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The Neutrosophic Numbers of the form N = a+bI have been defined by W. B. Vasantha Kandasamy and F. Smarandache in 2003 [see B2], and they were interpreted as: "a" is the determinate part of the number N, and "bI" is the indeterminate part of the number N by F. Smarandache in 2014 [see B3]. For the neutrosophic statistics "I" is a subset. Neutrosophic Statistics is the analysis of events described by the Neutrosophic Probability. Neutrosophic Probability is a generalization of the classical probability and imprecise probability in which the chance that an event A occurs is t% true - where t varies in the subset T, i% indeterminate - where i varies in the subset I, and f% false - where f varies in the subset F. In classical probability the sum of all space probabilities is equal to 1, while in Neutrosophic Probability it is equal to 3. In Imprecise Probability: the probability of an event is a subset T in [0, 1], not a number p in [0, 1], what’s left is supposed to be the opposite, subset F (also from the unit interval [0, 1]); there is no indeterminate subset I in imprecise probability [see B9]. The function that models the Neutrosophic Probability of a random variable x is called Neutrosophic distribution: NP(x) = ( T(x), I(x), F(x) ), where T(x) represents the probability that value x occurs, F(x) represents the probability that value x does not occur, and I(x) represents the indeterminate / unknown probability of value x [see B3].
Comparison between Neutrosophic Statistics (NS) and Interval Statistics (IS) We show that NS and IS are different, and in many applications the NS is more general than IS. NS is not reduced to only using neutrosophic numbers in statistical applications, as some people asserted, but it is much broader. NS deals with all types of indeterminacy, while IS deals only with indeterminacy that can be represented by intervals. However, not all indeterminacies (uncertainties) may be represented by intervals. Below we present several advantages of applying NS over IS: - Neutrosophic Statistics is based on Set Analysis, while Interval Statistics on Interval Analysis, therefore the Interval Statistics is a particular case of the Neutrosophic Statistics that uses all types of sets, not only intervals. - The numerical neutrosophic numbers permit the reduction of indeterminacy through operations, while the intervals increase the indeterminacy (see a simple example: let N1 = 4+3I, N2 = 5-2I, where indeterminacy I = [0,1]; using NS one has: N1 + N2 = 9 + I = [9, 10]; using IS one has: N1 + N2 = [4, 7] + [3, 5] = [7, 12]; clearly the indeterminacy or the real data point being in [7, 12], is bigger than the real data point being in [9, 10]). - Instead of intervals, for specific applications NS uses hesitant sets {discrete finite sets of numbers}, which make the calculations easier and reduce the indeterminacy (for example, if the real values may be 0.4, 7.9, or 41.5 (not sure which ones), instead of taking the interval [0.4, 41.5] as in IS, it is easier in NS to take just the hesitant set {0.4, 7.9, 41.5} of cardinal 3). - NS deals with sample or population whose size is not well-known. - NS deals with sample or population which contain individuals that only partially belong to the sample/population and others whose appurtenance is unknown. - NS deals with sample or population individuals whose degree of appurtenance to the sample or population may be outside of the interval [0, 1], as in neutrosophic overset (degree > 1), underset (degree < 0), and in general neutrosophic offset (both appurtenance degrees, > 1 and < 0, for various individuals) [see B4]. - Neutrosophic (or Indeterminate) Data is a vague, unclear, incomplete, partially unknown, conflicting indeterminate data. - NS also deals with refined neutrosophic data used in the Big Data. - NS may employ partially indeterminate curves. - NS also uses Thick Functions (as intersections of curves, that may not be represented by intervals) as probability distributions [see B3]. - Neutrosophic Probability Distribution (NPD) of an event (x) to occur is represented by three curves: NPD(x) = (T(x), I(x), F(x)), where T(x) represent the chance that the event E occurs, I(x) the indeterminate-chance that the event E occurs or not, and F(x) the chance that the event x does not occur. With T(x), I(x), F(x) being classical or neutrosophic (unclear, approximate, thick) functions – depending on each application, and T(x) + I(x) + F(x) in [0, 3] {see B9}. - Diagrams, histograms, pictographs, line/bar/cylinder graphs, plots with neutrosophic data (not represented by intervals) [see B9]. - Not well-known (or completely unknown): the mean, variance, standard deviation, probability distribution function, and other statistic - The Qualitative Data is represented by a finite discrete neutrosophic label set, instead of a label interval. - You cannot use Interval Statistics or Interval (Imprecise) Probability to compute the probability of a die on a cracked surface, or coin on a crack surface, or s defect die or coin [see B9]. We deal with indeterminacy with respect to the probability or statistics space or space's elements, indeterminacy with respect to the observer that evaluates the event, indeterminacy with respect to the event [4]. You cannot approximate the indeterminacy from these examples by using some interval, so you need neutrosophic probability and statistics that deal with all types of indeterminacies. - In conclusion: we cannot represent all types of indeterminacies by intervals.
More than 100 papers, nine books, one PhD thesis, and five international scientific seminars have been published or presented on neutrosophic statistics, including many journals by Elsevier and Springer of high impact factor.
References Books B1. Florentin Smarandache: A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability and Statistics (sixth edition). InfoLearnQuest, 1998 - 2007, 156 p. http://fs.unm.edu/eBook-Neutrosophics6.pdf B2. W. B. Vasantha Kandasamy, Florentin Smarandache, Fuzzy Cognitive Maps and Neutrosophic Cognitive Maps, Xiquan, Phoenix, 211 p., 2003, http://fs.unm.edu/NCMs.pdf B3. Florentin Smarandache: Introduction to Neutrosophic Statistics. Sitech & Education Publishing, 2014, 124 p. http://fs.unm.edu/NeutrosophicStatistics.pdf B4. Florentin Smarandache: Neutrosophic Overset, Neutrosophic Underset, and Neutrosophic Offset. Similarly for Neutrosophic Over-/Under-/Off- Logic, Probability, and Statistics. Pons Editions, Brussels, 2016, 168 p. http://fs.unm.edu/NeutrosophicOversetUndersetOffset.pdf B5. Maikel Leyva Vázquez, Florentin Smarandache: Neutrosofía: Nuevos avances en el tratamiento de la incertidumbre. Pons Editions, Bruselas, 2018, 74 p. http://fs.unm.edu/NeutrosofiaNuevosAvances.pdf B6. Tatiana Veronica Gutierrez Quinonez, Fabian Andres Espinoza, Ingrid Kathyuska Giraldo, Angel Steven Asanza, Mauricio Daniel Montenegro: Estadistica y Probabilidades: Una Vision Neutrosofica desde el Aprendizaje Basado en Problemas en la Construccion del Conocimiento. Pons Editions, Bruselas, 2020, 131 p. http://fs.unm.edu/EstadisticaYProbabilidadNeutrosofica.pdf B7. F. Smarandache, Neutrosophic Statistics vs. Classical Statistics, section in Nidus Idearum / Superluminal Physics, Vol. 7, third edition, p. 117, 2019, http://fs.unm.edu/NidusIdearum7-ed3.pdf . B8. F. Smarandache, Nidus Idearum de Neutrosophia (Book Series), Editions Pons, Brussels, Belgium, Vols. 1-7, 2016-2019; http://fs.unm.edu/ScienceLibrary.htm B9. F. Smarandache, Introduction to Neutrosophic Measure, Neutrosophic
Integral, and Neutrosophic Probability, Sitech Publishing House, Craiova, 2013,
http://fs.unm.edu/NeutrosophicMeasureIntegralProbability.pdf
PhD Thesis PhD1. Rafif Alhabib: Formulation of the classical probability and some probability distributions due to neutrosophic logic and its impact on Decision Making. PhD Thesis in Arabic, held under the supervision of Dr. M. M. Ranna, Dr. H. Farah, Dr. A. A. Salama, Faculty of Science, Department of Mathematical Statistics, University of Aleppo, Syrian Arab Republic, 2019. http://fs.unm.edu/NS/FormulationOfTheClassicalProbability-PhDThesis.pdf Scientific Presentations SP1. Muhammad Aslam, Testing wind speed using
Neutrosophic Weibull distribution,
Articles 1. Florentin Smarandache: Operators on Single-Valued Neutrosophic Oversets, Neutrosophic Undersets, and Neutrosophic Offsets. Journal of Mathematics and Informatics, Vol. 5, 2016, 63-67. 2. Florentin Smarandache: Interval-Valued Neutrosophic Oversets, Neutrosophic Undersets, and Neutrosophic Offsets. International Journal of Science and Engineering Investigations, Vol. 5, issue 54, 2016, Paper ID: 55416-01, 4 p. 3. Nouran M. Radwan, M. Badr Senousy, Alaa El Din M. Riad: Approaches for Managing Uncertainty in Learning Management Systems. Egyptian Computer Science Journal, vol. 40, no. 2, May 2016, 10 p.
4. Muhammad
Aslam: A
Variable Acceptance Sampling Plan under Neutrosophic Statistical Interval Method. Symmetry 2019,
11, 114, DOI: 10.3390/sym11010114. 5. Soumyadip Dhar, Malay K. Kundu: Accurate segmentation of complex document image using digital shearlet transform with neutrosophic set as uncertainty handing tool. Applied Soft Computing, vol. 61, 2017, 412–426. 6. B. Kavitha, S. Karthikeyan, P. Sheeba Maybell: An ensemble design of intrusion system for handling uncertainty using Neutrosophic Logic Classifier. Knowlwdge-Based Systems, vol. 28, 2012, 88-96. 7. Muhammad Aslam: A new attribute sampling plan using neutrosophic statistical interval method. Complex & Intelligent Systems, 6 p. DOI: 10.1007/s40747-018-0088-6
8. Muhammad
Aslam, Nasrullah Khan, Mohammed Albassam: Control
Chart for Failure-Censored Reliability Tests under Uncertainty Environment. Symmetry 2018,
10, 690, DOI: 10.3390/sym10120690.
9. Muhammad
Aslam, Nasrullah Khan, Ali Hussein AL-Marshadi: Design
of Variable Sampling Plan for Pareto Distribution Using Neutrosophic Statistical
Interval Method. Symmetry 2019,
11, 80, DOI: 10.3390/sym11010080. 10. Jun Ye, Jiqian Chen, Rui Yong, Shigui Du: Expression and Analysis of Joint Roughness Coefficient Using Neutrosophic Number Functions. Information, Volume 8, 2017, 13 pages. 11. Jiqian Chen, Jun Ye, Shigui Du, Rui Yong: Expressions of Rock Joint Roughness Coefficient Using Neutrosophic Interval Statistical Numbers. Symmetry, Volume 9, 2017, 7 pages. 12. Adrian Rubio-Solis, George Panoutsos: Fuzzy Uncertainty Assessment in RBF Neural Networks using neutrosophic sets for Multiclass Classification. Presented at 2014 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE) July 6-11, 2014, Beijing, China, 8 pages. 13. Pierpaolo D’Urso: Informational Paradigm, management of uncertainty and theoretical formalisms in the clustering framework: A review. Information Sciences, 400–401 (2017), pp. 30-62, 33 pages.
14. Muhammad
Aslam, Mohammed Albassam: Inspection
Plan Based on the Process Capability Index Using the Neutrosophic Statistical
Method. Mathematics 2019,
7, 631, DOI: 10.3390/math7070631. 15. Mirela Teodorescu, Florentin Smarandache, Daniela Gifu: Maintenance Operating System Uncertainties Approached through Neutrosophic Theory. 8 p.
16. Muhammad
Aslam, Rashad A. R. Bantan, Nasrullah Khan: Monitoring
the Process Based on Belief Statistic for Neutrosophic Gamma Distributed Product. Processes 2019,
7, 209, DOI: 10.3390/pr7040209. 17. Rafael Rojas-Gualdron, Florentin Smarandache, Carlos Diaz-Bohorquez: Application of The Neutrosophical Theory to Deal with Uncertainty in Supply Chain Risk Management. AGLALA 2019; 10 (2): 1-19. 18. Florentin Smarandache, Gheorghe Savoiu: Neutrosophic Index Numbers: Neutrosophic Logic Applied In The Statistical Indicators Theory. Critical Review, Vol. XI, 2015, pp. 67-100. 19. Murat Kirisci, Necip Simsek: Neutrosophic normed spaces and statistical convergence. Journal of Analysis, 11 April 2020, DOI: 10.1007/s41478-020-00234-0. 20. S.K. Patro: The Neutrosophic Statistical Distribution: More Problems, More Solutions. 17 p. 21. Deepesh Kunwar, Jayant Singh, Florentin Smarandache: Neutrosophic statistical evaluation of migration with particular reference to Jaipur. Octogon Mathematical Magazine, vol. 26, no. 2, October 2018, 560-568. 22. Deepesh Kunwar, Jayant Singh, Florentin Smarandache: Neutrosophic statistical techniques to find migration pattern in Jaipur. Octogon Mathematical Magazine, vol. 26, no. 2, October 2018, 583-592. 23. Muhammad Aslam, Osama H. Arif, Rehan Ahmad Khan Sherwani: New Diagnosis Test under the Neutrosophic Statistics: An Application to Diabetic Patients. Hindawi, BioMed Research International, Volume 2020, Article ID 2086185, 7 pages; DOI: 10.1155/2020/2086185. 24. Jose L. Salmeron, Florentin Smarandache: Processing Uncertainty and Indeterminacy in Information Systems success mapping. 13 p., arXiv:cs/0512047v2. 25. Wenzhong Jiang, Jun Ye, Wenhua Cui: Scale Effect and Anisotropic Analysis of Rock Joint Roughness Coefficient Neutrosophic Interval Statistical Numbers Based on Neutrosophic Statistics. Journal of Soft Computing in Civil Engineering, 2-4 / 2018, 62-71; DOI: 10.5281/zenodo.3130240.
26. Muhammad
Aslam, P. Jeyadurga, Saminathan Balamurali, Ali Hussein Al-Marshadi: Time-Truncated
Group Plan under aWeibull Distribution based on Neutrosophic Statistics. Mathematics 2019,
7, 905; DOI: 10.3390/math7100905 27. A.A. Salama, M. Elsayed Wahed, Eman Yousif: A Multi-objective Transportation Data Problems and their Based on Fuzzy Random Variables. Neutrosophic Knowledge, vol. 1, 2020, 41-53; DOI: 10.5281/zenodo.4269558. 28. Philippe Schweizer: Uncertainty: two probabilities for the three states of neutrosophy. International Journal of Neutrosophic Science (IJNS), Volume 2, Issue 1, 2020, 18-26; DOI: 10.5281/zenodo.3989350.
29. Carlos
N. Bouza-Herrera, Mir Subzar: Estimating
the Ratio of a Crisp Variable and a Neutrosophic Variable. International
Journal of Neutrosophic Science (IJNS), Volume 11, Issue 1, 2020, 9-21; DOI:
10.5281/zenodo.4275712 30. Angel Carlos Yumar Carralero, Darvin Manuel Ramirez Guerra, Giorver Perez Iribar: Analisis estadistico neutrosofico en la aplicacion de ejercicios fisicos en la rehabilitacion del adulto mayor con gonartrosis. Neutrosophic Computing and Machine Learning, Vol. 13, 1-9, 2020; DOI: https://zenodo.org/record/3901770. 31. Alexandra Dolores Molina Manzo, Rosa Leonor Maldonado Manzano, Blanca Esmeralda Brito Herrera, Johanna Irene Escobar Jara: Analisis estadistico neutrosofico de la incidencia del voto facultativo de los jovenes entre 16 y 18 anos en el proceso electoral del Ecuador. Neutrosophic Computing and Machine Learning, Vol. 11, 9-14, 2020; DOI: https://zenodo.org/record/3474439. 32. Johana Cristina Sierra Morán, Jenny Fernanda Enríquez Chuga, Wilmer Medardo Arias Collaguazo And Carlos Wilman Maldonado Gudiño:Neutrosophic statistics applied to the analysis of socially responsible participation in the community , Neutrosophic Sets and Systems, vol. 26, 2019, pp. 19 -28. DOI: 10.5281/zenodo.3244232 33. Paúl Alejandro Centeno Maldonado, Yusmany Puertas Martinez, Gabriela Stephanie Escobar Valverde, and Juan Danilo Inca Erazo: Neutrosophic statistics methods applied to demonstrate the extra-contractual liability of the state from the Administrative Organic Code, Neutrosophic Sets and Systems, vol. 26, 2019, pp. 29-34. DOI: 10.5281/zenodo.3244262 34. S. K. Patro, F. Smarandache: The Neutrosophic Statistical Distribution - More Problems, More Solutions, Neutrosophic Sets and Systems, vol. 12, 2016, pp. 73-79. doi.org/10.5281/zenodo.571153 35. Lilia Esther Valencia Cruzaty, Mariela Reyes Tomalá, Carlos Manuel Castillo Gallo and Florentin Smarandache, A Neutrosophic Statistic Method to PredictTax Time Series in Ecuador, Neutrosophic Sets and Systems, vol. 34, 2020, pp. 33-39. DOI: 10.5281/zenodo.3843289; http://fs.unm.edu/NSS/NeutrosophicStatisticMethod.pdf
36. Somen Debnath: Neutrosophication
of statistical data in a study to assess the knowledge, attitude and symptoms
on reproductive tract infection among women. Journal
of Fuzzy Extension & Applications (JFEA),
Volume 2, Issue 1, Winter 2021, 33-40; DOI: 10.22105/JFEA.2021.272508.1073. 37. Muhammad Aslam, Rashad A.R. Bantan, Nasrullah Khan: Design of tests for mean and variance under complexity-an application to rock measurement data. Elsevier: Measurement, Volume 177, June 2021, 109312; DOI: 10.1016/j.measurement.2021.109312. 38. O.H. Arif, Muhammad Aslam: A new sudden death chart for the Weibull distribution under complexity. Springer: Complex & Intelligent Systems (2021); DOI: 10.1007/s40747-021-00316-x. 39. Nasrullah Khan, Muhammad Aslam, Asma Arshad, Ambreen Shafqat: Tracking Temperature Under Uncertainty Using EWMA-MA Control Chart. Springer: Journal of Metrology Society of India (2021); DOI: 10.1007/s12647-021-00436-2. 40. Muhammad Aslam: Analyzing wind power data using analysis of means under neutrosophic statistics. Springer: Soft Computing (2021); DOI: 10.1007/s00500-021-05661-0. 41. Muhammad Aslam: On Testing Autocorrelation in Metrology Data Under Indeterminacy. Springer: Journal of Metrology Society of India (2021); DOI: 10.1007/s12647-021-00429-1. 42. Muhammad Aslam, Nasrullah Khan: Normality Test of Temperature in Jeddah City Using Cochran’s Test Under Indeterminacy. Springer: Journal of Metrology Society of India (2021); DOI: 10.1007/s12647-020-00428-8. 43. Muhammad Aslam, Gadde Srinivasa Rao, Nasrullah Khan, Liaquat Ahmad: Two-stage sampling plan using process loss index under neutrosophic statistics. Taylor&Francis: Communications in Statistics - Theory and Methods (2020); DOI: 10.1080/03610918.2019.1702212. 44. Ali Hussein Al-Marshadi, Ambreen Shafqat, Muhammad Aslam, Abdullah Alharbey: Performance of a New Time-Truncated Control Chart for Weibull Distribution Under Uncertainty. Atlantis Press: International Journal of Computational Intelligence Systems, Volume 14, Issue 1, 2021, 1256 - 1262; DOI: 10.2991/ijcis.d.210331.001. 45. Muhammad Aslam: Testing average wind speed using sampling plan for Weibull distribution under indeterminacy. Nature: Scientific Reports, 11, Article number: 7532 (2021); DOI: 10.1038/s41598-021-87136-8. 46. Muhammad Aslam, G. Srinivasa Rao, Nasrullah Khan: Single-stage and two-stage total failure-based group-sampling plans for the Weibull distribution under neutrosophic statistics. Springer: Complex & Intelligent Systems, 7, 891–900 (2021); DOI: 10.1007/s40747-020-00253-1. 47. Muhammad Aslam, G. Srinivasa Rao, Ambreen Shafqat, Liaquat Ahmad, Rehan Ahmad Khan Sherwani: Monitoring circuit boards products in the presence of indeterminacy. Elsevier: Measurement, Volume 168, 15 January 2021, 108404; DOI: 10.1016/j.measurement.2020.108404. 48. Mohammed Albassam, Nasrullah Khan, Muhammad Aslam: Neutrosophic D’Agostino Test of Normality: An Application to Water Data. Hindawi: Journal of Mathematics - Theory, Algorithms, and Applications within Neutrosophic Modelling and Optimisation, 2021, Article ID 5582102, 5 pages; DOI: 10.1155/2021/5582102. 49. Mohammed Albassam: Radar data analysis in the presence of uncertainty. Taylor&Francis: European Journal of Remote Sensing, 54:1, 140-144, 2021; DOI: 10.1080/22797254.2021.1886597. 50. Muhammad Aslam: A new goodness of fit test in the presence of uncertain parameters. Springer: Complex & Intelligent Systems, 7, 359–365, 2021; DOI: 10.1007/s40747-020-00214-8. 51. Abdullah M. Almarashi, Muhammad Aslam: Process Monitoring for Gamma Distributed Product under Neutrosophic Statistics Using Resampling Scheme. Hindawi: Journal of Mathematics: Soft Computing Algorithms Based on Fuzzy Extensions, Volume 2021, Article ID 6635846, 12 pages; DOI: 10.1155/2021/6635846. 52. Muhammad Aslam: A study on skewness and kurtosis estimators of wind speed distribution under indeterminacy. Springer: Theoretical and Applied Climatology, 143, 1227–1234, 2021; DOI: 10.1007/s00704-020-03509-5. 53. Muhammad Aslam, Ali Algarni: Analyzing the Solar Energy Data Using a New Anderson-Darling Test under Indeterminacy. Hindawi: International Journal of Photoenergy, Volume 2020, Article ID 6662389, 6 pages; DOI: 10.1155/2020/6662389. 54. Muhammad Aslam: Forecasting of the wind speed under uncertainty. Nature: Sc. Rep., Volume 10 (2020). 55. Azhar Ali Janjua, Muhammad Aslam, Naheed Sultana: Evaluating the relationship between climate variability and agricultural crops under indeterminacy. Springer: Theoretical and Applied Climatology, Volume 142, pages 1641–1648 (2020); DOI: 10.1007/s00704-020-03398-8. 56. Rehan Ahmad Khan Sherwan, Mishal Naeem, Muhammad Aslam, Muhammad Ali Raza, Muhammad Abid, Shumaila Abbas: Neutrosophic Beta Distribution with Properties and Applications. University of New Mexico: Neutrosophic Sets and Systems, Vol. 41, 209-214, 2021; DOI: 10.5281/zenodo.4625715. 57. Muhammad Aslam, Ambreen Shafqat, Mohammed Albassam, Jean-Claude Malela-Majika, Sandile C. Shongwe: A new CUSUM control chart under uncertainty with applications in petroleum and meteorology. PLoS ONE 16(2): e0246185, 2021; DOI: 10.1371/journal.pone.0246185. 58. Muhammad Aslam: Monitoring the road traffic crashes using NEWMA chart and repetitive sampling. Taylor&Francis: International Journal of Injury Control and Safety Promotion, Volume 28, 2021 - Issue 1, 39-45; DOI: 10.1080/17457300.2020.1835990. 59. Muhammad Aslam: Analysing Gray Cast Iron Data using a New Shapiro-Wilks test for Normality under Indeterminacy. Taylor&Francis: International Journal of Cast Metals Research, Volume 34, 2021 - Issue 1, 1-5; DOI: 10.1080/13640461.2020.1846959. 60. Ishmal Shahzadi, Muhammad Aslam, Hussain Aslam: Neutrosophic Statistical Analysis of Income of YouTube Channels. University of New Mexico: Neutrosophic Sets and Systems, Vol. 39, 101-106, 2020. 61. Nasrullah Khan, Muhammad Aslam, P. Jeyadurga, S. Balamurali: Monitoring of production of blood components by attribute control chart under indeterminacy. Nature: Sc. Rep., volume 11 (2021). 62. Muhammad Aslam, Rashad A.R. Bantan: A study on measurement system analysis in the presence of indeterminacy. Elsevier: Measurement, Volume 166, December 2020, 108201; DOI: 10.1016/j.measurement.2020.108201. 63. Muhammad Aslam, Rashad A. R. Bantan, Nasrullah Khan: Design of NEWMA np control chart for monitoring neutrosophic nonconforming items. Springer: Soft Computing, Volume 24, 16617–16626 (2020); DOI: 10.1007/s00500-020-04964-y. 64. M. Albassam, Muhammad Aslam: Monitoring Non-Conforming Products Using Multiple Dependent State Sampling Under Indeterminacy-An Application to Juice Industry. IEEE Access, vol. 8, pp. 172379-172386, 2020; DOI: 10.1109/ACCESS.2020.3024569. 65. Ahmed Ibrahim Shawky , Muhammad Aslam, Khushnoor Khan: Multiple Dependent State Sampling-Based Chart Using Belief Statistic under Neutrosophic Statistics. Hindawi: Journal of Mathematics, Volume 2020, Article ID 7680286, 14 pages; DOI: 10.1155/2020/7680286. 66. Muhammad Aslam: Introducing Grubbs’s test for detecting outliers under neutrosophic statistics - An application to medical datas. Science Direct: Journal of King Saud University - Science, Volume 32, Issue 6, September 2020, 2696-2700; DOI: 10.1016/j.jksus.2020.06.003. 67. Muhammad Aslam: A New Sampling Plan Using Neutrosophic Process Loss Consideration. MDPI: Symmetry, 2018, 10 (5), 132; DOI: 10.3390/sym10050132. 68. Muhammad Aslam, Osama H. Arif: Testing of Grouped Product for the Weibull Distribution Using Neutrosophic Statistics. MDPI: Symmetry, 2018, 10 (9), 403; DOI: 10.3390/sym10090403. 69. Muhammad Aslam, Nasrullah Khan, Muhammad Zahir Khan: Monitoring the Variability in the Process Using Neutrosophic Statistical Interval Method. MDPI: Symmetry, 2018, 10 (11), 562; DOI: 10.3390/sym10110562. 70. Muhammad Zahir Khan, Muhammad Farid Khan, Muhammad Aslam, Abdur Razzaque Mughal: Design of Fuzzy Sampling Plan Using the Birnbaum-Saunders Distribution. MDPI: Mathematics, 2019, 7 (1), 9; DOI: 10.3390/math7010009. 71. Muhammad Aslam, Ali Hussein Al-Marshadi: Design of Sampling Plan Using Regression Estimator under Indeterminacy. MDPI: Symmetry, 2018, 10 (12), 754; DOI: 10.3390/sym10120754. 72. Muhammad Zahir Khan, Muhammad Farid Khan, Muhammad Aslam, Seyed Taghi Akhavan Niaki, Abdur Razzaque Mughal: A Fuzzy EWMA Attribute Control Chart to Monitor Process Mean. MDPI: Information, 2018, 9 (12), 312; DOI: 10.3390/info9120312. 73. Muhammad Aslam, Nasrullah Khan, Mohammed Albassam: Control Chart for Failure-Censored Reliability Tests under Uncertainty Environment. MDPI: Symmetry, 2018, 10 (12), 690; DOI: 10.3390/sym10120690. 74. Muhammad Aslam, Mohammed Albassam: Application of Neutrosophic Logic to Evaluate Correlation between Prostate Cancer Mortality and Dietary Fat Assumption. MDPI: Symmetry, 2019, 11 (3), 330; DOI: 10.3390/sym11030330. 75. Muhammad Aslam, Mansour Sattam Aldosari: Inspection Strategy under Indeterminacy Based on Neutrosophic Coefficient of Variation. MDPI: Symmetry, 2019, 11 (2), 193; DOI: 10.3390/sym11020193. 76. Muhammad Aslam: A Variable Acceptance Sampling Plan under Neutrosophic Statistical Interval Method. MDPI: Symmetry, 2019, 11 (1), 114; DOI: 10.3390/sym11010114. 77. Muhammad Aslam, Nasrullah Khan, Ali Hussein Al-Marshadi: Design of Variable Sampling Plan for Pareto Distribution Using Neutrosophic Statistical Interval Method. MDPI: Symmetry, 2019, 11 (1), 80; DOI: 10.3390/sym11010080. 78. Muhammad Aslam, Rashad A. R. Bantan, Nasrullah Khan: Design of S2N—NEWMA Control Chart for Monitoring Process having Indeterminate Production Data. MDPI: Processes, 2019, 7 (10), 742; DOI: 10.3390/pr7100742. 79. Muhammad Aslam, Ali Hussein Al-Marshadi, Nasrullah Khan: A New X-Bar Control Chart for Using Neutrosophic Exponentially Weighted Moving Average. MDPI: Mathematics, 2019, 7 (10), 957; DOI: 10.3390/math7100957. 80. Muhammad Aslam, P. Jeyadurga, Saminathan Balamurali, Ali Hussein Al-Marshadi: Time-Truncated Group Plan under a Weibull Distribution based on Neutrosophic Statistics. MDPI: Mathematics, 2019, 7 (10), 905; DOI: 10.3390/math710090557. 81. Muhammad Aslam, Osama Hasan Arif: Classification of the State of Manufacturing Process under Indeterminacy. MDPI: Mathematics, 2019, 7 (9), 870; DOI: 10.3390/math7090870. 82. Muhammad Aslam, Mohammed Albassam: Inspection Plan Based on the Process Capability Index Using the Neutrosophic Statistical Method. MDPI: Mathematics, 2019, 7 (7), 631; DOI: 10.3390/math7070631. 83. Muhammad Aslam, Rashad A. R. Bantan, Nasrullah Khan: Monitoring the Process Based on Belief Statistic for Neutrosophic Gamma Distributed Product. MDPI: Processes, 2019, 7 (4), 209; DOI: 10.3390/pr7040209. 84. Muhammad Aslam: Product Acceptance Determination with Measurement Error Using the Neutrosophic Statistics. Hindawi: Advances in Fuzzy Systems, Volume 2019, Article ID 8953051, 8 pages; DOI: 10.1155/2019/8953051. 85. Muhammad Aslam, Rashad A. R. Bantan, Nasrullah Khan: Design of a New Attribute Control Chart Under Neutrosophic Statistics. Springer: International Journal of Fuzzy Systems, Volume 21, 433–440 (2019); DOI: 10.1007/s40815-018-0577-1. 86. Muhammad Aslam, Osama H. Arif: Test of Association in the Presence of Complex Environment. Hindawi: Complexity, Volume 2020, Article ID 2935435, 6 pages; DOI: 10.1155/2020/2935435. 87. Mohammed Albassam, Nasrullah Khan,Muhammad Aslam: The W/S Test for Data Having Neutrosophic Numbers: An Application to USA Village Population. Hindawi: Complexity, Volume 2020, Article ID 3690879, 8 pages; DOI: 10.1155/2020/3690879. 88. Muhammad Aslam, Osama H. Arif, Rehan Ahmad Khan Sherwani: New Diagnosis Test under the Neutrosophic Statistics: An Application to Diabetic Patients. Hindawi: BioMed Research International, Volume 2020, Article ID 2086185, 7 pages; DOI: 10.1155/2020/2086185. 88. Muhammad Aslam, Ali Hussein Al-Marshadi: Design of a Control Chart Based on COM-Poisson Distribution for the Uncertainty Environment. Hindawi: Complexity, Volume 2019, Article ID 8178067, 7 pages; DOI: 10.1155/2019/8178067. 89. Muhammad Aslam, Osama H. Arif: Multivariate Analysis under Indeterminacy: An Application to Chemical Content Data. Hindawi: Journal of Analytical Methods in Chemistry, Volume 2020, Article ID 1406028, 6 pages; DOI: 10.1155/2020/1406028. 90. Muhammad Aslam, Abdulmohsen Al-Shareef, Khushnoor Khan: Monitoring the temperature through moving average control under uncertainty environment. Nature: Sc. Rep., Volume 10, Article number: 12182 (2020). 91. Muhammad Aslam: Design of Sampling Plan for Exponential Distribution Under Neutrosophic Statistical Interval Method. IEEE Access, vol. 6, pp. 64153-64158, 2018; DOI: 10.1109/ACCESS.2018.2877923. 92. Muhammad Aslam: Control Chart for Variance Using Repetitive Sampling Under Neutrosophic Statistical Interval System. IEEE Access, vol. 7, pp. 25253-25262, 2019; DOI: 10.1109/ACCESS.2019.2899020. 93. Muhammad Aslam, M. Azam, M. Albassam: Sampling Plan Using Process Loss Index Using Multiple Dependent State Sampling Under Neutrosophic Statistics. IEEE Access, vol. 7, pp. 38568-38576, 2019; DOI: 10.1109/ACCESS.2019.2906408. 94. Naeem Jan, Muhammad Aslam, Kifayat Ullah, Tahir Mahmood, Jun Wang: An approach towards decision making and shortest path problems using the concepts of interval-valued Pythagorean fuzzy information. Wiley: International Journal of Intelligent Systems, Volume 34, Issue 10, October 2019, 2403-2428. 95. Muhammad Aslam: Attribute Control Chart Using the Repetitive Sampling Under Neutrosophic System. IEEE Access, vol. 7, 2019, 2163-3536; DOI: 10.1109/ACCESS.2019.2895162. 96. Muhammad Aslam, R. A. R. Bantan, N. Khan: Design of a Control Chart for Gamma Distributed Variables Under the Indeterminate Environment. IEEE Access, vol. 7, pp. 8858-8864, 2019; DOI: 10.1109/ACCESS.2019.2891005. 97. Muhammad Aslam, Muhammad Ali Raza: Design of New Sampling Plans for Multiple Manufacturing Lines Under Uncertainty. Springer: International Journal of Fuzzy Systems, volume 21, 978–992 (2019); DOI: 10.1007/s40815-018-0560-x. 98. Muhammad Aslam: A New Failure-Censored Reliability Test Using Neutrosophic Statistical Interval Method. Springer: International Journal of Fuzzy Systems, volume 21, 1214–1220 (2019); DOI: 10.1007/s40815-018-0588-y. 99. Muhammad Aslam: Neutrosophic analysis of variance: application to university students. Springer: Complex & Intelligent Systems, volume 5, 403–407 (2019); DOI: 10.1007/s40815-018-0588-y. 100. Muhammad Aslam, Mohammed Albassam: Presenting post hoc multiple comparison tests under neutrosophic statistics. Elsevier: Journal of King Saud University - Science, Volume 32, Issue 6, September 2020, 2728-2732; DOI: 10.1016/j.jksus.2020.06.008. 101. Muhammad Aslam, Mansour Sattam Aldosari: Analyzing alloy melting points data using a new Mann-Whitney test under indeterminacy. Elsevier: Journal of King Saud University - Science, Volume 32, Issue 6, September 2020, 2831-2834; DOI: 10.1016/j.jksus.2020.07.005. 102. Muhammad Aslam: On detecting outliers in complex data using Dixon’s test under neutrosophic statistics. Elsevier: Journal of King Saud University - Science, Volume 32, Issue 3, April 2020, 2005-2008; DOI: 10.1016/j.jksus.2020.02.003. 103. Muhammad Aslam: A new attribute sampling plan using neutrosophic statistical interval method. Springer: Complex & Intelligent Systems, 5, 365–370 (2019); DOI: 10.1007/s40747-018-0088-6. 104. Muhammad Aslam, Saminathan Balamurali, Jeyadurga Periyasamypandian, Ali Hussein Al-Marshadi: Plan for Food Inspection for Inflated-Pareto Data Under Uncertainty Environment. IEEE Access, vol. 7, 164186-164193, 2019; DOI: 10.1109/ACCESS.2019.2951019. 105. Muhammad Aslam, R. A. R. Bantan, N. Khan: Design of X-Bar Control Chart Using Multiple Dependent State Sampling Under Indeterminacy Environment. IEEE Access, vol. 7, pp. 152233-152242, 2019; DOI: 10.1109/ACCESS.2019.2947598. 106. Muhammad Aslam: Introducing Kolmogorov-Smirnov Tests under Uncertainty: An Application to Radioactive Data. American Chemical Society: ACS Omega 2020, 5, 1, 914-917; DOI: 10.1021/acsomega.9b03940. 107. Muhammad Aslam: Design of the Bartlett and Hartley tests for homogeneity of variances under indeterminacy environment. Taylor&Francis: Journal of Taibah University for Science, Volume 14, 2020, Issue 1; DOI: 10.1080/16583655.2019.1700675. 108. Muhammad Aslam, Muhammad Ali Raza, Liaquat Ahmad: Acceptance sampling plans for two-stage process for multiple manufacturing lines under neutrosophic statistics. IOS Press: Journal of Intelligent & Fuzzy Systems, vol. 37, no. 6, pp. 7839-7850, 2019; DOI: 10.3233/JIFS-182849. 109. Muhammad Kashif, Hafiza Nida, Muhammad Imran Khan, Muhammad Aslam: Decomposition of Matrix under Neutrosophic Environment. University of New Mexico: Neutrosophic Sets and Systems, vol. 30, 143-148, 2019. 110. Muhammad Aslam, Nasrullah Khan: A new variable control chart using neutrosophic interval method-an application to automobile industry. University of New Mexico: Journal of Intelligent & Fuzzy Systems, vol. 36, no. 3, pp. 2615-2623, 2019; DOI: 10.3233/JIFS-181767. 111. N Khan, L Ahmad, M Azam, M Aslam, F Smarandache, Control Chart for Monitoring Variation Using Multiple Dependent State Sampling Under Neutrosophic Statistics, in the book Neutrosophic Operational Research (eds. F. Smarandache, M. Abdel-Basset), Springer, pp 55-70, 10 September 2021, https://link.springer.com/chapter/10.1007/978-3-030-57197-9_4. 112. Rehan Ahmad Khan Sherwani, Muhammad Aslam, Muhammad Ali, Raza Muhammad, Farooq Muhammad, Abid Muhammad Tahir, Neutrosophic Normal Probability Distribution—A Spine of Parametric Neutrosophic Statistical Tests: Properties and Applications, in the book Neutrosophic Operational Research (eds. F. Smarandache, M. Abdel-Basset), Springer, pp 153-169, 10 September 2021, https://link.springer.com/chapter/10.1007/978-3-030-57197-9_8. 113. Rehan Ahmad Khan Sherwani, Muhammad Aslam, Huma Shakeel, Kamran Abbas, Farrukh Jamal, Neutrosophic Statistics for Grouped Data: Theory and Applications, in the book Neutrosophic Operational Research (eds. F. Smarandache, M. Abdel-Basset), Springer, pp 263-289, 10 September 2021, https://link.springer.com/chapter/10.1007/978-3-030-57197-9_14.
114. Arif, O.H., Aslam, M. A new sudden death chart for the Weibull distribution
under complexity. Complex Intell. Syst. 7, 2093–2101 (2021),
118. Florentin Smarandache, Foundation of Appurtenance and Inclusion Equations for Constructing the Operations of Neutrosophic Numbers Needed in Neutrosophic Statistics (revised). Prospects for Applied Mathematics and Data Analysis (PAMDA), Vol. 03, No. 01, PP. 29-48, 2023, https://fs.unm.edu/NS/AppurtenanceInclusionEquations-revisited.pdf
S1. History of Neutrosophic Set, Logic, Probability and Statistics and their Applications, Mathematics and Statistics Departments, King Abdulaziz University, Jeddah, Saudi Arabia, 19 December 2019.
S2.
Neutrosophic
Set and Logic / Interval Neutrosophic Set and Logic / Neutrosophic Probability
and Neutrosophic Statistics / Neutrosophic Precalculus and Calculus / Symbolic
Neutrosophic Theory / Open Challenges of Neutrosophic Set,
lecture series, by F. Smarandache, Nguyen Tat Thanh University, Ho Chi Minh
City, Vietnam, 31st May - 3th June 2016.
S3.
Neutrosophic
Set and Logic / Interval Neutrosophic Set and Logic / Neutrosophic Probability
and Neutrosophic Statistics / Neutrosophic Precalculus and Calculus / Symbolic
Neutrosophic Theory / Open Challenges of Neutrosophic Set, by
F. Smarandache, Ho Chi Minh City University of Technology (HUTECH), Ho Chi Minh
City, Vietnam, 30th May 2016.
S4.
Neutrosophic
Set and Logic / Interval Neutrosophic Set and Logic / Neutrosophic Probability
and Neutrosophic Statistics / Neutrosophic Precalculus and Calculus / Symbolic
Neutrosophic Theory / Open Challenges of Neutrosophic Set,
lecture series, by F. Smarandache, Vietnam national University, Vietnam
Institute for Advanced Study in Mathematics, Hanoi, Vietnam, lecture series, 14th May
– 26th May 2016. S5. Foundations of Neutrosophic Logic, Set, Probability and Statistics and their Applications in Science. n-Valued Refined Neutrosophic Set, Logic, Probability and Statistics, by F. Smarandache, Universidad Complutense de Madrid, Facultad de Ciencia Matematicas, Departamento de Geometria y Topologia, Instituto Matematico Interdisciplinar (IMI), Madrid, Spain, 9th July 2014.
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