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Fuzzy Logic, Neutrosophic Logic, and Applications

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Title: Fuzzy Logic, Neutrosophic Logic, and Applications


1
Fuzzy Logic, Neutrosophic Logic, and Applications
  • Florentin Smarandache
  • University of New Mexico
  • Gallup, NM 87301, USA
  • E-mail smarand_at_unm.edu
  • M. Khoshnevisan
  • Griffith University, Gold Coast
  • Queensland 9726, Australia
  • E-mail m.khoshnevisan_at_mailbox.edu.au

2
Contents
  • Introduction to Non-Standard Analysis
  • Survey of Tripartion Logics and Combination Rules
    up to Intuitionistic Fuzzy Logic
  • Neutrosophic Components
  • Neutrosophic Logic - a Generalization of the
    Intuitionistic Fuzzy Logic
  • Differences between Intuitionistic Fuzzy Logic
    (IFL) and Neutrosophic Logic (NL)
  • Operations on Classical Sets
  • Neutrosophic Logic Connectors
  • Generalizations of Other Logics
  • Applications of Fuzzy and Neutrosophic Logics to
    Finance

3
1. Introduction to Non-Standard Analysis
  • - in 1960s Abraham Robinson has developed the
    non-standard analysis
  • - formally, x is said to be infinitesimal if and
    only if for all positive integers n one has ?x? lt
    1/n
  • - by -a one denotes a monad, i.e. a set of
    hyper-real numbers in non-standard analysis
  • (-a) a-x x?R, x is infinitesimal,
  • and similarly b is a monad
  • (b) bx x?R, x is infinitesimal
  • - combining the two previous mentioned
    definitions one gets, what well call, a binad of
    -c
  • (-c)c-x x?R, x is infinitesimal ?
    cx x?R, x is infinitesimal
  • - the left and right borders of a non-standard
    interval -a, b are vague, imprecise,
    themselves being the non-standard (sub)sets (-a)
    and (b) as defined above
  • - by extension let inf -a, b -a and sup
    -a, b b

4
2. Survey of tripartition logics and combination
rules up to intuitionistic fuzzy logic
  • - Hoopers rule of combination of evidence
    (1680s), which was a Non-Bayesian approach to
    find a probabilistic model
  • idea of tripartition (truth, falsehood,
    indeterminacy) appeared in 1764 when J. H.
    Lambert investigated the credibility of one
    witness affected by the contrary testimony of
    another
  • - Lukasiewicz considered three values (1, 1/2,
    0), Post considered m values in between the
    wars
  • - Koopman in 1940s introduced the notions of
    lower and upper probability, followed by Good,
    and Dempster (1967) gave a rule of combining two
    arguments
  • - Shafer (1976) extended it to the
    Dempster-Shafer Theory of Belief Functions (DST)
    by defining the Belief and Plausibility functions
    and using the rule of inference of Dempster for
    combining two evidences proceeding from two
    different sources. Belief function is a
    connection between fuzzy reasoning and
    probability. DST is a generalization of the
    Bayesian Probability (Bayes 1760s, Laplace
    1780s) this uses the mathematical probability in
    a more general way, and is based on probabilistic
    combination of evidence in artificial
    intelligence.

5
2. Survey of tripartition logics and combination
rules up to intuitionistic fuzzy logic
  • in Lambert there is a chance p that the witness
    will be faithful and accurate, a chance q that he
    will be mendacious, and a chance 1-p-q that he
    will simply be careless apud Shafer (1986).
    Whence three components accurate, mendacious,
    careless, which add up to 1.
  • - The many-valued logic was replaced by Goguen
    (1969) and Zadeh (1975) with an Infinite-Valued
    Logic (of continuum power, as in the classical
    mathematical analysis and classical probability)
    called Fuzzy Logic, where the truth-value can be
    any number in the closed unit interval 0, 1.
    The Fuzzy Set was introduced by Zadeh in 1965.
  • - Van Fraassen introduced the supervaluation
    semantics in his attempt to solve the sorites
    paradoxes, followed by Dummett (1975) and Fine
    (1975). They all tripartitioned, considering a
    vague predicate which, having border cases, is
    undefined for these border cases. They all
    tripartitioned, considering a vague predicate
    which, having border cases, is undefined for
    these border cases. Van Fraassen took the vague
    predicate heap and extended it positively to
    those objects to which the predicate definitively
    applies and negatively to those objects to which
    it definitively doesnt apply. The remaining
    objects border was called penumbra. A sharp
    boundary between these two extensions does not
    exist for a soritical predicate. Inductive
    reasoning is no longer valid too if S is a
    sorites predicate, the proposition
    ?n(San?San1) is false.

6
2. Survey of tripartition logics and combination
rules up to intuitionistic fuzzy logic
  • Thus, the predicate Heap (positive
    extension) true, Heap (negative extension)
    false, Heap (penumbra) indeterminate.
  • - Narinyani (1980) used the tripartition to
    define what he called the indefinite set
  • - Atanassov (1982) continued on tripartition and
    gave five generalizations of the fuzzy set,
    studied their properties and applications to the
    neural networks in medicine
  • a) Intuitionistic Fuzzy Set (IFS)
  • Given an universe E, an IFS A over E is a set of
    ordered triples ltuniverse_element,
    degree_of_membership_to_A(M), degree_of_non-member
    ship_to_A(N)gt such that MN ? 1 and M, N ? 0,
    1. When M N 1 one obtains the fuzzy set,
    and if M N lt 1 there is an indeterminacy I
    1-M-N.
  • b) Intuitionistic L-Fuzzy Set (ILFS)
  • Is similar to IFS, but M and N belong to a fixed
    lattice L.
  • c) Interval-valued Intuitionistic Fuzzy Set
    (IVIFS)
  • Is similar to IFS, but M and N are subsets of 0,
    1 and sup M sup N ? 1.
  • d) Intuitionistic Fuzzy Set of Second Type
    (IFS2)
  • Is similar to IFS, but M2 N2 ? 1. M and N are
    inside of the upper right quarter of unit circle.
  • e) Temporal IFS
  • Is similar to IFS, but M and N are functions of
    the time-moment too.

7
3. Neutrosophic Logic Components
  • Lets consider the non-standard finite numbers 1
    1?,
  • where 1 is its standard part and ? its
    non-standard part,
  • and 0 0-?, where 0 is its standard part and
    ? its non-standard part. Then, we call -0,
    1 a non-standard unit interval.
  • Let T, I, F be standard or non-standard real
    subsets of -0, 1 ,
  • with sup T tsup, inf T tinf,
  • sup I isup, inf I iinf,
  • sup F fsup, inf F finf,
  • and nsup tsupisupfsup,
  • ninf tinfiinffinf.
  • T, I, F are called neutrosophic components, which
    represent the truth value, indeterminacy value,
    and falsehood value respectively referring to
    neutrosophy, neutrosophic logic, neutrosophic
    set, neutrosophic probability and statistics.
  • The sets T, I, F are not necessarily intervals,
    but may be any real sub-unitary subsets
    discrete or continuous single-element, finite,
    or (countably or uncountably) infinite union or
    intersection of various subsets etc.
  • They may also overlap. The real subsets could
    represent the relative errors in determining t,
    i, f (in the case when the subsets T, I, F are
    reduced to points).

8
3. Neutrosophic Logic Components
  • This representation is closer to the human mind
    reasoning. It characterizes/catches the
    imprecision of knowledge or linguistic
    inexactitude received by various observers
    (thats why T, I, F are subsets - not necessarily
    single-elements), uncertainty due to incomplete
    knowledge or acquisition errors or stochasticity
    (thats why the subset I exists), and vagueness
    due to lack of clear contours or limits (thats
    why T, I, F are subsets and I exists in
    particular for the appurtenance to the
    neutrosophic sets).

9
4. Neutrosophic Logic a Generalization of the
Intuitionistic Fuzzy Logic
  • Definition
  • A logic in which each proposition is estimated to
    have the percentage of truth in a subset T, the
    percentage of indeterminacy in a subset I, and
    the percentage of falsity in a subset F, where T,
    I, F are defined above, is called Neutrosophic
    Logic.
  • We use a subset of truth (or indeterminacy, or
    falsity), instead of a number only, because in
    many cases we are not able to exactly determine
    the percentages of truth and of falsity but to
    approximate them for example a proposition is
    between 0.30-0.40 true and between 0.60-0.70
    false, even worst between 0.30-0.40 or 0.45-0.50
    true (according to various analyzers), and 0.60
    or between 0.66-0.70 false.
  • The subsets are not necessary intervals, but any
    sets (discrete, continuous, open or closed or
    half-open/half-closed interval, intersections or
    unions of the previous sets, etc.) in accordance
    with the given proposition.
  • A subset may have one element only in special
    cases of this logic.
  • Constants (T, I, F) truth-values, where T, I, F
    are standard or non-standard subsets of the
    non-standard interval -0, 1 , where ninf
    inf T inf I inf F ? -0, and nsup sup T
    sup I sup F ? 3.
  • Atomic formulas a, b, c, .
  • Arbitrary formulas A, B, C, .

10
4. Neutrosophic Logic a Generalization of the
Intuitionistic Fuzzy Logic
  • Therefore, we finally generalize the
    intuitionistic fuzzy logic to a transcendental
    logic, called neutrosophic logic where the
    interval 0, 1 is exceeded, i.e. , the
    percentages of truth, indeterminacy, and falsity
    are approximated by non-standard subsets not by
    single numbers, and these subsets may overlap and
    exceed the unit interval in the sense of the
    non-standard analysis also the superior sums and
    inferior sum, nsup sup T sup I sup F ?
    -0, 3 , may be as high as 3 or 3, while ninf
    inf T inf I inf F ? -0, 3 , may be as low
    as 0 or 0.
  • Lets borrow from the modal logic the notion of
    world, which is a semantic device of what the
    world might have been like. Then, one says that
    the neutrosophic truth-value of a statement A,
    NLt(A) 1 if A is true in all possible worlds
    (syntagme first used by Leibniz) and all
    conjunctures, that one may call absolute truth
    (in the modal logic it was named necessary truth,
    Dinulescu-C?mpina (2000) names it intangible
    absolute truth ), whereas NLt(A) 1 if A is
    true in at least one world at some conjuncture,
    we call this relative truth because it is
    related to a specific world and a specific
    conjuncture (in the modal logic it was named
    possible truth).
  • Similarly for absolute and relative falsehood and
    absolute and relative indeterminacy.
  • The neutrosophic inference inference Dezert
    (2002), especially for plausible and paradoxist
    information, is still a subject of intense
    research today.

11
5. Differences between Neutrosophic Logic and
Intuitionistic Fuzzy Logic
  • a) Neutrosophic Logic can distinguish between
    absolute truth (truth in all possible worlds,
    according to Leibniz) and relative truth (truth
    in at least one world), because NL(absolute
    truth)1 while NL(relative truth)1. This has
    application in philosophy (see the neutrosophy).
    Thats why the unitary standard interval 0, 1
    used in IFL has been extended to the unitary
    non-standard interval -0, 1 in NL. Similar
    distinctions for absolute or relative falsehood,
    and absolute or relative indeterminacy are
    allowed in NL.
  • b) In NL there is no restriction on T, I, F
    other than they are subsets of -0, 1, thus
  • -0 ? inf T inf I inf F ? sup T sup
    I sup F ? 3. This non-restriction allows
    paraconsistent, dialetheist, and incomplete
    information to be characterized in NL i.e. the
    sum of all three components if they are defined
    as points, or sum of superior limits of all three
    components if they are defined as subsets can be
    gt1 (for paraconsistent information coming from
    different sources) or lt 1 for incomplete
    information, while that information can not be
    described in IFL because in IFL the components T
    (truth), I (indeterminacy), F (falsehood) are
    restricted either to tif1 or to t2 f2 ? 1,
    if T, I, F are all reduced to the points t, i, f
    respectively, or to sup T sup I sup F 1 if
    T, I, F are subsets of 0, 1.
  • Some researchers normalize the
    paraconsistent and incomplete information, but
    this procedure is not always suitable.
  • c) In NL the components T, I, F can be
    non-standard subsets included in the unitary
    non-standard interval -0, 1, not only standard
    subsets included in the unitary standard interval
    0, 1 as in IFL.
  • d) NL, like dialetheism, can describe paradoxes,
    NL(paradox) (1, I, 1), while IFL can not
    describe a paradox because the sum of components
    should be 1 in IFL.

12
5. Differences between Neutrosophic Logic and
Intuitionistic Fuzzy Logic
  • e) The logical operators in IFL (and
    similarly the set connectors in IFS) are defined
    with respect to T and F components only, and
    whats left up to 1 is considered the
    Indeterminacy. In NL (and similarly in NS) they
    are defined with respect to any of the three
    components.
  • f) Component I, indeterminacy, can be
    split into more subcomponents in order to better
    catch the vague information we work with, and
    such, for example, one can get more accurate
    answers to the Question-Answering Systems
    initiated by Zadeh (2003). In Belnaps
    four-valued logic (1977) indeterminacy is split
    into Uncertainty (U) and Contradiction (C).

13
6. Operations with Classical Sets
  • We need to present these set operations in
    order to be able to introduce the neutrosophic
    connectors.
  • Let S1 and S2 be two (unidimensional) real
    standard or non-standard subsets included in the
    non-standard interval -0, 8) then one defines
  • 6.1 Addition of Sets
  • S1?S2 x?xs1s2, where s1?S1 and s2?S2,
  • with inf S1?S2 inf S1 inf S2, sup S1?S2 sup
    S1 sup S2
  • and, as some particular cases, we have
  • a?S2 x?xas2, where s2?S2
  • with inf a?S2 a inf S2, sup a?S2 a
    sup S2.
  • 6.2 Subtraction of Sets
  • S1?S2 x?xs1-s2, where s1?S1 and s2?S2.
  • For real positive subsets (most of the cases will
    fall in this range) one gets
  • inf S1?S2 inf S1 - sup S2, sup S1?S2 sup S1 -
    inf S2
  • and, as some particular cases, we have
  • a?S2 x?xa-s2, where s2?S2,
  • with inf a?S2 a - sup S2, sup a?S2 a -
    inf S2
  • also 1?S2 x?x1-s2, where s2?S2,
  • with inf 1?S2 1 - sup S2, sup 1?S2 100
    - inf S2.

14
6. Operations with Classical Sets
  • 6.3 Multiplication of Sets
  • S1?S2 x?xs1?s2, where s1?S1 and s2?S2.
  • For real positive subsets (most of the cases will
    fall in this range) one gets inf S1?S2 inf S1 ?
    inf S2, sup S1?S2 sup S1 ? sup S2
  • and, as some particular cases, we have
  • a?S2 x?xa?s2, where s2?S2,
  • with inf a?S2 a ? inf S2, sup a?S2 a ?
    sup S2
  • also 1?S2 x?x 1?s2, where s2?S2,
  • with inf 1?S2 1? inf S2, sup 1?S2 1 ?
    sup S2.
  • 6.4 Division of a Set by a Number
  • Let k ??, then S1?k x?xs1/k, where s1?S1.

15
7. Neutrosophic Logic Connectors
  • One uses the definitions for neutrosophic
    probability and neutrosophic set operations.
  • Similarly, there are many ways to construct such
    connectives according to each particular problem
    to solve here we present the easiest ones
  • One notes the neutrosophic logic values of the
    propositions A1 and A2 by NL(A1) ( T1, I1, F1 )
    and NL(A2) ( T2, I2, F2 ) respectively.
  • For all neutrosophic logic values below if,
    after calculations, one obtains values lt 0 or gt
    1, one replaces them by 0 or 1 respectively.
  • 7.1. Negation
  • NL(?A1) ( 1?T1, 1?I1, 1?F1 ).
  • 7.2. Conjunction
  • NL(A1 ? A2) ( T1?T2, I1?I2, F1?F2 ).
  • (And, in a similar way, generalized for n
    propositions.)
  • 7.3 Weak or inclusive disjunction
  • NL(A1 ? A2) ( T1?T2?T1?T2, I1?I2?I1?I2,
    F1?F2?F1?F2 ).
  • (And, in a similar way, generalized for n
    propositions.)

16
7. Neutrosophic Logic Connectors
  • 7.4 Strong or exclusive disjunction
  • NL(A1 ? A2)
  • ( T1? (1?T2) ?T2? (1?T1) ?T1?T2?
    (1?T1) ? (1?T2),
  • I1 ? (1?I2) ?I2 ? (1?I1) ?I1
    ? I2 ? (1?I1) ? (1? I2),
  • F1? (1?F2) ?F2? (1? F1) ?F1? F2
    ? (1?F1) ? (1?F2) ).
  • (And, in a similar way, generalized for n
    propositions.)
  • 7.5 Material conditional (implication)
  • NL(A1 ? A2) ( 1?T1?T1?T2, 1?I1?I1?I2,
    1?F1?F1?F2 ).
  • 7.6 Material biconditional (equivalence)
  • NL(A1 ? A2) ( (1?T1?T1?T2) ?
    (1?T2?T1?T2),
  • (1? I1? I1? I2) ?
    (1?I2? I1 ? I2),
  • (1?F1?F1? F2) ?
    (1?F2?F1? F2) ).
  • 7.7 Sheffer's connector
  • NL(A1 A2) NL(?A1 ? ?A2) ( 1?T1?T2,
    1?I1?I2, 1?F1?F2 ).
  • 7.8 Peirce's connector

17
8. Generalizations
  • When all neutrosophic logic set components are
    reduced to one element, then
  • tsup tinf t, isup iinf i, fsup finf
    f, and nsup ninf n tif, therefore
    neutrosophic logic generalizes
  • - the intuitionistic logic, which supports
    incomplete theories (for 0 lt n lt 1 and i0, 0 ?
    t, i, f ? 1)
  • - the fuzzy logic (for n 1 and i 0, and 0 ?
    t, i, f ? 1)
  • from "CRC Concise Concise Encyclopedia of
    Mathematics", by Eric W. Weisstein, 1998, the
    fuzzy logic is "an extension of two-valued logic
    such that statements need not to be True or
    False, but may have a degree of truth between 0
    and 1"
  • - the intuitionistic fuzzy logic (for n1)
  • - the Boolean logic (for n 1 and i 0, with t,
    f either 0 or 1)
  • - the multi-valued logic (for 0 ? t, i, f ? 1)
  • definition of ltmany-valued logicgt from "The
    Cambridge Dictionary of Phylosophy", general
    editor Robert Audi, 1995, p. 461 "propositions
    may take many values beyond simple truth and
    falsity, values functionally determined by the
    values of their components" Lukasiewicz
    considered three values (1, 1/2, 0). Post
    considered m values, etc. But they varied in
    between 0 and 1 only. In the neutrosophic logic
    a proposition may take values even greater than 1
    (in percentage greater than 100) or less than 0.

18
8. Generalizations
  • - the paraconsistent logic, which support
    conflicting information (for n gt 1 and i 0,
    with both t, f lt 1)
  • - the dialetheism, which says that some
    contradictions are true (for t f 1 and i 0
    some paradoxes can be denoted this way too)
  • - the faillibilism, which says that uncertainty
    belongs to every proposition (for i gt 0)
  • Compared with all other logics, the neutrosophic
    logic and intuitionistic fuzzy logic introduce a
    percentage of "indeterminacy" - due to unexpected
    parameters hidden in some propositions, or
    unknowness, but neutrosophic logic let each
    component t, i, f be even boiling over 1
    (overflooded), i.e. be 1, or freezing under 0
    (underdried), i.e. be 0 in order to be able to
    make distinction between relative truth and
    absolute truth, and between relative falsity and
    absolute falsity in philosophy.

19
9. Applications of Fuzzy and Neutrosophic Logics
to Finance
  • Example 9.1 Reconciliation between theoretical
    and market prices of long-term options contracts.
  • The neutrosophic probability approach makes a
    distinction between relative sure event, event
    that is true only in certain world(s) NP (rse)
    1, and absolute sure event, event that is true
    for all possible world(s) NP (ase) 1. Similar
    relations can be drawn for relative impossible
    event / absolute impossible event and
    relative indeterminate event / absolute
    indeterminate event. In case where the truth-
    and falsity-components are complimentary i.e.
    they sum up to unity, and there is no
    indeterminacy and one is reduced to classical
    probability. Therefore, neutrosophic probability
    may be viewed as a generalization of classical
    and imprecise probabilities.
  • When a long-term option priced by the collective
    action of the market players is observed to be
    deviating from the theoretical price, three
    possibilities must be considered
  • (1) The theoretical price is obtained by an
    inadequate pricing model, which means that the
    market price may well be the true price,
  • (2) An unstable rationalization loop has taken
    shape that has pushed the market price of the
    option out of sync with its true price, or
  • (3) The nature of the deviation is indeterminate
    and could be due to (a) or (b) or a
    super-position of both (a) and (b) and/or due to
    some random white noise.

20
9. Applications of Fuzzy and Neutrosophic Logics
to Finance
  • However, it is to be noted that in none of these
    three possible cases are we referring to the
    efficiency or otherwise of the market as a whole.
    The market can only be as efficient as the
    information it gets to process. Therefore, if the
    information about the true price of the option is
    misleading (perhaps due to an inadequate pricing
    model), the market cannot be expected to process
    it into something useful after all, the markets
    cant be expected to pull jack-rabbits out of
    empty hats!
  • With T, I, F as the neutrosophic components, let
    us now define the following events
  • H p p is the true option price determined by
    the theoretical pricing model and
  • M p p is the true option price determined by
    the prevailing market price
  • Then there is a t chance that the event (H ? Mc)
    is true, or corollarily, the corresponding
    complimentary event (Hc ? M) is untrue, there is
    a f chance that the event (Mc ? H) is untrue, or
    corollarily, the complimentary event (M ? Hc) is
    true and there is a i chance that neither (H ?
    Mc) nor (M ? Hc) is true/untrue i.e. the
    determinant of the true market price is
    indeterminate. This would fit in nicely with
    possibility (c) enumerated above that the
    nature of the deviation could be due to either
    (a) or (b) or a super-position of both (a) and
    (b) and/or due to some random white noise.
  • Illustratively, a set of AR1 models used to
    extract the mean reversion parameter driving the
    volatility process over time have coefficients of
    determination in the range say between 50-70,
    then we can say that t varies in the set T (50 -
    70).

21
9. Applications of Fuzzy and Neutrosophic Logics
to Finance
  • If the subjective probability assessments of
    well-informed market players about the weight of
    the current excursions in implied volatility on
    short-term options lie in the range say between
    40-60, then f varies in the set F (40 - 60).
    Then unexplained variation in the temporal
    volatility driving process together with the
    subjective assessment by the market players will
    make the event indeterminate by either 30 or
    40. Then the neutrosophic probability of the
    true price of the option being determined by the
    theoretical pricing model is NP (H ? Mc) (50
    70), (40 60), 30, 40. The DSmT (acronym for
    Dezert-Smarandache Theory) can be used in cases
    like these to fuse the conflicting sources of
    information and arrive at a correct and
    computable probabilistic assessment of the true
    price of the long-term option.
  • Example 9.2 Extension of the MASS model as a
    cost-optimal relative allocation of facilities
    technique by the incorporation of neutrosophic
    statistics and the DSmT combination rule.

22
9. Applications of Fuzzy and Neutrosophic Logics
to Finance
  • The original CRAFT-type models for cost-optimal
    relative allocation of facilities technique as
    well as its later extensions are primarily
    deterministic in nature. A Modified Assignment
    (MASS) model (first proposed by Bhattacharya and
    Khoshnevisan in 2003) follows the same iterative,
    deterministic logic. However, some amount of
    introspection will reveal that the facilities
    layout problem is basically one of achieving best
    interconnectivity by optimal fusion of spatial
    information. In that sense, the problem may be
    better modeled in terms of mathematical
    information theory whereby the best layout is
    obtainable as the one that maximizes relative
    entropy of the spatial configuration. Going a
    step further, one may hypothesize a neutrosophic
    dimension to the problem. Given a DSmT type
    combination rule, the layout optimization problem
    may be framed as a normalized basic probability
    assignment for optimally comparing between
    several alternative interconnectivities. The
    neutrosophic argument can be justified by
    considering the very practical possibility of
    conflicting bodies of evidence for the structure
    of the load matrix possibly due to conflicting
    assessments of two or more design engineers.
  • If for example we consider two mutually
    conflicting bodies of evidence ?1 and ?2,
    characterized respectively by their basic
    probability assignments ?1 and ?2 and their cores
    k (?1) and k (?2) then one has to look for the
    optimal combination rule which maximizes the
    joint entropy of the two conflicting information
    sources. Mathematically, it boils down to the
    general optimization problem of finding the value
    of min? H (?) subject to the constraints
    that the marginal basic probability assignments
    ?1 (.) and ?2 (.) are obtainable by the summation
    over each column and summation over each row
    respectively of the relevant information matrix
    the sum of all cells of the information matrix is
    unity.

23
9. Applications of Fuzzy and Neutrosophic Logics
to Finance
  • Example 9.3 Conditional probability of actually
    detecting a financial fraud a neutrosophic
    extension to the application of Benfords
    first-digit law
  • In an earlier paper (Kumar and Bhattacharya,
    2002), we had proposed a Monte Carlo adaptation
    of Benfords first-digit law. There has been some
    research already on the application of Benfords
    law to financial fraud detection. However, most
    of the practical work in this regard has been
    concentrated in detecting the first digit
    frequencies from the account balances selected on
    basis of some known audit sampling method and
    then directly comparing the result with the
    expected Benford frequencies. We have voiced
    slight reservations about this technique in so
    far as that the Benford frequencies are
    necessarily steady state frequencies and may not
    therefore be truly reflected in the sample
    frequencies. As samples are always of finite
    sizes, it is therefore perhaps not entirely fair
    to arrive at any conclusion on the basis of such
    a direct comparison, as the sample frequencies
    wont be steady state frequencies.
  • However, if we draw digits randomly using the
    inverse transformation technique from within
    random number ranges derived from a cumulative
    probability distribution function based on the
    Benford frequencies then the problem boils down
    to running a goodness of fit kind of test to
    identify any significant difference between
    observed and simulated first-digit frequencies.
    This test may be conducted using a known sampling
    distribution like for example the Pearsons ?²
    distributions. The random number ranges for the
    Monte Carlo simulation are to be drawn from a
    cumulative probability distribution function
    based on the following Benford probabilities
    given in Table I below.

24
9. Applications of Fuzzy and Neutrosophic Logics
to Finance
  • The first-digit probabilities can be best
    approximated mathematically by the log-based
    formula as was derived by Benford P (First
    significant digit d) log10 1 (1/d).
  • 9.4. Computational Algorithm (first proposed by
    Kumar and Bhattacharya in 2002)
  • Define a finite sample size n and draw a sample
    from the relevant account balances using a
    suitable audit sampling procedure


  • Perform a continuous Monte Carlo run of length ?
    ? (1/2?)2/3 grouped in epochs of size n using a
    customized MS-Excel spreadsheet. Derivation of ?
    and other statistical issues have been discussed
    in detail in our earlier paper (Kumar and
    Bhattacharya, 2002)
  • Test for significant difference in sample
    frequencies between the first digits observed in
    the sample and those generated by the Monte Carlo
    simulation by using a goodness of fit test
    using the ?² distribution. The null and
    alternative hypotheses are as follows
  • H0 The observed first digit frequencies
    approximate a Benford distribution
  • H1 The observed first digit frequencies do not
    approximate a Benford distribution

25
9. Applications of Fuzzy and Neutrosophic Logics
to Finance
  • This statistical test will not reveal whether or
    not a fraud has actually been committed. All it
    does is establishing at a desired level of
    confidence that the accounting data may not be
    naturally occurring (if H0 can be rejected).
    However, given that H1 is accepted and H0 is
    rejected, it could possibly imply any of the
    following events
  • I. There is no manipulation - occurrence of a
    Type I error i.e. H0 rejected when true.
  • II. There is manipulation and such manipulation
    is definitely fraudulent.
  • III. There is manipulation and such manipulation
    may or may not be fraudulent.
  • IV. There is manipulation and such manipulation
    is definitely not fraudulent.
  • 9.5. Neutrosophic extension using DSmT
    combination rule
  • Neutrosophic probabilities are a generalization
    of classical and fuzzy probabilities and cover
    those events that involve some degree of
    indeterminacy. It provides a better approach to
    quantifying uncertainty than classical or even
    fuzzy probability theory. Neutrosophic
    probability theory uses a subset-approximation
    for truth-value as well as indeterminacy and
    falsity values. Also, this approach makes a
    distinction between relative true event and
    absolute true event the former being true in
    only some probability sub-spaces while the latter
    being true in all probability sub-spaces.
    Similarly, events that are false in only some
    probability sub-spaces are classified as
    relative false events while events that are
    false in all probability sub-spaces are
    classified as absolute false events. Again, the
    events that may be hard to classify as either
    true or false in some probability sub-spaces
    are classified as relative indeterminate events
    while events that bear this characteristic over
    all probability sub-spaces are classified as
    absolute indeterminate events.

26
9. Applications of Fuzzy and Neutrosophic Logics
to Finance
  • In classical probability n_sup ? 1 while in
    neutrosophic probability n_sup ? 3, where we
    have n_sup as the upper bound of the probability
    space. In cases where the truth and falsity
    components are complimentary, i.e. there is no
    indeterminacy, the components sum to unity and
    neutrosophic probability is reduced to classical
    probability as in the tossing of a fair coin or
    the drawing of a card from a well-shuffled deck.
    Coming back to our original problem of financial
    fraud detection, let E be the event whereby a
    Type I error has occurred and F be the event
    whereby a fraud is actually detected. Then the
    conditional neutrosophic probability NP (F Ec)
    is defined over a probability space consisting of
    a triple of sets (T, I, U). Here, T, I and U are
    probability sub-spaces wherein event F is t
    true, i indeterminate and u untrue
    respectively, given that no Type I error
    occurred.
  • The sub-space T within which t varies may be
    determined by factors such as past records of
    fraud in the organization, propensity to commit
    fraud by the employees concerned, and
    effectiveness of internal control systems. On the
    other hand, the sub-space U within which u varies
    may be determined by factors like personal track
    records of the employees in question, the
    position enjoyed and the remuneration drawn by
    those employees. For example, if the magnitude of
    the embezzled amount is deemed too frivolous with
    respect to the position and remuneration of the
    employees involved. The sub-space I within which
    i varies is most likely to be determined by the
    mutual inconsistency that might arise between the
    effects of some of the factors determining T and
    U. For example, if an employee is for some
    reason really irked with the organization, then
    he or she may be inclined to commit fraud not so
    much to further his or her own interests as to
    harm. The DSmT combination rule can be used in
    such a circumstance to remove the mutual
    inconsistency in the factors deciding T and U.

27
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