Neutrosophic Sets and Systems Neutrosophic Sets and Systems

: In this paper we exemplify the types of Plithogenic Probability and respectively Plithogenic Statistics. Several applications are given. The Plithogenic Probability of an event to occur is composed from the chances that the event occurs with respect to all random variables (parameters) that determine it. Each such a variable is described by a Probability Distribution (Density) Function, which may be a classical, (T,I,F)-neutrosophic, I-neutrosophic, (T,F)-intuitionistic fuzzy, (T,N,F)-picture fuzzy, (T,N,F)-spherical fuzzy, or (other fuzzy extension) distribution function. The Plithogenic Probability is a generalization of the classical MultiVariate Probability. The analysis of the events described by the plithogenic probability is the Plithogenic Statistics.


Introduction
The Plithogeny, as generalization of Dialectics and Neutrosophy, and then its applications to Plithogenic Set/Logic/Probability/Statistics (as generalization of fuzzy, intuitionistic fuzzy, neutrosophic set/logic/probability/statistics) [1,2] were introduced by Smarandache in 2017.
Plithogeny is the genesis or origination, creation, formation, development, and evolution of new entities from dynamics and organic fusions of contradictory and/or neutrals and/or non-contradictory multiple old entities.

Neutrosophic (or Indeterminate) Data
Neutrosophic (or Indeterminate) Data is a vague, unclear, incomplete, partially unknown, conflicting indeterminate data.The neutrosophic data can be metrical, or categorical, or both.Plithogenic Variate Data summarizes the associations (or inter-relationships) between Neutrosophic variables.While Neutrosophic Variable is a variable (or function, operator), that deals with neutrosophic data) either in its arguments or in its values, or in both.The problem to solve may have many dimensions, therefore multiple measurements and observations are needed since there are many sides to the problem, not only one.Neutrosophic variables may be: dependent; independent; or partially dependent, partially independent, and partially indeterminate.(i.e.unknown if dependent or independent).The data's attributes (features, functions etc.) are investigated by survey-based techniques within the frame of Neutrosophic Conjoint Analysis (which includes the choice based conjoint and the adaptive choice-based conjoint.)Indeterminacy may occur at the level of attributes as well.We may thus deal with neutrosophic (indeterminate, unclear, partially known etc.) attributes [3][4][5][6][7][8][9][10][11][12][13][14].

Classical MultiVariate Analysis vs. Plithogenic Variate Analysis
The Classical MultiVariate Analysis (MVA) studies a system, which is characterized by many variables, or one may call it a system-of-systems.The variables, i.e. the subsystems, and the system as a whole are also classical (i.e. they do not deal with indeterminacy).Many classical measurements are needed, and the classical relations between variables to be determined.This system-of-systems is generally represented by a surrogate approximate model.
The Plithogenic Variate Analysis (PVA) is an extension of of the classical MultiVariate Analysis, where indeterminate data or procedures, that are called neutrosophic data and respectively neutrosophic procedures, are allowed.Therefore PVA deals with neutrosophic/ indeterminate variables, neutrosophic/indeterminate subsystems, and neutrosophic/indeterminate system-of-systems as a whole.
Therefore the Plithogenic Variate Analysis studies a neutrosophic/indeterminate system as a whole, characterized by many neutrosophic/indeterminate variables (i.e.neutrosophic/indeterminate sub-systems), and many neutrosophic/indeterminate relationships.Hence many neutrosophic measurements and observations are needed.
The Plithogenic Variate Analysis requires complex computations, hence it is more complicated than the Classical MultiVariate Analysis due to the neutrosophic (indeterminate) data it deals with; nonetheless the PVA better reflects our world, giving results nearer to real-life situation.With the dramatic increase of computers power this complexity is overcome.
The Plithogenic Variate Analysis elucidates each attribute of the data, using various methods, such as: regression/factor/cluster/path/discriminant/latent (trait or profile)/multilevel analysis / structural equation/recursive partition/redundancy/ constrained correspondence/ artificial neural networks, multidimensional scaling, and so on.
The Plithogenic UniVariate Analysis (PUVA) comprises the procedures for analysis of neutrosophic/indeterminate data that contains only one neutrosophic/indeterminate variable.

Plithogenic Probability
The Plithogenic Probability of an event to occur is composed from the chances that the event occurs with respect to all random variables (parameters) that determine it.
The Plithogenic Probability, based on Plithogenic Variate Analysis, is a multi-dimensional probability ("plitho" means "many", synonym with "multi").We may say that it is a probability of sub-probabilities, where each sub-probability describes the behavior of one variable.We assume that the event we study is produced by one or more variables.
Each variable is represented by a Probability Distribution (Density) Function (PDF).where T is the chance that the event occurs, and F is the chance that the event do not occur, with T, F [0, 1], 0 ≤ T+ F ≤ 1, then we have a Plithogenic Intuitionistic Fuzzy Probability.(v) If all PDFs are in the Picture Fuzzy Set style, i.e. of the form (T, N, F), where T is the chance that the event occurs, N is the neutral-chance of the event to occur or not, and F is the chance that the event do not occur, with T, N, F  [0, 1], 0 ≤ T+N+F ≤ 1, then we have a Plithogenic Picture Fuzzy Probability.(vi) If all PDFs are in the Spherical Fuzzy Set style, i.e. of the form (T, H, F), where T is the chance that the event occurs, H is the neutral-chance of the event to occur or not, and F is the chance that the event do not occur, with T, H, F 

Subclasses of Plithogenic
then we have a Plithogenic Picture Fuzzy Probability.(vii) In general, if all PDFs are in any (fuzzy-extension set) style, then we have a Plithogenic (fuzzy-extension) Probability.

(viii)
If some PDFs are in one of the above styles, while others are in different styles, then we have a Plithogenic Hybrid Probability.
All the above sub-classes of plithogenic probability may be refined this way.

Convergence from MultiVariate to UniVariate Analysis
In order to be able to make a decision, we need to convert from Plithogenic (MultiVariate) Probability and Statistics to Plithogenic UniVariate Probability and Statistics.Actually we need to fusion (combine) all variables and obtain a single cumulative variable.
The Classical Probability Space is complete, i.e. all possible event that may occur are known.For example, let's consider a soccer game between teams A and B. The classical probability space is CPS = {A wins, tie game, B wins}.
The Neutrosophic Probability Space is in general incomplete, i.e. not all possible events are known, and there also are events that are only partially known.In our world, most real probability spaces are neutrosophic.
Example.Considering the same soccer game, the neutrosophic probability space NPS = {A wins, tie game, B wins, interrupted game, etc.}, "interrupted" means that due to some unexpected weather conditions, or to a surprising terrorist attack on the stadium, etc. the game is interrupted and rescheduled (this has happened in our world many times).

Florentin Smarandache, Plithogenic Probability & Statistics are generalizations of MultiVariate Probability & Statistics
In order to convert from multi-probability to uni-probability, we apply various logical operators (conjunctions, disjunctions, negations, implications, etc. and their combinations, depending on the application to do and on the expert) on the multi-probability.
Such applications are presented towards the end of the paper.

Plithogenic Statistics
Plithogenic Statistics (PS) encompasses the analysis and observations of the events studied by the Plithogenic Probability.
Plithogenic Statistics is a generalization of classical MultiVariate Statistics, and it is a simultaneous analysis of many outcome neutrosophic/indeterminate variables, and it as well is a multi-indeterminate statistics.

Subclasses of Plithogenic Statistics are:
- 9. Plithogenic Refined Statistics are, similarly, the most general form of statistics that studies the analysis and observations of the events described by the Plithogenic Refined Probability.

Applications of Plithogenic Probability
We retour our 2017 example [1] and pass it through all sub-classes of Plithogenic Probability.
In the Spring 2021 semester, at The University of New Mexico, United States, in a program of Electrical Engineering, Jenifer needs to pass four courses in order to graduate at the end of the semester: two courses of Mathematics (Second Order Differential Equations, and Stochastic Analysis), and two courses of Mechanics (Fluid Mechanics and Solid Mechanics).What is the Plithogenic Probability that Jenifer will graduate?
Her chances of graduating are estimated by the university's advisors.
There are four variables (courses), v1, v2, v3, v4 respectively, that generate four probability distributions.We consider the discrete probability distribution functions.
[ For the continuous ones, it will be similar.]

Classical MultiVariate Probability (CMVP)
The advisers have estimated that CMVP(Jenifer) = (0.5, 0.6, 0.8, 0.4), which means that Jenifer has 50% chance to pass the Second Order Differential Equations class, 60% chance of passing the Stochastic Analysis class (both as part of Mathematics), and 80% chance of passing the Fluid Mechanics class, and 40% chance of passing the Solid Mechanics class (both as part of Mechanics).

Plithogenic Hybrid Probability (PHP)
PHP(Jenifer) = ( 0.5; (0.7, 0.1, 0.4); (0.1, 0.2); 0.4 or 0.3 ), which means that Jenifer has 50% chance to pass the Second Order Differential Equations class; 70% chance of passing and 10% indeterminate-chance and 40% chance of failing the Stochastic Analysis class (both as part of Mathematics); and 10% chance of passing and 20% of failing the Fluid Mechanics class; and 40% or 30% chance of passing the Solid Mechanics class (both as part of Mechanics).
We have mixed herein: the fuzzy, neutrosophic, intuitionistic fuzzy, and indeterminate above cases.
(ii) Let's change the example and assume that for Jenifer to graduate she needs to pass at least one class among the four.Now we use the disjunction operator: or fuzzy disjunction (t-conorm).Therefore, Jenifer's chance of graduating is CMVP(Jenifer) = max{0.5,0.6, 0.8, 0.4} = 0.8, or 80% chance.
(iii) Let's change again the example and assume that for Jenifer to graduate she needs to pass at least one class of Mathematics and at least one class of Mechanics.Then, we use a mixture of conjunctions and disjunctions: CMVP(Jenifer) = = min{max{0.5,0.6}, max{0.8,0.4}} = min{0.6,0.8} = 0.6, or 60% chance.

Corresponding Applications of Plithogenic Statistics
A prospective is made on the university student population, that was enrolled this semester, in order to determine the chance of the average students to graduate.
Let's take a random sample of the university's student population in order to investigate what's the chance of graduating for an enrolled average student.
By inference statistics, we estimate the population's average student to be similar to the sample's average student.
We may have a classical random sample, i.e. the sample size is known and all sample individuals belong 100% of the population -i.e. the individuals are full-time students; or a neutrosophic random sample {i.e. the sample size may be unknown or only approximately known), and some or all individuals may only partially belonging to the population (for example part-time students), or may have taken some extra classes above the norm.
Even the university's student population is a neutrosophic population, since the number of students changes almost continuously (some students drop, others enroll earlier or later), and not all students are 100% enrolled: there are full-time, part-time, and even over-time (i.e.students enrolled in more than the required full time number or credit hour classes).
In a classical population, the population size is known, and all population individuals belong 100% to the population.
Let T = truth, with T belongs to [0,1], be the chance to graduate, I = indeterminate, with I belongs to [0,1], be the indeterminate-chance to graduate, and F = falsehood, with F belongs to [0,1], be the chance not to graduate, where 0 ≤ T + I + F ≤ 3.
Make the average of all sample students, assuming the sample size is n ≥ 2, T j T j T j I j I j I j F j F j F j where p, r, s ≥ 0 are integers, and p + r + s ≥ 1.
All refined neutrosophic sub-components ( ),1 k p, ( ),1 r, ( ),1 s, Then, the average of PRNPs of the sample students is: And we get the sample average students' plithogenic refined probability to (graduate, indeterminate graduate, not graduate).
For the cases when one or two among T, I, F are missing, we simply discard them.An average student is not among the best, not among the worst.Let's consider Jenifer is an average student, whose plithogenic probabilities have been obtained after sampling and computing the average of plithogenic probabilities of all its students -since we have already her data.
An average student has 40% chance to graduate, 90% indeterminate chance of graduating, and 50% chance not to graduate.

Plithogenic Indeterminate Statistics
An average student has (40% or 50%) chance of graduating, 90% indeterminate chance of graduating, and (50% or unknown) chance of not graduating.

Plithogenic Intuitionistic Fuzzy Statistics
An average student has 40% chance to graduate and 50% chance not to graduate.

Florentin Smarandache ,
Plithogenic Probability & Statistics are generalizations of MultiVariate Probability & Statistics Florentin Smarandache, Plithogenic Probability & Statistics are generalizations of MultiVariate Probability & Statistics For the Plithogenic Refined Neutrosophic Probabilities, the average is a straight-forward extension.Let the student Sj, 1 ≤ j ≤ n, have the Plithogenic Refined Neutrosophic Probability: Florentin Smarandache, Plithogenic Probability & Statistics are generalizations of MultiVariate Probability & Statistics