A class of universal approximators of real continuous ...

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A class of universal approximators of real continuous functions revisited C. Siettos · F. Giannino L. Russo · S. Cuomo

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Abstract We revisit the celebrated theorem that it is possible to approximate with any accuracy any real continuous function in the case of certain classes of systems defined by general rules. In other words, these systems are universal approximators. We review the key works that have proved this concept, highlighting their limitations and providing yet another proof that it is not restricted by the Gaussian expression of the membership functions. We also show how one can go inversely to approximate these systems with a series of polynomials. This provides us with analytical relations of these kind of systems which can facilitate a series of important analysis tasks such as stability analysis and design of controllers. Keywords Approximation of Continuous function · Complex systems · Chebyshev polynomials 1 Introduction Let X ⊆ IRn be the set of points x = {x1 , x2 , . . . , xn } and Y ⊆ IRm the set of points y = {y1 , x2 , . . . , ym }. In this paper we focus on maps y = f (x),

x ∈ X, y ∈ Y

Constantinos Siettos School of Applied Mathematics and Physical Sciences, National Technical University of Athens, Greece. E-mail: [email protected] Francesco Giannino Dipartimento di Agraria, University of Naples Federico II, Naples, Italy Lucia Russo Consiglio della Ricerca, Naples, Italy Salvatore Cuomo Dipartimento di Matematica e Applicazioni, University of Naples Federico II, Naples, Italy Corresponding authors: C. Siettos and S. Cuomo

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that describe relational systems defined by general rules. Here we schematically consider this kind of system by defining a space: F := (R, D) where R is a basic rule, D are the data. On F it is possible to define: i) F U a process to apply R to D (input) data; ii) IF an inference process; iii) DF a process that gives (output) data. For maps defined in this sense, we can adapt nice notions about the approximation of continuous functions. By a practical point of view F is a fuzzy system, denoted in the following as F-system. More in details, a F-system contains the unit F U , the rule-knowledge base which contains the fuzzy rules and the parameters for the definition of fuzzy sets and their membership functions, the inference engine IF and the unit DF (Zimmermann, 2001, Pappis and Siettos, 2005). Below we briefly explain the reasons behind our research. The extraction of mathematical models from first principles in the form of Ordinary and/or Partial Differential Equations for the efficient description of real world problems and processes is not a trivial task due the inherent complexity and ambiguity of our ability to translate mathematically the observed dynamics. In these cases one turns into techniques from time-series analysis to build black-box models in the form of both linear models such as autoregressive (AR), autoregressive exogenous and autoregression moving average (ARMAX) and nonlinear ones such as artificial neural networks. A methodology which relies on this basis is the celebrated Grange-Causality analysis first proposed by Clive Granger (Nobel Prize in Economics in 2003) in 1969 (Granger, 1969). The method fits a vector autoregressive model to find causal directed relations between time series. Its applications are numerous especially in financial systems (Granger, 2009) and neuroscience (Seth et al., 2015). However these approaches offer just a black-box representation of the governing systems dynamics, hence limiting our capability in integrating a-priori qualitative knowledge and expertise that could be available in describing system behavior and dynamics. Fuzzy Logic as introduced by Lotfi Zadeh back with two papers in 1965 and 1973 came to bridge that missing component, i.e. turning qualitative knowledge into mathematical models. In his seminal papers, Zadeh stated that as the complexity of a system increases, our ability to make precise and yet significant statements about its behavior diminishes until a threshold is reached beyond which precision and significance (or relevance) become almost mutually exclusive characteristics. Since then research in fuzzy logic theory and the development of applications based on fuzzy logic has been constantly increased. Theory and applications go hand-in-hand covering a wide range of disciplines. The theory of fuzzy sets is directed towards various disciplines of mathematics and statistics, including fuzzy operators (Pradera, et al., 2002); fuzzy relations (Pedrycz and Vasilakos, 2002), nonclassical logic (Biacino and Gerla, 2002; Novak, 2002), algebra (Di Nola, et al., 2002); topology (Albrecht, 2003). Other disciplines include economics and

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management (Lee, 1990; Gil-Lafuente, 2005); robotics (Ruan, et al., 2003); transportation engineering (Liang and Chen, 2009), food engineering (Perott et al., 2006), environmental science (Esogbue et al., 1992), nuclear engineering (Kunsch an Fortemps, 2002), neuroscience (Reyna et al., 2011; Balasubramanian and Obeid, 2011), neural networks (Flores, J.J., et. al. 2015), biology (Blanco, et al., 2002), pharmacology (Kilic, 2002) and epidemiology (Massad et al., 2008) to name just a few. For a review on fuzzy sets and systems please see (Pappis and Siettos, 2005). The power of F-systems comes from the fact that they are universal approximates in the sense that they can approximate in any accuracy any continuous nonlinear function according to a theorem presented by Wang and Mendel (1992). In this proof, Wang assumed a Gaussian form of the membership functions of the fuzzy sets, product inference and centroid defuzzification. In 1994 Kosko relaxed the hypothesis of the Gaussian form by replacing triangles, trapezoids and any other fuzzy sets by rectangles and introducing the concept of weighted additive functions. A critical review can be found in Klement et al. (1999). Ying (1998) proved that Tagaki-Sugeno F-systems are universal approximators. Ying and Chen (1999) provided necessary conditions for single-input single-output F-systems and a class of typical multiple-input single-output F-systems as universal approximators for continuous functions defined on compact domains with arbitrarily small uniform approximation error bounds. Zeng et al., 2000 gave the sufficient conditions for TS systems. Kreinovich et al. (2000) proved that F-systems are universal approximators for smooth functions and their derivatives. Here we give yet another proof that multiple input single output F-systems can approximate in any accuracy real continuous function and show also the inverse task, i.e. how one can derive analytical relations to approximate F-systems. The proof does not limit to Gaussian fuzzy sets and the extension to multiple input multiple output systems is straightforward.

2 F -systems as Universal Approximators: the proof Let X ⊆ IRn and Y ⊆ IRm as in the previous section. Conceptually, in the space F := (R, D) a system is an input-output map of the form y = f (x),

x ∈ X, y ∈ Y

(1)

Maps in F, through some functions called membership functions, are defined in the interval [0, 1] to linguistic/qualtitative inputs called fuzzy sets as negative, positive or large and small. The rule base incorporates the relations between the input-output variables, usually in the form of If-Then implications. As practical example, in a computing system for a (fuzzy) controller with n inputs and m outputs, a (fuzzy) rule reads as follows: Rule i): if x1 is FAi and x2 is FAi then:

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y1 is FCi where FAi and FCi are linguistic terms describing the input-output variables whose membership functions are defined by the fuzzy sets A and B. The fuzzy implication executes several fuzzy operations for the inference of the fuzzy outputs from the fuzzy inputs. Finally, maps defined in DF fuzzy outputs on the vector y. The most common used techniques in DF is the centroid, which calculates the center of mass of the resulting fuzzy space as obtained by the impact of all rules that are activated by a certain input as: R yµFC (y, x)dy (2) f (x) = RB µ (y, x)dy B FC The derived membership function µFC (y, x) depends on x, the membership function of the input the corresponds to the fuzzy set FAi and the inference method, i.e.: µFC (y) = g(y, x, Ψ, IF ) Ψ = {µFAi , i = 1, 2, . . . , n} and IF is the inference method. Given the membership function and the inference method µFC (y) is by construction a continuous function of x. In addition, due to the construction of the membership functions of fuzzy sets FCi , the inferred membership function µFC (y, x) is finite on the open set B ⊆ IRm , hence each µFC (y, x) has a bounded support set that belongs to B. In what follows we prove some important properties of the fuzzy map y = f (x). Proposition 1 Let A ⊆ IRn , B ⊆ IR and the fuzzy function f : A → B. Then the fuzzy function f (x) given by Eq. (1) is continuous in A. Proof. Let x, x0 ∈ A with (x 6= x0), and the modulus of continuity of the function of yµFC (y, x) in the set B be ω(δ, yµFC (y, x)). Because every µFC (y, x) is finite, then the function yµFC is bounded in B; hence ω(δ, yµFC (y, x)) → 0 when δ → 0. Hence: R R yρFC (y, x0 )dy yµFC (y, x)dy B − RB = f (x) − f (x0 ) = R µ (y, x)dy ρ (y, x0 )dy B FC B FC R yτFC (y, x, x0 )dy = RB , τ (y, x, x0 )dy B FC with: τFC (y, x, x0 ) = µFC (y, x) − ρFC (y, x0 ) and R yτ (y, x )dy R |yτ (y, x )|dy FC 0 FC 0 ≤ |f (x) − f (x0 )| = RB ≤ RB τ (y, x )dy |τ (y, x )|dy F 0 F 0 C C B B R ω(|x − x0 |, yτFC )dy ≤ B R = |τ (y)|dy B FC

A class of universal approximators of real continuous functions revisited

=

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ω(|x − x0 |, yτFC ) R (sup y − inf y) → 0 when x → x0 . y∈B |τ (y)| y∈B B FC

So the function f : A → B is continuous. t u

Proposition 2 Let A ⊆ IRn , B ⊆ IR and ΦA is the set of all fuzzy maps f : A → B. Then ΦA is aLlinear space of A on B concerning the operations of (a) addition defined by h σ = h, with h(x) N= f (x) + σ(x), ∀x ∈ A , and (b) the multiplication of numbers defined by λ f = φ, with φ(x) = λf (x) ∀x ∈ A. Proof. For the proof of the above propositions the following properties should hold true: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

f1 + f2 ∈ ΦA ∀f1 , f2 ∈ ΦA f1 + f2 = f2 + f1 , ∀f1 , f2 ∈ ΦA f1 + (f2 + f3 ) = (f1 + f2 ) + f3 , ∀f1 , f2 ∈ ΦA f + 0 = f ∀f ∈ ΦA f + (−f ) = 0 ∀f ∈ ΦA λf ∈ ΦA ∀λ ∈ IR, f ∈ ΦA (λ + µ)f = λf + µf ∀λ, µ ∈ IR, f ∈ ΦA λ(f1 + f2 ) = λf1 + λf2 ∀λ ∈ IR, f1 , f2 ∈ ΦA λ(µf ) = (λµ)f ∀λ, µ ∈ IR, f ∈ ΦA 1f = f ∀f ∈ ΦA

It is obvious that properties 2,3,4,5,7,8,9,10 hold true for all F-systems of the form of Eq. (3). For the first property we have: R R zρFC (z, x)dz yµFC (y, x)dy , f2 (x) = RB f1 (x) = RB µ ρ (z, x)dz (y, x)dy B FC B FC Hence, R

R R R ρFC (z, x)dz + B zρFC (z, x)dz B µFC (y, x)dy B R R f1 (x)+f2 (x) = µ (y, x)dy B ρFC (z, x)dz B FC (3) From the mean value theorem we have that B

yµFC (y, x)dy

Z ∃ς, σ ∈ B :

Z

and

Z

B

z∈B

z∈B

Z ρFC (z, x) = σρFC (ς)(sup z − inf z)

zρFC (z, x)dz = σ B

dz = ρFC (ς)(sup z − inf z)

ρFC (z, x)dz = ρFC (ς) B

B

z∈B

Importing (4) in (3) and setting w = y + σ, Eq. (3) becomes:

z∈B

(4)

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R wτFC (w, x)dw f1 (x) + f2 (x) = RB τ (w, x)dw B FC

with

τFC (w) = µFC (w − σ, x).

(5)

Because Eq. (5) has the same form with Eq. (2) we get that f1 (x)+f2 (x) ∈ ΦA . For the 6-th property we have: R R yµFC (y, x)dy wτFC (w, x)dw B λf (x) = λ R = B , w = λy. τFC (w, x)dw µ (y, x)dy B FC Hence λf (x) ∈ ΦA . t u

So the set ΦA is a linear space in B. The above proposition can be extended in a straightforward manner in Rn × IRm , assuming independence of the input and output variables. In this case the inference rules read: : if x1 is FA1 and x2 is FA2 and, . . . , xn is FAn then y1 is FC1 if x1 is FA0 1 and x2 is FA0 2 and, . . . , xn is FA0 n then y2 is FC2 ... ... ... ... ... (µ) (µ) (µ) if x1 is FA1 and x2 is FA2 and, . . . , xn is FAn then ym is FCm . Lemma 1 The set of all fuzzy maps f : A → B, ΦA , is a metric linear space. Proof. From proposition 2, we have that ΦA is a linear space. ∀f1 , f2 , f3 ∈ ΦA a real number is defined as d(f1 , f2 ) = |f1 − f2 |. It can be easily proved that the number d(f1 , f2 ) satisfies the following properties: (p1 ) d(f1 , f2 ) = 0 ⇔ f1 = f2 . (p2 ) d(f1 , f2 ) = d(f2 , f1 ). (p3 ) d(f1 , f2 ) ≤ d(f1 , f3 ) + d(f3 , f2 ). In fact, relation (p3 ) expresses the known triangle inequality. Consequently, the number d consists a metric of ΦA and so the linear space is a metric linear space. t u

Proposition 3 The metric linear space ΦA is an algebra in A i.e. the following hold: (a) f1 + f2 ∈ ΦA , (b) f1 f˙2 ∈ ΦA , and (c) f ∈ ΦA , ∀f, f1 , f2 ∈ ΦA and λ ∈ IR. Proof. As the space ΦA is linear, isR IR, conditions (a) and (c) hold true R zρ (z,x)dz yµ (y,x)dy f, f1 , f2 ∈ ΦA and λ ∈ IR. Let f1 (x) = RB µFFC(y,x)dy and f2 (x) = RB ρFFC(z,x)dz B B C C with f1 , f2 ∈ ΦA . Then: R R yµFC (y, x)dy B zρFC (z, x)dz B f1 (x)f2 (x) = R · R (6) µ (y, x)dy ρ (z, x)dz B FC B FC

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Implement the mean value theorem as in the proof of proposition 1, Eq. (6) reads: R wτFC (w, x)dw f1 (x)f2 (x) = RB , w = σy, τFC (w, x) = µFC (w/σ, x) τ (w, x)dw B FC

(7)

Hence, f1 f2 ∈ ΦA , ∀f1 , f2 ∈ ΦA and the space P hiA is an algebra in A. t u

Proposition 4 The linear space of fuzzy functions ΦA (a) separates the points in A, i.e. ∀x1 , x2 ∈ A, x1 6= x2 , ∃f ∈ ΦA : f (x1 ) 6= f (x2 ) and (b) it does not become zero in any point in IRn , i.e. ∀x ∈ A∃f ∈ ΦA : f (x) 6= 0. Proof. (a) Let that ∃x1 , x2 ∈ A : x1 6= x2 ⇒ f (x1 ) = f (x2 ). Then R R yρFC (y, x2 )dy yµFC (y, x1 )dy B R = RB ⇒ µ (y, x1 )dy ρ (y, x2 )dy B FC B FC

⇒

σµFC (ς)(supy∈B y − inf y∈B y) rρFC (ζ)(supy∈B y − inf y∈B y) = ⇒ µFC (ς)(supy∈B y − inf y∈B y) ρFC (ζ)(supy∈B y − inf y∈B y) Z

⇒σ=r⇒

Z yρFC (y, x2 )dy ⇒ µFC (y, x1 ) = ρFC (y, x2 )

yµFC (y, x1 )dy = B

B

(8) But: µFC = g(y, x1 , Ψ, IF ) and ρFC = g(y, x2 , Ψ, IF ) with Ψ = {µFAi , i = 1, 2, . . . , n} and IF is the inference method. Using the same membership function and the same inference method Eq. (8) gives x1 = x2 , which contradicts the initial assumption that x1 6= x2 . (b) Because every membership function µ(y, x1 ) is non-negative ∀y ∈ B, choosing y so that y ∈ IR+ , then ∃f ∈ ΦA : f (x) > 0, ∀x ∈ A. Hence the set ΦA does not become zero in any point in A. t u

Theorem 1 Every fuzzy function f of ΦA can be approximated by accuracy with a real continuous function.

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Proof. The linear metric space ΦA . (a) is an algebra in A (proposition 3), (b) separates the points in A, and (c) does not become zero in any point of A (proposition 4). t u Hence according to the Stone-Weierstrass Theorem (Rudin, 1964), ΦA consists of all real functions in A. Based on the above, given a fuzzy continuous function f on A, and > 0, there is a real function q : A → B : |f (x) − q(x)| ≤ . 3 Approximating F -systems with Polynomials According to the Weierstrass Theorem (Rudin, 1964) , if f is a continuous non-linear function in [a, b], the there is a polynomial P (x) such that: ∀x ∈ [a, b], ∃ > 0 : ||P (x) − f (x)|| ≤ .

(9)

Hence every f ∈ C(a, b) and thus every F-system can be approximated with any accuracy from a series of polynomials, i.e.: f (x) =

∞ X

ci pi (x)

i=0

where pi (x) are continuous polynomials of i-th order. However for any practical means of particular interest is the use of finite number of polynomials for the approximation of f (x), i.e. as: prm (x) =

∞ X

ci pi (x)

i=0

In this case the aim is the calculation of the minimum order m so that the polynomial prm (x) to adequately approximate f (x). The desired accuracy of approximation, say dopt , can be expressed in terms of maximum variance of prm (x) from f (x), that is dopt = min{max|f (x) − prm (x)|}. As proposed in (Siettos et al., 2002) an optimal choice is to use Chebyshev polynomials of first kind. Chebyshev polynomials of n-th order are given by (Draper and Smith, 1981; Achieser, 1992): Tn (x) = cos[n cos−1 (x)] Let us consider a two-input one-output F-system: A ⊆ IR2 , B ⊆ IR and a continuous fuzzy map f : A → B. Then the fuzzy map can be approximated as a generalized Fourier series of the form (Siettos et al., 1999) f (x1 , x2 ) =

∞ X ∞ X r=1 s=1

crs φr (x1 )θs (x2 ),

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where crs are the generalized Fourier coefficients, φr (x1 ) ∈ C(a, b) are orthogonal weighted basis functions subject to the weight function w(x1 ), θs (x2 ) ∈ C(c, d) are orthogonal weighted basis functions subject to the weight function u(x2 ). It can be proved that the coefficients crs are given by: Z crs =

d

u(x2 )θs (x2 ) c

hZ

b

i f (x1 , x2 )w(x1 )φr (x1 )dx1 dx2

(r, s = 1, 2, · · · )

a

Chebyshev polynomials are orthogonal in [−1, 1] subject to the weight functions w(x) = √

1 . 1 − x2

4 Conclusion In this work we revised the celebrated theorem that F-systems are universal approximators. We provided another proof that is not limited to Gaussian fuzzy sets. The proof applied for Mamdani-type F-systems using centroid as defuzzification technique. We also showed the inverse task based on the Weierstrass Theorem, i.e. how one can use (Chebyshev) polynomials to approximate F-systems and therefore derive analytical expressions that can be used for stability analysis and the design of controllers based on nonlinear theory tools.

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