Elements of the Kopula (eventological copula) theory (English version)

THE XIV CONFERENCE ON FAM AND EVENTOLOGY OF MULTIVARIATE STATISTICS, KRASNOYARSK, SIBERIA, RUSSIA, 2015 Elements of the Kopula (eventological copula) theory Oleg Yu. Vorobyev Institute of Mathematics and Computer Science Siberian Federal University Krasnoyarsk mailto:[email protected] https://www.sfu-kras.academia.edu/OlegVorobyev Abstract. New in the probability theory and eventology theory, the concept of Kopula (eventological copula) is introduced1 . The theorem on the characterization of the sets of events by Kopula is proved, which serves as the eventological pre-image of the well-known Sclar’s theorem on copulas (1959). The Kopulas of doublets and triplets of events are given, as well as of some N -sets of events.777 Keywords. Eventology, probability, Kolmogorov event, event, set of events, Kopula (eventological copula), Kopula characterizing a set of events. 1 Introduction Long time ago and little by little, the incentive for this work materialized in the theory of sets of events, eventology [5], where the need to locate the classes of event-probability distributions (e.p.d.) of the sets of events (s.e.), which were so arbitrary and spacious to be able without let or hindrance to deal with the relationships between pairs, triples, quadruplets, etc., of events, in other words, to understand the structure of statistical dependencies and generalities between events from some s.e. A similar need is perhaps the only one that has always fueled the development of the probability theory and statistics, which in one way or another are theories of studying and evaluating the structures of statistical dependencies and generalities in the distributions of sets of events. The classical copula theory [4, 2, 3, 1], existing since the 50s of the last century, allows us to construct classes of joint distribution functions that have given marginal distribution functions. In eventology, the theory of sets of events, proposed in the paper the theory of Kopula (eventological copula) allows us to solve a similar problem — c 2015 Oleg Yu. Vorobyev ⃝ Oleg Vorobyev (ed.), Proc. XIV FAMEMS’2015, Krasnoyarsk: SFU 1 To distinguish the quite differently defined notion of an eventological copula from the classical concept of copula in the sense of Sklar (1959), the following radical terminology with capital “K” is used: Kopula = eventological copula; N -Kopula = eventological N -copula. 777 Editing the text of November 12, 2017. to build classes of e.p.d’s of sets of events whose events happen with given probabilities of marginal events. 1.1 General statement of the problem of the Kopula theory We formulate the general statement of the problem of the N -Kopula theory for N -sets of events. If in the classical theory the copula is the tool for selecting some family of joint d.f.’s of a set of random variables from the set of all d.f.’s with given marginal d.f.’s, then in the eventological theory the Kopula is the tool for selecting a family of e.p.d.’s of the 1st kind of the set events from the set of all e.p.d.’s with the given probabilities of marginal events. However, unlike the classical d.f.’s the functions of e.p.d.’s of the 1st kind of the N -s.e. X are functions that are defined as the sets {p(X//X), X ⊆ X} (1.1) of all its 2N values, probabilities of the 1st kind p(X//X), on the set of all subsets of this N -s.e. So, let’s clarify, Kopula is the tool for selecting a family of sets of the form (1.1) from the set of all sets with given probabilities of marginal events. To specify a family of sets of 2N probabilities of the 1st kind (1.1), it is necessary and sufficient to specify a family of sets from the 2N − 1 parameters, since all the probabilities in each set must be nonnegative and give in the sum of one. And to specify a family of sets of 2N probability values of the 1st kind (1.1) with given probabilities of marginal events that form the X-set p̆ = {px , x ∈ X}, (1.2) it is necessary and sufficient to specify a family of sets from the 2N − N − 1 parameters, since for each collection there must be another N constraints for events x ∈ X: ∑ p(X//X) = px . (1.3) x∈X⊆X VOROBYEV 79 Therefore, “to define the family of functions of e.p.d.’s of the 1st kind with given probabilities of marginal events” means “to define a family of sets from the 2N −N −1 parameters” as sets of functions of these probabilities. Eventological theory should solve this problem with the help of a convenient tool, the Kopula, which allows us to define a family of sets from the 2N − N − 1 parameters as sets of functions from marginal probabilities, which in turn can be made dependent on a number of auxiliary parameters. In general, the N -Kopula in the eventological theory is an instrument for defining the family of probability distributions of the 1st kind of the N s.e., in the form of a family of 2N -set of functions of the probabilities of their N marginal events. 1.2 Prolegomena of the Kopula theory The main results of this paper are presented in a rather rigorous mathematical manner. And although the definitions, statements and proofs are provided with examples and illustrations, in order to visualize the ideas underlying the Kopula theory, in my opinion, a number of preliminary explanations in a less strict context, which are collected in several prolegomena, may be necessary. If in some set of events X, some events from the subset X ⊆ X are replaced by their complements, then we get a new set of events X(c|X) = X + (X − X)(c) , which is called the X-phenomenon of s.e. X. The set of all such X-phenomena for X ⊆ X is called the 2X -phenomenon-dom of s.e. X. In [9] a rather distinct theory of set-phenomena and the phenomenon-dom of some s.e. Similarly, the theory of set-phenomena [9] defines the phenomena and phenomenon-dom of the set p̆ = { px , x ∈ X } of the probabilities of marginal events x ∈ X: the X-phenomenon p̆(c|X) = { px , x ∈ X } + { 1 − px , x ∈ X − X } of the set of marginal probabilities p̆ is obtained by replacing the marginal probabilities px by complementary marginal probabilities 1 − px when x ∈ X − X. Prolegomenon 1 (set-phenomenon of a set of events The main conclusion of the above theory is obvious: probability distribution of the s.e. X characterizes the probability distribution and the set of marginal probabilities of each set-phenomenon from its 2X -phenomenon-dom. and a set of probabilities of events). Prolegomenon 2 (set-phenomenal transformation). 1) For each pair X(c|X) , X(c|Y ) set-phenomena of the set of events X their probability distributions are related to each other set-phenomenal transformations. 2) In each pair p̆(c|X) , p̆(c|Y ) set-phenomena of the set of marginal probabilities p̆ events from X are also interconnected by set-phenomenal transformations. The event x ∈ X is called half-rare [9] if the probability px = P(x) with which it happens is not more than half: px 6 1/2. If all events from the s.e. X are half-rare, we speak of a set of half-rare events, or a half-rare s.e. Prolegomenon 3 (sets of half-rare events and its Kopulas). 1) It is not difficult to guess that for any s.e. X, in the 2X phenomenon-dom of the sets of events X(c|X) , X ⊆ X, and in the X-phenomenon-dom of the sets of its marginal probabilities p̆(c|X) , X ⊆ X, there is always a half-rare set-phenomenon. If, in addition, there are no events in X happening with probability 1/2, then such a half-rare set-phenomenon is unique. 2) A Kopula of some family of half-rare N -s.e.’s is generated by 2N functions from half-rare variables defined on the half-hypercube [0, 1/2]N with values from [0, 1] that are continued by the set-phenomenal transformations of the half-rare variables to the corresponding half-hypercubes, all together completely filling the unit hypercube. Prolegomenon 4 (an invariance of the copula with Our task is to construct a Kopula of a family of arbitrary (unordered) sets of half-rare events, i.e. a 1-function, the arguments of which form an unordered set of probabilities of their marginal events. Therefore, it is natural to require such a function to be invariant with respect to the order of its arguments; with respect to the order of events in these sets. In other words, it is natural to consider this 1-function as a function of a set of arguments, rather than a vector of arguments with ordered components, as it is usually assumed. respect to the order of half-rare events). Prolegomenon 5 (insertable sets of half-rare events Two insertable sets of half-rare events for a given set of half-rare events X = {x0 } + X with the frame half-rare event x0 ∈ X, happening with the highest probability among all events from X, are two sets of half-rare events X ′ = {x0 }(∩)X and X ′′ = {xc0 }(∩)X that partition the set of other events X = X − {x0 } into two: X = X ′ (+)X ′′ . The events of one of them, X ′ , are contained in the frame half-rare event x0 , and the events of the other, X ′′ , are contained in its complement xc0 = Ω − x0 . and a frame half-rare event). Prolegomenon 6 (the insertable sets of events and conditional e.p.d.’s of a set of events with respect Conditional e.p.d.’s of the 1st kind of one s.e. X with respect to the other s.e. Y are defined in the traditional way [5]. However, until now attempts to define such a “conditional” s.e., which would have given to the frame event and its complement). 80 THE XIV FAMEMS’2015 CONFERENCE a conditional e.p.d. of the 1st kind, turned out to be completely impractical [13]. The concept of two insertable s.e.’s in a frame event is a well-defined “ersatz” of such “conditional” s.e.’s. The e.p.d.’s of this “ersatz”, although they do not coincide with two conditional e.p.d.’s of the 1st kind with respect to the frame event and its complement, but they are fully characterized by them. The converse is also true: the e.p.d.’s of two frame s.e.’s characterize the corresponding two conditioned e.p.d.’s of the 1st kind. Prolegomenon 7 (a frame method of constructing A frame method constructs a Kopula of a family of arbitrary sets of half-rare events on the basis of a conditional scheme by means of a recurrence formula via conditional e.p.d.’s concerning the frame event and its complement. A recurrent formula associates this Kopula with two Kopulas of families of their insertable sets of half-rare events of smaller dimension, which are characterized by the corresponding conditional e.p.d.’s of the 1st kind (see Prolegomenon 6). a Kopula of an arbitrary set of half-rare events). Prolegomenon 8 (a set-phenomenal transformation of To construct the Kopula of a family of arbitrary s.e.’s it is enough to construct the Kopula of the family of their halfrare set-phenomena and apply a set-phenomenal transformation to this Kopula. a half-rare Copula to an arbitrary one). Prolegomenon 9 (Cartesian representation of the It follows from the Prolegomenon 4 that the Cartesian representation of the N -Kopula in RN should be a symmetric function of N ordered variables, marginal probabilities of events from N -s.e. X, which is defined on the N -dimensional unit hypercube [0, 1]N . The Cartesian representation of the N -Kopula is based on the fact that its symmetric image takes the same values on all permutations of its arguments, that is, is defined by the permutation of N events group. Moreover, the value of such a symmetric function on an arbitrary N -vector w̄ = {w1 , ..., wN } ∈ [0, 1]N is equal to the value of the N -Kopula on an ordered X-set of marginal probabilities of half-rare events p̆ = {px , x ∈ X}, the ordered half-rare projection of the N -vector w̄, the order of the variables in which is defined by an N -permutation πw̄ that has the components of the half-rare projection w̄∗ in decreasing order, where { wn , wn 6 1/2, ∗ (1.4) wn = 1 − wn , wn > 1/2 N -Kopula in RN ). are components of the N -vector w̄∗ of the half-rare projection of the N -vector w̄, n = 1, ..., N . As a result, the ordered half-rare X is the set of marginal probabilities p̆ = p̆(w̄), on which the N -Kopula takes the same value as a symmetric function on w̄, is given by the formula p̆(w̄) = πw̄ (w̄∗ ), (1.5) which defines the Cartesian representation of the N Kopula in RN for each w̄ ∈ [0, 1]N . 2 The Kopula: definition, theorem and the simplest Kopulas We consider the general probability space of Kolmogorov events (Ω, A℧ , P), some particular probability space of events(Ω, A, P) and the N -set of events (N -s.e.) X ⊂ A with the event-probability distribution (e.p.d.2 ) of the 1st kind p(X) = {p(X//X) : X ⊆ X}, and of the second kind pX = {pX//X : X ⊆ X}, which, recall, are related to each other by the Mobius inversion formulas: ∑ p(Y //X), pX//X = X⊆Y p(X//X) = ∑ (−1)|Y |−|X| pY //X . X⊆Y Definition 1 (set-phenomena of a s.e. and its Every N -s.e. X ⊂ A generates its own 2 -phenomenon-dom, defined as a 2N -family { } 2(c|X) = X(c|X) , X ⊆ X , (2.1) phenomenon-dom). X composed of N -s.e. in the form X(c|X//X) = X(c|X) = X + (X − X)(c) ⊂ A, which for each X ⊆ X is called its set-phenomen [9], more precisely, X-phenomen, where X (c) = {xc : x ∈ X} is an М-complement of the s.e. X ⊆ X. We also recall that probabilities of the second kind   ∩ x = P(x) px = p{x} = P  x∈{x}⊆X are probabilities of marginal events from {x} ⊆ X (marginal probabilities), probabilities of the second kind   ∩ z  = P(x ∩ y) pxy = p{x,y} = P  z∈{x,y}⊆X 2 The abbreviations: e.p.d. and e.c.d. are used for the event-probability distribution and for the event-covariance distribution. VOROBYEV 81 are probabilities of double intersections of events from {x, y} ⊆ X, and probabilities of the second kind   ∩ x pZn = P  x∈Zn ⊆X are probabilities of n-intersections of events from Zn ⊆ X, where |Zn | = n; Definition 2 (set-phenomena of the set of probabilities of events from a s.e. and its phenomendom). The N -set of probabilities of events from Let } { ΨX = ψ ψ : [0, 1]⊗X → R+ 0 (2.6) be the family of all the nonnegative bounded numerical functions on the X-hypercube. Definition 3 (normalized function on the Xhypercube). A function ψ ∈ ΨX is called normalized on the X-hypercube if for each w̆ ∈ [0, 1]⊗X ) ∑ ( ψ w̆(c|X//X) = 1, (2.7) X⊆X X p̆ = {px : x ∈ X} also generates its 2X -phenomen-dom, the 2N -totality { } 2(c|p̆) = p̆(c|X//X) , X ⊆ X , (2.2) composed of N -sets in the form { } p̆(c|X//X) = pz : z ∈ X(c|X) , and defined for X ⊆ X as the N -set of probabilities of events from X-phenomenon X(c|X) of the s.e. X where for pz ∈ p̆(c|X//X) { px , z = x ∈ X, pz = 1 − px z = xc ∈ X (c) . In particular, for X = X p̆(c|X//X) = {px : x ∈ X} = p̆. ψ: x [0, 1] → R+ 0 (2.3) x∈X a nonnegative bounded numerical function defined on the set-product [8], X-hypercube ⊗ [0, 1]x . [0, 1]⊗X = x∈X Arguments of ψ form the N -set which generates its own 2X -phenomenon-dom, the 2N -totality { } 2(c|w̆) = w̆(c|X//X) , X ⊆ X (2.4) where for wz ∈ w̆(c|X//X) { wx , wz = 1 − wx z = x ∈ X, z = xc ∈ X (c) . x∈X⊆X i.e., the sum of its values on x-halves of N -sets of arguments from the 2X -phenomenon-dom 2(c|w̆) is wx . Denote by     ) ∑ ( Ψ0X = ψ ∈ ΨX : ψ w̆(c|X//X) = 1; w̆ ∈ [0, 1]⊗X   the family of functions, normalized on the Xhypercube; and by     ) ∑ ( Ψ1X = ψ ∈ ΨX : ψ w̆(c|X//X) = wx ; w̆ ∈ [0, 1]⊗X   x∈X⊆X the family of 1-functions on the X-hypercube. Лемма 1 (properties of 1-functions on the square ). A strict inclusion is fair: {x, y}- Ψ1{x,y} ⊂ Ψ0{x,y} . w̆ = {wx : x ∈ X} ∈ [0, 1]⊗X of N -sets of arguments: { } w̆(c|X//X) = wz : z ∈ X(c|X) Definition 4 (a 1-function on the X-hypercube ). A function ψ ∈ ΨX is called a 1-function on the X-hypercube if for all w̆ ∈ [0, 1]⊗X x-marginal equalities are satisfied for all x ∈ X: ( ) ∑ ψ w̆(c|X//X) = wx , (2.8) X⊆X We denote by ⊗ i.e., the sum of its values on all the N -sets of arguments from 2X -phenomenon-dom 2(c|w̆) is one. (2.5) Proof. In other words, the lemma states: 1) if ψ ∈ Ψ1{x,y} is a 1-function on the {x, y}-square then ψ ∈ Ψ0{x,y} is a normalized function on the {x, y}square; 2) among the normalized functions from Ψ0{x,y} there is one which is not a 1-function. But this is obvious, as it is confirmed by the following simple examples. First, indeed, for the doublet of events X = {x, y} by the definition of a 1-function, we have ψ(wx , wy ) + ψ(wx , 1 − wy ) = wx , (2.9) 82 THE XIV FAMEMS’2015 CONFERENCE However, it is not a 1-function, since ψ(wx , wy ) + ψ(wx , 1 − wy ) = (wx + wy )/4+ (wx + 1 − wy )/4 = wx /2 + 1/4 ̸= wx , ψ(wx , wy ) + ψ(1 − wx , wy ) = (wx + wy )/4+ (1 − wx + wy )/4 = wy /2 + 1/4 ̸= wy . The lemma is proved. Of course, in the general case, for an arbitrary s.e. X the same lemma is fulfilled. Lemma 2 (properties of 1-functions on the hypercube ). A strict inclusion is fair: X- Ψ1X ⊂ Ψ0X . Proof is similar. Note 1 (a representation of a 1-function on the X-hypercube in the form of 2|X| -set of functions). Figure 1: The graph of the Cartesian representation of the normalized function ψ(wx , wy ) = (wx + wy )/4 on the {x, y}-square from Ψ0{x,y} which is not a 1-function. ψ(wx , wy ) + ψ(1 − wx , wy ) = wy , (2.10) ψ(1 − wx , 1 − wy ) + ψ(wx , 1 − wy ) = 1 − wy , (2.11) ψ(1 − wx , 1 − wy ) + ψ(1 − wx , wy ) = 1 − wx . (2.12) The sums (2.9) and (2.12) as well as the sums (2.10) and (2.11) as a result give ψ(wx , wy ) + ψ(wx , 1 − wy )+ +ψ(1 − wx , 1 − wy ) + ψ(wx , 1 − wy ) = 1, (2.13) i.e., ψ ∈ Ψ0{x,y} is a normalized function on the {x, y}-square. Second, the function (see its graph in Fig. 13 ) ψ(wx , wy ) = (wx + wy )/4 is normalized on the {x, y}-square, since ψ(wx , wy ) + ψ(wx , 1 − wy )+ ψ(1 − wx , wy ) + ψ(1 − wx , 1 − wy ) = = (wx + wy )/4 + (wx + 1 − wy )/4+ (1 − wx + wy )/4 + (1 − wx + 1 − wy )/4 = 1. 3 In this figure and others, which illustrate the doublets of events, the map of this function on a unit square is shown under the graph in conditional colors where the white color corresponds to the level 1/4. Any 1-function ψ ∈ Ψ1X on the X-hypercube [0, 1]⊗X for each w̆ ∈ [0, 1]⊗X is represented in the form of 2|X| -set of the following functions: { } ψ(w̆) = ψX (w̆), X ⊆ X = { ( ) } (2.14) = ψ w̆(c|X//X) , X ⊆ X . Definition 5 (Kopula ). The 1-functions K Ψ1X ⊂ ΨX is called |X|-Kopulas4 of the s.e. X. ∈ As well as every 1-function (2.14), any |X|-Kopula of the s.e. X can be represented for w̆ ∈ [0, 1]⊗X in the form of 2|X| -set of the following functions: { } K(w̆) = KX (w̆), X ⊆ X = { ( ) } (2.15) = K w̆(c|X//X) , X ⊆ X . Note 2 (characteristic properties of Kopula). Each Kopula K has two characteristic properties 1) Kopula is nonnegative: ( ) K w̆(c|X//X) > 0 (2.16) for X ⊆ X, since by definition K ∈ ΨX ; 2) Kopula ia satisfied x-marginal equalities: ( ) ∑ K w̆(c|X//X) = wx (2.17) x∈X⊆X for x ∈ X, since by definition K ∈ Ψ1X ; From (2.17) by Lemma 2 a normalization of the Kopula follows: ( ) ∑ K w̆(c|X//X) = 1. X⊆X 4 see the footnote 1 on page 78. probabilistic (2.18) VOROBYEV 83 From (2.16) and (2.18) terrace-by-terrace probabilistic normalization of the Kopula follows: ( ) 0 6 K w̆(c|X//X) 6 1 (2.19) In addition, from (2.20) and from the fact that the |X|-Kopula is a 1-function, (2.21) follows for all x ∈ X. Therefore, the function p is a e.p.d. of the 1st kind of the s.e. X with the X-set of marginal probabilities p̆. The theorem is proved. for X ⊆ X. 2.2 Convex combination of Kopulas 2.1 Characterization of a set of events by Kopula The eventological analogue and the preimage of the well-known Sklar theorem on copulas [4] is the following theorem. Theorem 1 (characterization of a s.e. by Let p = {p(X//X) : X ⊆ X} be the e.p.d. of the 1st kind of the s.e. X with X-set of probabilities of marginal events p̆ = {px : x ∈ X} ∈ [0, 1]⊗X . Then there is a |X|-Kopula K ∈ Ψ1X that defines a family of e.p.d.’s of the 1st find of the s.e. X. This family contains the e.p.d. p, when Kopula’s arguments coincide with p̆. In other words, the such Kopula that foe all X ⊆ X ( ) p(X//X) = KX (p̆) = K p̆(c|X//X) . (2.20) A Lemma 3 (convex combination of Kopulas ). convex combination of an arbitrary set of Kopulas of one and the same s.e. is its Kopula too. Proof without tricks. Let X be a s.e., and K1 , . . . , Kn (2.24) Kopula ). Conversely, for any X-set of probabilities of marginal ✿✿✿✿✿✿✿✿✿✿ events p̆ ∈ [0, 1]⊗X and any |X|-Kopula K ∈ Ψ1X , the function p = {p(X//X) : X ⊆ X}, defined by formulas (2.20) for X ⊆ X, is an e.p.d. of the 1st kind, which characterizes the s.e. X with given X-set of the probabilities of marginal events p̆. Proof is a direct consequence of the properties of e.p.d. of the 1st kind of the s.e. X and the |X|-Kopula. First, if the e.p.d. of the 1st kind p = {p(X//X) : X ⊆ X} of some s.e. X with the X-set of marginal probabilities p̆ = {px : x ∈ X} is defined, then from properties of probabilities of the 1st kind it follows that for x ∈ X ∑ px = p(X//X), (2.21) x∈X⊆X i.e., the function K, defined by the e.p.d. of the 1st kind p and formulas (2.3), satisfies x-marginal equalities for x ∈ X: ∑ K(p̆(c|X//X) ) px = (2.22) x∈X⊆X (required for being a 1-function:: K ∈ Ψ1X ) and serves as the |X|-Kopula. Second, if the function K is the |X|-Kopula, then by Lemma 1: K ∈ Ψ1X ⊂ Ψ0X , i.e., it is normalized and, consequently, by (2.20) the function p is normalized too: ∑ 1= p(X//X). (2.23) X⊆X be some set of its Kopulas. Let us prove that their convex combination K= n ∑ α i Ki (2.25) i=1 (where, of course, α1 + . . . + αn = 1, αi > 0, i = 1, . . . , n) is also a Kopula. For this it suffices to prove that K is a 1-function. In other words, that for x ∈ X ∑ K(p̆(c|X//X) ) = px . (2.26) x∈X⊆X Since each Kopula from the set (2.24) has a property of a 1-function, then for x ∈ X we get what is required: ∑ K(p̆(c|X//X) ) = x∈X⊆X n ∑ αi = i=1 = n ∑ ∑ Ki (p̆(c|X//X) ) = (2.27) x∈X⊆X αi px = px . i=1 Corollary 1 (convex combination of Kopulas ). For every set of events X the space of 1-functions Ψ1X , as well as the space of its Kopulas, is a convex manifold. 2.3 Kopula of free variables: A computational aspect Without set-phenomenon transformations and variable transformations, analytic work on sets of half-rare events (s.h-r.e.’s) (see [9]) and sets of their marginal probabilities, is unlikely to be effective. However, in specific calculations at first, because of their unaccustomedness, these compulsory wisdoms can cause misunderstandings, leading to errors. Therefore, it is useful, in order to avoid unnecessary stumbling during calculations, to 84 THE XIV FAMEMS’2015 CONFERENCE introduce separate notation for half-rare marginal probabilities events from s.h-r.e’s. X and its setphenomena, that is, probabilities that are not greater than half, in order to distinguish them from free marginal probabilities, to the values of which there are no restrictions. So, we will talk about half-rare variables (hr variables) and free variables, assigning special notation to them5 : p̆ = {px , x ∈ X} ∈ [0, 1/2]⊗X −X-set of half-rare variables, p̆(c|X//X) ∈ [0, 1/2]⊗X ⊗ (1/2, 1]⊗X−X −X-renumbering p̆, w̆ = {wx , x ∈ X} ∈ [0, 1]⊗X (2.28) −X-set of free variables, w̆(c|X//X) ∈ [0, 1]⊗X p̆ = {px , py } ∈ [0, 1/2]x ⊗ [0, 1/2]y −X-set of half-rare variables, pxy (px , py ) ∈ [0, min{px , py }] −half-rare function of half-rare variables, (2.29) x w̆ = {wx , wy } ∈ [0, 1] ⊗ [0, 1] y −X-set of free variables, wxy (wx , wy ) ∈ [0, min{wx , wy }] −free function of free variables. (phenomenon replacement between half-rare and For every X ⊂ X phenomenon replacement of half-rare variables p̆ ∈ [0, 1/2]⊗X by free variables w̆ ∈ [0, 1]⊗X and vise-versa is defined for X ⊆ X by mutually inverse formulas of the set-phenomenon transformation of the form: free variables). wx ... wx wx ... wx Let’s construct the 1-Kopula K ∈ Ψ1X of a family of e.p.d.’s of monoplet of events X = {x} with the e.p.d. of the 1st kind { } p(X//{x}), X ⊆ {x} = {p(∅//{x}), p(x//{x})} and {x}-monoplet of marginal probabilities {px }, where px = P(x) = p(x//{x}). In other words, let’s construct a 1-function on the unit X-segment, i.e., a such nonnegative bounded numerical function that for x ∈ X and always interpreting them as probabilities of events. In particular, for the half-rare doublet X = {x, y} we have: p̆ = p̆(c|X//X) =  w̆(c|∅//X) ,     . . . ,    w̆(c|X//X) , =       ...,    (c|X//X) w̆ , 2.4 Kopula for a monoplet of events K : [0, 1] → [0, 1], −X-renumbering w̆, Note 3 accepted (see, for example, paragraph 11.7): { wx , wx 6 1/2, px = (2.31) 1 − wx , wx > 1/2. > 1/2, x ∈ X, 6 1/2, x ∈ X, > 1/2, x ∈ X − X, (2.30) 6 1/2, x ∈ X where for x ∈ X the following agreement is always 5 Just remember [9], that the formula of X-renumbering any X-set of probabilities of events has the form for X ⊆ X: p̆(c|X//X) = {px , x ∈ X} + {1 − px , x ∈ X − X}. ∑ x∈X⊆X ( ) K w̆(c|X//X) = wx . Since for X ⊆ X = {x} ( ) K w̆(c|X//{x}) = { K (1 − wx ) , X = ∅, K (wx ) , X = {x}, then a marginal and global normalization of the function K are written as: K(wx ) = wx , (2.32) K(wx ) + K(1 − wx ) = 1, and the global normalization obviously follows from the marginal one, which agrees with Lemma 2; and from the marginal normalization it follows that the 1-copula K of an arbitrary monoplet of events X = {x} is defined for free variables w̆ = {wx } ∈ [0, 1]x by a one formula: K(w̆) = K(wx ) = wx , which provides two values on each penomenon-dom by “free” formulas: ( ) K w̆(c|X//{x}) = { K (1 − wx ) = 1 − wx , X = ∅, = K (wx ) = wx , X = {x}. (2.33) 2(c|w̆) - (2.34) and the e.p.d. of the 1st kind of this monoplet with {x}-monoiplet of probabilities of events p̆ = {px } ∈ [0, 1]x are defined for half-rare variables by the 1Kopula (2.33) for X ⊆ {x} by exactly the same “halfrare” formulas: ( ) p(X//{x}) = K p̆(c|X//{x}) = { (2.35) K (1 − px ) = 1 − px , X = ∅, = K (px ) = px , X = {x}. VOROBYEV 85 2.5 Kopulas for a doublet of events Let’s construct an example of 2-Kopulas K ∈ Ψ1X of families of a doublet of events X = {x, y}, in other words, let’s construct on the unit {x, y}-square the such nonnegative bounded numerical functions K : [0, 1]⊗{x,y} → [0, 1], that for all z ∈ {x, y} ( ) ∑ K w̆(c|Z//{x,y}) = wz . z∈Z⊆{x,y} Since each 2-set of arguments w̆ ∈ [0, 1]x ⊗ [0, 1]y generates 2{x,y} -phenomenon-dom 2(c|w̆) = {w̆, w̆(c|{x}) , w̆(c|{y}) , w̆(c|∅) }, (2.36) composed from forth its set-phenomena w̆ = w̆(c|{x,y}//{x,y}) = {wx , wy }, w̆(c|{x}//{x,y}) = {wx , 1 − wy }, w̆(c|{x}//{x,y}) = {1 − wx , wy }, (2.37) w̆(c|∅//{x,y}) = {1 − wx , 1 − wy }, then K (w̆) = K(wx , wy ), ) K w̆(c|{x}//{x,y}) = K(wx , 1 − wy ), ( ) K w̆(c|{y}//{x,y}) = K(1 − wx , wy ) ( ) K w̆(c|∅//{x,y}) = K(1 − wx , 1 − wy ), ( and normalizations for every w̆ ∈ [0, 1]⊗{x,y} are written as: K(wx , wy ) + K(wx , 1 − wy ) = wx , K(wx , wy ) + K(1 − wx , wy ) = wy , K(wx , wy ) + K(wx , 1 − wy )+ +K(1 − wx , wy ) + K(1 − wx , 1 − wy ) = 1. The e.p.d. of the 1st kind of doublet of events is defined by the 2-Kopula for X ⊆ {x, y} in half-rare variables by general formulas: ( ) p(X//{x, y}) = K p̆(c|X//{x,y}) =  K (1 − px , 1 − py ) , X = ∅,    K (p , 1 − p ) , X = {x}. x y =  K (1 − p , p ) , X = {y}, x y    (2.38) K (px , py ) , X = {x, y},  1 − px − py + pxy (p̆), X = ∅,    p − p (p̆), X = {x}. x xy =  py − pxy (p̆), X = {y},    pxy (p̆), X = {x, y}, where pxy (p̆) is functional parameter that has a sense of probability of double intersection. This e.p.d. of the 1st kind of doublet of events in the free functional parameters and variables (after replacement (2.31)) has the form: ( ) p(X//{x, y}) = K w̆(c|X//{x,y}) =  w + wy − 1 + wxy (1 − wx , 1 − wy ),    x    wx > 1/2, wy > 1/2 ⇔ X = ∅,         wx − wxy (wx , 1 − wy ),      wx 6 1/2, wy > 1/2 ⇔ X = {x}, =    wy − wxy (1 − wx , wy ),      wx > 1/2, wy 6 1/2 ⇔ X = {y},           wxy (wx , wy ),    wx 6 1/2, wy 6 1/2 ⇔ X = {x, y}. (2.39) 2.6 Kopula for a doublet of independent events The simplest example of a 1-function on a {x, y}square is the so-called independent 2-Kopula, which for free variables w̆ ∈ [0, 1]⊗{x,y} is defined by the formula: K (w̆) = wx wy . (2.40) This provides it on each 2(c|w̆) -phenomenon the following four values: ( ) K w̆(c|{x,y}//{x,y}) = wx wy , ( ) K w̆(c|{x}//{x,y}) = wx (1 − wy ), ( ) K w̆(c|{y}//{x,y}) = (1 − wx )wy , ( ) K w̆(c|∅//{x,y}) = (1 − wx )(1 − wy ). (2.41) Indeed, the so-defined independent 2-Kopula is a 1function because ) ∑ ( K w̆(c|X//{x,y}) = wx wy + wx (1 − wy ) = wx , x∈X⊆{x,y} ∑ ( ) K w̆(c|X//{x,y}) = wx wy + (1 − wx )wy = wy . y∈X⊆{x,y} The e.p.d. of the 1st kind of doublet of independent events with the {x, y}-set of probabilities of events p̆ is defined by four values of the independent 2-Kopula (2.40) on its 2(c|p̆) -penomenon-dom by general formulas in half-rare variables (see Fig. 2), 86 THE XIV FAMEMS’2015 CONFERENCE i.e., for X ⊆ {x, y}: ( ) p(X//{x, y}) = K p̆(c|X//{x,y}) =  (1 − px )(1 − py ), X = ∅,    p (1 − p ), X = {x}. x y =  (1 − p )p , X = {y}, x y    px py , X = {x, y}. (2.42) substitution of variables and not get confused in the calculations:  wxy (wx , wy ) ,      w̆ ∈ [0, 1/2]x ⊗ [0, 1/2]y ,       wx − wxy (wx , 1 − wy ) ,      w̆ ∈ [0, 1/2]x ⊗ (1/2, 1]y , K (w̆) =   wy − wxy (1 − wx , wy ) ,    w̆ ∈ (1/2, 1]x ⊗ [0, 1/2]y ,         wx + wy − 1 + wxy (1 − wx , 1 − wy ) ,  w̆ ∈ (1/2, 1]x ⊗ (1/2, 1]y ; (2.44) and again with restrictions in the form of the familiar “human” inequalities6 :  wxy (wx , wy ) ,      wx 6 1/2, wy 6 1/2,       wx − wxy (wx , 1 − wy ) ,      wx 6 1/2, 1/2 < wy , K (w̆) =   wy − wxy (1 − wx , wy ) ,     1/2 < wx , wy 6 1/2,        wx + wy − 1 + wxy (1 − wx , 1 − wy ) ,    1/2 < wx , 1/2 < wy (2.45) where Figure 2: Graphs of Cartesian representation of the 2-Kopula of a family of e.p.d.’s of an independent half-rare doublet of events {x, y}; probabilities of the 1st kind are marked by different colors : p(xy) (aqua), p(x) (lime), p(y) (yellow) и p(∅) (fuchsia). 2.7 2-Kopula of free variables: A computational aspect With the phenomenal substitution (2.30) half-rare 2-Kopula as a function of the free variables takes the equivalent form: ) (  wxy w̆(c|X//X) ,      w̆ ∈ [0, 1/2]x ⊗ [0, 1/2]y ,     ( )   wx − wxy w̆(c|{x}//X) ,      w̆ ∈ [0, 1/2]x ⊗ (1/2, 1]y , K (w̆) = (2.43) ( )   wy − wxy w̆(c|{y}//X) ,     w̆ ∈ (1/2, 1]x ⊗ [0, 1/2]y ,     ) (    wx + wy − 1 + wxy w̆(c|∅//X) ,    w̆ ∈ (1/2, 1]x ⊗ (1/2, 1]y . We rewrite this formula a pair of times, in order to understand the properties of the phenomenon 0 6 wxy (wx , wy ) 6 min{wx , wy }, 0 6 wxy (wx , 1−wy ) 6 min{wx , 1−wy }, 0 6 wxy (1−wx , wy ) 6 min{1−wx , wy }, (2.46) 0 6 wxy (1−wx , 1−wy ) 6 min{1−wx , 1−wy } are the Fréchet inequalities for wxy as the halfrare probability of double intersection of half-rare events: either half-rare events x or y, or their halfrare complements, when events x or y are not halfrare. Note that the conditional formulas (2.43), (2.44) and (2.45) can not be rewritten as four unconditional formulas, because these conditions are in the right, and not in the left. This is explained exclusively by the properties of the phenomenon replacement of half-rare variables by free ones (2.30), which, for this reason, leads to formulas that are convenient for calculations. Note 4 (half-rare 2-Kopula of free variables of an With the functional parameter wxy (wx , wy ) = wx wy , which corresponds to the probability of double intersection of independent doublet of events). 6 probabilistic normalization: “0 6 ... 6 1” is assumed by default. VOROBYEV 87 independent events x and y happening with probabilities wx and wy , and means, of course, that the all the following four equations are satisfied: wxy (wx , wy ) = wx wy , wxy (wx , 1 − wy ) = wx (1 − wy ), wxy (1 − wx , wy ) = (1 − wx )wy , wxy (1 − wx , 1 − wy ) = (1 − wx )(1 − wy ); from (2.45) it follows that a half-rare 2-Kopula from free variables of the family of e.p.d.’s of the independent doublet of events X = {x, y} with X-sets of free marginal probabilities w̆ = {wx , wy } ∈ [0, 1]⊗X has the same view on all 2(c|w̆) phenomenon-doms: K (w̆) = wx wy . (2.47) 2.8 Upper 2-Kopula of Fréchet An example of a 1-function on a {x, y}-square is the so-called upper 2-Kopula of Fréchet, which suggests the probabilities of a double intersection to be its upper Fréchet boundary. In other words, the only functional free parameter in (2.39) is: + wxy (w̆) = wxy (w̆) = min{wx , wy }. (2.48) The upper 2-Kopula of Fréchet from free variables w̆ ∈ [0, 1]⊗{x,y} is defined by the formulas: ( ) p(X//{x, y}) = K w̆(c|X//{x,y}) =  wx + wy − 1 + min{1 − wx , 1 − wy },      wx > 1/2, wy > 1/2 ⇔ X = ∅,       wx − min{wx , 1 − wy },    wx 6 1/2, wy > 1/2 ⇔ X = {x}, =   wy − min{1 − wx , wy },    wx > 1/2, wy 6 1/2 ⇔ X = {y},        min{wx , wy },    wx 6 1/2, wy 6 1/2 ⇔ X = {x, y}. (2.49) After simple transformations, these formulas provide the upper 2-Kopula of Fréchet on each 2(c|w̆) -phenomenon-dom the following four values of free variables: ( ) p(X//{x, y}) = K w̆(c|X//{x,y}) =  min{wx , wy },      wx > 1/2, wy > 1/2 ⇔ X = ∅,        max{0, wx + wy − 1},    wx 6 1/2, wy > 1/2 ⇔ X = {x}, =   max{0, wx + wy − 1},     wx > 1/2, wy 6 1/2 ⇔ X = {y},       min{wx , wy },    wx 6 1/2, wy 6 1/2 ⇔ X = {x, y},  + wxy (wx , wy ),      wx > 1/2, wy > 1/2 ⇔ X = ∅,      −   wxy (wx , wy ),    wx 6 1/2, wy > 1/2 ⇔ X = {x}, = −   wxy (wx , wy ),     wx > 1/2, wy 6 1/2 ⇔ X = {y},      +   (wx , wy ), wxy    wx 6 1/2, wy 6 1/2 ⇔ X = {x, y}, (2.50) which, as can be seen, are defined by the upper and lower Fréchet boundaries of the probability of double intersecyion, depending on the combination of the values of the free variables. The same four values from the half-rare variables have the form: ( ) p(X//{x, y}) = K p̆(c|X//{x,y}) =  min{1 − px , 1 − py }, X = ∅,      max{0, px − py }, X = {x}, =   X = {y}, max{0, py − px },    min{px , py }, X = {x, y}. (2.51) If px > py , this formula takes the form: p(X//{x, y}) = K  1 − px , X      px − py , X =   0, X     py , X ( ) p̆(c|X//{x,y}) = = ∅, = {x}, = {y}, = {x, y}. (2.52) 88 THE XIV FAMEMS’2015 CONFERENCE And if px < py , then this formula takes the form: ( ) p(X//{x, y}) = K p̆(c|X//{x,y}) =  1 − py , X = ∅,      0, (2.53) X = {x}, =   py − px , X = {y},     py , X = {x, y}. So the definite upper 2-Kopula of Fréchet is indeed a 1-function, due to the fact that when wx > wy ) ∑ ( K w̆(c|X//{x,y}) = wy + (wx − wy ) = wx , x∈X⊆{x,y} ∑ ( ) K w̆(c|X//{x,y}) = wy + 0 = wy , (2.54) y∈X⊆{x,y} and when wx < wy ) ∑ ( K w̆(c|X//{x,y}) = wx + 0 = wx , x∈X⊆{x,y} ∑ ( ) (2.55) K w̆(c|X//{x,y}) = wx + (wy − wx ) = wy . y∈X⊆{x,y} Figure 4: Graphs of the Cartesian representation of the lower 2-Kopula of Fréchet of a family of e.p.d.’s of half-rare doublet of events {x, y}; probabilities of the 1st kind are marked by different colors: p(xy) (aqua), p(x) (lime), p(y) (yellow) и p(∅) (fuchsia). which suggests the probabilities of a double intersection to be its upper Fréchet boundary. In other words, the only functional free parameter in (2.39) is: − (w̆) = max{0, wx + wy − 1}. wxy (w̆) = wxy (2.56) The lower 2-Kopula of Fréchet from free variables w̆ ∈ [0, 1]⊗{x,y} is defined by the formulas: Figure 3: Graphs of the Cartesian representation of the upper 2-Kopula of Fréchet of a family of e.p.d.’s of half-rare doublet of events {x, y}; probabilities of the 1st kind are marked by different colors: p(xy) (aqua), p(x) (lime), p(y) (yellow) и p(∅) (fuchsia). 2.9 Lower 2-Kopula of Fréchet A once more example of a 1-function on a {x, y}square is the so-called lower 2-Kopula of Fréchet, ( ) p(X//{x, y}) = K w̆(c|X//{x,y}) =  wx + wy − 1 + max{0, 1 − wx − wy },     wx > 1/2, wy > 1/2 ⇔ X = ∅,        wx − max{0, wx − wy },    wx 6 1/2, wy > 1/2 ⇔ X = {x}, =   wy − max{0, wy − wx },     wx > 1/2, wy 6 1/2 ⇔ X = {y},        max{0, wx + wy − 1},    wx 6 1/2, wy 6 1/2 ⇔ X = {x, y}. (2.57) After simple transformations, these formulas provide the lower 2-Kopula of Fréchet on each 2(c|w̆) -phenomenon-dom the following four values VOROBYEV 89 of free variables: ( ) p(X//{x, y}) = K w̆(c|X//{x,y}) =   max{0, wx + wy − 1},    wx > 1/2, wy > 1/2 ⇔ X = ∅,        min{wx , wy },    wx 6 1/2, wy > 1/2 ⇔ X = {x}, =   min{wx , wy },     wx > 1/2, wy 6 1/2 ⇔ X = {y},        max{0, wx + wy − 1},    wx 6 1/2, wy 6 1/2 ⇔ X = {x, y},  − wxy (wx , wy ),      w > 1/2, wy > 1/2 ⇔ X = ∅,  x     +   wxy (wx , wy ),   wx 6 1/2, wy > 1/2 ⇔ X = {x}, = +   wxy (wx , wy ),      wx > 1/2, wy 6 1/2 ⇔ X = {y},    −   wxy (wx , wy ),    wx 6 1/2, wy 6 1/2 ⇔ X = {x, y}, (2.58) which, as can be seen, are also defined by the upper and lower Fréchet boundaries of the probability of double intersection only in other combinations of the values of the free variables. The same four values from the half-rare variables have the more simple form: ( ) p(X//{x, y}) = K p̆(c|X//{x,y}) =  max{0, 1−px −py }, X = ∅,      min{px , 1−py }, X = {x}, =   X = {y}, min{1−px , py },    (2.59) max{0, px +py −1}, X = {x, y},  1−px −py , X = ∅,      px , X = {x}, =   py , X = {y},     0, X = {x, y}, So the definite lower 2-Kopula of Fréchet is indeed a 1-function, due to the fact that for all half-rare variables ) ∑ ( K p̆(c|X//{x,y}) = px + 0 = px , x∈X⊆{x,y} ∑ ( ) K w̆(c|X//{x,y}) = py + 0 = py . y∈X⊆{x,y} (2.60) Figure 5: Graphs of the Cartesian representation of the convex combination of upper and lower 2-Kopulas of Fréchet of a family of e.p.d.’s of half-rare doublet of events {x, y} with functional weight parameter (2.63) in the formula (2.61); probabilities of the 1st kind are marked by different colors: p(xy) (aqua), p(x) (lime), p(y) (yellow) и p(∅) (fuchsia). 2.10 Convex combinations of the lower, upper and independent 2-Kopulas of Fréchet A rather general example of a 1-function on a {x, y}-square is the convex combinations of the upper, lower, and independent 2-Kopulas ofwise Fréchet, which propose the probabilities of a pair intersection to become a convex combination (this is allowed by the lemma 3) of its upper and lower Fréchet boundaries, as well as the probability of double intersection of independent events with some functional weighting parameter. 2.10.1 Convex combination of the lower and upper 2-Kopulas of Fréchet A convex combination of the lower and upper 2Kopula of Fréchet can be ensured by the unique functional free parameter wxy (w̆) in (2.39), in which the probability of double intersection is computed by the following formula: wxy (w̆) = + − (w̆) (w̆) + (1+α)/2wxy = (1−α)/2wxy (2.61) where α = α(w̆) ∈ [−1, 1] is an arbitrary function on [0, 1]⊗{x,y} with values from [−1, 1], and − wxy (w̆) = max{0, wx + wy − 1}, + (w̆) = min{w̆} wxy (2.62) 90 THE XIV FAMEMS’2015 CONFERENCE are the lower and upper Fréchet-boundaries of probability of double intersection. − (w̆) For α = −1, the probability wxy (w̆) = wxy coincides with the lower Fréchet boundary of half-rare marginal probabilities; for α = 1, the + probability wxy (w̆) = wxy (w̆) coincides with the upper Fréchet boundary of marginal probabilities. Unfortunately, these are the properties of a convex combination such that for α = 0 the probability of double intersection is equal to half of the sum of its lower and upper Fréchet boundaries: − + wxy (w̆) = (wxy (w̆) + wxy (w̆))/2 (see Figure6), and not an independent 2-Kopula, no matter how much we want it. This “blunder” of the convex combination can easily be corrected by conjugation of two convex combinations, as done below. In Fig. 5 it is a graph of this 2-Kopula for a deliberately intricate weight function with values from [−1, 1]: α = α(w̆) = sin(15(wx − wy )). (2.63) Kopulas of Fréchet. The conjugation of these two convex combinations can be ensured by the unique functional free parameter wxy (w̆) in (2.39) by the following conjugation formula for two convex combinations: wxy (w̆) =  + wxy (w̆) max{w̆}(1+α),     α 6 0;  = ( )  +  w ( w̆) max{ w̆}(1−α)+α ,  xy    α > 0, (2.64) where α = α(w̆) ∈ [−1, 1] is an arbitrary function on [0, 1]⊗{x,y} with values from [−1, 1], and + wxy (w̆) = min{w̆} (2.65) is the upper Fréchet-boundary of probability of double intersection. For α = 0, the probability wxy (w̆) = min{w̆} max{w̆} = wx wy coincides with the probability of double intersection of independent events; for α = −1, the probability wxy (w̆) = 0 coincides with the lower Fréchet-boundary of half-rare marginal probabilities; for α = 1, the + (w̆) coincides with the probability wxy (w̆) = wxy upper Fréchet-boundary of marginal probabilities. In Fig. 7 it is a graph of this 2-Kopula for the same weight function with values from [−1, 1] as in the previous example. α = α(w̆) = sin(15(wx − wy )). (2.66) 3 The frame method of construction of Kopula 3.1 Inserted sets of events and conditional e.p.d.’s Definition 6 (inserted s.e.’s). For each pais of s.e.’s X and Y with the joint e.p.d. {p(X + Y //X + Y)}, X ⊆ X , Y ⊆ Y} Figure 6: Graphs of the Cartesian representation of the convex combination of upper and lower 2-Kopulas of Fréchet of a family of e.p.d.’s of half-rare doublet of events {x, y} with the constant functional weight parameter α(w̆) = 0 in the formula (2.61); probabilities of the 1st kind are marked by different colors: p(xy) (aqua), p(x) (lime), p(y) (yellow) и p(∅) (fuchsia). 2.10.2 The conjugation of two convex combinations of the independent one with the lower and upper 2-Kopula of Fréchet We construct two convex combinations of the the independent 2-copula and the lower and upper 2- (3.1) for every Y ⊆ Y the Y -inserted s.e.’s, generated by X , in the frame s.e. Y are s.e.’s, which are denoted by X (∩Y //Y) , and defined as the following M-intersection7 : X (∩Y //Y) = X (∩) {ter(Y //Y)} = = {x ∩ ter(Y //Y), x ∈ X } (3.2) and have the e.p.d., which coincides with the projection of the joint e.p.d. (3.1) for fixed Y ⊆ Y and every X ⊆ X : p(X (∩) {ter(Y //Y)}) = p(X + Y //X + Y)}. 7 M-intersection is an intersection by Minkowski. (3.3) VOROBYEV 91 probabilities of the terraced events, from which the Y -pseudo-distributions are composed, are normalized not by unity, but by the probabilities of the corresponding frame terraces events p(Y //Y). And the sum of the normalizing constants by Y ⊆ Y is obviously equal to one. Note 5 (symmetry of inserted and frame s.e.’s). In Definition 6 the s.e. X and Y can always be swapped, i.e., to take the s.e. X on a role of the frame one, and the s.e. Y to take on a role of s.e., that generates X-inserted s.e.’s for every X ⊆ X : Y (∩X//X ) = Y (∩) {ter(X//X )} = = {y ∩ ter(X//X ), y ∈ Y}. (3.6) Note 6 (M-sum of the all inserted s.e.’s). The Msum8 of the all Y -inserted s.e.’s X (∩Y //Y) for Y ⊆ Y forms the given s.e. X : (∑ ) X (∩Y //Y) = X = Y ⊆Y Figure 7: Graphs of the Cartesian representation of the convex combination of lower and independent, and independent and upper 2Kopulas of Fréchet of a family of e.p.d.’s of half-rare doublet of events {x, y} with the functional weight parameter (2.66) in the formula (2.61); probabilities of the 1st kind are marked by different colors: p(xy) (aqua), p(x) (lime), p(y) (yellow) и p(∅) (fuchsia). Definition 7 (event-probabilistic pseudo- distribution of an inserted s.e.). For each Y ⊆ Y the Y -inserted s.e. X (∩Y //Y) = X (∩) {ter(Y //Y)} (3.4) with the e.p.d. (3.3) has the event-probabilistic Y pseudo-distribution, which is defined as a set of probabilitieso of terraced events that coincide with probabilities from the e.p.d. (3.3) for all X ⊆ X excepting X = ∅: p(Y ) (X + Y //X + Y) = { p(X + Y //X + Y), X ̸= ∅, = p(Y //X + Y) − 1 + p(Y //Y), X = ∅, (3.5)   p(X + Y / /X + Y), X ̸ = ∅,  ∑ = p(Y //Y) − p(X + Y //X + Y), X = ∅.   X̸=∅ X⊆X The sum of all probabilities from every Y -pseudodistribution (3.5) is p(Y //Y) = P(ter(Y //Y)), the probability of a terraced event, generated by the frame s.e. Y, in which the given s.e. X (∩Y //Y) is inserted. Thus, the only difference of e.p.d.’s of Y -inserted s.e.’s from their event-probabilistic Y -pseudodistributions, lies in the fact that the sums of the = X (∩∅//Y) (+) ... (+) X (∩Y//Y) . | {z } (3.7) 2|Y| Note 7 (charcterization of Y -inserted s.e.’s by The e.p.d. of Y -inserted s.e. X with every Y ⊆ Y has a form for X ⊆ X : ( ) p X(∩){ter(Y //Y)}//X (∩Y //Y) = { p(X + Y //X + Y), ∅ ̸= X ⊆ X , = (3.8) p(Y //X + Y) + 1 − p(Y //Y), X = ∅, { p(X//X | Y //Y)p(Y //Y), ∅ ̸= X ⊆ X , = 1 − (1 − p(∅//X | Y //Y))p(Y //Y), X = ∅ conditional e.p.d.’s of the 1st kind). (∩Y //Y) where for every Y ⊆ Y p(X//X | Y //Y) = p(X + Y //X + Y) , p(Y //Y) (3.9) i.e., the probabilities of the 1st kind, forming for X ⊆ X the Y -conditional e.p.d. of the 1st kind of the s.e. X with respect to terraced event ter(Y //Y) generated by the s.e. Y. In other words, Y -inserted s.e. X (∩Y //Y) for Y ⊆ Y are characterized by formulas (3.8) and Y conditional e.p.d.’s of the 1st kind of the s.e. X with respect to the terraced event ter(Y //Y), generated by the s.e. Y. Note 8 (mutual characterization of conditional e.p.d.’s of the 1st kind and pseudo-distributions of inserted s.e.’s). 8 M-sum The connection between each is a sum by Minkowski. 92 THE XIV FAMEMS’2015 CONFERENCE Y -pseudo-distribution of the Y -inserted s.e. with the corresponding Y -conditional e.p.d. looks simpler. It is sufficient for each fixed Y ⊆ Y to normalize all its probabilities of “inserted” terraced events by the probability of a terraced event p(Y //Y) in order to obtain corresponding to the Y -conditional probabilities regarding the fact that the corresponding frame terraces event ter(Y //Y) happened. As a result, we have the following obvious inversion formulas: p(X//X | Y //Y) = p (Y ) p(Y ) (X + Y //X + Y) , p(Y //Y) (3.10) (X + Y //X + Y) = p(X//X | Y //Y)p(Y //Y). Note 9 (about the appropriateness of the concept of inserted s.e.’s). It would seem, why develop a theory of inserted s.e.’s, pseudo-distributions of which are simply characterized by conditional e.p.d.’s. Is not it better to instead practice the theory of conditional e.p.d., especially since this theory has long had excellent recommendations in many areas. However, in eventology, as the theory of events, which prefers to work directly with sets of events, there is one rather serious objection. The fact is that conditional e.p.d., as any e.p.d. in eventology, there must be a set of some events, in this case, a set of well-defined “conditional events”. But until now it has not been possible to give a satisfactory definition of the “conditional event”, except for my impractical definition in [13]. So, the inserted s.e.’s are a completely satisfactory “surrogate” definition of the sets of “conditional events”. Such that e.p.d.’s of inserted s.e.’s, although they do not coincide with the desired conditional e.p.d.’s, but are associated with them by well-defined mutualinverse transformations. As a result, inserted s.e.’s play the role of a convenient eventological tool for working with conditional e.p.d.’s of a one set of events regarding terrace events generated by another set of events. Example 1 (two inserted s.e.’s in a frame monoplet). Let in formulas (3.1) the s.e. X is an arbitrary set, and the s.e. Y = {y} is a frame monoplet of events, which have the joint e.p.d. in a form: {p(X + Y //X + {y}), X ⊆ X , Y ⊆ {y}}. (3.11) Then there is the {y}-inserted s.e. and the ∅inserted s.e.: X (∩{y}//{y}) = {x ∩ y, x ∈ X }, X (∩∅//{y}) = {x ∩ y c , x ∈ X }. (3.12) These inserted s.e.’s are characterized for every of two subsets of the monoplet Y = {y} ⊆ {y} and Y = ∅ ⊆ {y} by formulas (3.3) and by two corresponding e.p.d.’s {p(X//X + {y})}, X ⊆ X }, {p(X + {y}//X + {y})}, X ⊆ X }. (3.13) which by formulas (3.5) define two Y -pseudodistributions for X ⊆ X : p({y}) (X + {y}//X + {y}) = { p(X + {y}//X + {y}), X ̸= ∅, = p({y}//X + {y}) − 1 + p({y}//{y}),X = ∅,(3.14) { p(X + {y}//X + {y}), X ̸= ∅, = p({y}//X + {y}) − 1 + py , X = ∅. p(∅) (X//X + {y}) = { p(X//X + {y}), X ̸= ∅, = p(∅//X + {y}) − 1 + p(∅//{y}), X = ∅, (3.15) { p(X//X + {y}), X ̸= ∅, = p(∅//X + {y}) − py , X = ∅, where py = P(y) is a probability of the frame event y ∈ {y}. First of all, note that the sum of the probabilities of terraced events from the {y}-pseudo-distribution (3.14) is py , and the probabilities of the ∅-pseudodistribution (3.15) is 1 − py ; and secondly, that these two pseudo-distributions define a joint e.p.d. of the s.e. X and the monoplet {y}, i.e., e.p.d. of the s.e. X + {y}, which is related to them by fairly obvious formulas for Z ⊆ X + {y}: { p({y}) (Z//X + {y}), y ∈ Z, (3.16) p(Z//X + {y}) = (∅) p (Z//X + {y}), y ̸∈ Z. The formulas (3.16) are recurrent, connecting the e.p.d. of s.e. X + {y} with two pseudo-distributions of the inserted s.e. X ′ = X (∩{y}//Y) and X ′′ = X (∩∅//Y) whose power is less by one. The inversion formulas (3.10) allow recurrence formulas (3.16) to express the e.p.d. of X + {y} via the conditional e.p.d. with respect to one of its events y ∈ X + {y} and its complements y c = Ω − y: p(Z//X + {y}) =  p(X//X | {y}//{y})py , Z = X + {y},     (3.17) X ⊆ X; =  p(X//X | ∅//{y})(1 − py ), Z = X,    X ⊆ X, where Z ⊆ X + {y}. Note that these formulas, like (3.16), can be used recursively to express the e.p.d. of s.e. X + {y} through two conditional e.p.d.’s of the s.e. X whose power is less by one. VOROBYEV 93 3.2 Inserted and conditional Kopulas of a family of sets of events with respect to the set of events Definition 8 (inserted Kopulas). The N -Kopulas of Y -inserted N -s.e.’s X (∩Y //Y) = X (∩) {ter(Y //Y)} = = { x ∩ ter(Y //Y), x ∈ X } , where (∩Y //Y) (3.18) (3.19) (∩Y //Y) //X ) = p̆(Y ) = p̆(c|X } { }(3.20) { ) , x ∈ X = P(x ∩ ter(Y / /Y)), x ∈ X = p(Y x is the set of probabilities of “inserted” marginal events from the X (∩Y //Y) , and { } (∩Y //Y) //X (∩Y //Y) ) p̆(c|X = px(Y ) , x ∈ X + } { (3.21) ) + p(Y //X ) − p(Y x ,x ∈ X − X are X-phenomena of the set of “inserted” marginal probabilities p̆(Y ) . We also need to define an inserted Y -pseudoKopula with respect to the s.e. Y, which characterizes the Y -pseudo-distribution of the Y -inserted s.e. X (∩Y //Y) , inserted into the terraces event ter(Y //Y) generated by the s.e. Y. Although the Y -pseudo-Kopula is not a Kopula, i.e., is not a 1-function, it has properties very reminiscent of the Kopula properties. Definition 9 (inserted pseudo-Kopulas). The Y pseudo-Kopula of the Y -pseudo-distribution of Y inserted s.e. X (∩Y ) = X (∩Y //Y) = X (∩) {ter(Y //Y)} (3.22) with respect to the s.e. Y is a such function K(Y ) on X -hypercube with sides [0, p(Y //Y)] that 1) is non-negative: ( ) (∩Y ) //X(∩Y ) ) K(Y ) w̆(c|X >0 for w̆(c|X (∩Y ) //X(∩Y ) ) x∈X⊆X (3.24) where which for each Y ⊆ Y are defined (see Definition 6) as intersections by Minkowsi of the s.e. X with terraced events ter(Y //Y), generated by the s.e. Y, are called the Y -inserted N -Kopulas with respect to the s.e. Y. Such Y -inserted N -Kopulas characterizes e.p.d.’s of the 1st kind of Y -inserted N -s.e.’s by formulas for X ⊆ X p(X(∩){ter(Y //Y)}) = ( ) (∩Y //Y) //X (∩Y //Y) ) = K(Y ) p̆(c|X , 2) satisfies the Y -marginal equalities for x ∈ X: ( ) ∑ (∩Y ) (c|X (∩Y ) //X(∩Y ) ) //X(∩Y ) ) K(Y ) w̆(c|X = wx∩ter(Y //X ) (3.23) ∈ [0, p(Y //Y)]⊗X для X ⊆ X; w̆(c|X (∩Y ) //X (∩Y ) ) = } { (c|X (∩Y ) //X (∩Y ) ) ,x ∈ X wx∩ter(Y //X ) (3.25) is a X-phenomenon of the X -set of marginal probabilities of the Y -pseudo-distribution of Y inserted s.e. X (∩Y ) , i.e., (c|X (∩Y ) //X (∩Y ) ) wx∩ter(Y //X ) = { wx∩ter(Y //X ) , = p(Y //Y) − wx∩ter(Y //X ) , x ∈ X, x ∈ X − X. (3.26) From (3.24) and (3.26) it follows the probabilistic Y normalization of pseudo-Kopula: ( ) ∑ (∩Y ) //X (∩Y ) ) K(Y ) w̆(c|X = p(Y //Y). (3.27) X⊆X And from (3.23) and (3.27) it follows the terraceby-terrace probabilistic Y -normalization of pseudoKopula: ( ) (∩Y ) //X (∩Y ) ) 0 6 K(Y ) w̆(c|X 6 p(Y //Y) (3.28) for X ⊆ X. Such Y -pseudo-Kopulas characterize the Y -pseudodistribution (3.5) of Y -inserted s.e.’s X (∩Y ) by formulas for X ⊆ X where p(Y ) (X + Y //X + Y) = ( ) (∩Y //Y) //X (∩Y //Y) ) = K(Y ) p̆(c|X , (3.29) (∩Y //Y) //X (∩Y //Y) ) = p̆(Y ) = p̆(c|X { } { }(3.30) ) = p(Y = P(x ∩ ter(Y //Y)), x ∈ X x ,x ∈ X is a set of Y -marginal probabilities, coinciding with the set of marginal probabilities of Y -inserted s.e.’s X (∩Y ) , and } { (∩Y //Y) //X (∩Y //Y) ) p̆(c|X = px(Y ) , x ∈ X + { } (3.31) + p(Y //X ) − px(Y ) , x ∈ X − X are X-phenomena probabilities p̆(Y ) . of the set Y -marginal Definition 10 (conditional Kopulas). The N Kopulas, characterizing Y -inserted e.p.d.’s of the 94 THE XIV FAMEMS’2015 CONFERENCE 1st kind of the N -s.e. X with respect to the terraced event ter(Y //Y), generated by the s.e. Y, i.e., e.p.d.’s of the 1st kind, defined by joint e.p.d. X and Y by formulas with fixed Y ⊆ Y for X ⊆ X : p(X//X | Y //Y) = p(X + Y //X + Y) , p(Y //Y) (3.32) are called the Y -conditional N -Kopulas of the N s.e. X with respect to the terraced event ter(Y //Y), generated by the s.e. Y. Such Y -cvonditional N -Kopulas characterize the Y conditional e.p.d. of the 1st kind (3.32) by formulas for X ⊆ X : ( ) p(X//X | Y //Y) = K|Y p̆(c|X//X |Y //Y) , (3.33) where { } p̆(c|X //X |Y //Y) = p̆|Y = p|Y , x ∈ X = x { } = P(x ∩ ter(Y //Y))/p(Y //Y), x ∈ X (3.34) is a set of conditional marginal probabilities of events x ∈ X with respect to the terraced event ter(Y //Y), and p̆(c|X//X |Y //Y) = { } { } |Y = p|Y , x ∈ X + 1 − p , x ∈ X − X x x (3.35) are X-phenomenon of the set of conditional marginal probabilities p̆|Y . Note 10 (connection between conditional and “inserted” marginal probaabilities). Conditional marginal probabilities are connected with “inserted” marginal probabilities for x ∈ X by the formula of conditional probability: p|Y x = 1 p(Y ) , p(Y //Y) x (3.36) since “inserted” marginal probabilities (3.20) are probabilities of intersections of events x ∈ X with the terraced event ter(Y //Y). The connection between the corresponding set of conditional “inserted” marginal probabilities we shall write in the similar way: 1 p̆(Y ) , p(Y //Y) (3.37) (∩Y //Y) 1 //X (∩Y //Y) ) = p̆(c|X . p(Y //Y) p̆|Y = p̆(c|X//X |Y //Y) Note 11 (connection between conditional Kopulas From Definition and inserted pseudo-Kopulas). 10 of conditional Kopula and Definition 9 of inserted pseudo-Kopula with respect to the s.e. Y, and also from the formula (3.37) it follows the simple inversion formulas that connect conditional Kopulas and inserted Pseudo-Kopulas of the family of sets of events X for X ⊆ X : ( ) K|Y p̆(c|X//X |Y //Y) = ( ) 1 K(Y ) p(Y //Y)p̆(c|X//X |Y //Y) , = p(Y //Y) (3.38) ( K(Y ) p̆(c|X (∩Y //Y) = p(Y //Y) K ( |Y //X (∩Y //Y) ) ) = ) 1 c|X (∩Y //Y) //X (∩Y //Y) ) ( p̆ . p(Y //Y) Note 12 (two formulas of full probability for a Kopula). The Kopula K of s.e. X is expressed through Y -conditional Kpulas K|Y for Y ⊆ Y by the usual formula of full probability: ( ) ∑ ( ) K p̆(c|X//X ) = K|Y p̆(c|X//X |Y //Y) p(Y //Y). (3.39) Y ⊆Y From (3.39) and (3.38) we obtain an analogue of the formula of total probability — the representation of the Kopula of s.e. X in the form of sum of Y pseudo-Kopulas by Y ⊆ Y: ( ) ∑ ( ) (∩Y //Y) //X (∩Y //Y) ) K p̆(c|X//X ) = K(Y) p̆(c|X . (3.40) Y ⊆Y Note 13 (Kopula of a sum of sets). A Kopula of sum X + Y of two s.e.’s X and Y characterizes their joint e.p.d. of the 1st kind and by definition has the form ( ) p(X + Y //X + Y) = K p̆(c|X+Y //X +Y) , (3.41) where p̆(c|X+Y //X +Y) = {px , x ∈ X} + {py , y ∈ Y }+ (3.42) +{1 − px , x ∈ X − X} + {1 − py , y ∈ Y − Y } is the (X + Y )-phenomenon of the set of marginal probabilities p̆(c|X +Y//X +Y) = {px , x ∈ X } + {py , y ∈ Y} (3.43) for the sum X + Y. From previous formulas (3.32), (3.33), and (3.29) for a inserted pseudo-Kopula and conditional Kopula we obtain formulas ( ) K p̆(c|X+Y //X +Y) = ( ) (3.44) (∩Y //Y) //X (∩Y //Y) ) = K(Y) p̆(c|X , ( ) K p̆(c|X+Y //X +Y) = ( ) ( ) = K|Y p̆(c|X//X |Y //Y) K p̆(c|Y //Y) , (3.45) VOROBYEV 95 that for each Y ⊆ Y connect the Kopula of sum X + Y with the product of Y -conditional Kopula X with respect to Y and the value of Kopula Y at Y phenomenon; and also with the Y -inserted pseudoKopula of X which is inserted in the terraced event ter(Y //Y), generated by Y. 3.3 Theory of the frame method for constructing Kopula The basis of the frame method of constructing Kopula is a rather simple idea of composing an arbitrary N -s.e. X using the recurrence frame formula: X = {x0 , x1 , ..., xN −1 } = {x0 } + X , (3.46) where (N −1)-s.e.’s ( ) X = X − {x0 } = {x1 , ..., xN −1 } = X ′ (+) X ′′ (3.47) are composed from two (N − 1)-s.e.’s X ′ and X ′′ by set-theoretic operation of M -union9 and defined as the inserted s.e.’s in the frame monoplet {x0 } by the following formulas: X ′ = X (∩{x0 }//{x0 }) = = {x0 } (∩) X = {x0 ∩ x1 , ..., x0 ∩ xN −1 }, (3.48) X ′′ = X (∩∅//{x0 }) = = {xc0 } (∩) X = {xc0 ∩ x1 , ..., xc0 ∩ xN −1 }. This simple idea allows us to find the recurrent frame formulas for the N -Kopula of s.e. X as functions of the set of marginal probabilities p̆ = {p0 , p1 , . . . , pN −1 }. The frame method relies on formulas (3.16) and (3.17) and also correspondingly on (3.44) and (3.45), and constructs two recurrent formulas: ( ) ({x }) (∅) KX (p̆) = Recursion1 KX ′ 0 (p̆), KX ′′ (p̆) , (3.49) ( ) |{x } |∅ KX (p̆) = Recursion2 KX ′ 0 (p̆), KX ′′ (p̆), p0 , (3.50) for the N -Kopula of N -s.e. X through known probability p0 of the event x0 and together with it through two known inserted pseudo-(N−1)-Kopulas (see Definition 9), i.e., through pseudo-(N − 1)Kopulas of inserted (N − 1)-s.e.’s X ′ and X ′′ in the frame monoplet {x0 }, either through two known conditional (N −1)-Kopulas (see Definition 10) with respect to the frame monoplet {x0 } of the same X ′ and X ′′ . 9 М-intersection and М-union are an intersection and union od sets by Minkowski (see details in [5]). Note, that for the sake of brevity in the formulas (3.49) and (3.50) we use the following abbreviations, of course, given that X = X + {x0 }: ( ) KX (p̆) = K p̆(c|X+Y //X +{x0 }) , ( ) (∩{x0 }//{x0 }) ({x }) //X ′ ) KX ′ 0 (p̆) = K({x0 }) p̆(c|X , ( ) (∩∅//{x0 }) (∅) //X ′′ ) KX ′′ (p̆) = K(∅) p̆(c|X , ( ) |{x } KX ′ 0 (p̆) = K|{x0 } p̆(c|X//X |{x0 }//{x0 }) , ( ) |∅ KX ′′ (p̆) = K|∅ p̆(c|X//X |∅//{x0 }) . (3.51) Note 14 (about term “frame”). Although in formulas (3.46) and (3.47) only monoplet {x0 } is a frame set, we shall call frame (with respect to this monoplet) the s.e. X itself, construcyed from two inserted s.e.’s X ′ = X (∩{x0 }//{x0 }) and X ′′ = X (∩∅//{x0 }) , more hoping to clarify understanding than to cause misunderstandings. Note 15 (recurrent formulas of the frame method). Getting rid of abbreviations (3.51) and using (3.44) and (3.45), we write the recurrent formulas of the frame method (3.49) and (3.50) in the expanded form: ( ) K p̆(c|X+Y //X +{x0 }) = ( )  (∩{x0 }//{x0 }) //X ′ ) ({x }) (c|X  , Y = {x0 }, (3.52) K 0 p̆ = ( )  K(∅) p̆(c|X (∩∅//{x0 }) //X ′′ ) , Y = ∅, ( ) K p̆(c|X+Y //X +{x0 }) = ( )  |{x0 } (c|X//X |{x0 }//{x0 })  p̆ p0 , Y = {x0 }, (3.53) K = ( )  K|∅ p̆(c|X//X |∅//{x0 }) (1 − p ), Y = ∅. 0 Although formulas (3.46) and (3.47) are satisfied for any s.e., but in the proposed frame method (3.49) and (3.50) we use only half-rare s.e. (s.hr.e.) [9]. This, however, does not detract from the generality of its application, since the setphenomenon transformations are any s.e. can be obtained from its half-rare projection [9]. We will make the following useful Note 16 (any half-rare s.e. is composed by the frame method from two inserted s.e.’s which are always half- If the frame s.e. X in (3.46) ans (3.47) is half-rare, i.e., its marginal probabilities from p̆ = {p0 , p1 , ..., pN −1 } are not more than half, for example: rare). 1/2 > p0 > p1 > ... > pN −1 , (3.54) 96 THE XIV FAMEMS’2015 CONFERENCE then the both inserted s.e.’s X ′ = {x′1 , ..., x′N −1 } and X ′′ = {x′′1 , ..., x′′N −1 }, and together with them and the s.e. X are also half-rare by its Definition (3.48). In other words, their marginal probabilities from p̆′ = {p′1 , ..., p′N −1 } и p̆′′ = {p′′1 , ..., p′′N −1 } do not exceed the corresponding marginal probabilities events from the frame s.e. X: p1 > p′1 , ..., pN −2 p1 > p′′1 , ..., pN −2 > p′N −1 , > p′′N −1 , (3.55) and marginal probabilities from p̆N −1 = {p1 , ..., pN −1 } are half-rare by definition. Thus, any half-rare N -s.e. is composed by the frame method with the formula (3.47) from two inserted (N − 1)-s.e.’s X ′ and X ′′ , which are required to be half-rare. Lemma 4 (about independent half-rare s.e.’s, constructed by the frame method from two inserted That in the family of half-rare s.e.’s X with sets of marginal probabilitiues p̆, constructed by the frame method from two inserted half-rare s.e.’s X ′ and X ′′ , there was an independent half-rare s.e., it is necessary so that the sets of marginal probabilities are related to the marginal probabilities of the frame s.e. X = {x0 } + X = {x0 } + (X ′ (+)X ′′ ) by the following way: } { p̆′ = p′1 , . . . , p′N −1 half-rare s.e.’s). = { p1 p0 , . . . , pN −1 p0 } , } { ′′ p̆ = p′′1 , . . . , p′′N −1 = { p1 (1 − p0 ), . . . , pN −1 (1 − p0 ) } ; (3.56) and sufficient so that the e.p.d. of the 1st kind of inserted s.e.’s X ′ and X ′′ to be calculated from the formulas for X ⊆ X : ) ( ∩ ∩ ′ ′ ′c ′ = p(X //X ) = P x x x′ ∈X ′ p(X ′′ //X ′′ ) = P x′′ ∈X ′′ x′′ ∩ x′′ ∈X ′′ −X ′′  ∏ ∏  (1 − px ), p (1 − p ) x 0   x∈X −X x∈X ∏ =  (1 − px ) + p0 ,  (1 − p0 ) x′′c ) (3.59) The sfficiency follows from (3.57)10 and formulas that connect the e.p.d. of the 1st kind of frame s.e. X with the e.p.d. of the 1st kind of inserted s.e.’s X ′ and X ′′ , which have the form for X ⊆ X : p(X + Y //X + {x0 }) =  ′ ′   p(X //X ),  ′   p(∅//X ) − 1 + p0 , =   p(X ′′ //X ′′ ),    p(∅//X ′′ ) − p , 0 Y = {x0 }, X = ̸ ∅, Y = {x0 }, X = ∅, (3.60) Y = ∅, X = ̸ ∅, Y = ∅, X = ∅. Demanding (3.60) to perform sufficient conditions (3.57), we get p(X + Y //X + {x0 }) = ∏  ∏  (1−px ), Y = {x0 }, X ⊆ X , px p0     x∈X x∈X−X (3.61) = ∏ ∏    (1−px ),Y = ∅, X ⊆ X . px  (1−p0 ) x∈X x∈X−X As a result, for the s.e. X we have the e.p.d. of the 1st kind of independent events: p(X + Y //X + {x0 }) = ∏ ∏ (1 − px ), px = p(Z//X) = x∈Z (3.62) x∈X−Z where { X + {x0 }, Y = {x0 }, X ⊆ X , Z= X, Y = ∅, X ⊆ X . (3.63) The lemma is proved. x∈X ∩ p′n = P(x0 ∩ xn ) = pn p0 , p′′n = P(xc0 ∩ xn ) = pn (1 − p0 ). x′ ∈X ′ −X ′  ∏ ∏  (1 − px ), X ̸= ∅, p p x 0   x∈X −X x∈X ∏ =  (1 − px ) + 1 − p0 , X = ∅,  p0 ( Proof. The necessity is obvious, since the inserted marginal probabilities of the independent s.e. X are probabilities of double intersections of independent events which have the required form for n = 1, ..., N − 1: (3.57) = X ̸= ∅, X = ∅, x∈X where X ′ = { x′ , x ∈ X } = { x0 ∩ x, x ∈ X } ⊆ X ′ , (3.58) X ′′ = { x′′ , x ∈ X } = { xc0 ∩ x, x ∈ X } ⊆ X ′′ . 4 The Kopula theory for monoplets of events Theory of the Kopula of monoplets of events (1Kopula) seemed to be completed by the formula (2.35). This formula defines the 1-Kopula of an arbitrary monoplet of events {x} with {x}monoplet of marginal probabilities p̆ = {px } ∈ [0, 1]x in the unique form: K (p̆) = K (px ) = px , 10 By (4.1) the way, the necessary condition also follows from (3.57). VOROBYEV 97 which which provides 2 values on each 2(c|p̆) phenomenon-dom by general formulas for X ⊆ {x}: ( ) K p̆(c|X//{x}) = { (4.2) K (1 − px ) = 1 − px , X = ∅, = K (px ) = px , X = {x}. However, the formula (4.2) can be generalized in the following simple way: ( ) K p̆(c|X//{x}) = { (4.3) K∅ (1 − px ) = 1 − K{x} (px ) , X = ∅, = K{x} (px ) , X = {x}, {x} where K variables: is any function such that in half-rare K{x} : [0, 1/2] → [0, 1/2], (4.4) and in free variables: K{x} : [0, 1] → [0, 1]. ( ) ′′ K′′ p̆(c|S//{s }) = { K′′ (1 − ps′′ ) = 1 − ps′′ , = K′′ (ps′′ ) = ps′′ , s′ = x ∩ y ⊆ x, s′′ = xc ∩ y ⊆ xc , ps′ + ps′′ = py 6 px 6 1/2 6 1 − px . Consequently, the 1-Kopulas of inserted monoplets (5.4) and (5.5) are bound by the restriction on the sum of their marginal probabilities: (5.8) ps′ + ps′′ = py , and depend on only one parameter: ps′ ∈ [0, py ]. Ω (5.9) x s′c ∩ x = y c ∩ x s′ = x ∩ y In order to construct by the frame method the p̆ordered frame half-rare doublet of events ✿✿✿✿✿✿✿✿ ( ) X = {x, y} = {x} + X = {x} + X ′ (+)X ′′ (5.1) with the X-set of marginal probabilities p̆ = {px , py }, where s′′ = x ∩ y c Ω y s′′c ∩ xc = y c ∩ xc s′c ∩ x = y c ∩ x (5.2) let’s suppose that we have at our disposal two halfrare inserted monoplets of events with known 1-Kopulas: ( ) ′ K′ p̆(c|S//{s }) = { K′ (1 − ps′ ) = 1 − ps′ , S = ∅, = K′ (ps′ ) = ps′ , S = {s′ }, (5.7) Consequently, the 1-Kopulas of inserted monoplanes (5.4) and (5.5) are bound by the sum of their marginal probabilities: 5.1 The frame method for constructing a half-rare doublet of events X ′ = {x ∩ y} = {s′ } и X ′′ = {xc ∩ y} = {s′′ }, (5.6) and also because of the p̆-ordering assumption (5.2), we get that 5 The Kopula theory for doublets of events 1/2 > px > py > 0, (5.5) By Definition of inserted monoplets (5.3) (see Fig. 8) (4.5) In this case, the 1-Kopula (4.2) is an important special case of 1-Kopula (4.3) when K{x} (px ) = px . This case corresponds to a uniform marginal distribution function on the unit interval in the theory of the classical copula [4]. S = ∅, S = {s′′ }, s′ = x ∩ y s′′ = x ∩ y c (5.3) s′′c ∩ xc = y c ∩ xc (5.4) | {z x }| {z c x } Figure 8: Venn diagrams of the frame half-rare doublet of events X = {x, y}, 1/2 > px > py (up), and two inserted monoplets X ′ = {s′ } and X ′′ = {s′′ } (down) agreed with the frame doublet X in the following sense:y = s′ + s′′ и s′ ⊆ x, s′′ ⊆ xc . 98 THE XIV FAMEMS’2015 CONFERENCE Ω y We get the following formulas: p(xy//{x, y}) = p(s′ //X ′ ) = ps′ , s′c ∩ y = xc ∩ y p(x//{x, y}) = p(∅//X ′ ) − 1 + px = px − ps′ , (5.10) p(y//{x, y}) = p(s′′ //X ′′ ) = py − ps′ , p(∅//{x, y}) = p(∅//X ′′ ) − px = 1 − py − px + ps′ . These formulas express the e.p.d. of the 1st kind of the p̆-ordered half-rare frame doublet of events X = {x, y} through the e.p.d. of the 1st kind of inserted monoplets X ′ and X ′′ , and the probability of frame event x, and, in the final result, through their marginal probabilities px and py , and marginal probability ps′ of the inserted monoplet X ′ = {s′ } = {x ∩ y}. The formulas (5.10) express values of the 2-Kopula of p̆-ordered doublet X = {x, y} through 1-Kopulas of inserted monoplets X ′ = {s′ } and X ′′ = {s′′ }. Rewrite this in a form of an explicit recurrent formula: ( ) p(X//{x, y}) = KX p̆(c|X//{x,y}) =  KX ′ (ps′ ), X = {x, y},     KX ′ (1 − ps′ ) − 1 + px , X = {x}, (5.11) =   KX ′′ (ps′′ ), X = {y},    KX ′′ (1 − ps′′ ) − px , X = ∅. Considering (2.34) and (5.9), we will continue: ( ) p(X//{x, y}) = KX p̆(c|X//{x,y}) =  p s′ , X = {x, y},     px − ps′ , (5.12) X = {x}, =   X = {y}, ps′′ ,   1 − px − ps′′ , X = ∅,  ps′ , X = {x, y},     px − ps′ , X = {x}, = (5.13)   ′ p − p , X = {y}, y s    1 − px − py + ps′ , X = ∅. We note, by the way, that the restriction (5.9) by the assumption of p̆-ordered (5.2) is a special case of Fréchet-inequalities: 0 6 ps′ 6 p+ xy = min{px , py } = py . Note 17 (5.14) (frame mathod for otherwise p̆-ordered half- For rare doublet of events). half-rare doublet of events otherwise p̆-ordered ( ) X = {y, x} = {y} + X = {y} + X ′ (+)X ′′ , (5.15) where X ′ = {y ∩ x} = {s′ } и X ′′ = {y c ∩ x} = {s′′ }, (5.16) s′ = y ∩ x s′′ = y Ω s ′′c c c ∩y =x ∩y c ∩x x c s′c ∩ y = xc ∩ y s′ = y ∩ x s′′ = y c ∩x s′′c ∩ y c = xc ∩ y c | {z y }| {z c y } Figure 9: Venn diagrams of the frame, otherwise ordered half-rare doublet events X = {y, x}, 1/2 > py > px (up), and two inserted monoplets X ′ = {s′ } and X ′′ = {s′′ } (down), agreed with the frame doublet X in the following sense:x = s′ + s′′ и s′ ⊆ y, s′′ ⊆ y c . with the X-set of marginal probabilities p̆ = {py , px }, где 1/2 > py > px , the assumptions (5.18) take symmetrical form: s′ = y ∩ x ⊆ y, s′′ = y c ∩ x ⊆ y c . (5.17) By Definition of inserted monoplets (5.16) (см. Рис. 9) s′ = x ∩ y ⊆ y, s′′ = x ∩ y c ⊆ y c , (5.18) and also because of the assumption and also because of the assumption of another p̆-ordering, we get that ps′ + ps′′ = px 6 py 6 1/2 6 1 − py . (5.19) Consequently, 1-copulas of inserted monoplanes are connected by a restriction on the sum of their marginal probabilities: ps′ + ps′′ = px , (5.20) and depend on only one parameter: ps′ ∈ [0, px ]. (5.21) VOROBYEV 99 By the assumptions, the following formulas are valid: p(xy//{y, x}) = p(s′ //X ′ ) = ps′ , p(y//{y, x}) = p(∅//X ′ ) − 1 + py = py − ps′ , (5.22) p(x//{y, x}) = p(s′′ //X ′′ ) = px − ps′ , p(∅//{y, x}) = p(∅//X ′′ ) − py = 1 − px − py − ps′ . These formulas express the e.p.d. of the 1st kind otherwise p̆-ordered half-rare frame doublet of events X through the e.p.d. of the 1st kind of inserted monoplets X ′ and X ′′ , and the probability of frame event y, and, in the final result, through own marginal probabilities px и py , and the marginal probabilities of inserted monoplet X ′ = {s′ } = {x ∩ y} (see Fig. 9). The formulas (5.22) express values of the 2-Kopula of p̆-ordered doublet X = {y, x} through 1-Kopulas of inserted monoplets X ′ = {s′ } and X ′′ = {s′′ }. Rewrite this in the form of explicit recurrent formula: ( ) p(X//{y, x}) = KX p̆(c|X//{y,x}) =  KX ′ (ps′ ), X = {x, y},     (5.23) ′ ′ KX (1 − ps ) − 1 + py , X = {y}, =  KX ′′ (ps′′ ), X = {x},    KX ′′ (1 − ps′′ ) − py , X = ∅. Continue: ( ) p(X//{x, y}) = KX p̆(c|X//{x,y}) =  p s′ , X = {x, y},     py − ps′ , X = {y}, =  ′′ p , X = {x}, s    1 − px − ps′′ , X = ∅,  ps′ ,    p − p ′ , y s =  p − p x s′ ,    1 − px − py + ps′ , X X X X = {x, y}, = {y}, = {x}, = ∅. (5.27) ( ) K p̆(c|X+Y //X +{x}) = ( )  |{x} (c|X//X |{x}//{x})  p̆ px , Y = {x}, (5.28) K = ( )  K|∅ p̆(c|X//X |∅//{x}) (1 − p ), Y = ∅. x In the formulas (5.27) the pseudo-Kopulas K({x}) and K(∅) of inserted monoplets X ′ and X ′′ , correspondingly, are defined by the first and the second pairs of probabilities from (5.13) correspondingly, i.e., by formulas: ( ) (∩{x}//{x}) //X ′ ) K({x}) p̆(c|X = { (5.29) p s′ , X = {x, y}, = px − ps′ , X = {x}, ( ) (∩∅//{x}) //X ′′ ) K(∅) p̆(c|X = { py − ps′ , X = {y}, = 1 − px − py + ps′ , X = ∅, (5.30) where, for example, for X = {y} {y}(∩{x}//{x}) = {y ∩ x} ⊆ X ′ = {s′ }, {y}(∩∅//{x}) = {y ∩ xc } ⊆ X ′′ = {y − s′ }, (5.31) (5.24) and the corresponding sets of marginal probabilities of inserted monoplets X ′ and X ′′ have the form (∩{x}//{x}) //X ′ ) = {ps′ }, p̆(c|{y} (5.25) We note, as above, that the restriction (5.21) by the assumption of another p̆-ordering is a special case of Fréchet inequalities: 0 6 ps′ 6 p+ xy = min{px , py } = px . doublet X = {x, y} = {x} + {y} = {x} + X : ( ) K p̆(c|X+Y //X +{x}) = ( )  (∩{x}//{x}) //X ′ ) ({x}) (c|X  p̆ , Y = {x}, K = ( )  K(∅) p̆(c|X (∩∅//{x}) //X ′′ ) , Y = ∅, (5.26) 5.2 The frame method: recurrent formulas for a half-rare doublet of events The formulas (5.11), as well as formulas (5.13), can be rewrite in the form of special cases of recurrent formulas (3.52) and (3.53) from Note 15 for the p̆(c|{y} (∩∅//{x}) //X ′′ ) (5.32) = {py − ps′ }. In the formulas (5.28) the conditional Kopulas K|{x} and K|∅ are defined by the first and the second pairs of probabilities from (5.13), normalized by px and by 1 − px correspondingly, i.e. by the formulas: ( ) K|{x} p̆(c|X//X |{x}//{x}) = { 1 (5.33) ps′ , X = {x, y}, = p1x ′ px (px − ps ), X = {x}, ( ) K|∅ p̆(c|X//X |∅//{x}) = { 1 (py − ps′ ), X = {y}, x = 1−p 1 ′ 1−px (1 − px − py + ps ), X = ∅. (5.34) 100 THE XIV FAMEMS’2015 CONFERENCE The corresponding sets of marginal conditional probabilities of events y ∈ X with respect to the frame terraced events ter({x}//{x}) = x и ter(∅//{x}) = xc correspondingly have the form: { } p s′ (c|X //X |{x}//{x}) p̆ = , px { } (5.35) py − ps′ (c|X //X |∅//{x}) p̆ = . 1 − px The e.p.d. of the 1st kind of independent triplet of events X with the X-set of probabilities of events p̆ is defined by 23 values of the independent 3-Kopula (6.1) on 2(c|p̆) -phenomenon-dom by the general formulas of half-rare variables, i.e., for X ⊆ {x, y}: Remind, that Fréchet restrictions on the functional parameter ps′ = ps′ (px , py ) for p̆-ordered half-rare doublet of events X = {x, y} have the form: 6.2 Three-dimensional maps of the independent 3-Kopula 0 6 ps′ 6 p+ xy = min{px , py } = py , (5.36) and for otherwise p̆-ordered half-rare doublet of events X = {y, x} have the form: 0 6 ps′ 6 p+ xy = min{px , py } = px . ( ) ∏ ∏ (1 − px ). (6.3) p(X//X) = K p̆(c|X//X) = px x∈X x∈X−X In Fig. 10 it is shown the results of visualization of the three-dimensional graph of independent 3Kopula (8.1) of the triplet X = {x, y, z}, defined on the cube [0, 1]3 , in projections on planes, which are orthogonal to the axis py . (5.37) 6 The Kopula theory for triplets of events 6.1 Independent 3-Kopula First, without the frame method, which is not required here, consider the simplest example of a 3-Kopula K ∈ Ψ1X of the (N − 1)-set of events X = {x, y, z}, i.e., a 1-function on the unit Xcube. In other words, construct such a nonnegative bounded numerical function 6.3 The frame method for constructing a half-rare triplet of events K : [0, 1]⊗X → [0, 1], that for all z ∈ X ∑ x∈X⊆X ( K w̆ (c|X//X) ) In order to construct by the frame method the p̆-ordered frame half-rare triplet of events X = ✿✿✿✿✿✿✿ {x, y, z} with the X-set of marginal probabilities p̆ = {px , py , pz }, where = wx . Such a simple example of a 1-function on X-cube is so-called independent (N − 1)-Kopula, which for all free variables w̆ = {wx , wy , wz } ∈ [0, 1]x ⊗ [0, 1]y ⊗ [0, 1]z = [0, 1]⊗X is defined by the formula: K (w̆) = wx wy wz , (6.1) that provides it on each 2(c|w̆) -phenomenon-dom the following 23 values: ( ) ∏ ∏ (1 − wx ) K w̆(c|X//X) = wx (6.2) x∈X x∈X−X for X ⊆ X. Indeed, as in the case of the doublet of events this function is a 1-function, since for all x ∈ X ) ( ∏ ∑ ∏ wz (1 − wz ) = wx . x∈X⊆X z∈X z∈X−X Figure 10: The visualization of projections of the same three-dimensional map of Cartesian representation of independent 3-Kopula of the triplet X = {x, y, z} on the unit cube in conditional colors with values of marginal probaboility py = 0.1, ..., 0.9, 1.0, where the white color corresponds to points in which probabilities of all terraced events are 1/8. The orientation of axes: (px , pz ) = (horizontal, vertical). 1/2 > px > py > pz > 0, (6.4) let’s suppose that X = {x} + {y, z} = {x} + (X ′ (+)X ′′ ) (6.5) and in our disposal we have two inserted half-rare doublets of events X ′ = {s′ , t′ } и X ′′ = {s′′ , t′′ }, with the known 2-Kopulas (see Fig. 11) which by the definition satisfy the following inclusions: s′ = x ∩ y ⊆ x, t′ = x ∩ z ⊆ x, s′ ∪ t′ ⊆ x, (6.6) s′′ = xc ∩ y ⊆ xc , t′′ = xc ∩ z ⊆ xc, s′′ ∪ t′′ ⊆ xc . VOROBYEV Ω 101 xc ∩ s′′c ∩ t′′c In view of this, we obtain the formulas: y p(xyz//X) = p(s′ t′ //X ′ ), p(xy//X) = p(s′ //X ′ ), s′′ ∩ t′′c s′ ∩ t′c x∩ s′c ∩ t′c x p(xz//X) = p(t′ //X ′ ), p(x//X) = p(∅//X ′ ) − 1 + px , s′ ∩ t ′ s′′ ∩ t′′ p(yz//X) = p(s′′ t′′ //X ′′ ), p(y//X) = p(s′′ //X ′′ ), p(z//X) = p(t′′ //X ′′ ), p(∅//X) = p(∅//X ′′ ) − px , s′c ∩ t′ s ′′c ∩t ′′ z Ω s′ x∩s′c∩t′c s′′ s′ ∩ t′c ′ s ∩t xc∩s′′c∩t′′c s′′ ∩ t′′c ′ ′′ s ∩t s′c ∩ t′ | {z p(s′ //X ′ ) + p(s′ t′ //X ′ )+ + p(s′′ //X ′ ) + p(s′′ t′′ //X ′ ) = py , s′′c ∩ t′′ t′′ }| x that express the e.p.d. of the 1st kind of frame halfrare triplet of events X through the e.p.d. of the 1st kind of inserted half-rare doublets X ′ and X ′′ , and the probability of frame event x. In the language of e.p.d. of the 1st kind assumptions (6.6) mean that ′′ t′ (6.11) {z c } x Figure 11: Venn diagrams of the frame half-rare trip0let of events X = {x, y, z}, 1/2 > px > py > pz (up), and two inserted doublets X ′ = {s′ , t′ } and X ′′ = {s′′ , t′′ } (down), agreed with the frame triplet X in the following sense:y = s′ + s′′, z = t′ + t′′ и s′ ∪ t′ ⊆ x, s′′ ∪ t′′ ⊆ xc . p(t′ //X ′ ) + p(s′ t′ //X ′ )+ + p(t′′ //X ′ ) + p(s′′ t′′ //X ′ ) = pz , (6.12) or the same in the language of probabilities events: ps′ + ps′′ = py , pt′ + pt′′ = pz , (6.13) In view of the assumptions made (see Fig. 11) In addition, the third pair of inclusions under the assumptions (6.6) means that ter(xyz//X) = x∩ y ∩ z = s′ ∩ t′ = ter(s′ t′ //X ′ ), ter(xy//X) = x∩ y ∩ z c = s′ ∩ t′c = ter(s′ //X ′ ), ps′ + pt′ − ps′ t′ 6 px , ps′′ + pt′′ − ps′′ t′′ 6 1 − px , ter(xz//X) = x∩ y c ∩ z = s′c ∩ t′ = ter(t′ //X ′ ), (6.7) ter(yz//X) = xc ∩ y ∩ z = s′′ ∩ t′′ = ter(s′′ t′′ //X ′′ ), ter(y//X) = xc ∩ y ∩ z c = s′′ ∩ t′′c = ter(s′′ //X ′′ ), ter(z//X) = xc ∩ y c ∩ z = s′′c ∩ t′′ = ter(t′′ //X ′′ ), these 6 terraced events are defined. All of them are generated by the frame half-rare triplet X, with the exception of two terraced events ter(x//X) = x ∩ y c ∩ z c , ter(∅//X) = xc ∩ y c ∩ z c , where ps′ t′ = p(s′ t′ //X ′ ) = P(s′ ∩ t′ ), ps′′ t′′ = p(s′′ t′′ //X ′′ ) = P(s′′ ∩ t′′ ) ter(∅//X) = xc − s′′ ∪ t′′ , Taking into account the Fréchet inequalities the restrictions (6.13) and (6.14) are equivalent to the following inequalities for 4 parameters ps′ , pt′ , ps′ t′ и ps′′ t′′ of inserted doublets X ′ and X ′′ : (6.8) 0 6 p s′ 6 p y , 0 6 p t′ 6 p z , (6.9) p− s′ t ′ − ps′′ t′′ 6 ps′ t′ 6 p+ s′ t ′ , (6.16) 6 ps′′ t′′ 6 p+ s′′ t′′ , where p− s′ t′ = max{0, ps′ + pt′ − px }, or, equivalently, ter(x//X) = ter(∅//X ′ ) − xc , ter(∅//X) = ter(∅//X ′′ ) − x. (6.15) are probabilities of double intersections of events from the inserted doublets X ′ and X ′′ . that are defined by the formulas: ter(x//X) = x − s′ ∪ t′ , (6.14) p+ s′ t′ = min{ps′ , pt′ }, (6.10) p− s′′ t′′ = max{0, px + py + pz − 1 − ps′ − pt′ }, p+ s′′ t′′ = min{py − ps′ , pz − pt′ }, (6.17) 102 THE XIV FAMEMS’2015 CONFERENCE are the lower and upper Frechet-boundaries of probabilities of double intersections of inserted doublets X ′ and X ′′ with respect to the frame monoplet {x}. Let’s write the formulas (6.11), using only these 4 parameters and remembering the restrictions (6.16): p(xyz//X) = ps′ t′ , p(xy//X) = ps′ − ps′ t′ , p(xz//X) = pt′ − ps′ t′ , where, for example, for X = {y, z} {y, z}(∩{x}//{x}) = {y ∩ x, z ∩ x} ⊆ X ′ = {s′ , t′ }, (6.23) {y, z}(∩∅//{x}) = {y ∩ xc , z ∩ xc } ⊆ X ′′ = {s′′ , t′′ }, and the corresponding sets of marginal probabilities of inserted doublets X ′ и X ′′ have the form p̆(c|{y,z} (∩{x}//{x}) p̆(c|{y,z} p(x//X) = px − ps′ − pt′ + ps′ t′ , (6.18) p(yz//X) = ps′′ t′′ , p(y//X) = py − ps′ − ps′′ t′′ , p(z//X) = pz − pt′ − ps′′ t′′ , p(∅//X) = 1 − px − py − pz + ps′ + pt′ + ps′′ t′′ . 6.4 The frame method: recurrent formulas for a half-rare triplet of events The formulas (6.18) as well as the formulas (6.11) can be written in the form of special cases of recurrence formulas (3.52) and (3.53) from Note 15 for the triplet X = {x, y, z} = {x} + {y, z} = {x} + X : ( ) K p̆(c|X+Y //X +{x}) = ( )  (∩{x}//{x}) //X ′ ) ({x}) (c|X  p̆ , Y = {x}, (6.19) K = ( )  K(∅) p̆(c|X (∩∅//{x}) //X ′′ ) , Y = ∅, ( ) K p̆(c|X+Y //X +{x}) = ( )  |{x} (c|X//X |{x}//{x})  p̆ px , Y = {x}, (6.20) K = ( )  K|∅ p̆(c|X//X |∅//{x}) (1 − p ), Y = ∅. x In the formulas (6.19) the inserted pseudoKopulas K({x}) and K(∅) are defined by the first and the second four probabilities from (6.18) correspondingly, i.e., by the formulas: ( ) (∩{x}//{x}) //X ′ ) K({x}) p̆(c|X =  ps′ t′ , X = {y, z},    p ′ − p ′ ′ , (6.21) X = {y}, s st =  ′ ′ ′ X = {z},  pt − ps t ,  px − ps′ − pt′ + ps′ t′ , X = ∅, ( ) (∩∅//{x}) //X ′′ ) K(∅) p̆(c|X =   ps′′ t′′ , X = {y, z},     ′ ′′ ′′  X = {y}, py − ps − ps t , (6.22) = pz − pt′ − ps′′ t′′ , X = {z},    1 − px − py − pz +    +p ′ + p ′ + p ′′ ′′ , X = ∅, s t s t (∩∅//{x}) //X ′′ ) //X ′′ = {ps′ , pt′ }, ) = {p − p ′ , p − p ′ }. y s z t (6.24) In the formulas (6.20) the conditional Kopulas K|{x} and K|∅ are defined by the first and the second four probabilities from (6.18), normalized by px and by 1 − px correspondingly, i.e., by the formulas: ( ) K|{x} p̆(c|X//X |{x}//{x}) =  1  ′ ′ X = {y, z},  px ps t ,    1 (p ′ − p ′ ′ ), (6.25) X = {y}, s st = p1x  ′ ′ ′ X = {z},  px (pt − ps t ),   1 − 1 (p ′ + p ′ − p ′ ′ ), X = ∅, s t st px ( ) K|∅ p̆(c|X//X |∅//{x}) =  1 ′′ ′′ X = {y, z},   1−px ps t ,   1  ′ ′′ ′′ (p − p − p ), X = {y},  s s t  1−px y (6.26) 1 = 1−px (pz − pt′ − ps′′ t′′ ), X = {z},   1    1−px (1 − px − py − pz )+   1 (ps′ + pt′ + ps′′ t′′ ), X = ∅. + 1−p x The corresponding sets of marginal conditional probabilities of events y, z ∈ X with respect to the frame terraced events ter({x}//{x}) = x и ter(∅//{x}) = xc correspondingly have the from: } { ps′ pt′ , , p̆(c|X //X |{x}//{x}) = px px { } (6.27) py − ps′ pz − pt′ p̆(c|X //X |∅//{x}) = . , 1 − px 1 − px Remind, that the four functional parameters ps′ , pt′ , ps′ t′ and ps′′ t′′ in the recurrent formulas (6.19), (6.20), and also in the formulas for pseudoKopulas (6.21), (6.22), and the conditional Kopulas (6.25), (6.26), obey the Frechet-constraints (6.16). 7 The Kopula theory for quadruplets of events 7.1 The frame method for constructing a half-rare quadruplet of events In order by the frame method to construct the p̆ordered frame half-rare quadruplet of events X = ✿✿✿✿✿✿✿✿ VOROBYEV 103 {x, y, z, v} with the X-set of marginal probabilities p̆ = {px , py , pz , pv }, where (7.1) 1/2 > px > py > pz > pv > 0, let’s suppose that X = {x} + {y, z, v} = {x} + (X ′ (+)X ′′ ) (7.2) and we have two inserted half-rare triplets of events X ′ = {s′ , t′ , u′ } и X ′′ = {s′′ , t′′ , u′′ }, with the known 3-Kopulas, which by definition satisfy the following inclusions (see Fig. 12): s′ = x ∩ y ⊆ x, t′ = x ∩ z ⊆ x, u′ = x ∩ v ⊆ x, s′ ∪ t′ ∪ u′ ⊆ x, (7.3) s′′ = xc ∩ y ⊆ xc , t′′ = xc ∩ z ⊆ xc , u′′ = xc ∩ v ⊆ xc , s′′ ∪ t′′ ∪ u′′ ⊆ xc . Ω ter(x//X) y ∩xc ter(xy//X) ter(xyz//X) Let us dwell in more detail on Fréchet-restrictions for 11 = 24 − 4 − 1 functional parameters of a Kopula of quadruplet of events, to derive the recurrent sequence of such Fréchet-restrictions, which begins with Fréchet-restrictions for a doublet of events (5.36), continues with Fréchetrestrictions for a triplet of events (6.16), and should be supported by Fréchet-restrictions for parameters of a Kopula of quadruplet of events X = {x, y, z, v} and so on. To this end, we first recall Fréchet-restrictions for parameters of Kopulas of a doublet and a triplet of events. 7.2.1 Fréchet-restrictions for a doublet of events For a Kopula of doublet of events, the Fréchetrestrictions of a 1 = 22 − 2 − 1 parameter of inserted monoplets X ′ and X ′′ have the form: ter(∅//X) y ∩x 7.2 Recurrent Fréchet-restrictions in the frame method ter(y//X) ter(xyv//X) ter(yz//X) ter(xyzv//X) 0 6 p s′ 6 p y . ter(yv//X) (7.4) ter(yzv//X) 7.2.2 Fréchet-restrictions for a triplet events ter(xz//X) ter(xzv//X) ter(xv//X) ter(z//X) ter(zv//X) ter(v//X) v ∩xz ∩xc z ∩x | {z x Ω v ∩xc }| s′c∩t′c∩u′c − xc s′ {z c } x s′′c∩t′′c∩u′′c − x For a Kopula of doublet of events, the Fréchetrestrictions of 4 = 23 − 3 − 1 parameters of inserted doublets X ′ and X ′′ have the form: 0 6 p s′ 6 p y , 0 6 p t′ 6 p z , s′′ s′∩t′c∩u′c s′∩t′∩u′c s′′∩t′′c∩u′′c s′∩t′c∩u′ s′′∩t′′∩u′′c s′∩t′∩u′ s′c∩t′∩u′c s′c∩t′∩u′ p− s′ t ′ − ps′′ t′′ s′′∩t′′c∩u′′ 6 ps′′ t′′ 6 p+ s′′ t′′ , p− s′ t′ = max{0, ps′ + pt′ − px }, s′′c∩t′′∩u′′c s′′c∩t′′∩u′′ s′′c∩t′′c∩u′′ p+ s′ t′ = min{ps′ , pt′ }, t′ | u′ t′′ {z x }| u′′ {z c x (7.5) where s′′∩t′′∩u′′ s′c∩t′c∩u′ 6 ps′ t′ 6 p+ s′ t ′ , } Figure 12: Venn diagrams of the frame half-rare quadruplet of events X = {x, y, z, v}, 1/2 > px > py > pz > pv (up), and two inserted triplets X ′ = {s′ , t′ , u′ } and X ′′ = {s′′ , t′′ , u′′ } (down), agreed with the frame quadruplet X in the following sense: y = s′ + s′′, z = t′ + t′′, v = u′ + u′′ и s′ ∪ t′ ∪ u′ ⊆ x, s′′ ∪ t′′ ∪ u′′ ⊆ xc . The recurrent formulas, which express the e.p.d. of the 1st kind of p̆-ordered half-rare quadruplet of events X through the e.p.d. of the 1st kind of two inserted triplets X ′ and X ′′ , follow from the general recurrent formulas (3.52) and (3.53) in Note 15 as well as in cases of a doublet and a triplet of events. And therefore, and because of the cumbersomeness, these formulas are not represented here, but are only illustrated by Venn diagrams (see Fig. 12). p− s′′ t′′ = max{0, px + py + pz − 1 − ps′ − pt′ }, (7.6) p+ s′′ t′′ = min{py − ps′ , pz − pt′ }, are the lower and upper Fréchet-boundaries probabilities of double intersections of events from inserted doublets X ′ and X ′′ with respect to the frame monoplet {x}. The case of triplet of events gives a new level of Fréchet-restrictions (the two last Fréchetboundaries in (7.6)), when probabilities of double intersections of events from inserted doublets have Fréchet-boundaries that depend not only on marginal probabilities of the triplet, but and on inserted marginal probabilities on which, in turn, the usual Fréchet-restrictions mentioned above are imposed. 104 THE XIV FAMEMS’2015 CONFERENCE 7.2.3 Fréchet-restrictions for a quadruplet of events For a Kopula of quadruplet of events the Fréchetrestrictions of 11 = 24 − 4 − 1 parameters of the inserted triplet X ′ and X ′′ have the form: 0 6 ps′ 6 py , 0 6 pt′ 6 pz , 0 6 pu′ 6 pv , + p− s′ t′ 6 ps′ t′ 6 ps′ t′ , + p− s′ u′ 6 ps′ u′ 6 ps′ u′ , p− t ′ u′ − ps′′ t′′ p− s′′ u′′ p− t′′ u′′ − ps′ t′ u′ p− s′′ t′′ u′′ 6 6 6 6 6 6 pt′ u′ 6 p+ t ′ u′ , ′′ ′′ ps t 6 p+ s′′ t′′ , ′′ ′′ ps u 6 p+ s′′ u′′ , + pt′′ u′′ 6 pt′′ u′′ , ps′ t′ u′ 6 p+ s′ t ′ u ′ , ps′′ t′′ u′′ 6 p+ s′′ t′′ u′′ , (7.7) The such Fréchet-restrictions and Fréchetboundaries for a doublet (7.4), a triplet (7.5,7.6), a quadruplet (7.7,7.8) of events and so on, will call the recurrent Fréchet-restrictions and recurrent Fréchet-boundaries. 7.3 The frame method: recurrent formulas for a half-rare quadruplet of events where p− s′ t ′ p+ s′ t ′ − ps′′ t′′ p+ s′′ t′′ p− s′ u ′ p+ s′ u ′ − ps′′ u′′ p+ s′′ u′′ p− t ′ u′ p+ t ′ u′ p− t′′ u′′ p+ t′′ u′′ p− s′ t ′ u ′ p+ s′ t ′ u ′ − ps′′ t′′ u′′ p+ s′′ t′′ u′′ The all Fréchet-restrictions in the considered frame methods for a doublet, a triplet and a quadruplet of events differ from the usual Fréchet-restrictions, which are functions of only corresponding marginal probabilities. They differ in that they have a recurrent structure. When, as the power of intersections of inserted events increases, the Fréchet-boundaries of their probabilities are functions of Fréchet-boundaries for probabilities of intersections of lower power. The recurrent formulas for Kopula of a quadruplet of events immediately can be written in the form of special cases of recurrence formulas (3.52) and (3.53) from Note 15 for the quadruplet X = {x, y, z, v} = {x} + {y, z, v} = {x} + X : = max{0, ps′ + pt′ − px }, = min{ps′ , pt′ }, = max{0, px + py + pz − 1 − ps′ − pt′ }, = min{py − ps′ , pz − pt′ }, ( ) K p̆(c|X+Y //X +{x}) = ( )  (∩{x}//{x}) //X ′ ) ({x}) (c|X  p̆ , Y = {x}, K = ( )  K(∅) p̆(c|X (∩∅//{x}) //X ′′ ) , Y = ∅, = max{0, ps′ + pu′ − px }, = min{ps′ , pu′ }, = max{0, px + py + pv − 1 − ps′ − pu′ }, = min{py − ps′ , pv − pu′ }, = max{0, pt′ + pu′ − px }, = min{pt′ , pu′ }, = max{0, px + pz + pv − 1 − pt′ − pu′ }, = min{pz − pt′ , pv − pu′ }, = max{0, ps′ t′ + ps′ u′ + pt′ u′ − 2px }, = min{ps′ t′ , ps′ u′ , pt′ u′ }, = max{0, ps′′ t′′ + ps′′ u′′ + pt′′ u′′ − 2(1 − px )}, = min{ps′′ t′′ , ps′′ u′′ , pt′′ u′′ } (7.8) are the lower and upper Fréchet-boundaries of probabilities of double and triple intersections of events from the inserted triplets X ′ and X ′′ with respect to the frame monoplet {x}. The case of a quadruplet of events gives the following level of Fréchet-restrictions (the four last Fréchet-boundaries in (7.8)), when probabilities of triple intersections of events from inserted triplets have Fréchet-boundaries that depend directly not so much on marginal probabilities as on inserted probabilities of double intersections, on which, in turn, Fr’echet-restrictions of the previous level, mentioned above, are imposed. (7.9) ( ) K p̆(c|X+Y //X +{x}) = ( )  |{x} (c|X//X |{x}//{x})  p̆ px , Y = {x}, (7.10) K = ( )  K|∅ p̆(c|X//X |∅//{x}) (1 − p ), Y = ∅. x In the formulas (7.9) the inserted pseudo-Kopulas K({x}) and K(∅) are defined by the octuples of probabilities, i.e., by the formulas: ( ) (∩{x}//{x}) //X ′ ) K({x}) p̆(c|X =  ps′ t′ u′ , X      ′ ′ ′ ′ ′ p − p , X  st stu     ′ u′ − ps′ t′ u′ , p X s     ′ ′ ′ ′ ′  p − p , X stu  tu = ps′ − ps′ t′ − ps′ u′ + ps′ t′ u′ , X    ′ ′ ′ ′ ′ ′ ′ ′ pt − ps t − pt u + ps t u , X     pu′ − ps′ u′ − pt′ u′ + ps′ t′ u′ , X      px − ps′ − pt′ − pu′ +    +ps′ t′ + ps′ u′ + pt′ u′ − ps′ t′ u′ , X = {y, z, v}, = {y, z}, = {y, v}, = {z, v}, (7.11) = {z}, = {v}, = {v}, = ∅, VOROBYEV ( ) (∩∅//{x}) //X ′′ ) K(∅) p̆(c|X =  X  ps′′ t′′ u′′ ,    ′′ ′′ ′′ ′′ ′′ p − p , X  s t s t u     ′′ u′′ − ps′′ t′′ u′′ , p X s     ′′ ′′ ′′ ′′ ′′  pt u − ps t u , X     ′− p − p  y s     −ps′′ t′′ − ps′′ u′′ + ps′′ t′′ u′′ , X    p − p ′ − z t = −ps′′ t′′ − pt′′ u′′ + ps′′ t′′ u′′ , X      pv − pt′ −     −ps′′ u′′ − pt′′ u′′ + ps′′ t′′ u′′ , X      1 − px − py − pz − pv +      +ps′ + pt′ + pu′ −     −ps′′ t′′ − ps′′ u′′ − pt′′ u′′ +    +ps′′ t′′ u′′ , X 105 = {y, z, v}, = {y, z}, = {y, v}, = {z, v}, = {y}, (7.12) = {z}, = {v}, = ∅, where, for example, for X = {y, z, v} {y, z, v}(∩{x}//{x}) = {y ∩ x, z ∩ x, v ∩ x} ⊆ X ′ , (7.13) {y, z, v}(∩∅//{x}) = {y ∩ xc , z ∩ xc , v ∩ xc } ⊆ X ′′ , and the corresponding sets of marginal probabilities of inserted triplets X ′ and X ′′ have the form ( ) K|∅ p̆(c|X//X |∅//{x}) =  1 ps′′ t′′ u′′ , X   x  1−p  1  ′′ ′′ ′′ ′′ ′′ (p − p ), X  s t s t u 1−px    1  ′′ ′′ (p − p X  s u s′′ t′′ u′′ ), 1−px    1  ′′ ′′ ′′ ′′ ′′ X  1−px (pt u − ps t u ),    1  ′   1−px (py − ps )−   1  (ps′′ t′′ + ps′′ u′′ − ps′′ t′′ u′′ ), X − 1−p  x    1 (p − p ′ )− z t = 1−px1  − 1−px (ps′′ t′′ + pt′′ u′′ − ps′′ t′′ u′′ ), X     1  ′  1−px (pv − pt )−   1   − 1−px (ps′′ u′′ + pt′′ u′′ − ps′′ t′′ u′′ ), X   1  1 − 1−p (py + pz + pv )+  x   1   + 1−px (ps′ + pt′ + pu′ )−    1   (ps′′ t′′ + ps′′ u′′ + pt′′ u′′ )+ − 1−p  x   1 + 1−px ps′′ t′′ u′′ , X = {y}, (7.16) = {z}, = {v}, = ∅, The corresponding sets of marginal conditional probabilities of events y, z ∈ X with respect to the frame terraced events ter({x}//{x}) = x and ter(∅//{x}) = xc correspondingly have the form: } { ps′ pt′ pu′ , , , p̆(c|X //X |{x}//{x}) = px px px }(7.17) { py − ps′ pz − pt′ pv − pu′ . , , p̆(c|X //X |∅//{x}) = 1 − px 1 − px 1 − px Recall that 11 functional parameters ps′ , pt′ , pu′, ps′ t′ , ps′ u′ , pt′ u′ , (∩{x}//{x}) //X ′′ ) p̆(c|{y,z,v} = {ps′ , pt′ , pu′ }, (7.14) ps′′ t′′ , ps′′ u′′ , pt′′ u′′ , ps′ t′ u′ , ps′′ t′′ u′′ (∩∅//{x}) //X ′′ ) p̆(c|{y,z,v} = {py − ps′ , pz − pt′ , pv − pu′ }. In the formulas (7.10) the condirional Kopulas K|{x} and K|∅ are defined by the octuples of probabilities that are normalized by px and by 1−px correspondingly, i.e., by the formulas: ( ) K|{x} p̆(c|X//X |{x}//{x}) = 1 ps′ t′ u′ , X    p1x   ′ ′ ′ ′ ′ X   px (ps t − ps t u ),   1  ′ ′ ′ ′ ′ (p − p ), X su stu  px    1  ′ ′ ′ ′ ′ X  px (pt u − ps t u ),    1 (p ′ − p ′ ′ − p ′ ′ + p ′ ′ ′ ), X s st su stu = p1x  ′ ′ ′ ′ ′ (p − p − p + p t st tu s′ t′ u′ ), X  px    1   px (pu′ − ps′ u′ − pt′ u′ + ps′ t′ u′ ), X    1 − p1 (ps′ + pt′ + pu′ )+  x     + p1x (ps′ t′ + ps′ u′ + pt′ u′ )−    1 − px ps′ t′ u′ , X = {y, z, v}, = {y, z}, = {y, v}, = {z, v}, (7.18) in the recurrent formulas (7.11, 7.12) and (7.15, 7.16) obey the Fréchet-restrictions (7.7) and Fréchet-boundaries (7.8). 8 The Kopula theory for a set of events 8.1 Independent N -Kopula = {y, z, v}, = {y, z}, = {y, v}, = {z, v}, = {z}, (7.15) = {v}, = {v}, = ∅, First, without the frame method, which is not required here, let’s consider the simplest example of the N -Kopula K ∈ Ψ1X of an N -set of events X, i.e., a 1-function on the unit X-hypercube. In other words, we construct a nonnegative bounded numerical function K : [0, 1]⊗X → [0, 1], that for all z ∈ X ∑ x∈X⊆X ( ) K w̆(c|X//X) = wx . 106 THE XIV FAMEMS’2015 CONFERENCE A such simplest example of a 1-function on the unit X-hypercube is the so-called independent N -Kopula which for all free variables w̆ ∈ [0, 1]⊗X is defined by the formula: ∏ K (w̆) = wx , (8.1) x∈X that provides it on each 2(c|w̆) -phenomenon-dom the following 2N values: ( ) ∏ ∏ wx (1 − wx ) K w̆(c|X//X) = (8.2) x∈X x∈X−X for X ⊆ X. Indeed11 as in the case of doublet of events this function is a 1-function, since for all x∈X ) ( ∏ ∑ ∏ (1 − wz ) = wx . wz x∈X⊆X z∈X z∈X−X The e.p.d. of the 1st kind of ondependent N -s.e. X with the X-set of probabilities of events p̆ is defined by 2N values of the independent N -Kopula (8.1) on the 2(c|p̆) -phenomenon-dom by the general formulas of half-rare variables, i.e., for X ⊆ {x, y}: ( ) ∏ ∏ (1 − px ). (8.3) p(X//X) = K p̆(c|X//X) = px x∈X We write out more detailed formulas for corresponding pseudo-(N −1)-Kopulas: ( ) ( ) (∩{x0 }//{x0 }) ′ ′ //X ′ ) K({x0 }) p̆(c|X = K({x0 }) p̆(c|X //X ) = { p(X ′ //X ′ ), X ′ ̸= ∅, = ′ p(∅//X ) − 1 + p0 , X ′ = ∅, (8.6) ( The general recurrent formulas (3.52, 3.53) of the frame method for constructing a Kopula of a set of events are derived in Note 15. Recall these formulas: ( ) K p̆(c|X+Y //X +{x0 }) = ( )  (∩{x0 }//{x0 }) //X ′ ) ({x }) (c|X  , Y = {x0 }, (8.4) K 0 p̆ =  (∅)( (c|X (∩∅//{x0 }) //X ′′ ))  K p̆ , Y = ∅, ′′ ) ( ) ) = K(∅) p̆(c|X ′′ //X ′′ ) = and for conditional (N −1)-Kopulas: ( ) K|{x0 } p̆(c|X//X |{x0 }//{x0 }) = { 1 p(X ′ //X ′ ), X′ = ̸ ∅, = p10 ′ ′ (p(∅/ /X ) − 1 + p ), X = ∅, 0 p0 (8.7) K ( |∅ = p̆ { ) (c|X//X |∅//{x0 }) = 1 ′′ ′′ 1−p0 p(X //X ), 1 ′′ 1−p0 (p(∅//X ) − p0 ), X ′′ = ̸ ∅, ′′ X = ∅, where X ′ = X (∩{x0 }//{x0 }) = X(∩){x0 } = {x ∩ x0 , x ∈ X} и X ′′ = X (∩∅//{x0 }) = X(∩){xc0 } = {x ∩ xc0 , x ∈ X} для X ⊆ X . 8.3 Recurrent formulas for Fréchet-boundaries and Fréchet-restrictions Now let us consider recurrent formulas for Fréchet-boundaries и Fréchet-restrictions and for the 2N −N−1 functional parameters of an N -Kopula of N -set of events X = {x0 , x1 , . . . , xN−1 } = ) (c|X+Y //X +{x0 }) K p̆ = ( )  |{x0 } (c|X//X |{x0 }//{x0 })  K p̆ p0 , Y = {x0 }, (8.5)  = ( )  K|∅ p̆(c|X//X |∅//{x0 }) (1 − p ), Y = ∅, 0 (∩∅//{x0 }) //X K(∅) p̆(c|X { p(X ′′ //X ′′ ), X ′′ ̸= ∅, = ′′ p(∅//X ) − p0 , X ′′ = ∅, x∈X−X 8.2 The frame method for constructing a half-rare set of events ( monoplet {x0 }, or through two known conditional (N − 1)-Kopulas (see Definition 10) with respect to the frame monoplet {x0 } for the same X ′ and X ′′ . = {x0 } + {x1 , . . . , xN−1 } = = {x0 } + X = = {x0 } + (X ′ (+)X ′′ ), (8.8) where X ′ = {x0 ∩ x1 , . . . , x0 ∩ xN−1 }, X ′′ = {xc0 ∩ x1 , . . . , xc0 ∩ xN−1 } which express the N -Kopula of N -s.e. X = X + {x0 }, where X = X ′ (+)X ′′ , through the known probability p0 of the event x0 and together with it either two known inserted pseudo-(N − 1)-Kopulas (see Definition 9), i.e., pseudo-(N − 1)-Kopulas of inserted (N − 1)-s.e.’s X ′ and X ′′ in the frame are inserted (N −1)-s.e.’s, and 11 Perhaps this statement deserves to be called a lemma, which, incidentally, is not difficult to prove. is the X-set of probabilities of marginal events from X, i.e., pn = P(xn ), n = 0, 1, . . . , N −1. p̆(c|X//X) = {p0 , p1 , . . . , pN−1 } (8.9) (8.10) VOROBYEV 107 Judging by the form of Fréchet-boundaries и Fréchet-restrictions for a doublet, a triplet and a quadruplet of events, collected in paragraph 7.2, these Fréchet-restrictions consists of two groups, such that one of them, which refers to the parameters of the inserted (N − 1)-s.e. X ′ , consists of 2N−1−1 Fréchet-restrictions, and the other, which refers to the parameters of the inserted (N −1)-s.e. X ′′ , consists of 2N−1 −(N −1)−1 Fréchet-restrictions. And, as it should: and so on. Note 19 (recurrent formulas for Fréchetboundaries and Fréchet-restrictions). Probabilities of n-intersections (n = 2, ..., N−1) of events from the inserted s.e.’s X ′ and X ′′ have the recurrent Fréchetrestrictions (see paragraph 7.2) that are written by denotations from Note 18 by the following way with respect to X ′ : + ′ ′ p− X ′ //X ′ 6 pXn //X 6 pX ′ //X ′ , n N 2 −N − 1 = (2 N−1 −1)+(2 N−1 −(N −1)−1). (8.11) The first group, related to the inserted (N − 1)-s.e. X ′ , contains Fréchet-restrictions for probabilities of the second kind ( ) ∩ ′ x , pX ′ //X ′ = P (8.12) x′ ∈X ′ that are numbered by nonempty subsets X ′ ̸= ∅ of inserted (N−1)-s.e. X ′ (the number od such subsets: 2N−1 − 1); the second group, related to the inserted (N −1)-s.e. X ′′ , contains Fréchet-restrictions for the such probabilities of the second kind: ( ) ∩ x′′ , pX ′′ //X ′′ = P (8.13) x′′ ∈X ′′ ′′ ′′ that are numbered by subsets X ⊆ X with the power |X ′′ | > 2 (number of such subsets: 2N−1−(N − 1)−1). Note 18 (denotations for subsets of fixed power). To more conveniently represent the recurrent Fréchet-restrictions, agree to denote Xn′ ⊆ X ′ ⇐⇒ |Xn′ | = n, X ′′n ⊆ X ′′ ⇐⇒ |X ′′n | = n. (8.14) the subsets consisting of n events. In this notation, for example, the X–set of marginal probabilities p̆(c|X//X) is written as the X-set probabilities of the second kind that are numbered by monoplets of events X1 = {x}, x ∈ X: {p0 , p1 , . . . , pN−1 } = {pX1 //X , X1 ⊆ X}. (8.15) The set of probabilities of double intersection of events x ∈ X, i.e., the set of probabilities of the second kind that are numbered by doublets, has the form: {p{x,y}//X , {x, y} ⊆ X} = {pX2 //X , X2 ⊆ X}. (8.16) And the set of probabilities of triple intersections of events x ∈ X, i.e., the set of probabilities of the second kind that are numbered by triplets, has the form: {p{x,y,z}//X , {x, y, z} ⊆ X} = {pX3 //X , X3 ⊆ X} (8.17) n (8.18) where     ∑ ′ ′) , = max (p − p p− 0, p − ′ ′ x x X / /X Xn //X n−1   ′ ′ Xn (8.19) −1 ⊆Xn { } ′ ′ ′ ′ p+ X ′ //X ′ = ′min ′ pXn−1 //X , Xn−1 ⊆ X . Xn−1 ⊆Xn n are recurrent the lower and upper Fréchetboundaries. And the lower Fréchet-boundary we can write somewhat differently after simple transformations:     ∑ pXn′ −1 //X ′ −(n−1)px . (8.20) p− ′ / Xn /X ′ = max0,  ′ ′ Xn−1 ⊆Xn Similar look the recurrent Fréchet-restrictions with respect to the inserted s.e. X ′′ : + ′′ ′′ p− X ′′ //X ′′ 6 pX n //X 6 pX ′′ //X ′′ , n n (8.21) where     ∑ (1−px −pX ′′n−1 //X ′′ ) , p− X ′′n //X ′′ = max0, 1−px −  ′′ ′′ X n−1 ⊆X n p+ X ′′ //X ′′ = n min X ′′n−1 ⊆X ′′n (8.22) { } pX ′′n−1 //X ′′ , X ′′n−1 ⊆ X ′′ . are recurrent the lower and upper Fréchetboundaries. And the lower Fréchet-boundary we can write somewhat differently after simple transformations: p− X ′′n //X ′′ =     (8.23) ∑ = max 0, pX ′′n−1 //X ′′ −(n−1)(1−px ) .  ′′  ′′ X n−1 ⊆X n It remains to write out more N−1 recurrent Fréchetrestrictions on probabilities of marginal events from the inserted s.e. X ′ , i.e., on probabilities of the second kind that are numbered by monoplets X1′ ⊆ X ′ : 0 6 pX1′ //X ′ 6 pX1 //X , (8.24) 108 THE XIV FAMEMS’2015 CONFERENCE which are restricted by marginal probabilities of events from the (N − 1)-s.e. X and which together with the recurrent Fréchet-restrictions (8.18, 8.21) form the all totality of recurrent Fréchetrestrictions. This totality consists of 2N − N − 1 restrictions. And recurrent the lower and upper Fréchet-boundaries in these restrictions are defined by recurrent formulas (8.19, 8.22). 9 Parametrization of functional parameters of Kopula by Fréchet-correlations of inserted events The parametrization of functional parameter pt′ by the double Fréchet-correlation is similar: pt′ (px , py , pz ) = { px pz − Korxz Kov− xz , Korxz < 0, = px pz + Korxz Kov+ xz , Korxz > 0 { px pz + Korxz px pz , Korxz < 0, = px pz + Korxz (pz − px pz ), Korxz > 0. (9.7) 9.2 Inserted triple covariances and Fréchet-correlations Let’s consider parametrization on an example of functional parameters ps′ , pt′ , ps′ t′ и ps′′ t′′ of 3Kopula of the p̆-ordered half-rare triplet X = {x, y, z}, which in the frame method is constructed from two inserted pseudo-2-Kopulas. We recall first that an absolute triple Fréchetcorrelation [5] of three events x, y and z is defined similarly to the double one:  Kovxyz  , Kovxyz < 0,  |Kov− xyz | (9.8) Korxyz =  xyz  Kov+ , Kov > 0, xyz Kov xyz 9.1 Parametrization of functional parameters ps′ and pt′ The Fréchet-restriction of the functional parameter ps′ = ps′ (px , py , pz ) 0 6 ps′ 6 py , (9.1) that in the frame method has a sense of probability of double intersection of events x and y: ps′ = pxy//X = P(x ∩ y), (9.2) is baced on the notion of Fréchet-correlation [5]  Kovxy  , Kovxy < 0,  |Kov− xy | Korxy = (9.3)   Kovxy , Kov > 0, + xy Kov xy where Kovxy = P(x ∩ y) − P(x)P(y) (9.4) is a covariance of events x and y, and Kov− xy = max{0, px + py − 1} − px py = −px py , Kov+ xy = min{px , py } − px py = py − px py (9.5) are its the lower and upper Fréchet-boundaries. From Definition (9.3) we get the parametrization of functional parameter ps′ by the double Fréchetcorrelation on the following form: ps′ (px , py , pz ) = { px py − Korxy Kov− xy , Korxy < 0, = px py + Korxy Kov+ xy , Korxy > 0 { px py + Korxy px py , Korxy < 0, = px py + Korxy (py − px py ), Korxy > 0. (9.6) where Kovxyz = P(x ∩ y ∩ z) − P(x)P(y)P(z) (9.9) is the triple covariance of events x, y and z, and Kov− xyz = max{0, px + py + pz − 2} − px py pz = = −px py pz , Kov+ xyz = min{px , py , pz } − px py pz = (9.10) = pz − px py pz are its absolute the lower and upper Fréchetboundaries. The definition of the inserted triple Fréchetcorrelation differs of the definition of absolute one (9.8) in that its the lower and upper Fréchetboundaries must depend on the e.p.d. of the inserted doublets X ′ and X ′′ . So they differ from absolute Fréchet-boundaries (9.10) and have the form (9.19), where p− s′ t′ = max{0, ps′ + pt′ − px }, p+ s′ t′ = min{ps′ , pt′ }, p− s′′ t′′ = max{0, px + py + pz − 1 − ps′ − pt′ }, (9.11) p+ s′′ t′′ = min{py − ps′ , pz − pt′ }, are the lower and upper Fréchet-boundaries of probabilities of double intersections of events from inserted doublets X ′ and X ′′ with respect to the frame monoplet {x}, which should serve inserted the lower and upper Fréchet-boundaries of probabilities of triple intersections (9.20) of events from the triplet X = {x} + (X ′ (+)X ′′ ). However, as might be expected, these Fréchetboundaries are not always ready to serve as VOROBYEV 109 the lower and upper Fréchet-boundaries for probabilities of triple intersections. For this reason, it is necessary to modify the definitions of two inserted triple covariances and, respectively, — inserted the lower and upper Fréchet-boundaries of these covariances. The first modification of definitions (see Fig. ({x}) 22,23,24,25). For brevity, we denote p⋆ = (∅) = (1 − px )py pz . Two inserted triple px py pz , p⋆ covariance are defined by the firmulas:  [ ] ({x}) ({x}) +  , p⋆ ∈ p− ps′ t′ −p⋆ s′ t ′ , p s′ t ′ , ({x}) ({x}) − Kov({x}) , p⋆ < p− xyz = ps′ t′ −p⋆ s′ t′ ,   + ({x}) ({x}) + ps′ t′ −p⋆ , ps′ t′ < p⋆ , brevity of the formulas:  [ ] {x} {x} −  p⋆ , if p⋆ ∈ p+  s′ t ′ , p s′ t ′ ,     ({x}) p0 = −p{x} −p− (p − )(p+ ⋆ s′ t ′ s′ t ′ s′ t ′ )  , p− ′ t′ + ({x}) + −  s p +p −2p  ⋆ ′ ′ ′ ′  s t s t  if p{x} ̸∈ [p+ , p− ] , ⋆ s′ t′ s′ t′ (∅) p0 Two inserted triple covariances are defined by the formulas: ({x}) (9.12)  ] [ − (∅) (∅) +  ps′′ t′′ −p⋆ , p⋆ ∈ ps′ t′ , ps′ t′ , (∅) (∅) − − Kov(∅) xyz = ps′′ t′′ −p⋆ , p⋆ < ps′′ t′′ ,   + (∅) (∅) ps′′ t′′ −p⋆ , p+ s′′ t′′ < p⋆ , Kov({x}) xyz = ps′ t′ − p0 (∅) Kov(∅) xyz = ps′′ t′′ − p0 , (9.13)  ] [ ({x}) ({x}) + +  , p⋆ ∈ p− ps′ t′ −p⋆ s′ t ′ , p s′ t ′ , ({x}) ({x}) +({x}) Kovxyz = p− , p⋆ < p− s′ t′ −p⋆ s′ t′ ,   + ({x}) ({x}) + ps′ t′ −p⋆ , ps′ t′ < p⋆ , (9.16) and inserted the lower and upper Fréchetboundaries of these covariances — by the formulas: ({x}) , ({x}) , −({x}) Kovxyz = p− s′ t′ − p0 and inserted the lower and upper Fréchetboundaries of these covariances — by the formulas:  [ ] ({x}) ({x}) − +  , p⋆ ∈ p− ps′ t′ −p⋆ s′ t ′ , p s ′ t ′ , ({x}) ({x}) Kov−({x}) = p− , p⋆ < p− xyz s′ t′ −p⋆ s′ t ′ ,   + ({x}) ({x}) + ps′ t′ −p⋆ , ps′ t′ < p⋆ , (9.15)  [ ] −  p∅⋆ , если p∅⋆ ∈ p+  s′ t ′ , p s′ t ′ ,     + − = −p(∅) (p − ⋆ )(ps′ t′ −ps′ t′ ) s′′ t′′  p− , (∅) − +  s′′ t′′ + −2p p +p  ⋆  s′′ t′′ ] s′′ t′′ [  если p∅ ̸∈ p+ , p− , ⋆ s′ t ′ s′ t ′ Kov+({x}) = p+ xyz s′ t′ − p0 (9.17) −(∅) Kovxyz = p− s′′ t′′ − + Kov+(∅) xyz = ps′′ t′′ − (∅) p0 , (∅) p0 . ⋆⋆⋆ For any modification definitions of two inserted Fréchet-correlations look in the usual way:  Kov({x}) xyz  , Kov({x})  xyz < 0,  Kov−({x}) xyz ({x}) Korxyz = ({x})   xyz  Kov+({x}) , Kov({x}) > 0, xyz Kovxyz (9.18)  (∅) −  ps′′ t′′ −p⋆ , (∅) − Kov−(∅) xyz = ps′′ t′′ −p⋆ ,   + (∅) ps′′ t′′ −p⋆ ,  (∅) +  ps′′ t′′ −p⋆ , +(∅) − Kovxyz = ps′′ t′′ −p(∅) ⋆ ,   + (∅) ps′′ t′′ −p⋆ , ] [ (∅) + p⋆ ∈ p− s′ t′ , ps′ t′ , (∅) p⋆ < p− s′′ t′′ , (∅) + ps′′ t′′ < p⋆ , (9.14) ] (∅) + p⋆ ∈ p− s′ t ′ , p s′ t ′ , (∅) − p⋆ < ps′′ t′′ , (∅) p+ s′′ t′′ < p⋆ , [ The second modification of definitions (see Fig. 26,27,28,29). We introduce some notation for Kor(∅) xyz =     Kov(∅) xyz −(∅) Kovxyz (∅)   xyz  Kov+(∅) , Kovxyz , Kov(∅) xyz < 0, Kov(∅) xyz > 0. 9.3 Parametrization of functional parameters ps′ t′ and ps′′ t′′ The Fréchet-restriction of two functional parameters ps′ t′ = ps′ t′ (px , py , pz ) и ps′′ t′′ = ps′′ t′′ (px , py , pz ) + p− s′ t′ 6 ps′ t′ 6 ps′ t′ , + p− s′′ t′′ 6 ps′′ t′′ 6 ps′′ t′′ , (9.19) 110 THE XIV FAMEMS’2015 CONFERENCE that in the frame method have a sense of probabilities of triple intersections of events: ps′ t′ = P(x ∩ y ∩ z), ps′′ t′′ = P(xc ∩ y ∩ z), (9.20) is based on the notion of the inserted triple Fréchetcorrelation. From definitions (9.18) and (9.11) we get the parametrization of functional parameter ps′ t′ of the in the inserted triple Fréchet-correlation Kor({x}) xyz following form: ps′ t′ (px , py , pz ) =  ({x}) −({x})  px py pz − Korxyz Kovxyz ,  ({x})  Korxyz < 0, = +({x})   px py pz + Kor({x}) ,  xyz Kovxyz   ({x}) Korxyz > 0, (9.21) The parametrization of functional parameter ps′′ t′′ of the inserted triple Fréchet-correlation Kor(∅) xyz follows from the same definitions (9.18) and (9.11): ps′′ t′′ (px , py , pz ) =  (∅) px py pz − Korxyz Kov−(∅)  xyz ,    Kor({x}) xyz < 0, = +(∅)   px py pz + Kor(∅)  xyz Kovxyz ,   ({x}) Korxyz > 0, • the above parametrization algorithm for functional parameters of 3-Kopula extends to the parametrization of the functional parameters of N -Kopulas by inserted Fréchetcorrelations of higher orders. 10 Examples of Kopulas of some families of sets of events 10.1 Examples of different 2-Kopulas with a functional parameter within Frechet boundaries Consider in Fig.’s12 13, 14, 15, and 16 a number of examples of 2-Kopulas of doublets of half-rare events X = {x, y} and its set-phenomena X(c|x) = {x, y c }, X(c|y) = {xc , y}, and X(c|xy) = {xc , y c }, each of which is characterized by its own functional parameter P(x ∩ y) = pxy (wx , wy ), lying within the Fréchet boundaries: 0 6 pxy (wx , wy ) 6 min{wx , wy }. (10.1) Upper 2-Kopula of Fréchet (embedded): pxy (wx , wy ) = min{wx , wy }. (10.2) (9.22) Note 20 (about parametrization of functional parameters of 3-Kopula by Fréchet-correlations). The parametrization of the four functional parameters ps′ , pt′ , ps′ t′ and ps′′ t′′ of 3-Kopula of the p̆-ordered half-rare triplet X = {x, y, z} by two double Fréchet-correlations Korxy and Korxz (9.3) and by two inserted triple Fréchet-correlations Kor({x}) and Kor(∅) xyz xyz (9.18) has the following advantages. Each of four Fréchet-correlations is a numerical characteristics of dependency of events with values from fixed interval [−1, +1]. And these values clearly indicate the proximity to Fréchet-boundaries and to indepedent 3-Kopula. The value “−1” indicates to the lower Fréchetboundary, the value “+1” — to the upper Fréchetboundary, and the value “0” — to independent events. For example, the equality of all these four Fréchet-correlations to zero determines a family of independent 3-Kopulas. Advantages of the proposed idea of parametrization of functional parameters of 3-Kopula are that Independent 2-Kopula of Fréchet: pxy (wx , wy ) = wx wy . Lower 2-Kopula of Fréchet (minimum-intersected): pxy (wx , wy ) = max{0, wx + wy − 1}. (10.4) Half-independent 2-Kopula: pxy (wx , wy ) = wx wy /2. (10.5) Half-embedded 2-Kopula: pxy (wx , wy ) = min{wx , wy }/2. (10.6) Arbitrary-embedded 2-Kopula: pxy (wx , wy ) = = min{wx , wy }(1 + sin(15(wx − wy )))/2. (10.7) Continuously-arbitrary-embedded 2-Kopula: pxy (wx , wy ) = = wx wy + (α(wx , wy ) − wx wy )β(wx , wy ), • an each Fréchet-correlation can take arbitrary value from [−1, +1] without any connection with the values of the other three Fréchetcorrelations; (10.3) (10.8) 12 In each figure, below the graph, maps of these 2-Kopulas on unit squares in conditional colors are shown too, where the white color corresponds to the points at which the probabilities of terraced events are 1/4. VOROBYEV 111 where α(wx , wy ) = = min{wx , wy }(1 + sin(15(wx − wy )))/2, √ β(wx , wy ) = 4 (1/2 − wx )(1/2 − wy ). (10.9) Figure 14: Cartesian representations of 2-Kopulas of doublets of halfrare events X = {x, y} and its set-phenomena X(c|x) , X(c|y) и X(c|xy) (from left to right): half-independent (up), arbitrary-embedded (lower), and continuously-arbitrary-embedded (down). 2-Kopula of Frank, θ ∈ R − {0}: Figure 13: Cartesian representations of 2-Kopulas of doublets of halfrare events X = {x, y} and its set-phenomena X(c|x) , X(c|y) и X(c|xy) corresponding to Frechet Kopula (from up to down): upper (embedded), independent, lower (minimum-intersected). pxy (wx , wy ) = ] [ (exp(−θwx ) − 1)(exp(−θwy ) − 1) 1 = − log 1 + . θ exp(−θ) − 1 (10.13) 2-Kopula of Gumbel, θ ∈ [1, ∞): 10.2 Examples of 2-Kopulas with a functional parameter corresponding to some classical copulas In Fig.s 17, 18, 20, 21, and 19 it is shown 2-Kopulas of doublets of half-rare events X = {x, y} and its setphenomena X(c|x) , X(c|y) , and X(c|xy) , corresponding to some classical copulas. 2-Kopula of Ali-Mikhail-Haq, θ ∈ [−1, 1): pxy (wx , wy ) = wx wy = . 1 − θ(1 − wx )(1 − wy ) (10.10) pxy (wx , wy ) = [ ( )1/θ ] (10.14) . = exp − (− log(wx ))θ + (− log(wy ))θ 10.3 Examples of 3-Kopulas, functional parameters of which serve Fréchet-correlations of events In Fig.’s13 22, 23, 24, and 25 it is shown 3-Kopulas of triplets of half-rare events X = {x, y, z}, functional parameters of which serve Fréchet-correlations in the first modification of definitions; and in Fig.’s 26, 27, 28, and 29 — in the second modification of definitions (see paragraph 9.2). 2-Kopula of Clayton, θ ∈ [−1, ∞) − {0}: pxy (wx , wy ) = }]−1/θ [ { . = max wx−θ + wy−θ − 1; 0 11 Appendix (10.11) 2-Kopula of Joe, θ ∈ [1, ∞): pxy (wx , wy ) = [ ]1/θ (10.12) = 1− (1−wx )θ+(1−wy )θ−(1−wx )θ (1−wy )θ . 11.1 Abbreviations in the Kopula theory Consider the universal probability space (Ω, A℧ , P) and one of its subject-name realizations, a partial 13 Each figure shows maps of these 3-copulas on a cube in conditional colors, where the white color corresponds to the points at which the probabilities of terraced events are 1/8. 112 THE XIV FAMEMS’2015 CONFERENCE Figure 15: Cartesian representations of 2-Kopulas of doublets of halfrare events X = {x, y} and its set-phenomena X(c|x) , X(c|y) и X(c|xy) corresponding to Frechet Kopula (from up to down): from upper to independent, i.e., for nonnegative Fréchet-correlations θ = 1, 0.75, 0.50, 0.25, 0.10, 0.05, 0. Figure 16: Cartesian representations of 2-Kopulas of doublets of halfrare events X = {x, y} and its set-phenomena X(c|x) , X(c|y) и X(c|xy) corresponding to Frechet Kopula (from up to down): from independent to lower, i.e., for non-positive Fréchet-correlations θ = 0, −0.05, −0.10, −0.25, −0.50, −0.75, −1. probability space (Ω, A, P). The elements of the sigma-algebra A℧ are the universal Kolmogorov events x℧ ∈ A℧ , and the elements of the sigma- algebra A — events x ∈ A, which serve as names of universal Kolmogorov events x℧ (see in details [8]). VOROBYEV 113 Figure 17: Cartesian representations of 2-Kopulas of doublets of halfrare events X = {x, y} and its set-phenomena X(c|x) , X(c|y) и X(c|xy) corresponding to Ali-Mikhail-Haq Kopula (from up to down): from nearupper (θ = 0.999) through independent (θ = 0) to lower (θ = −1.0). The notions that are relevant to a s.e. X ⊆ A for which it is convenient to use the following abbreviations: X = {x : x ∈ X} — a set of events (s.e.); Figure 18: Cartesian representations of 2-Kopulas of doublets of halfrare events X = {x, y} and its set-phenomena X(c|x) , X(c|y) и X(c|xy) corresponding to Clayton Kopula (from up to down): from near-upper (θ = 6.5) around pinked-independent (θ = 0.1, −0.1) to lower (θ = −1.0). p̆ = {px , x ∈ X} — an X-set of probabilities of events from X; (c|X) X = X(c|X//X) = {x : x ∈ X} + {xc : x ∈ X − X} — an X-phenomenon of X, X ⊆ X; X(c|X) = X(c|X//X) = X — an X-phenomenon of X equal to X; p̆(c|X//X) = {px , x ∈ X} + {1−px , x ∈ X−X} — an X-set of probabilities of events from X(c|X) , X ⊆ X; p̆(c|X//X) = p̆ — an X-set of probabilities of events from X(c|X) equal to p̆; p(X//X) — a value of e.p.d. of the 1st kind of X for X ⊆ X; ( K p̆(c|X//X) ) — a value of the Kopula of e.p.d. of the 1st kind of X for X ⊆ X; ) ( p(X//X) = K p̆(c|X//X) Figure 19: Cartesian representations of 2-Kopulas of doublets of halfrare events X = {x, y} and its set-phenomena X(c|x) , X(c|y) и X(c|xy) corresponding to Joe Kopula (from up to down): from independent (θ = 1.0) to near-lower (θ = 6.5). — the definition of e.p.d. of the 1st kind of X by its Kopula, X ⊆ X; s̆ = {sx , x ∈ X} ∈ [0, 1/2]⊗X — an X-set of half-rare variables; w̆ = {wx , x ∈ X} ∈ [0, 1]⊗X — an X-set of free variables. 11.2 Set-phenomenon renumbering a e.p.d. of the 1st kind and its Kopulas Lemma 5 (Set-phenomenon renumbering a e.p.d. of the 1st kind and its Kopulas). E.p.d. of the 1st kind and 114 THE XIV FAMEMS’2015 CONFERENCE Kor = −1 Kor = −0.5 Kor = −0.3 Kor = −0.1 Kor = 0 Kor = 0.1 Figure 20: Cartesian representations of 2-Kopulas of doublets of halfrare events X = {x, y} and its set-phenomena X(c|x) , X(c|y) и X(c|xy) corresponding to Frank Kopula (from up to down): from near-upper (θ = 6.5) around pinked-independent (θ = 0.1, −0.1) to near-lower (θ = −6.5). Kor = 0.3 Kor = 0.5 Kor = 1 Figure 22: The first modification of definitions. Cartesian representations of 3-Kopulas of triplets of half-rare events X = {x, y, z}, constructed by the frame method (6.18) with non-negative values of a single parameter (from up to down) Kor = −1, −0.5, −0.3, −0.1, 0, 0.1, 0.3, 0.5, 1, to which all four inserted Frechet-correlations are equal (see paragraph 9). The independent 3-Kopula is obtained for Kor = 0. Figure 21: Cartesian representations of 2-Kopulas of doublets of halfrare events X = {x, y} and its set-phenomena X(c|x) , X(c|y) и X(c|xy) corresponding to Gumbel Kopula (from up to down): from inndependent (θ = 1.0) to near-lower (θ = 6.5). Kopulas of the s.e. X and of its S-phenomena X(c|S) are connected by formulas of mutually inversion set-phenomenon renumbering for X ⊆ X и S ⊆ X: ( ) p X (c|S∩X) //X(c|S) = p((S∆X)c //X), ( ) (11.1) (c|S∩(S∆X)c ) p(X//X) = p ((S∆X)c ) //X(c|S) ; ( ) ( ) (c|S∩X) c //X(c|S) ) K p̆(c|X = K p̆(c|(S∆X) //X) , (11.2) ( ) ( ) c c|((S∆X)c )(c|S∩(S∆X) ) //X(c|S) ) (c|X//X) ( K p̆ = K p̆ . Proof follows immediately from the formulas of the set-phenomenon renumbering the terraced VOROBYEV 115 Kor = −1 Kor = 0 Kor = −0.9 Kor = 0.1 Kor = −0.8 Kor = 0.2 Kor = −0.7 Kor = 0.3 Kor = −0.6 Kor = 0.4 Kor = −0.5 Kor = 0.5 Kor = −0.4 Kor = 0.6 Kor = −0.3 Kor = 0.7 Kor = −0.2 Kor = 0.8 Kor = −0.1 Kor = 0.9 Kor = 0 Figure 23: The first modification of definitions. Cartesian representations of 3-Kopulas of triplets of half-rare events X = {x, y, z}, constructed by the frame method (6.18) with non-positive values of the parameter Kor = −1, −0.9, ..., −0.2, −0.1, 0 (from up to down), to which inserted Frechet-correlations are equal (see paragraph 9). The independent 3Kopula is obtained for Kor = 0. events of the 1st kind of the s.e. and of its setphenomena proved in [9]. Kor = 1 Figure 24: The first modification of definitions. Cartesian representations of 3-Kopulas of triplets of half-rare events X = {x, y, z}, constructed by the frame method (6.18) with non-negative values of the parameter Kor = 0, 0.1, 0.2, ..., 0.9, 1 (from up to down), to which inserted Frechetcorrelations are equal (see paragraph 9). The independent 3-Kopula is obtained for Kor = 0. 116 THE XIV FAMEMS’2015 CONFERENCE Kor = −1 Kor = 1, py = 0.0125, ...(0.0125)..., 0.125 Kor = −0.5 Kor = −0.3 Kor = 1, py = 0.125, ...(0.0125)..., 0.25 Kor = −0.1 Kor = 0 Kor = 1, py = 0.25, ...(0.0125)..., 0.375 Kor = 0.1 Kor = 0.3 Kor = 1, py = 0.375, ...(0.0125)..., 0.5 Figure 25: The first modification of definitions. Cartesian representations of 3-Kopulas of triplets of half-rare events X = {x, y, z}, constructed by the frame method (6.18) with the value of the parameter Kor = 1 and marginal probability py = 0.0125, ...(0.0125)..., 0.5 (from left to right, from up to down). 11.3 Useful denotations for a doublet of events that are invariant relative to the p̆-order The following special denotations for a doublet of events X = {x, y} and the X-set of marginal probabilities p̆ = {px , py } that are invariant relative to the p̆-order, are useful. Kor = 0.5 Kor = 1 Figure 26: The second modification of definitions. Cartesian representations of 3-Kopulas of triplets of half-rare events X = {x, y, z}, constructed by the frame method (6.18) with non-positive values of the single parameter Kor = −1, −0.5, −0.3, −0.1, 0, 0.1, 0.3, 0.5, 1 (from up to down), to which all four inserted Frechet-correlations are equal (see paragraph 9). The independent 3-Kopula is obtained for Kor = 0. X = {x, y} = {x↑ , x↓ }, p̆ = {px , py } = {p↑ , p↓ }, 1/2 > p↑ > p↓ , ↑ ↓ ↑ (11.3) ↓ w̆ = {wx , wy } = {w , w }, 1 > w > w , { {x}, px > py , ↑ {x } = max{X} = {y}, иначе; { {x}, px 6 py , {x↓ } = min{X} = {y}, иначе; { wx , w x > w y , w = max{w̆} = wy , иначе; { w x , w x 6 wy , w↓ = min{w̆} = wy , иначе; ↑ (11.4) (11.5) VOROBYEV 117 Kor = −1 Kor = 0 Kor = −0.9 Kor = 0.1 Kor = −0.8 Kor = 0.2 Kor = −0.7 Kor = 0.3 Kor = −0.6 Kor = 0.4 Kor = −0.5 Kor = 0.5 Kor = −0.4 Kor = 0.6 Kor = −0.3 Kor = 0.7 Kor = −0.2 Kor = 0.8 Kor = −0.1 Kor = 0.9 Kor = 0 Figure 27: The second modification of definitions. Cartesian representations of 3-Kopulas of triplets of half-rare events X = {x, y, z}, constructed by the frame method (6.18) with non-positive values of the single parameter Kor = −1, −0.9, ..., −0.2, −0.1, 0 (from up to down), to which inserted Frechet-correlations are equal (see paragraph 9). The independent 3-Kopula is obtained for Kor = 0. { px , px > py , p = max{p̆} = py , иначе; { } = max min{wx , 1 − wx }, min{wy , 1 − wy } , Kor = 1 Figure 28: The second modification of definitions. Cartesian representations of 3-Kopulas of triplets of half-rare events X = {x, y, z}, constructed by the frame method (6.18) with non-negative values of the parameter Kor = 0, 0.1, 0.2, ..., 0.9, 1 (from up to down), to which inserted Frechet-correlations are equal (see paragraph 9). The independent 3-Kopula is obtained for Kor = 0. ↑ In such invariant denotations, it is not difficult to write down the general recurrence formula for 118 THE XIV FAMEMS’2015 CONFERENCE Kor = 1, py = 0.0125, ...(0.0125)..., 0.125 Kor = 1, py = 0.125, ...(0.0125)..., 0.25 probabilities has the form: ( ) K p̆(c|X//X) = K(c|X//X) (w̆) , ( ) (11.8) K(c|X//X) (p̆) = K w̆(c|X//X) ; ( ) ( ) c c K(c|S//X) p̆(c|X//X) = K(c|X//X) p̆(c|(S ∆X) //X) ( ) (11.9) c c = K p̆(c|(S ∆X) //X) . For example, for X ⊆ X = {x, y} ( ) ( ) K(c|X//X) p̆(c|X//X) = K p̆(c|X//X) , ( ) ( ) c K(c|{x}//X) p̆(c|X//X) = K p̆(c|({y}∆X) //X) , ( ) ( ) (11.10) c K(c|{y}//X) p̆(c|X//X) = K p̆(c|({x}∆X) //X) , ( ) ( ) c K(c|∅//X) p̆(c|X//X) = K p̆(c|X //X) . 11.4 Recurrent properties of the p̆-ordering a half-rare s.e. 11.4.1 Recurrent properties of the p̆-ordering a half-rare doublets of events Kor = 1, py = 0.25, ...(0.0125)..., 0.375 Kor = 1, py = 0.375, ...(0.0125)..., 0.5 Figure 29: The second modification of definitions. Cartesian representations of 3-Kopulas of triplets of half-rare events X = {x, y, z}, constructed by the frame method (6.18) with the value of the parameter Kor = 1 and marginal probability py = 0.0125, ...(0.0125)..., 0.5 (from left to right, from up to down). the half-rare 2-Kopula of the doublet X, united combining both orders: ( ) ( ) K p̆(c|X//X) = K(c|X//X) p̆(c|X//X) =  ′ X = X,  xy (p̆)) , K (p  K′′ (p↓ − p (p̆)) , X = {x↓ }, (11.7) xy =  K′ (1 − pxy (p̆)) − 1 + p↑ , X = {x↑ },   )  ′′ ( K 1 − p↓ + pxy (p̆) − p↑ , X = ∅ where by Definition (11.3) } min{wx , 1 − wx }, min{wy , 1 − wy } , { } p↓ = min min{wx , 1 − wx }, min{wy , 1 − wy } . p↑ = max { The mutual set-phenomenon inversion of 2Kopulas of half-rare p̆ and free w̆ marginal Let us explain the role of p̆-ordering in the frame method using the example of constructing a 2Kopula of the p̆-ordered half-rare events X = {x, y} with X-set of marginal probabilities of events p̆ = {px , py }, that is, 1/2 > px > py :  X = {x, y},  pxy , p −p , ( )  X = {x}, x xy K′ p̆(c|X//{x,y}) =  p −p , X = {y}, y xy    1−px −py +pxy , X = ∅, (11.11) where, when selected as a function parameter pxy of the 1-Kopulas of inserted half-rare monoplates X ′ = {s′ } = {x ∩ y} and X ′′ = {s′′ } = {xc ∩ y}, are equal, respectively:  p s′ t ′ , S = {s′ , t′ },    ( ) p ′ −p ′ ′ , ′ S = {s′ }, s st K′ p̆(c|S//X ) =  ′ ′ ′ S = {t′ },  pt −ps t ,  1−ps′ −pt′ +ps′ t′ , S = ∅, ( ) ′′ K′′ p̆(c|S//X ) =  ps′′ t′′ ,    p − p ′ −p ′′ ′′ , y s s t =  ′ p − p −p t s′′ t′′ ,   z  1−py −pz +ps′ +pt′ +ps′′ t′′ , (11.12) S = {s′′ , t′′ }, S = {s′′ }, S = {t′′ }, S = ∅, under the assumption that inserted half-rare monoplets have “equally direct” p̆-orders: py > ps′ > pt′ , pz > ps′′ > pt′′ . (11.13) VOROBYEV 119 However, nothing prevents the emergence of two more “opposite p̆ orders” on the inserted half-rare monoplets: under the assumption that the inserted half-rare doublets have “equally direct” p̆-orders: py > pt′ > ps′ , pz > ps′′ > pt′′ . (11.14) py > ps′ > pt′ , pz > ps′′ > pt′′ . py > ps′ > pt′ , pz > pt′′ > ps′′ ; (11.15) However, nothing prevents the emergence of two more “opposite p̆ orders” on the inserted half-rare doublets: py > pt′ > ps′ , except for the “equally inverse” p̆-order py > pt′ > ps′ , pz > pt′′ > ps′′ , pz > ps′′ > pt′′ . (11.16) which can not be due to the consistency of the functional parameters, i.e., because py = ps′ + ps′′ > pt′ + pt′′ = pz . py > ps′ > pt′ , pz > pt′′ > ps′′ ; py > pt′ > ps′ , pz > pt′′ > ps′′ , Let us explain the role of p̆-ordering in the frame method using the example of constructing a 3Kopula of the p̆-ordered half-rare events X = {x, y, z} with X-set of marginal probabilities of events p̆ = {px , py , pz }, that is, 1/2 > px > py > pz : ( ) K p̆(c|X//{x,y,z}) =  ′  K (ps′ , pt′ ) , X = {x, y, z},    ′  ′ ′ K (ps , 1−pt ) , X = {x, y},     K′ (1−ps′ , pt′ ) , X = {x, z},    K′ (1−p ′ , 1−p ′ )−1+p , X = {x}, (11.18) s t x = ′ ′  X = {y, z}, K (py −ps′ , pz −pt′ ) ,    ′′   K (py −ps′ , 1−pz +pt′ ) , X = {y},    ′′  ′ ′ X = {z}, K (1−py +ps , pz −pt ) ,    ′′ K (1−py +ps′ , 1−pz +pt′ )−px , X = ∅, where, when selected as function parameters ps′ , pt′ , ps′ t′ and ps′′ t′′ and despite the fact that ps′′ = py − ps′ , pt′′ = pz − pt′ , the 2-Kopulas of inserted half-rare doublets X ′ = {s′ , t′ } = {x ∩ y, x ∩ z} and X ′′ = {s′′ , t′′ } = {xc ∩y, z c ∩z} are equal, respectively:  ps′ t′ , S = {s′ , t′ },     ( ) ′ ps′ −ps′ t′ , S = {s′ }, K′ p̆(c|S//X ) =  pt′ −ps′ t′ , S = {t′ },    1−ps′ −pt′ +ps′ t′ , S = ∅, ) ( (11.19) ′′ K′′ p̆(c|S//X ) =  ps′′ t′′ , S = {s′′ , t′′ },    p − p ′ −p ′′ ′′ , S = {s′′ }, y s s t =  pz − pt′ −ps′′ t′′ , S = {t′′ },    1−py −pz +ps′ +pt′ +ps′′ t′′ , S = ∅, (11.21) (11.22) except for the “equally inverse” p̆-order (11.17) 11.4.2 Recurrent properties of p̆-ordering the half-rare triplets of events (11.20) (11.23) which can not be due to the consistency of the functional parameters, i.e., because py = ps′ + ps′′ > pt′ + pt′′ = pz . (11.24) 11.4.3 Extending the frame method to p̆-non-ordered half-rare s.e.’s Above we outlined the frame method for constructing N -Kopulas of p̆-ordered half-rare ✿✿✿✿✿✿✿ N -s.e.’s. It remains to extend it to construct N Kopulas of p̆-disordered half-rare N -s.e.’s using ✿✿✿✿✿✿✿✿✿✿ the following technique, based on the obvious invariance property of permutations of events in s.e.: “as events from some s.e. do not order, the s.e. will not change "; and very useful in practical calculations. We denote by X∗ = {x∗0 , x∗1 , ..., x∗N −1 }, (11.25) — the p̆-ordered half-rare N -s.e., which consists ✿✿✿✿✿✿✿ from the same events, that an “arbitrary” p̆non-ordered half-rare N -s.e. ✿✿✿✿✿✿✿✿✿✿✿✿ X = {x0 , x1 , ..., xN −1 }, (11.26) i.e., X∗ = {x∗0 , x∗1 , ..., x∗N−1 } = {x0 , x1 , ..., xN−1 } = X, (11.27) but arranged in descending order of their probabilities. In other words, the X∗ -set of marginal probabilities p̆∗ = {p∗0 , p∗1 , ..., p∗N −1 }, (11.28) is such that 1/2 > p∗0 > p∗1 > ... > p∗N −1 (11.29) 120 THE XIV FAMEMS’2015 CONFERENCE where in the notation just introduced, assuming that we have available Kopulas of the p̆-ordered N -s.e’s p∗0 = max{px : x ∈ X}, p∗1 = max{px : x ∈ X − {x∗0 }}, ... p∗n+1 = max{px : x ∈ X − {x∗1 , ..., x∗n }}, ... X∗ = {x∗0 , x∗1 , ..., x∗N −1 } = X (11.30) for N = 1, 2. p∗N = max{px : x ∈ X − {x∗1 , ..., x∗N −1 }}. Example 2 ∗ ∗ Consequently, X -set of marginal probabilities p̆ which consists of the same probabilities that X-set of marginal probabilities p̆, i.e., (invariant formula for the 2-Kopula of a Let X = {x0 , x1 } be the p̆-non-ordered half-rare doublet of events. Then its 2-Kopula is calculated at each point p̆(c|X//{x0 ,x1 }) ∈ [0, 1]⊗X by the following formulas: half-rare doublet of events). p̆∗ = {p∗0 , p∗1 , ..., p∗N −1 } = {p0 , p1 , ..., pN −1 } = p̆, (11.31) but arranged in descending order. Now, to construct the N -Kopulas of the p̆disordered N -s.e X by the frame method it is ✿✿✿✿✿✿✿✿✿✿ sufficient to construct this N -Kopula of the p̆ordered N -s.e X∗ = X by this method, reasoning by ✿✿✿✿✿✿✿✿ (11.27) and (11.31) reasoning that Теперь для построения рамочным методом N -Копулы p̆-✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ неупорядоченного N -s.e. X достаточно построить этим методом N -Копулу p̆упорядоченного N -s.e. X∗ = X, reasoning by ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ virtue of (11.27) and (11.31), that K (p̆) = K (p̆∗ ) , (11.32) i.e., for X ⊆ X ( ) ( ) ∗ ∗ K p̆(c|X//X) = K p̆∗(c|X //X ) (11.33) X ∗ = {x∗ : x ∈ X} ⊆ X∗ (11.34) where are subsets of the p̆-ordered (N −1)-s.e. X∗ . ✿✿✿✿✿✿✿ Note 21 (properties of functions of an unordered set Equations (11.32) and (11.33) should not be regarded as a unique property of the Kopula invariance with respect to permutations of its arguments. This property is possessed by any Kopula, since it is a function of an unordered set of arguments. Therefore it is quite natural that the Kopula is invariant under permutations of the arguments, like any other such function. This property must be remembered only in practical calculations, when we volence-nolens must introduce an arbitrary order on a disordered set in order to be able to perform calculations. of arguments). Consider the examples of Kopulas of arbitrary, i.e., p̆-disordered, s.e.’s X = {x0 , x1 , ..., xN −1 } ( ) ( ) ∗ ∗ ∗ K p̆(c|X//{x0 ,x1 }) = K p̆(c|X //{x0 ,x1 }) =  ′ K (ps′ ) , X ∗ = {x∗0 , x∗1 },    K′′ (p − p ′ ) , (11.35) X ∗ = {x∗1 }, 1 s = ′  K (1 − ps′ ) − 1 + p0 , X ∗ = {x∗0 },    ′′ K (1 − p1 + ps′ ) − p0 , X ∗ = ∅, where X ′ = {s′ } = {x∗0 ∩ x∗1 }, X ′′ = {s′′ } = {(x∗0 )c ∩ x∗1 }. Example 3 (11.36) (invariant formula for the 3-Kopula of a Let X = {x0 , x1 , x2 } be the p̆-non-ordered half-rare triplet of events. Then its 3-Kopula is calculated at each point p̆(c|X//{x0 ,x1 ,x2 }) ∈ [0, 1]⊗X by the following formulas: half-rare triplet of events). ( ) ( ) ∗ ∗ ∗ ∗ K p̆(c|X//{x0 ,x1 ,x2 }) = K p̆(c|X //{x0 ,x1 ,x2 }) =  ′  K (ps′ , pt′ ) , X ∗ ={x∗0 , x∗1 , x∗2 },    ′  K (ps′ , 1−pt′ ) , X ∗ ={x∗0 , x∗1 },     ′  X ∗ = {x∗0 , x∗2 },  K (1−ps′ , pt′ ) ,  K′ (1−p ′ , 1−p ′ )−1+p , (11.37) X ∗ ={x∗0 }, s t 0 =  K′′ (p1 −ps′ , p2 −pt′ ) , X ∗ ={x∗1 , x∗2 },    ′′   X ∗ ={x∗2 }, K (1−p1 +ps′ , p2 −pt′ ) ,    ′′  X ∗ ={x∗1 },  K (p1 −ps′ , 1−p2 +pt′ ) ,   ′′ K (1−p1 +ps′ , 1−p2 +pt′ )−p0 , X ∗ = ∅, where when selecting as function parameters ps′ , pt′ , ps′ t′ и ps′′ t′′ and despite the fact that ps′′ = p1 − ps′ , pt′′ = p2 − pt′ , 2-Kopulas of the inserted half-rare doublets X ′ = {s′ , t′ } = {x∗0 ∩ x∗1 , x∗0 ∩ x∗2 } and X ′′ = {s′′ , t′′ } = {(x∗0 )c ∩ x∗1 , (x∗0 )c ∩ x∗2 } are equal VOROBYEV 121 respectively:   ps′ t′ , p ′ −p ′ ′ , ( )  ′ s st K′ p̆(c|S//X ) =  ′ ′ t′ , p −p t s    1−ps′ −pt′ +ps′ t′ , S = {s′ , t′ }, S = {s′ }, S = {t′ }, S = ∅, ) ( (11.38) ′′ K′′ p̆(c|S//X ) =  ps′′ t′′ , S = {s′′ , t′′ },    p − p ′ −p ′′ ′′ , S = {s′′ }, 1 s s t =  p2 − pt′ −ps′′ t′′ , S = {t′′ },    1−p1 −p2 +ps′ +pt′ +ps′′ t′′ , S = ∅. 11.5 Geometric interpretation of set-phenomenon renumberings For a subset of events V ⊆ X V -phenomenon renumbering of the terrace events, generated by (N − 1)-s.h-r.e. X, is based on the replacement of events from the subset V c = X − V by their complements: X(c|V ) = V + (V c )(c) = = {x, x ∈ V } + {xc , x ∈ V c }, (11.39) {x, y c } {y c } {y c } X(c|x) {xc , y c } X(c|∅) {x} ∅ ∅ {xc } {x} ∅ ∅ {xc } X(c|xy) {x, y} X(c|y) {y} {y} {xc , y} Figure 30: Geometric interpretation of a set-phenomenon renumbering the terraced events, generated by the doublet of half-rare events X = {x, y} = {x, y}(c|xy) , by reflections with respect to straight lines orthogonal to the coordinate axes and intersecting them at the points 1/2. In the form of unit squares (with the origin in the bottom left corner of each square), the four Venn diagrams of doublets of half-rare events X = X(c|xy) and its set-phenomena X(c|x) = {x, y c }, X(c|y) = {xc , y} and X(c|∅) = {xc , y c } are shown; the terrace events are marked with subsets of the doublet of half-rare events X or its set-phenomena, consisting of both half-rare events and its complements; on each diagram pairs of events, from which the doublet of half-rare events or its set-phenomena consists, are shaded (aqua). from which the mutually inverse set-phenomenon renumbering formulas follow: Look at the X-set (11.41) as a 2X -hyper-point from a half-rare 2N -vertex simplex ( ) ter X (c|V ∩X)//X(c|V ) = ter((V∆X)c//X), ( ) (11.40) (c|V ∩(V∆X)c ) ter(X//X) = ter ((V∆X)c ) //X(c|V ) S2 X =   (11.42)  ∑ = {p(X//X), X ⊆ X} : p(X//X) > 0, p(X//X)=1 ,   for V ⊆ X and X ⊆ X. Therefore, the V -phenomenon renumbering of the terrace events, generated by (N − 1)-s.h-r.e. X, by the formulas (11.40) is geometrically interpreted on the (N − 1)-dimensional Venn diagram of this s.e. as a reflection of the X-hypercube relative to those hyperplanes that are orthogonal to the x-axes numbered by the events x ∈ V c ⊆ X (see Fig. 30 for the doublet of events). 11.6 Projection of the 2X -simplex on the X-hypercube Take an arbitrary (N − 1)-s.e. X ⊆ A with e.p.d. of the 1st kind, which, as is known [5], is defined as the 2X -set of probabilities of terrace events of the 1st kind {p(X//X), X ⊆ X}. (11.41) X⊆X to each vertex of which the degenerate e.p.d. corresponds. In this e.p.d., as is known, only one of the 1st kind of probability, equal to one, is different from zero. Number the vertex 2X of the simplex S2X by the subset X ⊆ X. The degenerate e.p.d. of the 1st kind with p(X//X) = 1 corresponds to this vertex. And associate the vertex f rakX with the hypercube [0, 1] otimes f rakX , numbered by the f rakX -set: каждой вершине которого, соответствует вырожденное e.p.d., у которого, как известно, лишь одна вероятность of the 1st kind, равная единице, отлична от нуля. Занумеруйте подмножеством X ⊆ X вершину 2X -симплекса S2X , которой соответствует вырожденное e.p.d. of the 1st kind с p(X//X) = 1. And associate with it the vertex of X-hypercube [0, 1]⊗X , numbering by the following X-set: {ΥX//X (x), x ∈ X} (11.43) 122 THE XIV FAMEMS’2015 CONFERENCE where ΥX//X (x) = { 1, x ∈ X, 0, иначе, (11.44) are values of the indicator of subset X ⊆ X on events x ∈ X. Define a prohection pr : S2X → [0, 1]⊗X of the 2X -simplex S2X on X-hypercube [0, 1]⊗X by the following formula: pr({p(X//X), X ⊆ X}) = {px , x ∈ X} (11.45) where px = ∑ p(X//X)ΥX//X (x) = X⊆X = ∑ and the X-vertex of the 2X -simplex, i.e., the vertex enumerated by the subset X0 = X is projected into the X-set {1, ..., 1}, consisting of the unit probabilities of marginal events, in other words, projected into the X-vertex of the X-hypercube, i.e., to the vertex opposite to the origin: { 1, X = X, (11.48) {1, ..., 1} ∼ p(X//X) = 0, X ̸= X ⊆ X. In general, due to the linearity of the projection (11.45), the set of such points of the 2X -simplex that project into the same point of the X-hypercube is convex and forms a sub-simplex of smaller dimension. 11.7 Half-rare events on Venn (N −1)-diagram (11.46) p(X//X) x∈X⊆X is a convex combination of hypercube vertices, which, as known [5], is intepreted as the probability of event x ∈ X. With projection (11.45) vertices of the 2X -simplex maps to vertices of the X-hypercube, and edges map to its edges or diagonals (see [6], [7] and Fig. 32). Example 4 (projections of vertices of a 2X -simplex). For example, the vertex of the 2X -simplex enumerated by the subset X0 ⊆ X corresponds to the degenerate e.p.d. of the 1st kind with probabilities { 1, X = X0 , p(X//X) = 0, X0 ̸= X ⊆ X. From (11.46) you obtain that { 1, x ∈ X0 , px = ΥX0 //X (x) = 0, x ∈ X − X0 . Therefore, by (11.43) {px , x ∈ X} = {ΥX0 //X (x), x ∈ X} We will figure out how a Venn (N − 1)-diagram of an arbitrary (N −1)-s.e. is constructed on the basis of the projection (11.45), in which the role of the space of universal elementary events Ω is played by the unit (N − 1)–dimensional hypercube. Such a Venn (N − 1)-diagram puts terraced hypercubes generated by dividing a unit hypercube in half orthogonal to each of the N axes into a oneto-one correspondence with the terraced events generated by the given (N −1)-s.e. Take first (N − 1)-s.h-r.e. X and represent its Venn (N − 1)-diagram14 On which Ω is represented by a unit (N − 1)-dimensional hypercube that serves as ordered15 image of the X-hypercube ⊗ [0, 1]x , [0, 1]⊗X = (11.49) x∈X broken by hyperplanes orthogonal to x-axis and intersecting them at points 1/2 into 2N X-terraced hypercubes for X ⊆ X ⊗ ⊗ [0, 1/2]x (1/2, 1]x , (11.50) [0, 1]⊗ ter(X//X) = x∈X x∈X−X where each marginal half-rare event x ∈ X is represented as a x-half of X-hypercube containing the origin: is a vertex of the X-hypercube. [0, 1/2]x ⊗ [0, 1]⊗(X−{x}) , (11.51) X In particular, the ∅-vertex of the 2 -simplex, i.e., the vertex numbered by the subset X0 = ∅ is projected into the X-set {0, ..., 0}, consisting of the zero probabilities of marginal events, in other words, projected into the ∅-vertex of the X-hypercube, i.e., to the vertex located at the beginning coordinates: { 1, X = ∅, {0, ..., 0} ∼ p(X//X) = (11.47) 0, ∅ ̸= X ⊆ X; its complement xc = Ω − x is represented in the form of another x-half of X-hypercube that does not contain the origin: (1/2, 1]x ⊗ [0, 1]⊗(X−{x}) , 14 see (11.52) the Venn 2-diagram doublet of half-rare events in Fig. 32. role of the order of events in s.e. when working with their images in RN is discussed in [?]. 15 The VOROBYEV 123 and the X-terraced event ter(X//X) — as a Xterraced hypercube (11.50): ter(X//X) ∼ [0, 1]⊗ ter(X//X) . in particular, for the ∅-phenomenon X(c|∅) = X(c) and for ∅-vertex and X-vertex we have: (11.53) The formula (11.53) once again points to a oneto-one correspondence between the 2N -space of terraced hypercubes (11.50) from the Venn (N − 1)-diagram of (N − 1)-s.h-r.e. X and 2N -totality of terraced events, generated by X. {0, x ∈ X} = {0, ..., 0} ∈ [0, 1]⊗ter(∅//X (correspondence between the numbering of terraced hypercubes and vertices of 2X -simplex of e.p.d.’s of the 1st kind of (N − ) , ⊗ter(X(c) //X(c) ) {1, x ∈ X} = {1, ..., 1} ∈ [0, 1] (11.57) . {y} {x, y} p x > 1 − py If the correspondence between the terraced hypercubes and the terraced events looks natural, then for the sets of half-rare events X the correspondence between the terraced hypercubes and the numbering of the vertices of the 2X simplex projected into the corresponding vertices of the X-hypercube under the projection (11.45) is defined by the operation of the complement and requires a special Note 22 (c) ter(x//xy) 1 − p x > 1 − py ter(∅//xy) ter(y//yx) ter(∅//yx) 1 − py > p x 1 − p y > 1 − px py > p x py > 1 − p x ter(yx//yx) ter(x//yx) 1)-s.h-r.e. X on its On the Venn (N −1)-diagram of (N − 1)-set of half-rare events X, every X c -terraced c hypercube [0, 1]⊗ ter(X //X) contains the X-vertex of X-hypercube, into which corresponding X-vertex of 2X -simplex S2X of e.p.d.’s of the 1st kind of (N −1)-s.h-r.e. X for X ⊆ X is projected: Venn (N − 1)-diagram). {1, x ∈ X}+{0, x ∈ X − X} ∈ [0, 1]⊗ter(X c //X) ,(11.54) in particular, for ∅-vertex and X-vertex we have: {0, x ∈ X} = {0, ..., 0} ∈ [0, 1]⊗ter(X//X) , {1, x ∈ X} = {1, ..., 1} ∈ [0, 1]⊗ter(∅//X) . (11.55) Нетрудно догадаться, что (N−1)-диаграмма Венна произвольного сет-феномена м.пр.с́. отличается от (N −1)-диаграммы Венна самого X лишь перенумерацией террасных events по формулам из [9]. Сделаем It is not difficult to guess that the Venn (N − 1)diagram of an arbitrary set-phenomenon of s.h-r.e. differs from the Venn (N−1)-diagram of X itself only by renumbering terraced events using formulas from [9]. Let’s do Note 23 (Venn (N − 1)-diagram of set-phenomena of On the Venn (N − 1)diagram of the V -phenomenon X(c|V ) of (N − 1)set of half-rare events X, every V∆X-terraced hypercube [0, 1]⊗ ter(V∆X//X) contains the X-vertex of X-hypercube, into which the corresponding X-vertex of 2X -simplex S2X of e.p.d. of the 1st kind of (N − 1)-s.h-r.e. X for X ⊆ X and V ⊆ X is projected: a set of half-rare events). {1, x ∈ X}+{0, x ∈ X − X} ∈ [0, 1]⊗ter(V∆X//X), (11.56) ter(xy//xy) px > py ter(y//xy) 1 − px > py ∅ {x} Figure 31: The projection of a simplex (tetrahedron) of doublets events S2{x,y} on a unit {x, y}-square [0, 1]⊗{x,y} of its marginal probabilities p̆ = {px , py }. The X-вершины of this simplex are projected in corresponding X-vertices of {x, y}-square, X ⊆ {x, y}, and the all of e.p.d.’s of doublets of events with given {x, y}-set of probabilities of marginal events p̆ = {px , py } are projected in each point p̆ ∈ [0, 1]⊗{x,y} . In the left down quadrant (aqua) e.p.d.’s of the all of doublets of half-rare events of two kind are projected: 1/2 > px > py (unshaded) and px < py 6 1/2 (shaded triangle); in the remaining 3 quadrants e.p.d.’s of ∅-phenomena, {y}-phenomena and {x}-phenomena of doublets of half-rare events are projected. The half-rare doublets of the second kind: px < py 6 1/2, are projected in the shaded triangle of left down quadrant, and its set-phenomena — in shaded triangles of corresponding quadrants. The terraced events, generated by doublets of half-rare events {x, y}, are marked by the white formulas. 11.8 Set-phenomenon spectrum of functions on the X-hypercube Definition 11 (set-phenomenon spectrum of functions on the X-hypercube ). With each function ψ ∈ ΨX , defined on the X-hypercube, the 2N functions are connected. These functions are defined on the X-hypercube by formulas: ( ) ψX (w̆) = ψ w̆(c|X//X) for X ⊆ X. The family of the all such functions {ψX : X ⊆ X} is called the set-phenomenon X-spectrum of the function ψ. 124 THE XIV FAMEMS’2015 CONFERENCE {y} {x, y} px > 1 − p y {x} 1 − p x > 1 − py ∅ {y} ∅ 1 − py > p x 1 − py > 1 − p x py > p x py > 1 − p x {y, x} {x} {x} Figure 32: The projection of a simplex (tetrahedron) of doublets events S2{x,y} on a unit {x, y}-square [0, 1]⊗{x,y} of its marginal probabilities p̆ = {px , py }. The X-вершины of this simplex are projected in corresponding X-vertices of {x, y}-square, X ⊆ {x, y}, and the all of e.p.d.’s of doublets of events with given {x, y}-set of probabilities of marginal events p̆ = {px , py } are projected in each point p̆ ∈ [0, 1]⊗{x,y} . In the left down quadrant (aqua) e.p.d.’s of the all of doublets of half-rare events of two kind are projected: 1/2 > px > py (unshaded) and px < py 6 1/2 (shaded triangle); in the remaining 3 quadrants e.p.d.’s of ∅-phenomena, {y}-phenomena and {x}-phenomena of doublets of half-rare events are projected. The half-rare doublets of the second kind: px < py 6 1/2, are projected in the shaded triangle of left down quadrant, and its set-phenomena — in shaded triangles of corresponding quadrants. The terraced events, generated by doublets of half-rare events {x, y}, are marked by the white formulas. Let’s define for each X ⊆ X the terraced Xhypercube ⊗ [1 ] ⊗ [ 1) ter⊗ (X//X) = ,1 0, , 2 2 x∈X x∈X−X from which the X-hypercube is composed: ∑ [0, 1]⊗X = ter⊗ (X//X). X⊆X Lemma 6 (on a set-phenomenon X-spectrum of normalized function ). In order that the family of functions {θX : X ⊆ X} из ΨX is a set-phenomenon X-spectrum of some function normalized on the X-hypercube, it is necessary and sufficient that ∑ θX (w̆) = 1 (11.58) X⊆X for all w̆ ∈ [0, 1]⊗X . Proof. 1) If the family {θX : X ⊆ X} is a set-phenomenon X-spectrum of some normalized function, then by Definition 11 the equality (11.58) w̆ ∈ [0, 1/2)⊗X , w̆ ∈ ter⊗ (X//X), w̆ ∈ [1/2, 1]⊗X and show that the ψ is normalized on the Xhypercube. Indeed, noting that for an ) ( arbitrary X ⊆ X the equality ψ (w̆) = θX w̆(c|X//X) is ( (c|X//X) ) = θX (w̆) , we equivalent to the equality ψ w̆ obtain the required: X⊆X 1 − px > py ∅  ( )  θ∅ w̆(c|∅) ,   . . . ,  ) ( ψ (w̆) = θX w̆(c|X//X) ,  . . . ,    ( (c|X) ) , θX w̆ ∑ {x, y} {y} px > py is satisfied. 2) Let now the equality (11.58) is satisfied. Construct the function ψ on the Xhypercube by the following way ( ) ∑ ψ w̆(c|X//X) = θX (w̆) = 1. X⊆X Lemma 7 (on a set-phenomenon X-spectrum of the 1-function ). In order that the family of functions {θX : X ⊆ X} из ΨX is a set-pehomenon X-spectrum of some 1-function on the X-hypercube, it is necessary and sufficient that for each x ∈ X ∑ θX (w̆) = wx x∈X⊆X (11.59) for all w̆ ∈ [0, 1]⊗X . Proof. 1) If the family (1.3) is a set-phenomenon X-spectrum of some 1-function, then partial sums of functions from the family at x ∈ X ⊆ X равны wx : ∑ ψX (w̆) = wx x∈X⊆X for each w̆ ∈ [0, 1]⊗X by Definition 4. 2) Let now the equalities (1.4) are satisfied. Let’s construct the function ψ on X-hypercube by the following way  ( ) (c|∅)  , θ∅ w̆   . . . ,( ) ψ (w̆) = θX w̆(c|X//X) ,   ...,   ( )  θX w̆(c|X) , w̆ ∈ [0, 1/2)⊗X , w̆ ∈ ter⊗ (X//X), w̆ ∈ [1/2, 1]⊗X and show that ψ is a 1-function on the X-hypercube. Indeed, noting that for X ⊆ X the ( an arbitrary ) equality ψ (w̆) = θX w̆(c|X//X) is equivalent to ( ) the equality ψ w̆(c|X//X) = θX (w̆) , we obtain the required: ∑ x∈X⊆X ( ) ψ w̆(c|X//X) = ∑ x∈X⊆X θX (w̆) = wx . VOROBYEV 125 {0, 1} {x} {1, 1} xy 10 {0, 1, 1} {1, 1, 1} xyz xyz 100 000 ∅ {x} xy 00 xzy 100 ∅ xzy 000 yxz 010 yxz 000 zxy 010 yx 01 zxy 000 yzx 001 yx 00 yzx 000 zyx 001 zyx 011 yx 11 zyx 000 zyx 010 yzx 101 yx 10 yzx 100 zxy 011 zxy 001 yxz 110 {x,y} xy 11 xy 01 xzy 101 {y} {x,y} {px , py } = {0, 0} {1, 0} Figure 33: The projection of 22 -vertices simplex on a square, on which the scheme is superimposed, illustrating a connection of two permutations of events in a half-rare doublet with the 22 set-phenomena. Although this task is purely technical, but its solution opens the way for the application of the proposed Kopula (eventological copula) theory to the construction of the eventological theory of ordinary copulas that determine the xzy 001 {y} xyz xyz 110 010 {px , py , pz } = {0, 0, 1} {1, 0, 1} {0, 1, 0} {1, 1, 0} xyz xyz 101 001 {x,z} 12 Remaining behind the scenes In the text and, in particular, in the Appendix, the value of the p̆-ordering condition of the set of events is specified, which complicates the computational implementation of the above algorithms in the frame method of constructing Kopulas as set functions of the set of marginal probabilities. The reason for this complication lies in the properties of the set-functions, i.e., functions of a set that differ from the properties of arbitrary functions of several variables. The point is that the set-function of the set of marginal probabilities is necessarily a symmetric function of the marginal probability vector (Cartesian representation of Kopula, see Prolegomenon 9), to determine which it is sufficient to specify its values only on those vectors whose components are ordered, for example, in descending order, so that the remaining values can be determined by the appropriate permutations of the arguments. For example, the Cartesian representation of an N -Kopula in RN is sufficient to define on the 1/N ! part of the unit N -hypercube so that this representation becomes definite on the whole hypercube by continuing permutations of arguments. yxz 100 xzy 110 {z} xzy 010 yxz 011 yxz 001 zxy 110 zxy 100 yzx 011 yzx 010 zyx 101 zyx 111 zyx 100 zyx 110 yzx 111 yzx 110 zxy 111 zxy 101 yxz 111 {x,y,z} yxz 101 xzy 111 xzy 011 xyz xyz 111 011 {px , py , pz } = {0, 0, 0} {y,z} {1, 0, 0} Figure 34: These are not geometrical projections of 23 -vertices simplex on a cube, but two conditional schemes of these projections, which illustrate a connection of six permutations of events in a half-rare triplet of events with its 23 set-phenomena. The conditional scheme of the projection on the upper half of the cube is shown at the top, on the lower half — at the bottom. In the Venn diagram of half-rare events: x is the left, y is the right, and z is the lower half of the cube. joint distribution of a given set of marginal distributions. The author encountered this when developing the program code, which calculated all the illustrations for the Kopula examples. The problem is solved programmatically, but requires a detailed description of this solution (see Fig. 33 and 34), which, of course, together with the 126 eventological theory of copula deserves a separate publication. In conclusion, I can not resist the temptation to quote the formulation of the tenth Prolegomenon of the Kopula theory, which reveals the content of these my next publications. Prolegomenon 10 (Cartesian representation of the N -Kopula defines 2N classical copulas of N marginal uniform distributions on [0, 1]). 13 On the inevitable development of language This first work on the theory of the eventological copula is over at the end of July 2015. It sums up the work on the eventological theory of probabilities, raising the theory of Kopula to its apex. The work is written in a mathematical language, in which the state of the eventological theory was reflected precisely at the time when the author unexpectedly, but by the way, got a brilliant example of two statisticians from sociology and ecology, who immediately forced him to postpone polishing of the Kopula theory for almost a year in order to immediately immerse themselves in the destructive creation of a new unifying eventological theory of experience and chance by the agonizing fusion of two dual theories: the eventological theory of believabilities and the eventological theory of probabilities. Because of this, the mathematical language of this work is just a pretension to the eventological probability theory, which does not yet know that there is a very close twin that exists — the eventological theory of believabilities. Therefore, in the terminology of this work, those crucial changes in the basic concepts and notations that were invented to construct a unifying eventological theory did not find any worthy reflection. Of course, the new unifying theory suggests the development of the original mathematical language of dual Kopulas, one of which hosts the eventological probability theory, and the other — in the eventological believability theory. ⋆ The English version of this article was published on November 12, 2017. Therefore, my later works [12, 11, 10], which expand the eventological formalism, including the bra-ket formalism of the theory of experience and chance, are added to the list of references. THE XIV FAMEMS’2015 CONFERENCE References [1] C. Alsina, M.J. Frank, and B. Schweizer. Assocative functions: Triangular Norms and Copulas. World Scientific Publishing Co. Pte. Ltd., Singapore, 2006. [2] R.B. Nelsen. An introducvtion to copulas. Springer, New York, 1999. [3] R.B. Nelsen. 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