Switching by aggregation operators in
regime-switching models
Radko Mesiar1 2 , Jozef Komornı́k3 , Magda Komornı́ková12 , and Danuša Szökeová1
1
2
3
Faculty of Civil Engineering, Slovak University of Technology, Bratislava,
Slovakia mesiar, magda,szoke@math.sk
UTIA AV CR Prague, Czech Republic
Faculty of Management, Comenius University, Bratislava, Slovakia
Jozef.Komornik@fm.uniba.sk
Summary. A generalization of switching mechanism from a 1-dimensional operator
to k-dimensional operators is proposed and discussed. Two possible approaches based
on a transition function and k-dimensional aggregation operator are introduced.
Several examples are included.
Key words: shape function, switching mechanism, transition function, aggregation
operator
1 Structure of the model
Regime-switching models have been proposed and utilized by many authors investigating financial time series. A readable exposition is presented in [FD00], where the
classes of STAR (Smooth Transition Autoregressive) and LSTAR (Logistic STAR)
models have been introduced.
The following generalization GSTAR of 2-regimes LSTAR models was proposed
in [BKK04] for a time series yt :
yt = Φ1 (B)yt (1 − H(qt )) + Φ2 (B)yt H(qt )
(1)
where Φ1 (B), Φ2 (B) are autoregressive polynomials in the shift operator B, H is a
transition function, i.e., a non-decreasing surjective map of the values of a so-called
threshold variable qt to the interval [0, 1].
Similarly as in case of LSTAR models we assume that qt = yt−d for a suitable
delay d > 0 and H has the form
H(qt ) = h(γ(qt − c)) + 1/2
(2)
where
h : [−∞, ∞] → [−1/2, 1/2]
is a non-decreasing shape function (an odd bijection), c = H −1 (1/2) is the threshold
constant and γ is the smoothness parameter.
Examples of possible shape functions are:
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R.Mesiar, J.Komornı́k, M.Komornı́ková, D.Szökeová
a) logistic function (shifted to h(0) = 0):
h(x) =
1
1
−
1 + e−x
2
(3)
b) cubic spline function (see [BKK04]):
8
1
>
<− 2
h(x) =
1
x
4
>
:1
2
−
1
x3
108
x < −3
−3 ≤ x ≤ 3
x>3
(3a)
2 Switching by aggregation operators
A method of construction of regime-switching models, based on combinations of
shape functions with aggregation operators (for details on aggregation operators
see, e.g. [CMM02]) has been indicated in [Bog05] [KK05]. Recall that a function A :
I k → I, where I is a real interval, is called an aggregation
operator whenever it is non
decreasing in each coordinate and sup A (x) x ∈ I k = sup I, inf A (x) x ∈ I k =
inf I. Moreover, A is called idempotent whenever A (x, . . . , x) = x for each x ∈ I.
The idea is to find an optimal form of transition between two possible regimes
based on information contained in variables yt−1 , . . . , yt−k . Two possible suggested
approaches are:
Case i.
Transform values yt−1 , . . . , yt−k by a fixed transition transformation f : R → [0, 1]
into values ui = f (yt−i ), i = 1, . . . , k and then apply a continuous k-dimensional
aggregation operator A: [0, 1]k → [0, 1], i.e.,
F (yt−1 , . . . , yt−k ) = A(u1 , . . . , uk ) = A(f (yt−1 ), . . . , f (yt−k )) .
(4)
Case ii.
To apply first some continuous aggregation operator B: Rk → R to the observed
values yt−i , i = 1, . . . , k and transform the resulting output by means of a transition
function f : R → [0, 1], i.e.,
F (yt−1 , . . . , yt−k ) = f (B(yt−1 , . . . , yt−k )) .
(5)
Among genuine properties of our generalization we include the requirement of
the idempotency of applied aggregation operators A and B, reflecting the idea that
if yt−k = yt−k+1 = . . . = yt−1 = w, then the switching mechanism depends only on
the value w and the smooth transition function f . In such a case both approaches
(i) and (ii) coincide and collapse into a one-dimensional switching procedure, i.e.,
F (yt−1 , . . . , yt−d ) = F (w, . . . , w) = A(f (w), . . . , f (w)) =
= f (B(w, . . . , w)) = f (w) .
(6)
Moreover, in some distinguished cases of idempotent aggregation operators B:
Rk → R, B and f commute, f ◦ B = B ◦ f . Hence if A: [0, 1]k → [0, 1] is taken as
the restriction of B to the unit interval, A = B | [0, 1]k , we have
Switching by aggregation operators in regime-switching models
1173
B(f (yt−1 ), . . . , f (yt−k )) = f (B(yt−1 ), . . . , B(yt−k )) ,
i.e., both approaches (i) and (ii) coincide.
More details about commuting can be found in [MS03]. In our special case of
commuting of a k-dimensional continuous idempotent aggregation operator B and a
one-dimensional smooth transition function f can be viewed as a generalized Cauchy
equation, compare e.g., [BVD02], i.e., for all (x1 , . . . , xk ) ∈ Rk it holds
B(f (x1 ), . . . , f (xk )) = f (B(x1 , . . . , xk ))
(the original Cauchy equation is based on the sum operator B, f (x1 ) + f (x2 ) =
f (x1 + x2 )).
Among typical continuous idempotent aggregation operator on the real line we
recall some examples (for more details see [CMM02]):
- arithmetic mean M (x1 , . . . , xk ) =
1
k
- weighted means W (x1 , . . . , xk ) =
Pk
i=1
k
P
xi ;
i=1
k
P
wi xi , where the weights wi ∈ [0, 1],
i=1
wi = 1;
- OWA operators W ′ (x1 , . . . , xk ) =
k
P
wi x′i with weights as in the case of
i=1
weighted means, but with x′i as non-decreasing permutation of xi inputs, i.e.,
x′i ≤ . . . ≤ x′k (in this class we can find also M in and M ax operators, corresponding to the extreme cases w1 = 1 and wi = 0 otherwise, or wk = 1 and
wi = 0 otherwise);
- symmetric (Šipoš) and asymmetric (Denneberg) Choquet integral (more details
can be found in [Den94]).
Among all those operators, only projections (as special weighted means) and
order statistics operators (as special cases of OWA operators) commute with an
arbitrary transition function f . Observe that for a continuous aggregation operators
B, f ◦ B = B ◦ f hold for each transition function f iff B is a lattice polynomial
(see [MR04]). In all other cases, the two possible switching mechanism approaches
differ, in general.
Recall that the shape functions were supposed to be odd, and thus each transition function f is necessarily symmetric with respect to the point (a, 1/2), where
f (a) = 1/2. We expect similar properties from aggregation operators applied for
constructing a transition function F .
Definition 1. Let k ∈ N . Then F : Rk → [0, 1] is called a k-transition function
whenever it is continuous, non-decreasing, Ran F ⊇ ]0, 1[, and there is a constant
a ∈ R such that F (a, . . . , a) = 1/2 and for any X ∈ Rk , F (X) = 1 − F (2a − X)(i.e.,
graph of F is symmetric with respect to the point (a, . . . , a, 1/2)) . Constant a is
called the symmetry point of F .
Recall that, given a shape function g: [−∞, +∞] → [−1/2, 1/2] a 1-transition
function f can be introduced, fixing parameters γ > 0 and a ∈ R, by f (x) =
g(γ(x − a)) + 1/2. Vice versa, from any 1-transition function f with symmetry point
a we can deduce a shape function g putting g(y) = f (a + y/γ) − 1/2 for y ∈ R (and
1174
R.Mesiar, J.Komornı́k, M.Komornı́ková, D.Szökeová
g(−∞) = −1/2, g(+∞) = 1/2), where γ > 0 is a chosen parameter. Observe also
that each unary aggregation operator is just the identity function (differing only in
the domain), and thus for k = 1 both concepts i) and ii) of constructing 1-transition
function coincide. This is no more true for k > 1.
Theorem 1. For k > 1, F : Rk → [0, 1] is a k-transition function constructed by
concept i), i.e., F = A ◦ (f x . . . xf ), where A: [0, 1]k → [0, 1] is an idempotent
continuous aggregation operator and f : R → [0, 1] is a 1-transition function, if and
only if A is self-dual, i. e., A(1 − X) = 1 − A(X) for all X ∈ [0, 1]k . Moreover, the
symmetry point aF of F and af of f coincide, aF = af .
Proof. Let F be a k-transition function with symmetry point aF . Then
1/2 = F (aF , . . . , aF ) = A(f (aF ), . . . , f (aF )) = f (aF ),
and
f (x) = A(f (x), . . . , f (x)) = F (x, . . . , x) = 1 − F (2aF − x, . . . , 2aF − x) =
= 1 − A(f (2aF − x), . . . , f (2aF − x)) = 1 − f (2aF − x)
for all x ∈ R, i.e. aF is also symmetry point of f .
However, then for any (x1 , . . . , xk ) ∈ Rk we have
A(1 − f (x1 ), . . . , 1 − f (xk )) = A(f (2aF − x1 ), . . . , f (2aF − xk )) =
= F (2aF − x1 , . . . , 2aF − xk ) = 1 − F (x1 , . . . , xk ) =
= 1 − A(f (x1 ), . . . , f (xk )).
As far as Ranf ⊇ ]0, 1[ and A is continuous, this results to the self-duality of A.
Vice-versa, let f be a 1-transition function with symmetry point af and let
A: [0, 1]k → [0, 1] be a continuous idempotent self-dual aggregation operator.
Then F = A ◦ (f x . . . xf ) is non-decreasing, continuous, RanF ⊇ ]0, 1[. Moreover,
F (af , . . . , af ) = A(f (af ), . . . , f (af )) = A(1/2, . . . , 1/2) = 1/2 (due to the idempotency, but also self-duality of A) and for all (x1 , . . . , xk ) ∈ Rk it holds
F (x1 , . . . , xk ) = A(f (x1 ), . . . , f (xk )) = 1 − A(1 − f (x1 ), . . . , 1 − f (xk )) =
= 1 − A(f (2af − x1 ), . . . , f (2af − xk )) = 1 − F (2af − x1 , . . . , 2af − xk ),
i.e., F is a k-transition function with symmetry point af .
⊔
⊓
Self-dual aggregation operators are discussed, for example, in [CMM02]. Idempotent continuous self-dual aggregation operators can be built from continuous idempotent aggregation operators (for example, integrals of Choquet, Sugeno, Shilkret,
etc., based on some monotone normal set function vanishing in the empty set) by
means of several construction methods. [Sil79] has proposed, starting from a continuous idempotent aggregation operator C: [0, 1]k → [0, 1], to define a self-dual
continuous idempotent aggregation operator A: [0, 1]k → [0, 1] by
A(x1 , . . . , xk ) =
C(x1 , . . . , xk )
C(x1 , . . . , xk ) + C(1 − x1 , . . . , 1 − xk )
with the convention 0/0 = 1/2.
For a deeper discussion on construction of self-dual aggregation operators we recommend [SM04]. Typical examples of self-dual continuous idempotent aggregation
operators A: [0, 1]k → [0, 1] are:
Switching by aggregation operators in regime-switching models
1. A = W the weighted mean, i.e., A(x1 , . . . , xk ) =
k
P
1175
wi xi ;
i=1
2. symetric OWA operator (derived from any OWA operator C by Lapresta approach, A(X) = 21 (C(X) + 1 − C(1 − X)) ) A(x1 , . . . , xk ) =
k
P
wi x′i , see above,
i=1
where wi = wk−i+1 for all i = 1, . . . , k (note that arithmetic means is here a
special case when w1 = . . . = wk = 1/k);
3. Operators A(x1 , . . . , xk ) =
min(x1 ,...,xk )
min(x1 ,...,xk )+1−max(x1 ,...,xk )
=
x′k
x′k +1−x′1
x′1
x′1 +1−x′k
and
(with the convention 0/0 = 1/2, derived from M in
A(x1 , . . . , xk ) =
and M ax operators, respectively, by Silvert approach); the first operator is not
continuous in points (x1 , . . . , xk ) such that {0, 1} ⊆ {x1 , . . . , xk }.
Theorem 2. For k > 1, F : Rk → [0, 1] is a k-transition function with symmetry
point aF constructed by concept ii) by means of a strictly monotone function f , i.e.,
F = f ◦B, where f is a strictly increasing 1-transition function and B: Rk → [0, 1] is
a continuous idempotent aggregation operator, if and only if aF = af is the symmetry
point of f and B, i.e.,
B(ξ) = 2aF − B(2aF − ξ) for all ξ ∈ Rk .
Proof. Let F = f ◦ B is a k-transition function with symmetry point aF . Then
1/2 = F (aF , . . . , aF ) = f (B(aF , . . . , aF )) = f (aF ) and
f (x) = f (B(x, . . . , x)) = F (x, . . . , x) = 1 − F (2aF − x, . . . , 2aF − x) =
= 1 − f (B(2aF − x, . . . , 2aF − x)) = 1 − f (2aF − x),
i.e., aF is the symmetry point of f . Moreover, for any (x1 , . . . , xk ) ∈ Rk ,
f (B(x1 , . . . , xk )) = F (x1 , . . . , xk ) = 1 − f (B(2aF − x1 , . . . , 2aF − xk )) =
= f (2aF − B(2aF − x1 , . . . , 2aF − xk )).
Due to the strict monotonicity of f ,
B(x1 , . . . , xk ) = 2aF − B(2aF − x1 , . . . , 2aF − xk ).
Observe that the sufficiency is true for arbitrary 1-transition function f . Namely,
let f be a 1-transition function with symmetry point af and let B: Rk → R be a
continuous idempotent aggregation operator with symmetry point af . Put F = f ◦B.
Then F (af , . . . , af ) = f (B(af , . . . , af )) = f (af ) = 1/2, and for any (x1 , . . . , xk ) ∈
Rk ,
F (x1 , . . . , xk ) = f (B(x1 , . . . , xk )) = f (2af − B(2af − x1 , . . . , 2af − xk )) =
= 1 − f (B(2af − x1 , . . . , 2af − xk )) = 1 − F (2af − x1 , . . . , 2af − xk ),
i.e., F is a k-transition function with symmetry point af .
⊔
⊓
The symmetry of an aggregation operator B allows its simple representation.
Lemma 1. A continuous idempotent aggregation operator B: Rk → R has a as symmetry point if and only if there is a continuous idempotent odd aggregation operator
C: Rk → R, C(−ξ) = −C(ξ), such that
B(ξ) = a + C(ξ − d), ξ ∈ Rk .
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R.Mesiar, J.Komornı́k, M.Komornı́ková, D.Szökeová
Proof. To see the sufficiency, only symmetry of B with respect to the a has to be
checked. However, for any ξ ∈ Rk we have
B(2a − ξ) = a + C(a − ξ) = a − C(ξ − d) = 2a − (a + C(ξ − a)) = 2a − B(ξ).
For the necessity, put C(X) = B(X + a) − a. Then
C(−ξ) = B(−ξ + a) − a = 2a − B(2a − (−ξ + a)) − a = a − B(ξ + a) = −C(ξ).
The idempotency and continuity of C is obvious.
⊔
⊓
Typical examples of continuous idempotent odd aggregation operators C: Rk →
R are:
1. C = M the arithmetic mean,
2. Šipoš integral (i. e., symmetric Choquet integral) with respect to any fuzzy
measure µ, see [Sip79] [Den94] [Pap95].
3. C(x1 , . . . , xk ) = med(min xi , max xi , 0), where med denotes Median.
Conclusion
We have introduced and discussed a switching mechanism for regime-switching time
series models based on latest k observed values, thus generalizing STAR models corresponding to k = 1.Two approaches based on k−dimensional aggregation operators
and transition functions were given and their properties were discussed. Offered modelling of non-linear time series extends the backlog of possible models to be fitted to
real data.
Acknowledgement
The research summarized in this paper was partly supported by the Grants APVT20-003204, VEGA 1/3006/06 and GAČR 402/04/1026.
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