use Of Transformed Auxiliary Variable in Estimating the Finite Population Mean,” Biometri. J., Vol.41, Issue.5, pp. 627-636, 1999.
  [12] S. Hussain, M. Sharma & H. Chandra, “Optimum Exponential Ratio Type Estimator for Estimating the Population Mean,” J. Stat. Appl. Prob. Lett., Vol.8, Issue.2, pp.73-82, 2021.
  [13] S.K. Thompson, “Adaptive Cluster Sampling,” Journal of American Statistical Association, Vol.85, Issue.412, pp. 1050–1059, 1990.
  [14] S.K. Yadav, S. Mishra, S.S. Mishra & N. Chutiman, “Improved Ratio Estimators Of Population Mean In Adaptive Cluster Sampling,” Journal of Statistical Applications & Probability Letters, Vol.3, Issue.1, pp.01-06, 2016.
  [15] V. S. Sisodia, & V. K. Dwivedi, “A Modified Ratio Estimator using Coefficient of Variation of Auxiliary Variable,” J. Indian Soc. Agric. Statist., Vol.33, Issue.2, pp. 13-18, 1981.
  [16] W.G. Cochran, “The Estimation of the Yields of Cereal Experiments by Sampling for the Ratio of Grain to Total Produce,” Journal Agricultural Science, Vol.30, Issue.2, pp. 262-275, 1940.
  [17] Z. Yan and B. Tian, Ratio Method to The Mean Estimation using Co-Efficient of Skewness of Auxiliary Variable, ICICA, Part II, CCIS 106, pp.103-110,2010.

  Citation
  A.A. Bhat, M. Sharma, M.I.J. Bhat, S.H. Bhat, "Modified Ratio Type Estimator for the Estimation of Population Mean under Adaptive Cluster Sampling," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.9, Issue.4, pp.1-4, 2022

• Open Access   Article

  On the Performance Robustness of Artificial Neural Network Approaches and Gumbel Extreme Value Distribution for Prediction of Wind Speed

  Amr R. Kamel, Abdulaziz A. Alqarni, Moataz A. Ahmed

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.9 , Issue.4 , pp.5-22, Aug-2022

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  Engineers and scientists have historically focused on modeling and predicting extreme values of geophysical variables, such as wind, ocean surface waves, sea level, temperature, and river flow. In particular, the analysis of wind speed is crucial for different applications in maritime, environmental, civil engineering as well as for the preparation of mitigation and management of natural disasters. This topic is gaining interest in the field of renewable generation, from the viewpoint of assessing both wind power production and wind-tower mechanical reliability and safety. In order to generate trustworthy information for decision support, it is necessary to analyze data spanning extended time periods and recorded at diverse places. Such data must also be analyzed using a variety of techniques. In this paper, we study the Type I Gumbel extreme value (GEV) distribution using graphical and analytical methods for estimation its parameters and determine the best estimators of the unknown parameters of the GEV distribution which can be used to prediction the extreme wind speed. Also, Bayesian estimators of the parameters are derived by using Markov Chain Monte Carlo (MCMC) methods. Moreover, we proposed Bayesian GEV distribution approach to compare the Artificial Neural Networks (ANNs) based on Multi-Layer Perceptrons (MLPs) network for prediction of wind speed. The performance Robustness of the Bayesian GEV distribution approach and ANNs used in wind speed predication are evaluated by some model performance indicators viz, mean absolute error, mean relative error, root mean square error and correlation coefficient. The results showed that the proposed Bayesian GEV distribution approach is found to be better suited for prediction of wind speed.

  Key-Words / Index Term
  Artificial neural networks; Bayesian estimation; Extreme value theory; Gumbel extreme value distribution; MCMC; Robustness; Evaluation metrics; Wind speed prediction.

  References
  [1] Manwell, J. F., McGowan, J. G., and Rogers, A. L. Wind energy explained: theory, design and application. John Wiley & Sons. Chichester, UK, ?2010.
  [2] Sivanandam, S. N., and Deepa, S. N. Introduction to neural networks using Matlab 6.0. Tata McGraw-Hill Education, 2006.
  [3] Alharbi, A. A., Kamel, A.R., and Atia, S. A. A New Robust Molding of Heat and Mass Transfer Process in MHD Based on Adaptive-Network-Based Fuzzy Inference System. WSEAS Transactions on Heat and Mass Transfer, Vol.17, pp 80-96, 2022.
  [4] Kotz, S., and Nadarajah, S. Extreme value distributions: theory and applications. world scientific, 2000.
  [5] Lee, B. H., Ahn, D. J., Kim, H. G., and Ha, Y. C. (2012). An estimation of the extreme wind speed using the Korea wind map. Renewable energy, Vol.42, pp 4-10, 2012.
  [6] Mahdi, S., and Cenac, M. Estimating Parameters of Gumbel Distribution using the Methods of Moments, probability weighted Moments and maximum likelihood. Revista de Matemática: Teoríay Aplicaciones, Vol.12, pp 151-156, 2005.
  [7] Aydin, D., and ?eno?lu, B. Monte Carlo comparison of the parameter estimation methods for the two-parameter Gumbel distribution. Journal of Modern Applied Statistical Methods, Vol.14, 2015.
  [8] Dey, S., Mazucheli, J., and Nadarajah, S. Kumaraswamy distribution: different methods of estimation. Computational and Applied Mathematics, Vol.37, pp 2094-2111, 2018.
  [9] Alharbi, Y. S., and Kamel, A.R. Fuzzy System Reliability Analysis for Kumaraswamy Distribution: Bayesian and Non-Bayesian Estimation with Simulation and an Application on Cancer Data Set.? WSEAS Transactions on Biology and Biomedicine, Vol.19, pp118-139, ?2022.
  [10] Demirhan, H., and Kalaylioglu, Z. On the generalized multivariate Gumbel distribution. Statistics & Probability Letters, Vol.103, pp 93-99.? 2015.
  [11] Cordeiro, G. M., Nadarajah, S., and Ortega, E. M. The kumaraswamy gumbel distribution. Statistical Methods & Applications, Vol.21, pp 139-168, 2012.
  [12] Cooray, K. Generalized gumbel distribution. Journal of Applied Statistics, Vol.37, pp 171-179, 2010.
  [13] Azad, H. B., Mekhilef, S., and Ganapathy, V. G. Long-term wind speed forecasting and general pattern recognition using neural networks. IEEE Transactions on Sustainable Energy, Vol.5, pp 546-553, 2014.
  [14] El-Shanshoury, G. I., and Ramadan, A. A. Estimation of extreme value analysis of wind speed in the north-western coast of Egypt. Arab J Nucl Sci Appl, Vol.45, pp 265-274, ?2012.
  [15] Barhmi, S., Elfatni, O., and Belhaj, I. Forecasting of wind speed using multiple linear regression and artificial neural networks. Energy Systems, Vol.11, pp 935-946, 2020.
  [16] Chiodo, E., Fantauzzi, M., and Mazzanti, G. The Compound Inverse Rayleigh as an Extreme Wind Speed Distribution and Its Bayes Estimation. Energies, Vol.15, 2022.
  [17] Fu, J. Y., Li, Q. S., and Xie, Z. N. Prediction of wind loads on a large flat roof using fuzzy neural networks. Engineering Structures, Vol.28, pp 153-161, 2006.
  [18] McCulloch, W. S., and Pitts, W. A logical calculus of the ideas immanent in nervous activity. The bulletin of mathematical biophysics, Vol.5, pp 115-133, 1943.
  [19] Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E. Equation of state calculations by fast computing machines. The journal of chemical physics, Vol.21, pp 1087-1092, 1953.
  [20] Rumelhart, D. E. Hinton and R. Williams. Learning Internal Representations by Error Propagation, pp 318-362., 1986.
  [21] Lek, S., and Park, Y, S. Artificial Neural Networks. In: Jorgensen, S.E., Fath, B. (Eds.), Encyclopedia of Ecology. Elsevier, Amsterdam, pp. 237-245. 2008.
  [22] Perez-Garcia, A. N., Tornez-Xavier, G. M., Flores-Nava, L. M., Gómez-Castañeda, F., and Moreno-Cadenas, J. A. Multilayer perceptron network with integrated training algorithm in FPGA. In 2014 11-th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE) (pp. 1-6). IEEE.? 2014.
  [23] Deshpande, R. R. On the rainfall time series prediction using multilayer perceptron artificial neural network. International Journal of emerging technology and advanced engineering, Vol.2, pp 2250-2459, 2012.
  [24] Velo, R., López, P., and Maseda, F. Wind speed estimation using multilayer perceptron. Energy Conversion and Management, Vol.81, pp 1-9, 2014.
  [25] van Cranenburgh, S., and Kouwenhoven, M. An artificial neural network based method to uncover the value-of-travel-time distribution. Transportation, Vol.48, pp 2545-2583.? 2021.
  [26] Fosgerau, M. Using nonparametrics to specify a model to measure the value of travel time. Transportation Research Part A: Policy and Practice, Vol.41, pp 842-856, 2007.
  [27] Coles, S., Bawa, J., Trenner, L., and Dorazio, P. An introduction to statistical modeling of extreme values (Vol. 208, p. 208). London: Springer.? 2001
  [28] Kamel, A.R. Handling outliers in seemingly unrelated regression equations model, MSc thesis, Faculty of graduate studies for statistical research (FGSSR), Cairo University, Egypt, 2021.
  [29] Gumbel, E.J. Statistical Theory of Extreme Values and Some Practical Applications, a Series of Lectures, Vol. 33, pp.1–49, National Bureau of Standards, Applied Mathematics Series, US Government Printing Office, 1954.
  [30] Hansen, A. The three extreme value distributions: An introductory review. Frontiers in Physics, Vol. 8, pp 1-8,? 2020.
  [31] Bali, T. G. The generalized extreme value distribution. Economics letters, Vol. 79, pp 423-427, 2003.
  [32] Ramachandran, G. Properties of extreme order statistics and their application to fire protection and insurance problems. Fire Safety Journal, Vol. 5, pp 59-76, 1982.
  [33] Brown, B. G., and Katz, R. W. Regional analysis of temperature extremes: Spatial analog for climate change?. Journal of Climate, Vol. 8, pp 108-119, 1995.
  [34] Naess, A. Estimation of long return period design values for wind speeds. Journal of engineering mechanics, Vol. 124, pp 252-259, 1998.
  [35] Broussard, J. P., and Booth, G. G. The behavior of extreme values in Germany`s stock index futures: An application to intradaily margin setting. European Journal of Operational Research, Vol. 104, pp 393-402, 1998.
  [36] Xapsos, M. A., Summers, G. P., and Burke, E. A. Extreme value analysis of solar energetic proton peak fluxes. Solar Physics, Vol.183, pp 157-164, seller’s, 1998.
  [37] Harlow, D. G., Smith, R. L., and Taylor, H. M. Lower tail analysis of the distribution of the strength of load-sharing systems. Journal of Applied Probability, Vol. 20, pp 358-367,? 1983.
  [38] Kuru, G. A., Whyte, A. G. D., and Woollons, R. C. Utility of reverse Weibull and extreme value density functions to refine diameter distribution growth estimates. Forest ecology and management, Vol. 48(1-2), pp 165-174,? 1992.
  [39] Jeribi, A., Ghorbel, A. Forecasting developed and BRICS stock markets with cryptocurrencies and gold: generalized orthogonal generalized autoregressive conditional heteroskedasticity and generalized autoregressive score analysis. International Journal of Emerging Markets, in press. 2021. https://doi.org/10.1108/ijoem-06-2020-0688.
  [40] Abonazel, M. R., and Abd-Elftah, A. I. Forecasting Egyptian GDP using ARIMA models. Reports on Economics and Finance, Vol.5, pp 35-47, 2019.
  [41] Swain, J. J., Venkatraman, S., and Wilson, J. R. Least-squares estimation of distribution functions in Johnson`s translation system. Journal of Statistical Computation and Simulation, Vol. 29, pp 271-297, 1988.
  [42] Kao, J. H. Computer methods for estimating Weibull parameters in reliability studies. IRE Transactions on Reliability and Quality Control, pp 15-22,? 1958
  [43] Youssef, A. H., Kamel, A. R., and Abonazel, M. R. Robust SURE estimates of profitability in the Egyptian insurance market. Statistical Journal of the IAOS, Vol. 37, pp 1275-1287,? 2021.
  [44] Youssef, A. H., Kamel, A. R., and Abonazel, M. R. Efficiency Comparisons of Robust and Non-Robust Estimators for Seemingly Unrelated Regressions Model. WSEAS Transactions on Mathematics, Vol. 21, pp 218-244, 2022.
  [45] Almongy, H. M., and Almetwally, E. M. Robust estimation methods of generalized exponential distribution with outliers. Pakistan Journal of Statistics and Operation Research, pp 545-559, 2020.
  [46] Gauss, C. F. Least Squares method for the Combinations of Observations. Translated by J. Bertrand.? Mallet-Bachelier, Paris, 1955.
  [47] Kumar, K. Classical and Bayesian estimation in log-logistic distribution under random censoring. International Journal of System Assurance Engineering and Management, Vol. 9, pp 440-451, 2018.
  [48] Kamel, A.R. Markov Chain Monte Carlo Techniques: A Review and Comparison. Working paper, Faculty of Graduate Studies for Statistical Research, Cairo University, Egypt, 2022.
  [49] Hastings, W. K. Monte Carlo sampling methods using Markov chains and their applications.? Biometrika, Vol.57, pp. 97–109. 1970.
  [50] Gómez, Y. M., Bolfarine, H., and Gómez, H. W. Gumbel distribution with heavy tails and applications to environmental data. Mathematics and Computers in Simulation, Vol. 157, pp. 115-129, 2019.
  [51] Al-Imam, A. A Gateway Towards Machine Learning: Predictive Analytics and Neural Networks in IBM-SPSS (SPSS v. 24). Retrieved January, Vol. 3, 2019.?
  [52] Kamel, A. R., and Alqarni, M.A. A New Characterization of Exponential Distribution through Minimum Chi-Squared Divergence Principle. Journal of Advanced Research in Applied Mathematics and Statistics, Vol. 5, pp. 14-26, 2020.
  [53] Alizadeh, M., Mojarad, M., and Arfaeinia, H. Provide a Dynamic Routing Algorithm for MPLS Networks Using Fuzzy Filtering Approach. International Journal of Scientific Research in Network Security and Communication, Vol.9, Issue.1, pp.1-7, 2021.
  [54] Abonazel, M., and Rabie, A. The impact of using robust estimations in regression models: An application on the Egyptian economy. Journal of Advanced Research in Applied Mathematics and Statistics, Vol. 4, pp. 8-16.? 2019.
  [55] Abonazel, M., and Nourhan, E. Using the ARDL bound testing approach to study the inflation rate in Egypt. Economic consultant, Vol. 3, pp. 24-41.? 2020.
  [56] Mohamed S.A., and Mohamed S.M. Estimation Methods of Kumaraswamy Lindley Regression Model with Applications on Egyptian Stock Exchange Data. International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.9, Issue.1, pp.19-23, 2022.

  Citation
  Amr R. Kamel, Abdulaziz A. Alqarni, Moataz A. Ahmed, "On the Performance Robustness of Artificial Neural Network Approaches and Gumbel Extreme Value Distribution for Prediction of Wind Speed," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.9, Issue.4, pp.5-22, 2022

• Open Access   Article

  Modelling the Reliability of a Complex Repairable System (Aircraft Engine) Using Weibull Distribution Model and Gamma Distribution Model (A Case Study of Arik Airline Aircraft Engine)

  N.F. Adeleye, O. Fatoki

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.9 , Issue.4 , pp.23-32, Aug-2022

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  Abstract
  The reliability of a complex repairable system that is mended (but not replaced) after failure will frequently rely on the system`s age in chronological terms. A non-homogeneous Poisson process can be used to describe this age-dependent dependability if only minimum repairs are undertaken in order to prevent the immediate rate of system failure from being perturbed. This study examined the theoretical and practical consequences of the non-homogeneous Poisson process model for reliability and provides estimation, comparison, and goodness of fit techniques when the process has a Weibull reliability process model and a Gamma reliability process model. Applications of the Weibull model in the field of reliability and the Gamma model also in the area of reliability are discussed. There are two categories of systems: non-repairable systems and repairable systems. The term ``system`` used in this research work simply means a component. If the continuous operation of a system is desired, then in the former case, the system would be repairable or replaced by a new system upon failure. The aircraft engine is a complicated device that periodically needs maintenance and repair. This research discussed how the Weibull process model (a non-homogeneous Poisson process) and the Gamma distribution model can be used as a new approach to modelling aircraft engine life. The two models were compared to determine which model was the better model. This is to improve the quality of services provided by the aviation industries and to also reduce the rate of plane crash in the aviation industry. A methodology constructed on data generated from several systems was used to estimate the process parameters. A methodology constructed on data generated from several systems was used to estimate the process parameters. By analysing both data fit and forecasting accuracy, these models` overall capability is determined.

  Key-Words / Index Term
  Reliability, Complex System, System Failure, Non-Homogeneous Poisson Process (NHPP), Homogeneous Poisson Process (HPP), Weibull Model, Gamma Model, Non-repairable System, Repairable System, Total Reliability Period, Total Flying Time, Flying Hours, Aircraft Engine, Parameters.

  References
  [1] H. Ascher., & H. Feingold, “Repairable Systems Reliability: Modeling, Inference, Misconceptions and their causes,” Marcel Dekker NY 1984.
  [2] H. L. Crow, “Evaluating the reliability of repairable systems,” Annual Reliability and Maintainability Symposium, pp.275-279, 1990.
  [3] C. E. Love, & R. Guo, “An application of a bathtub failure model to imperfectly repaired systems data,” Quality and Reliability Engineering International 9, pp. 127-135, 1993.
  [4] H. L. Crow, “Confidence intervals on the reliability of repairable systems.” In Proceedings of the Annual Reliability and Maintainability Symposium, pp. 126-133, 1993.
  [5] G. Kleinert, G. Gregg, “Designed-in maintenance and cost Innovations,” AIM Yearbook, 1990.
  [6] E. E. Lewis, “Introduction to reliability engineering,” Wiley NY 1987.
  [7] G. Kleinert, “The economics of high thrust turbofan maintenance.” Aircraft Maintenance World, August 1990.
  [8] J. T. Duane, “Learning curve approach to reliability monitoring.” IEEE Transactions on Aerospace 2, pp. 563-566 1964.
  [9] T. R. Moss, “Uncertainties in reliability statistics.” Reliability Engineering and System Safety 34, pp.79-90 1991.
  [10] P. Andersen, O. Borgan, R. Gill, N. Keiding, “Statistical Based on counting processes.” Springer 1992.
  [11] R. D. Baker, “Some new tests for the power law process.” Technometrics 38. Pp.256-265 1996.
  [12] R. E. Barlow, A. W. Marsall, F. Proschan, “Properties of probability distributions with monotone hazard rate.” Annuals of mathematical statistics 34, pp. 375-389, 1993.
  [13] M. Berman, “Inhomogeneous and modulated Gamma processes.” Biometrika 68, pp. 143-152, 1981.
  [14] M. Berman, T. R. Turner, “Approximate point process likelihoods with GLIM.” Applied statistics 41, pp.31-38, 1992.
  [15] M. T. Boswell, “Estimating and testing trend in a stochastic process of Poisson type.” Annuals of Mathematical Statistics 37, pp.1564-1573, 1996.
  [16] M. Brown, F. Proschan, “Imperfect repair.” Journal of Applied Probability 20, pp.851-859, 1983.
  [17] D. R. Cox, “Some statistical methods connected with series of events with discussion.” Journal of the Royal Statistical Society, Series B, 17, pp.129-164, 1955.
  [18] D. R. Cox, P. W. Lewis, “The statistical Analysis of series of Events.” Methuen, London, 1996.
  [19] D. R. Cox, “The statistical analysis of dependencies in point processes.” In Stochastic Point Processes, Wiley, NY pp.55-66, 1972.
  [20] B. Jabir, “Evaluation of Average Outgoing Quality for Reliability Acceptance Sampling Plan Based on the Exponentiated Rayleigh Distribution.” International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.8, Issue.6, pp.40-45, 2021.
  [20] M. J. Crowder, A. C. Kimber, R. L. Smith, T. J. Sweeting, “Statistical Analysis of Reliability Data.” Chapman & Hall, Great Britain, 1991.
  [21] H. H. Sureendra, D. Seshachalam, K. R. Sudhindra, “Reliability Analysis of Solar Energy Resources Using Weibull Distribution for a Standalone System in India Context.” Vol.7, Issue.1, pp.64-68, 2020.

  Citation
  N.F. Adeleye, O. Fatoki, "Modelling the Reliability of a Complex Repairable System (Aircraft Engine) Using Weibull Distribution Model and Gamma Distribution Model (A Case Study of Arik Airline Aircraft Engine)," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.9, Issue.4, pp.23-32, 2022

• Open Access   Article

  Trend Analysis Forecast of Life Expectancy at Birth in Nigeria

  I.M. Ogolo

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.9 , Issue.4 , pp.33-40, Aug-2022

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  Abstract
  The current study aims to examine trends in Nigeria`s birth life expectancy. Four different trend models of time series, including linear, quadratic, exponential, and s-curve trend models, were fitted to examples of life expectancy at birth in Nigeria (annual data set from 1950 to 2021). The best trend model was then chosen using model accuracy metrics like Mean Absolute Percentage (MAPE), Mean Absolute Deviation (MAD), and Mean Squared Deviation (MSD). The results showed that in all three model accuracy approaches examined, the quadratic and s-curve trend models compete favorably. On the other hand, the s-curve trend model had the lowest MAD and MAPE and best fit the data. A six-year projection of cases of life expectancy at birth in Nigeria was created using the s-curve trend model, and it showed that cases of life expectancy at birth in Nigeria will rise (grow) over the following six years, from 2022 to 2027. We urge the government to take charge of enhancing the delivery of products and services, including the establishment of a health care infrastructure, the training of health professionals, immunization programs, and other preventative health care initiatives.

  Key-Words / Index Term
  Trend Models, Forecast, Life Expectancy, Birth, Time Series Analysis.

  References
  [1] A. Galor, H.S. Moay, “Infant and child mortality: Levels trend and data deficiencies,”. Disease and Mortality in Sub-Shraran, Africa, pp. 31-42, 2005.
  [2] B. Chen, Y. Hong, “Testing for smooth structural changes in time series models via nonparametric regression,”. Econoometric, Vol. 80, Issue. 3, pp.1157-1183, 2001.
  [3] T.O. Hasegama, “Average and healthy life expectancies and self-rated health in the European Country”. China Prefectural University of Health Science, Vol.51, Issue.2, pp.144-150, 2014.
  [4] L.V. Hoi, H.O. Phuc, T.V. Dung, N.T. Chuc, L. Lindholm, “Remaining life expectancy among older people in rural area socio-economic inequalities during a Period of Multiple Transitions,”. BMC Public Health. Vol.9, p.471, 2009.
  [5] L. Kyte, C. Wells, “Variation in life expectancy between rural and urban areas of England,”. Health Stat Q. Vol.46, pp.25-50, 2010.
  [6] R.M. Chetty, S.A. Abraham, S. Lin, B. Scuderi, N. Turner, A. Bergeron, D. Cutler, “The association between income and life expectancy in the United State, 2001-2014” JAMA. Vol.315, Issue.16, 1pp.750-766, 2016.
  [7] H. Waldron, “Trends in mortality differentials and life expectancy for male and social security-covered workers by soci-economic status. Soc Secur Bull. Vol.67, Issue.3, pp.1-28, 2007.
  [8] D.J. Strauss, P.J. Achon, R.M. Shavelle, “Estimation of future mortality rates and life expectancy in chronic medical conditions,”. J Insur Med. Vol.37, Issue.1, pp.20-34, 2005.
  [9] D. Lai, R.J. Hardy, S.P. Tsai, “Statistical analysis of the standardized mortality ratio and life expectancy,”. School of Public Health, University of Texas. Am j Epidemiol. Vol.143, Issue.8, pp.832-840, 1996.
  [10] L. Nkpordee, N. Wonu, “Application of time series analysis on the forecasting of the outbreak of malaria epidemic in Nigeria”. International Journal of Statistics and Applied Mathematics; Vol.3, Issue.6, pp.98-105, 2018.

  Citation
  I.M. Ogolo, "Trend Analysis Forecast of Life Expectancy at Birth in Nigeria," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.9, Issue.4, pp.33-40, 2022

• Open Access   Article

  Construction of Diophantine Triples Involving Hexagonal Pyramidal Numbers

  C. Saranya, K. Manjula

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.9 , Issue.4 , pp.41-43, Aug-2022

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  Abstract
  We scrutinize for three particular polynomials with integer coefficients to such an extent that the result of any two numbers expanded by a non-zero number (or polynomials with number coefficients) is a perfect square.

  Key-Words / Index Term
  Dio triples, pyramidal numbers, Hexagonal pyramidal number, polynomials, integers, perfect square.

  References
  [1] Beardon, A.F. and Deshpande, M.N., “Diophantine triples”, The Mathematical Gazette, vol.86. pp. 258-260, 2002.
  [2] Bugeaud, Y.Dujella, A. and Mignotte, M., “On the family of Diophantine triples” {K-1, K+1, 16K3-4K}, Glasgow Math. J. vol.49. pp. 333-344, 2007.
  [3] Carmichael, R.D. “Theory of numbers and Diophantine triples”, Dover Publications.
  [4] Deshpande, M.N. “Families of Diophantine triples”, Bulletin of the Marathwada Mathematical Society, vol 4. pp. 19-21, 2003.
  [5] Hua, L.K. “Introduction to the Theory of Numbers”, Springer-Verlag, Berlin-New York, 1982.
  [6] Janaki, G. and Saranya, C., “Construction of the Diophantine Triple involving Pentatope Number”, International Journal for Research in Applied Science & Engineering Technology, vol.6. Issue III, March 2018.
  [7] Janaki, G. and Saranya, C. “Special Dio 3-tuples for pentatope number”, Journal of Mathematics and Informatics, vol.11, Special issue, 119-123, Dec 2017.
  [8] Janaki, G. and Saranya, C., “Half companion sequences of special dio 3-tuples involving centered square numbers”, International Journal for Recent Technology and Engineering, vol.8, issue 3, 3843-3845, September 2019.
  [9] Saranya C., and Janaki G., “Some Non-extendable Diophantine Triples involving centered square numbers”, International Journal of Scientific Research in Mathematical and Statistical Sciences, vol 6, Issue 6, 105-107, December 2019.
  [10] Saranya C., and Achya. B., “Special Diophantine triples involving square pyramidal numbers”, Indian Journal of Advanced Mathematics, Volume 1, Issue 2, 27-29, October 2021.
  [11] Saranya C., and Mahalakshmi, E., “Dio-Triples involving pentagonal pyramidal numbers”, International Journal of Scientific Research in Mathematical and Statistical Sciences, vol 8, Issue 6, 45-48, December 2021.
  [12] Saranya C., and Achya. B., “Diophantine triples involving square pyramidal numbers”, Advances and Applications in Mathematical Sciences, Volume 21, Issue 3, 1541-1547, January 2022.

  Citation
  C. Saranya, K. Manjula, "Construction of Diophantine Triples Involving Hexagonal Pyramidal Numbers," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.9, Issue.4, pp.41-43, 2022

• Open Access   Article

  Mathematical Modeling of the Causes, Complications and Solutions of Diabetes Mellitus

  Kanee Goodfaith Leyira

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.9 , Issue.4 , pp.44-49, Aug-2022

  Abstract

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  Abstract
  This research study focused on studying the causes, complications and possible solutions of Diabetes Mellitus through mathematical modeling and computing simulation in order to evaluate the medical condition of diabetic patients. And to find suitable and optimal methods to control and manage blood glucose levels in the body. The study examined various types of diabetes mellitus and different clinical categories. We improved and modified on the existing mathematical models for diabetes mellitus.

  Key-Words / Index Term
  Diabetes Mellitus, Complications, Glucose, Insulin, Mathematical Modeling, Metabolism, Population

  References
  [1] World Health Organisation, ‘’Definition and diagnosis of diabetes mellitus and intermediate hyperglycemia’’, WHO, Geneva, 2016.
  [2] A. Kouidere, A. Labzai, H. Ferjouchia, O. Balatif, M. Rachik, ‘’A New Mathematical Modeling with Optimal Control Strategy for the Dynamics of Population of Diabetics and Its Complications with Effect of Behavioral Factors’, Hindawi Journal of Applied Mathematics, pp 12 Volume 2020.
  [3] W. Banzi, I. Kambutse, V. Dusabejambo, E. Rutaganda, F. Minani, J. Niyobuhungiro, L. Mpinganzima, J.M. Ntagand, ‘’Mathematical Modelling of Glucose-Insulin System and Test of Abnormalities of Type 2 Diabetic Patients’’ Hindawi International Journal of Mathematics and Mathematical Sciences, pp 12 Volume 2021.
  [4] ADA (American Diabetes Association), “Diagnosis and classification of diabetes mellitus,” Diabetes Care, vol. 32, no. 1, pp. S62–S69, 2009.
  [5] I. MacFarlane, M. Bliss, and J. Jackson, ‘’History of Diabetes’’, Textbook of Diabetes, 2nd edition, 1997.
  [6] R Yadav, Maya, ‘’A Mathematical Model For The Study Of Diabetes Mellitus’’, Journal of Physics: Conference Series, 2020.
  [7] K.R.D. Anuradha, ‘’A Mathematical Model of Glucose-Insulin regulation under the influence of externally ingested glucose (G-I-E model)’’, International Journal of Mathematics and Statistics Invention 4(5) pp 54-58, 2016.
  [8] A. A. Jan, ‘’Causes, Complications and Management of Diabetes Mellitus’’ Chronicle Journal of Food and Nutrition, 1[1] pp 002, 2017.
  [9] H. Al Ali, A. Daneshkhah, A. Boutayeb, and Z. Mukandavire, ‘’Examining Type 1 Diabetes Mathematical Models Using Experimental Data’’, International Journal of Environmental Research and Public Health, 19 (737), 2022.
  [10] S. Peric and T.M. Stulnig, ‘’Diabetes and COVID-19’’, Wien Klin Wochenschr, 132, pp 356–361 2020.
  [11] A.K. Singh, R. Gupta, A Ghosh, A. Misra, ‘’Diabetes in COVID-19: Prevalence, pathophysiology, prognosis and practical considerations’’, Diabetes Metabolism Syndrome Clinical Research Review, 14, pp 303–310, 2020.
  [12] O. E. Ifeanyi, ‘’An update on Diabetes Mellitus’’, International Journal of Current Research in Medical Sciences 4(6), pp 71-81, 2018.
  [13] T. Marius, I. Mihai, R. Mariana, A. Dragos, I. Catalin, ‘’Blood Glucose Control Using Differential Equations Systems for Patients with Diabetes Type 1,’’ World Wide Journal of Multidisciplinary Research and Development, 5(1), pp 54-59, 2019.
  [14] N. Fitriyah, M. W. Musthofa, P. P. Rahayu, ‘’Mathematics Model of Diabetes Mellitus Illness without Genetic Factors with Treatment’’ Kaunia Journal, vol xvii, No 1, pp 21-25, 2021.
  [15] P. Venkatesha, S. Abilash, A. S. Shreyakar, A. Chandran, ‘’Mathematical Modelling of Blood Glucose Level by Glucose Tolerance Test’’, International Journal of Science, Engineering and Management (IJSEM) Vol 2, Issue 11, 2017.
  [16] K.W. Bunonyo, T.Y. Bunonyo, ‘’Mathematical Modeling of The Effect of Epinephrine And Insulin On Blood Glucose Concentration’’ International Journal of Mathematics Trends and Technology Volume 67 Issue 8, pp 125-132, 2021.
  [17] R. Mishra1, K. Baishnabi, L. Sahu, J.S. A. Bania, ‘’Multilayer Perceptron Model- A Novel Methodology towards the Advance Determination of Diabetic Retinal Infirmity in Kids’’ International Journal of Scientific Research in Multidisciplinary Studies Vol.6, Issue.9, pp.77-85, 2020.
  [18] N. Y. Gharaibeh, “A novel approach for detection of microaneurysms in diabetic retinopathy disease from retinal fundus images,” Journal Computer and Information Science, Vol. 10, No. 1, 2017
  [19] T. Subhramaniyan, P. Sankar, P. Paulraj, M. G. Ragunathan, J. Jayanthi, ‘’Algebraic Approach to predict Diabetic Retinopathy: Evidence from Diabetes database’’ International Journal of Scientific Research in, Mathematical and Statistical Sciences Volume-5, Issue-5, pp.164-168, 2018.
  [20] World Health Organization, ‘’Improving diabetes outcomes for all, a hundred years on from the discovery of insulin’’, report of the Global Diabetes Summit. Geneva: 2021.
  [21] J. Hussain, Denghmingliani Zadeng. ‘’A mathematical model of glucose-insulin interaction’’, Science Vision, Volume 14 Issue No 2, 2014.
  [22] J. Hussain, Denghmingliani Zadeng, ‘’Mathematical modeling on the effect of exercise on the interactions of glucose and insulin’’ International Journal of Applied Engineering Research, ISSN 0973-4562, Volume 9, Number 22, pp. 16211- 16221, 2014.
  [23] I. I. Mohammed, I. I. Adamu, S. J. Barka, ‘’Mathematical Model for the Dynamics of Glucose, Insulin and ?-Cell Mass under the Effect of Trauma, Excitement and Stress’’, Modeling and Numerical Simulation of Material Science, 9, pp 71-96, 2019.
  [24] "Evolutionary Computation Access on Incremental Map Reduce for Mining Large Scale Data”, International Journal of Recent Technology and Engineering, 2019.
  [25] Sonia Akter, M.D. Sirajul Islam, M.D Haider Ali Biswas, Sajib Mandal. "Mathematical Model Applied to Monitoring the Glucose-Insulin Interaction inside the Body of Diabetes Patients”, GANIT: Journal of Bangladesh Mathematical Society, 2020.
  [26] Isa Ibrahim Mohammed, Ibrahim Isa Adamu, Seni James Barka. "Mathematical Model for the Dynamics of Glucose, Insulin and ?-Cell Mass under the Effect of Trauma, Excitement and Stress”, Modeling and Numerical Simulation of Material Science, 2019.
  [27] R Yadav, Maya. "A Mathematical Model For The Study Of Diabetes Mellitus”, Journal of Physics: Conference Series, 2020.

  Citation
  Kanee Goodfaith Leyira, "Mathematical Modeling of the Causes, Complications and Solutions of Diabetes Mellitus," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.9, Issue.4, pp.44-49, 2022

• Open Access   Article

  Integral solutions of an Exponential Diophantine Equation 25x+24y=z2

  C. Saranya, G. Yashvandhini

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.9 , Issue.4 , pp.50-52, Aug-2022

  Abstract

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  Citation

  Abstract
  An exponential Diophantine equation is an equation in which the exponents are integers. Numerous exponential Diophantine equations are solved by Catalan conjecture. In this paper, we search for non-negative integer solutions to the exponential Diophantine equation 25x+24y = z2 using Catalan Conjecture. Also, we discussed three theorems and a corollary with their integer solutions.

  Key-Words / Index Term
  Catalan conjecture, Exponential Diophantine equation, Integer solutions, Diophantine Equations.

  References
  [1] R.D.Carmichael, “History Of Theory Of Numbers & Diophantine Analysis”, Dover Publication, New York, 1959.
  [2] L.K.Hua, “Introduction To The Theory Of Numbers”, Springer-Verlag, Berlin-New York, 1982.
  [3] Ivan Niven, Herbert, S.Zuckerman & Hugh L.Montgomery, “An Introduction To The Theory Of Numbers”, John Wiley & Sons Inc, New York 2004.
  [4] Somchit Chotchaisthit, “On The Diophantine Equation , Where P Is A Prime Number”, American Jr. Of Mathematics & Sciences, Vol-1, Issue 1, Jan 2012.
  [5] G.Jeyakrishnan & G.Komahan, “On The Diophantine Equation ”, International Journal For Innovative Research In Science & Technology, Vol 3, Issue 9, 119-120, Feb 2017.
  [6] G.Jeyakrishnan & G.Komahan, “On The Diophantine Equation ”, Actaciencia Indica Mathematics, Vol 2, 195-196, 2016.
  [7] P.Saranya & G.Janaki, “On The Exponential Diophantine Equation ”, International Research Journal Of Engineering & Technology, Vol 4, Issue 11, 1042-1044, Nov 2017.
  [8] C.Saranya & G.Janaki, “Solution Of Exponential Diophantine Equation Involving Jarasandha Numbers”, Advances and Applications in Mathematical Sciences, Volume 18, Issue 12, 1625-1629, Oct 2019.

  Citation
  C. Saranya, G. Yashvandhini, "Integral solutions of an Exponential Diophantine Equation 25x+24y=z2," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.9, Issue.4, pp.50-52, 2022

• Open Access   Article

  Series Representations of Analytic Functions Defined In the Complex Plane

  Kanee Goodfaith Leyira

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.9 , Issue.4 , pp.53-58, Aug-2022

  Abstract

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  Citation

  Abstract
  In this study, we examine the Taylor and Laurent Series for Series Representations of Analytic Functions as Defined in the Complex Plane. We explained the methodology following the statement of some series, theorems, convergences and their proofs applying the Cauchy’s integral formula and Weierstrass M-Test. The results of the study are discussed through solving some examples. The study concludes that analytic functions as a power series brings us to Taylor and Laurent series as defined in the complex plane.

  Key-Words / Index Term
  Analytic Function, Complex Plane, Convergences, Power Series, Singularity.

  References
  [1] Dennis. G. Zill, ‘’Advanced Engineering Mathematics Sixth edition’’; Loyola Marymount University. Description: Burlington, MA: Jones & Bartlett Learning, USA, pp. 878-908, 2017.
  [2] John M. Howie, ‘’Complex Analysis’’. Springer Undergraduate Mathematics Series. Springer publication, 2003.
  [3] Vizender. Singh (). ‘’Complex Analysis’’. Guru Jambheshwar University of Science and Technology, HISR-125001.
  [4] Li Yongzhao, ‘’Series Representations for Analytic Functions’’. State Key Laboratory of Integrated Services Networks, Xidian University. Conference. China, pp.22,2011
  [5] Rodolfo. R. Rosales, In Andre. Nachbin; Jeremy Orlo? and Jorn. Dunkel. ‘’Complex analysis with applications’’. Spring lecture notes on May 20, 2020.
  [6] Jeremy. Orlo?, ‘’Taylor and Laurent series’’. Topic 7 Notes, 2020.
  [7] Perumalla Ramulu, Lakshma G. Reddy, ‘’Zeros of Analytic Functions with Restricted Coefficients,’’ International Journal of Scientific Research in Mathematical and Statistical Sciences. Vol. 7, Issue 2. pp.146-151,2020.
  [8] François Dubeau, Calvin Gnang, ‘’On the Chebyshev-Halley family of iteration functions and the n-th root computation problem’’. International Journal of Pure and Applied Mathematics, 85(6), 1051–1059, 2013.
  [9] François Dubeau, Calvin Gnang ‘‘Fixed Point and Newton’s Methods in the Complex Plane’ ’Journal of Complex Analysis, p 11, Volume 2018,
  [10] Daniel Alpay, ‘’A Complex Analysis Problem Book’’, Springer Science and Business Media LLC, 2016.
  [11] John M. Howie, ‘’Complex Analysis’’, Springer Science and Business Media LLC, 2003.
  [12] Sadri Hassani, ‘’Complex Series’’, Mathematical Methods, 2009
  [13] Handbook of Mathematics, 2015.
  [14] ‘’Elektrodynamik’’, Springer-Lehrbuch, 2007
  [15] Gohberg, "Elementary properties of holomorphic functions", Holomorphic Operator Functions of One Variable and Applications, 2009
  [16] N.D. Aparicio, "Elastic complex analysis and its applications in fracture mechanics", International Journal of Solids and Structures, 2000
  [17] Ravi P. Agarwal. "Laurent’s Series", An Introduction to Complex Analysis, 2011
  [18] Selçuk ?. Bayin "Mathematical Methods in Science and Engineering", Wiley, 2018
  [19] Rolf Busam, Eberhard Freitag. "Complex Analysis", Springer Science and Business Media LLC, 2009
  [20] Robert B. Burckel. "Classical Analysis in the Complex Plane", Springer Science and Business Media LLC, 2021
  [21] Ram M. Murty. "A Second Course in Analysis", Springer Science and Business Media LLC, 2022
  [22] Ravi P. Agarwal. "Singularities and Poles I", An Introduction to Complex Analysis, 2011
  [23] "Functions of a Complex Variable", Wiley, 2007

  Citation
  Kanee Goodfaith Leyira, "Series Representations of Analytic Functions Defined In the Complex Plane," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.9, Issue.4, pp.53-58, 2022