Geometry of generators of triangular norms and copulas

Kamila Houšková; Mirko Navara

doi:10.1515/demo-2024-0004

Article Open Access

Geometry of generators of triangular norms and copulas

Kamila Houšková and Mirko Navara

Published/Copyright: June 22, 2024

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information

From the journal Dependence Modeling Volume 12 Issue 1

Abstract

We study the relations between the shapes of strict triangular norms and their generators. The generators are not unique; to avoid this ambiguity, we introduce the class of balanced generators. We find conditions for their existence and a procedure of their computation. We formulate their influence on local properties of the corresponding triangular norms, in particular at the extremes of their domain. Many strict triangular norms are also copulas; thus, our results can also be applied to bivariate extreme value distributions described by some families of copulas.

Keywords: multiplicative generator; additive generator; balanced generator; Frank copula; Ali-Mikhail-Haq copula; extreme value distribution

MSC 2010: 03E72; 20M14; 54E70

1 Formulation of the problem

In this section, we present the basic notions, inspiration and motivation, and a rough formulation of the basic problem; its well-defined instances will be studied in the subsequent sections.

1.1 Basic notions concerning triangular norms and copulas

Triangular norms (t-norms for short) are used in fuzzy logics as the interpretation of a conjunction [2]. They are binary operations on the real interval [ 0 , 1 ] , which are commutative, associative, nondecreasing, and with neutral element 1. We restrict attention to t-norms T , which are continuous and, moreover, Archimedean, which means that T ( x , x ) < x whenever 0 < x < 1 . They are called strict if T ( x , y ) < T ( x , z ) whenever y < z and x > 0 , nilpotent otherwise. We refer to [5] for basics on t-norms.

Unlike commutativity, associativity is not directly interpretable in the graphs of binary operations.

Therefore, we often describe them using unary functions, called generators, and known associative operations, addition or multiplication. It is known [6,9] that each continuous Archimedean t-norm T has a decreasing additive generator, t : [ 0 , 1 ] → [ 0 , ∞ ] , and an increasing multiplicative generator, θ : [ 0 , 1 ] → [ 0 , 1 ] , which allow us to express it as

(1) T ( x , y ) = t − 1 ( t ( x ) + t ( y ) ) , if t ( x ) + t ( y ) ≤ t ( 0 ) , 0 , otherwise,

(2) = θ − 1 ( θ ( x ) θ ( y ) ) , if θ ( x ) θ ( y ) ≥ θ ( 0 ) , 0 , otherwise.

We always have t ( 1 ) = 0 and θ ( 1 ) = 1 . For strict t-norms, t : [ 0 , 1 ] → [ 0 , ∞ ] and θ : [ 0 , 1 ] → [ 0 , 1 ] are bijections; t ( 0 ) = ∞ , θ ( 0 ) = 0 , and only the first cases occur in (1) and (2). The generators of a given continuous Archimedean t-norm are not unique; additive generators are determined up to positive multiples and multiplicative generators up to positive powers [5].

(Binary) copulas are the joint cumulative distribution functions of binary random vectors whose marginal distributions are uniform on [ 0 , 1 ] . We refer to [12] for basics on copulas.

A strict t-norm is a copula iff its additive generator is convex [5]. It makes sense to speak of convexity in relation to additive generators because if one is convex, so are all others. This is not true for multiplicative generators, where convexity depends on the choice of a generator; some multiplicative generators of a t-norm may be convex, others concave, some neither of these, and they still generate the same t-norm.

Associative copulas are t-norms. Thus, many families of binary operations were introduced and studied independently as copulas and as t-norms. Copulas were used to estimate extreme value distributions [1,12], which are used to predict the risks of disasters, in nature as well as in economy; our investigation of their values near the bounds of the domains could contribute to these studies, too.

1.2 Inspiration

Many different proofs of the existence of generators of strict t-norms were published (see the bibliography in [11]). However, their applicability and interpretability is disappointing. In particular, knowing the t-norm seems to give no clue about the “shape” of a generator, and vice versa. For comparison, if we see a graph of a continuous function, we have a rough idea how the graphs of its derivative or integral may look like – where they are zero, increasing, decreasing, convex, etc. However, given a t-norm, we can say very little about its generators. Vice versa, given a generator, we cannot describe the graph of the corresponding t-norm in terms of convexity, etc. We contribute to attempts to fill in this gap. We enhance and extend the results of [3,4] by proving and demonstrating tools that admit visible links between t-norms and their generators.

As far as we know, there was only one result that allowed us to link the shape of a t-norm and its multiplicative generator [10]. We adapt its formulation from [11]:

Theorem 1

Let θ be a multiplicative generator of a strict t-norm T such that θ ′ ( 0 ) ∈ ] 0 , ∞ [ . Then,

(3) θ ( x ) = ∂ ∂ y T ( x , y ) y = 0 = lim y → 0 + T ( x , y ) y ,

for all x ∈ [ 0 , 1 ] whenever these expressions are defined.

Formula (3) means that, up to the scaling factor 1 ∕ y , the function x ↦ T ( x , y ) becomes a very good approximation of a multiplicative generator of T when y → 0 (Figure 1). Thus, the restriction of T to the line segment [ 0 , 1 ] × { y } (for small y > 0 ) gives a good approximation of one distinguished multiplicative generator of T . Other multiplicative generators of T are powers of this one and do not allow such an interpretation. We shall discuss this situation in more detail in Section 3.

$Figure 1 Generator θ 2 F {\theta }_{2}^{F} of the Frank t-norm T 2 F {T}_{2}^{F} (see Example 4), two of its powers (square and square root), and its approximation by (3).$

Figure 1

Generator θ 2 F of the Frank t-norm T 2 F (see Example 4), two of its powers (square and square root), and its approximation by (3).

1.3 Need of normalization of generators

To demonstrate the difficulties, let us consider the functions drawn in Figure 2. All functions in the left graph are different multiplicative generators of the same t-norm, the product t-norm T P ( x , y ) = x ⋅ y . All functions in the right graph are different multiplicative generators of another t-norm, the Einstein t-norm (Hamacher t-norm T 2 H , see Example 2). What is the “shape” distinguishing them? It should be a feature that is the same for all the generators inside each figure and expresses a difference between the left and the right graph. The “shape” does not make sense unless we choose a unique representative generator by some additional criteria.

Figure 2

Different powers of multiplicative generators of the product (left) and of the Einstein t-norm (right).

2 Different ways of normalization of generators

In this section, we discuss several ways of choosing a unique additive or multiplicative generator. One of these methods will be then studied in detail in Section 3. We also define several families of t-norms used in examples throughout this article.

2.1 Values of the additive generators at boundary points

All additive generators attain zero at 1 and the maximal value at 0. For nilpotent t-norms [5], the maximum is finite, and it can be chosen as any positive number. Thus, we have

Proposition 1

[8] Each nilpotent t-norm has a unique normed additive generator, t 1 , determined by the condition t 1 ( 0 ) = 1 and computed as t 1 ( x ) ≔ t ( x ) t ( 0 ) , where t is any of the t-norm’s additive generators.

Example 1

Yager t-norms with parameter λ ∈ ] 0 , ∞ [ are given by

T λ Y ( x , y ) = max ( 1 − ( ( 1 − x ) λ + ( 1 − y ) λ ) 1 ∕ λ , 0 ) .

They have normed additive generators

t λ Y ( x ) = ( 1 − x ) λ .

Some of them are drawn in Figure 3 (left).

Figure 3

Additive generators of a collection of Yager t-norms (see Example 1) for different values of parameters in the usual form with value 1 at 1 (left) and their multiples with value 1 at 1/2 (right).

This method of unification is often natural and useful. However, it can be used only for nilpotent t-norms.

2.2 Values of the additive generators at an internal point

Proposition 1 is not applicable to strict t-norms because t ( 0 ) = ∞ for all their additive generators. They are unbounded in any neighborhood of 0; thus, speaking of their “shape” near 0 is not very useful. We could ask whether they tend to ∞ “faster” or “slower” than the negative logarithm, but it would not help our geometrical intuition and understanding of the graph.

For both strict and nilpotent t-norms, we may choose a unique additive generator by fixing the value in any internal point of the interval ] 0 , 1 [ , (Figure 3 right).

Example 2

Hamacher t-norms (Ali-Mikhail-Haq copulas) are defined for parameter λ ∈ [ 0 , ∞ [ by

T λ H ( x , y ) = x y λ + ( 1 − λ ) ( x + y − x y ) .

For λ ∈ ] 0 , ∞ [ , they have multiplicative generators

(4) θ λ H ( x ) = x λ + ( 1 − λ ) x

and additive generators

(5) t λ H ( x ) = ln λ + ( 1 − λ ) x x ,

shown in Figure 4 (left). The functions t λ H ( x ) ∕ t λ H ( 1 ∕ 2 ) are additive generators of the same t-norms T λ H , but with a fixed value 1 at 1/2 (Figure 4, right).

For λ = 0 , we obtain the Hamacher product,

T 0 H ( x , y ) = x y x + y − x y , for ( x , y ) ≠ ( 0 , 0 ) , 0 , otherwise.

It has a multiplicative generator

θ 0 H ( x ) = exp x − 1 x

and an additive generator

t 0 H ( x ) = 1 − x x .

Figure 4

Additive generators of a collection of Hamacher t-norms (see Example 2) for different values of parameters in the usual form (left) and their multiples with value 1 at 1/2 (right).

Unfortunately, this method of unification leads to rather different pictures and does not seem to help our intuition concerning the properties of additive generators. Moreover, the choice of the fixed value and the internal point at which this value is attained make the process ambiguous, leading to rather different representations of the same t-norms.

2.3 Values of the multiplicative generators at boundary points

All multiplicative generators attain 1 at 1 and the minimal value at 0. For nilpotent t-norms [5], the minimum is positive and it can be chosen as any number from ] 0 , 1 [ because any positive power of a multiplicative generator is a multiplicative generator of the same t-norm. Thus, we have:

Proposition 2

For each r ∈ ] 0 , 1 [ , each nilpotent t-norm has a unique multiplicative generator, θ r , determined by the condition θ r ( 0 ) = r and computed as θ r ( x ) ≔ θ ( x ) p , where θ is any of its multiplicative generators and p = ln r ln t ( 0 ) .

Example 3

Yager t-norms (Example 1) have multiplicative generators

θ λ Y ( x ) = exp ( − ( 1 − x ) λ )

with fixed value exp ( − 1 ) at 0. Some of them are drawn in Figure 5 (left).

$Figure 5 Multiplicative generators of a collection of Yager t-norms (see Example 3) for different values of parameters in the usual form with value 1 ∕ e 1/{\rm{e}} at 1 (left) and their powers with value 1 ∕ e 1/{\rm{e}} at 1/2 (right).$

Figure 5

Multiplicative generators of a collection of Yager t-norms (see Example 3) for different values of parameters in the usual form with value 1 ∕ e at 1 (left) and their powers with value 1 ∕ e at 1/2 (right).

This method of unification can be used only for nilpotent t-norms.

2.4 Values of the multiplicative generators at an internal point

Proposition 2 is not applicable to strict t-norms because θ ( 0 ) = 0 for all their multiplicative generators.

For both strict and nilpotent t-norms, we can choose a unique multiplicative generator by fixing the value in any internal point of the interval ] 0 , 1 [ (Figure 5, right).

Example 4

Strict Frank t-norms (Frank copulas) with parameter λ ∈ ] 0 , ∞ [ are defined by

T λ F ( x , y ) = log λ 1 + ( λ x − 1 ) ( λ y − 1 ) λ − 1 , for λ ∈ ( 0 , ∞ ) \ { 1 } , x ⋅ y , for λ = 1 .

For λ ≠ 1 , they have multiplicative generators

(6) θ λ F ( x ) = λ x − 1 λ − 1 ,

(Figure 7, left). Properly chosen powers of these multiplicative generators attain 1/2 at 1/2 (Figure 7, middle).

Figure 6

Additive generators of a collection of Hamacher t-norms with derivatives ‒ 1 at 1 (see Example 5) for different values of parameters: the whole graphs (left) and details (right). These are additive generators of the same collection of t-norms as in Figure 4.

Once more, this method of unification leads to rather different pictures and does not seem to help our intuition concerning the properties of multiplicative generators. Furthermore, the choice of the fixed value and the internal point at which this value is attained make the process ambiguous, leading to rather different representations of the same t-norms.

2.5 Derivatives at 1

Assume that the additive generator t has a nonzero finite (left) derivative at 1. As is well known, all additive generators are its positive multiples, t r = r t , r > 0 , with derivatives t ′ ( x ) = r t ′ ( x ) . Among them, for r 1 = − 1 ∕ t ′ ( 1 ) > 0 , we obtain an additive generator t r 1 ( x ) = − t ( x ) ∕ t ′ ( 1 ) , which is the only one satisfying the property t r 1 ′ ( 1 ) = − 1 . Consequently, this condition can be used to select a unique additive generator.

Example 5

Hamacher t-norms (Example 2) have additive generators t λ H from (5). Their derivatives are

( t λ H ) ′ ( x ) = λ x ( ( λ − 1 ) x − λ ) , ( t λ H ) ′ ( 1 ) = − λ .

Functions t λ H ( x ) ∕ λ in Figure 6 are the additive generators of the same collection of t-norms, with derivatives − 1 at 1. From the whole graph, it does not seem that this condition is fulfilled; a detail of the graphs on the right shows that it is really so.

Figure 7

Collection of multiplicative generators of Frank t-norms from Example 4 for different values of parameters (left), their powers with value 1/2 at 1/2 (middle) or with derivatives 1 at 1 (right). All three figures show multiplicative generators of the same collection of t-norms.

For multiplicative generators, we may also fix the derivative at 1. This leads to a different representation of strict t-norms by unique unary functions.

Example 6

Strict Frank t-norms with parameter λ ∈ ] 0 , ∞ [ \ { 1 } , have multiplicative generators θ λ F from (6). Their derivatives are

(7) ( θ λ F ) ′ ( x ) = λ x ln λ λ − 1 , ( θ λ F ) ′ ( 1 ) = λ ln λ λ − 1 ∈ ] 0 , ∞ [ .

Properly chosen powers of these multiplicative generators have derivatives 1 at 1 (Figure 7, right).

The condition that the derivative at 1 is nonzero and finite is satisfied either by all additive generators of a t-norm or by none. The same holds for multiplicative generators. We formulate the consequences for the “shape” of the t-norm near the point ( 1 , 1 ) .

Theorem 2

Let T be a continuous Archimedean t-norm. From the following conditions, 1 and 2 are equivalent and they imply 3:

T has an additive generator whose derivative at 1 is nonzero, finite, and continuous.
T has a multiplicative generator whose derivative at 1 is nonzero, finite, and continuous.
T is differentiable at ( 1 , 1 ) (i.e., it has there a total differential; explicitly, its linear approximation in the neighborhood of ( 1 , 1 ) is ( x , y ) ↦ x + y − 1 ).

Remark 1

Theorem 2 operates with the existence of derivatives as follows: if the derivative of one generator exists and satisfies the assumptions, then the derivative of the other generator also exists. The choice of an additive and a multiplicative generator is arbitrary.

Proof

Let θ be a multiplicative generator of T . Then, t ( x ) = − ln θ ( x ) is its additive generator. If Condition 1 holds for some additive generator, it holds for all additive generators; thus, we may, without loss of generality, restrict attention to t . Analogously, we may restrict attention to θ in Condition 2. Their derivatives satisfy the relation

t ′ ( x ) = θ ′ ( x ) θ ( x ) .

In a neighborhood of 1, the denominator θ ( x ) is nonzero; thus, when one side is nonzero and finite, so is the other. This proves the equivalence of Conditions 1 and 2. In this case, also the inverses of the generators have nonzero finite derivatives at the corresponding points (the derivative of t − 1 at t ( 1 ) = 0 and the derivative of θ − 1 at θ ( 1 ) = 1 ). The derivatives of the inverses are continuous at these points because we can find a neighborhood in which the derivatives of generators differ from zero by more than some sufficiently small ε > 0 . Then, T is a composition of continuous functions with continuous derivatives; its differential follows from the boundary conditions for a t-norm.□

As a consequence, the contours of a t-norm satisfying the conditions of Theorem 2 are almost linear near the point ( 1 , 1 ) . Note that the linear approximation at ( 1 , 1 ) coincides (locally) with the Łukasiewicz t-norm, T L ( x , y ) = max ( x + y − 1 , 0 ) , which is nilpotent, but the theorem applies to some strict t-norms, too (Figure 9).

Figure 8

Multiplicative generators from Example 10 for different values of parameters (left) and the corresponding balanced generators (right).

Following [8], any continuous t-norm can be approximated by a strict t-norm with arbitrary precision. As an example, the minimum t-norm is the limit of strict Frank t-norms (Example 4) for parameter λ → 0 . It is not differentiable at ( 1 , 1 ) (and does not have a generator), although all Frank t-norms with λ > 0 satisfy the assumptions of Theorem 2. This makes the statement of Theorem 2 somewhat surprising.

Although fixing a derivative at 1 is not a usual way of choosing a unique generator, it may give us insight into its properties and can be useful in some cases. Its use is limited by the assumption that the derivative exists, is finite, and is nonzero.

2.6 Derivatives at 0

For strict t-norms, Proposition 1 does not apply; both the values of the additive generators and their derivatives are unbounded in any neighbourhood of 0. Thus, we concentrate on the multiplicative generators. Among them, we determine a balanced generator, which (if it exists) is unique.

Definition 1

[3] A multiplicative generator θ * of a strict t-norm is called a balanced generator if it has a nonzero finite (right) derivative at 0. This means

θ * ′ ( 0 ) = lim x → 0 + θ * ( x ) x ∈ ] 0 , ∞ [ ,

or, equivalently,

(8) lim x → 0 + θ * ( x ) x ⋅ θ * ′ ( 0 ) = 1 .

We reserve the notation θ * for balanced generators.

Example 7

Hamacher t-norms with parameter λ ∈ ] 0 , ∞ [ have multiplicative generators (4) with derivatives

( θ λ H ) ′ ( x ) = λ ( λ + ( 1 − λ ) x ) 2 , ( θ λ H ) ′ ( 0 ) = 1 λ ∈ ] 0 , ∞ [ .

These generators are balanced.

Example 8

Strict Frank t-norms with parameter λ ∈ ] 0 , ∞ [ \ { 1 } have multiplicative generators (6) with derivatives (7); hence,

( θ λ F ) ′ ( 0 ) = ln λ λ − 1 ∈ ] 0 , ∞ [ .

These generators are balanced.

Example 9

All generators constructed by Theorem 1 are balanced.

The notion of balanced generators is new, and it has some particularity. For example, we cannot put other limitation on the derivative at 0 than θ * ′ ( 0 ) ∈ ] 0 , ∞ [ ; this value is specific for any t-norm having a balanced generator. We devote Section 3 to balanced generators and their properties.

2.7 Other ways of normalization

Of course, there are other possibilities for selecting a unique generator. For example, we may fix its derivative at an internal point of the domain, provided that the derivative exists, is finite, and is nonzero. Even a geometric interpretation (using the angle of the tangent) exists, but it seems of little utility. Also, higher-order derivatives could be considered, but we did not find supporting arguments for their use. Besides, their existence requires stronger assumptions.

On the other hand, we could fix integrals over some intervals. These always exist for continuous functions, and there is also a geometrical interpretation by the area under curve. In this instance, the only unambiguous possibility seems to be the integral over the whole domain, [ 0 , 1 ] . This might be subject to future research. However, for additive generators, this integral may be infinite and therefore not pertinent.

3 Balanced generators

In the previous section, we introduced several methods of normalization of generators. Among them, balanced generators were introduced as an option. In this section, we study this new approach in more detail.

3.1 Method of finding balanced generators

We shall see that every strict t-norm has at most one balanced generator. A question arises: How to find the balanced generator if it exists, especially when we know of a multiplicative generator that is not balanced? We give an answer by formulating the following necessary condition for the existence of a balanced generator:

Theorem 3

Let T be a strict t-norm with a multiplicative generator θ . If T has a balanced generator, θ * , then it is of the form

(9) θ * = θ r ,

where

(10) r = lim x → 0 + θ ( x ) x θ ′ ( x ) ∈ ] 0 , ∞ [ .

Proof

Each multiplicative generator θ must be a positive power of (any other) multiplicative generator, i.e., of the form

(11) θ ( x ) = ( θ * ( x ) ) p ,

for some p ∈ ] 0 , ∞ [ (see [5]). We define

c ≔ θ * ′ ( 0 ) = lim x → 0 + θ * ( x ) x .

Then,

(12) c p = lim x → 0 + ( θ * ( x ) ) p x p = lim x → 0 + θ ( x ) x p .

Using L’Hôpital’s rule, we can also write

(13) c p = lim x → 0 + ( ( θ * ( x ) ) p ) ′ ( x p ) ′ = lim x → 0 + θ ′ ( x ) p x p − 1 .

Dividing (12) by (13), we obtain

1 = lim x → 0 + θ ( x ) x p p x p − 1 θ ′ ( x ) = lim x → 0 + p θ ( x ) x θ ′ ( x ) ,

which shows that 1 ∕ p = r in (9) and (10).□

As a consequence, every t-norm has at most one balanced multiplicative generator. The derivative of its p th power at 0 is 0 for p > 1 and ∞ for p < 1 (Figure 1).

Example 10

For λ > 0 , we define T λ as the t-norm with a multiplicative generator

θ λ ( x ) = 1 − cos π x λ 2

(Figure 8, left). The derivatives of the multiplicative generators are

θ λ ′ ( x ) = π 2 λ x λ − 1 sin π x λ , θ λ ′ ( 1 ) = 0 ,

so Theorem 2 is not applicable. Using the limit

(14) lim x → 0 sin π x λ π x λ = 1 ,

we obtain

θ λ ′ ( 0 ) = lim x → 0 + π 2 2 λ x 2 λ − 1 = 0 , if λ > 1 2 , ∞ , if λ < 1 2 , π 2 4 , if λ = 1 2 .

Among these generators, only that with λ = 1 2 is balanced.

Using (10), (14), and L’Hôpital’s rule, we obtain

r λ = lim x → 0 + θ λ ( x ) x θ λ ′ ( x ) = lim x → 0 + 1 − cos π x λ π λ x λ sin π x λ = lim x → 0 + 1 − cos π x λ π 2 λ x 2 λ = lim x → 0 + π λ x λ − 1 sin π x λ π 2 2 λ 2 x 2 λ − 1 = 1 2 λ .

For each λ , the function θ λ r λ is a balanced generator of T λ (Figure 8, right).

$Figure 9 Contour plot of the t-norm generated by 1 1 − ln x \frac{1}{1-\mathrm{ln}x} (left) and Hamacher product (right).$

Figure 9

Contour plot of the t-norm generated by 1 1 − ln x (left) and Hamacher product (right).

3.2 Properties of t-norms with balanced generators

If the derivative of a balanced generator is continuous at 0, the corresponding t-norm is differentiable at ( 0 , 0 ) and its total differential there is zero. We can say even more about its similarity to the product near this point (about the Taylor series centered in the origin):

Theorem 4

Let T be a strict t-norm with a balanced generator θ * . Then,

lim ( x , y ) → ( 0 , 0 ) + T ( x , y ) x y = θ * ′ ( 0 ) ,

where the limit is taken for x ≠ 0 , y ≠ 0 , i.e., with respect to the region ] 0 , 1 ] 2 .

Proof

We consider ( x , y ) in a neighborhood ] 0 , ε ] 2 . Then, T ( x , y ) ≤ min ( x , y ) ≤ ε and

lim z → 0 + θ * ( z ) z = θ * ′ ( 0 ) .

Substitution z = T ( x , y ) gives

lim ( x , y ) → ( 0 , 0 ) T ( x , y ) θ * ( T ( x , y ) ) = 1 θ * ′ ( 0 ) .

In the latter limit, we substitute from (2), θ * ( T ( x , y ) ) = θ * ( x ) θ * ( y ) and obtain

lim ( x , y ) → ( 0 , 0 ) T ( x , y ) θ * ( x ) θ * ( y ) = 1 θ * ′ ( 0 ) ,

which allows us to compute the limit from (4):

(15) lim ( x , y ) → ( 0 , 0 ) T ( x , y ) x y = lim ( x , y ) → ( 0 , 0 ) T ( x , y ) θ * ( x ) θ * ( y ) θ * ( x ) x θ * ( y ) y = ( θ * ′ ( 0 ) ) 2 θ * ′ ( 0 ) = θ * ′ ( 0 ) .□

This might be surprising if we note that θ * ′ ( 0 ) may be arbitrarily large. The obvious inequality T ( x , y ) ≤ min ( x , y ) shows that the approximation of T ( x , y ) by θ * ′ ( 0 ) x y may possibly be good only for small values of x and y , but it is a correct expression of its asymptotic behavior.

3.3 Application to copulas

Let ( X , Y ) be a two-dimensional absolutely continuous random vector. We denote by F ( X , Y ) its joint cumulative distribution function and by F X and F Y the marginal cumulative distribution functions; their inverses, F X − 1 and F Y − 1 are the quantile functions of random variables X and Y .

By Sklar’s theorem [12], binary operation T defined by

T ( x , y ) = F ( X , Y ) ( F X − 1 ( x ) , F Y − 1 ( y ) )

is a copula and all copulas arise this way. (There may be problems with definitions of values of the quantile functions at the bounds 0 and 1, but all copulas satisfy T ( x , 0 ) = T ( 0 , x ) = 0 , T ( x , 1 ) = T ( 1 , x ) = x .) Thus, copulas describe the dependence of random variables in a form independent of the marginal distributions.

For a copula T ,

lim ( x , y ) → ( 0 , 0 ) + T ( x , y ) x y = ∂ 2 T ( x , y ) ∂ x ∂ y ( x , y ) = ( 0 , 0 ) +

is the limit of its density at the origin. For a small positive ε , values F X − 1 ( ε ) , F Y − 1 ( ε ) are ε -quantiles, i.e., the probabilities P satisfy

P ( X ≤ F X − 1 ( ε ) ) = ε = P ( Y ≤ F Y − 1 ( ε ) ) .

For independent random variables, we would have the joint probability P ( X ≤ F X − 1 ( ε ) , Y ≤ F Y − 1 ( ε ) ) = ε 2 , but respecting the dependence, we need a more general formula

P ( X ≤ F X − 1 ( ε ) , Y ≤ F Y − 1 ( ε ) ) = F ( X , Y ) ( F X − 1 ( ε ) , F Y − 1 ( ε ) ) = T ( ε , ε ) .

This value represents the probability of simultaneous occurrence of extreme (= improbable) values, both with marginal probabilities ε . These are crucial in risk assessment. (The task may be to estimate the risk of rare events, such as natural disasters and bank crises, for example, T ( ε , ε ) may represent a risk of simultaneous floods on two tributaries, causing a flood down the river.) If the density has a limit, c , at ( 0 , 0 ) , we can estimate T ( ε , ε ) ≈ c ε 2 .

If T is an associative copula with a balanced generator θ * , (15) allows us to evaluate the asymptotic density at ( 0 , 0 ) as θ * ′ ( 0 ) , leading to an estimate of simultaneous occurrence of values below the ε -quantiles

P ( X ≤ F X − 1 ( ε ) , Y ≤ F Y − 1 ( ε ) ) = T ( ε , ε ) ≈ θ * ′ ( 0 ) ε 2 ,

in contrast to the value ε 2 obtained for the model with independent variables.

3.4 T-norms without balanced generators

When looking for a balanced generator according to Theorem 3, we do not need to check all the conditions in advance. It may happen that the limit (10) is 0 or ∞ ; then, the t-norm does not have a balanced generator. This has clear consequences for its properties in the neighborhood of the point ( 0 , 0 ) .

Convention 1

In this section, we assume ln 0 = − ∞ , exp ( − ∞ ) = 0 , 1 ∞ = 0 , and − 1 0 = − ∞ whenever the whole expression with these substitutions makes sense.

Example 11

Let us take the t-norm with a multiplicative generator

θ ( x ) = 1 1 − ln x .

Its derivative is

θ ′ ( x ) = 1 x ⋅ ( 1 − ln x ) 2 .

Formula (10) gives

r = lim x → 0 + θ ( x ) x θ ′ ( x ) = lim x → 0 + ( 1 − ln x ) = ∞ .

Therefore, this t-norm does not have a balanced generator. (We obtain r = ∞ for any multiplicative generator of this t-norm.) Using the inverse function (cf. Convention 1),

θ − 1 ( x ) = exp x − 1 x ,

we express the corresponding t-norm

T ( x , y ) = exp 1 ( 1 − ln x ) ( 1 − ln y ) − 1 1 ( 1 − ln x ) ( 1 − ln y ) = exp ( 1 − ( 1 − ln x ) ( 1 − ln y ) ) = x y exp ( − ( ln x ) ( ln y ) ) .

T-norm T is in Figure 9 (left) and its generator in Figure 10. Near the point ( 0 , 0 ) , it goes to 0 faster than any positive multiple of x y , cf. Theorem 4.

$Figure 10 Multiplicative generator exp ( x − 1 x ) \exp \left(\frac{x-1}{x}) of Hamacher product from Example 12 and its inverse, 1 1 − ln x \frac{1}{1-\mathrm{ln}x} , the generator from Example 11.$

Figure 10

Multiplicative generator exp ( x − 1 x ) of Hamacher product from Example 12 and its inverse, 1 1 − ln x , the generator from Example 11.

Example 12

Hamacher product (Example 2 and Figure 9, right) has a multiplicative generator

θ 0 H ( x ) = exp x − 1 x

(Figure 10), which is the inverse of the generator θ from Example 11. Hamacher product has no balanced generator because (10) gives

r = lim x → 0 + θ 0 H ( x ) x ( θ 0 H ) ′ ( x ) = lim x → 0 + exp ( x − 1 x ) x exp ( x − 1 x ) x 2 = lim x → 0 + x = 0 .

It is not differentiable at ( 0 , 0 ) (cf. Theorem 4), it approaches min ( x , y ) in its neighborhood. The Hamacher product occurs also as a special case of families of t-norms studied in the sequel. We presented it explicitly because of its simplicity.

Example 13

Dombi t-norms [5] with parameter λ > 0 ,

T λ D ( x , y ) = 1 1 + 1 x − 1 λ + 1 y − 1 λ 1 λ ,

have multiplicative generators

θ λ D ( x ) = exp − 1 − x x λ

with derivatives

( θ λ D ) ′ ( x ) = λ ⋅ 1 − x x λ exp − 1 − x x λ ( 1 − x ) x .

Formula (10) gives

r = lim x → 0 + θ λ D ( x ) x ( θ λ D ) ′ ( x ) = lim x → 0 + 1 − x λ ⋅ 1 − x x λ = 0 .

Dombi t-norms do not have balanced generators. (Hamacher product T 0 H from Example 12 is T 1 D .)

Example 14

Schweizer’s second family of t-norms [7] is defined for parameter λ > 0 by

T λ S 2 ( x , y ) = ( x − λ + y − λ − 1 ) − λ .

They are Clayton copulas [12]. (In [5], Schweizer-Sklar t-norms are defined for any λ ∈ [ − ∞ , ∞ ] with the opposite sign of parameter λ ; thus, our family corresponds to negative parameters in [5]; the other Schweizer-Sklar t-norms are not strict.)

T-norms from the Schweizer’s second family have multiplicative generators

θ λ S 2 ( x ) = exp ( 1 − x − λ ) ,

with derivatives

( θ λ S 2 ) ′ ( x ) = λ x − λ − 1 exp ( 1 − x − λ ) .

Formula (10) gives

r = lim x → 0 + θ λ S 2 ( x ) x ( θ λ S 2 ) ′ ( x ) = lim x → 0 + 1 λ x − λ = lim x → 0 + 1 λ x λ = 0 .

Schweizer’s second family of t-norms does not admit balanced generators. (Hamacher product T 0 H from Example 12 is T 1 S 2 .)

Although we presented several negative examples in detail, balanced generators exist for many families of t-norms found in the literature. With their use, we can get oriented in their influence on the “shape” of the corresponding t-norms. For example, Figure 11 shows two t-norms and their balanced generators; one convex and the other concave. It is clear which belongs to which of the 3D graphs of the aforementioned t-norms. Theorem 1 gives also a hint; look at the emphasized curves that represent T ( x , y ) for y = 1 ∕ 20 .

Figure 11

Top: 3D graphs of Frank t-norms with parameters 16 (left) and 1/16 (right). Bottom: Their balanced generators. Note that the balanced generators look similar to the red curves on the surfaces, drawn for a small second argument (1/20), scaled up by multiplication by 20.

4 Conclusion

We presented a critical overview of preceding ways of normalization of generators of t-norms and copulas. We proposed a new method, based on balanced generators. We collected some results that relate the shapes of (uniquely chosen) generators and the corresponding t-norms. Although they refer only to derivatives at endpoints of the domain, they help to understand the geometry of these objects and their properties near the extremes. These can be important in the study of (associative) copulas, used in risk assessment. Hence, our results contribute also to the knowledge of properties of mathematical models based on copulas.

Funding information: This work was supported by the CTU institutional support (Future Fund).
Author contributions: Both authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript. Both authors contributed equally during the research and preparation of this paper.
Conflict of interest: The authors state no conflict of interest.

References

[1] Genest, C., & Rivest, L.-P. (1989). A characterization of Gumbel’s family of extreme value distributions. Statistics & Probability Letters 8, 207–211. 10.1016/0167-7152(89)90123-5Search in Google Scholar

[2] Hájek, C. (1998). Metamathematics of Fuzzy Logic. Kluwer, Dordrecht. 10.1007/978-94-011-5300-3Search in Google Scholar

[3] Houšková, K. (2023). The Interplay of Triangular Norms and Their Generators. BSc. Thesis, Czech Technical University in Prague, Czech Republic. Search in Google Scholar

[4] Houšková, K., & Navara, M. (2023). Relations between the shapes of triangular norms and their generators. In: Durante, F., Saminger-Platz, S., Trutschnig, W., Vetterlein, T. (eds.) Copulas. Theory and Applications. 40th Linz Seminar on Fuzzy Set Theory (pp. 42–46), Linz, Austria: Johannes Kepler University. Search in Google Scholar

[5] Klement, E. P., Mesiar, R., & Pap, E. (2000). Triangular Norms. vol. 8 of Trends in Logic. Dordrecht, Netherlands: Kluwer Academic Publishers. 10.1007/978-94-015-9540-7Search in Google Scholar

[6] Ling, C. M. (1965). Representation of associative functions. Publ. Math. Debrecen, 12, 189–212. 10.5486/PMD.1965.12.1-4.19Search in Google Scholar

[7] Lowen, R. (2012). Fuzzy Set Theory: Basic Concepts, Techniques and Bibliography. Dordrecht: Springer Science & Business Media. Search in Google Scholar

[8] Marko, V., & Mesiar, R. (2001). Continuous Archimedean t-norms and their bounds. Fuzzy Sets Systems, 121, 183–190. 10.1016/S0165-0114(00)00032-4Search in Google Scholar

[9] Mostert, P. S., & Shields, A. L. (1957). On the structure of semi-groups on a compact manifold with boundary. Annals of Mathematics, II. Series, 65, 117–143. 10.2307/1969668Search in Google Scholar

[10] Navara, M., & Petrík, M. (2008). Two methods of reconstruction of generators of continuous t-norms. In: Magdalena, L., Ojeda-Aciego, M., Verdegay, J. L. (eds.), 12th International Conference Information Processing and Management of Uncertainty in Knowledge-Based Systems (pp. 1016–1021), Spain: Málaga. Search in Google Scholar

[11] Navara, M., & Petrík, M. (2020). Generators of fuzzy logical operations. In: Nguyen, H. T., Kreinovich, V. (eds.), Algebraic Techniques and Their Use in Describing and Processing Uncertainty, Studies in Computational Intelligence (vol. 878, pp. 89–112), Springer. DOI: https://www.10.1007/978-3-030-38565-1_8. Search in Google Scholar

[12] Nelsen, R. B. (1999). An Introduction to Copulas. Lecture Notes in Statistics (vol. 139), New York: Springer. 10.1007/978-1-4757-3076-0Search in Google Scholar

Received: 2023-11-20

Revised: 2024-03-24

Accepted: 2024-04-30

Published Online: 2024-06-22

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/demo-2024-0004

Keywords for this article

multiplicative generator; additive generator; balanced generator; Frank copula; Ali-Mikhail-Haq copula; extreme value distribution

Creative Commons

BY 4.0