FUZZY TRANSFORM: Application to Reef Growth Problem. Irina Perfilieva

Transcription

1 FUZZY TRANSFORM: Application to Reef Growth Problem Irina Perfilieva University of Ostrava Institute for research and applications of fuzzy modeling DAR Research Center Ostrava, Czech Republic

2 OUTLINE Basic Functions F-Transforms: Direct and Inverse Properties of F-Transforms F-Transform Numeric Integration F-Transform Euler and Euler-Cauchy Methods for the Cauchy Problem F-Transform Euler Method for the Cauchy Problem with Noise Reef Growth Model and the Sea Level History

3 1. BASIC FUNCTIONS The basic functions play the role of fuzzy points or granules. [a, b] a fixed interval, h = (b a)/(n 1), n 2 x k = a+h(k 1) nodes on [a, b], k = 1,..., n. Definition 1 Functions A 1 (x),..., A n (x) defined on [a, b], are basic if A k : [a, b] [0,1], A k (x k ) = 1 A k (x) = 0 if x (x k 1, x k+1 ) where x 1 = a, x n+1 = b A k (x) is continuous A k (x) ր on [x k 1, x k ] A k (x) ց on [x k, x k+1 ] n k=1 A k (x) = 1, x [a, b] A k (x k x) = A k (x k + x), x [a, b] A k+1 (x) = A k (x h), x [a, b]

4 BASIC FUNCTIONS Triangular shaped basic functions Sinusoidal shaped basic functions

5 DEFINITE INTEGRALS OF BASIC FUNCTIONS A 1 (x),..., A n (x) basic functions on [a, b] x2 a A 1 (x)dx = h 2 b x n 1 A n (x)dx = h 2 and for k = 2,..., n 1 xk+1 x k 1 A k (x)dx = h

6 2. F-TRANSFORMS: DIRECT AND INVERSE f(x) a continuous function on [a, b] A 1 (x),..., A n (x) basic functions. Definition 2 The n-tuple of real numbers [F 1,..., F n ] is an F-transform of f w.r.t. A 1 (x),..., A n (x) if F k = ba f(x)a k (x)dx ba A k (x)dx Denote the F-transform of f by We have F n [f] = [F 1,..., F n ]. x2 F 1 = 2 h x 1 f(x)a 1 (x)dx F k = 1 x k+1 h x k 1 f(x)a k (x)dx, 1. F n = 2 xn h x n 1 f(x)a n (x)dx k = 2,..., n

7 F-TRANSFORM OF A PARTIAL FUNCTION Let f(x) be known at nodes x 1,..., x l [a, b], and A 1 (x),..., A n (x) be basic functions. Then the F-transform components of f can be computed as follows: F k = lj=1 f(x j )A k (x j ) lj=1 A k (x j ) where 1 k n and n < l.

8 INVERSE F-TRANSFORM Definition 3 Let [F 1,..., F n ] be the F-transform of f(x) w.r.t. A 1 (x),..., A n (x). The function f F,n (x) = n k=1 F k A k (x) is called the inverse F-transform. Inverse F-transform special singleton model! Lemma 1 Let f(x) continuous function on [a, b], A (n) 1 (x),..., A(n) n (x) basic functions, n 2, {f F,n (x)} the sequence of inverse F-transforms w.r.t. given basic functions. Then for any ε > 0 there exists n ε such that for each n > n ε and for all x [a, b] f(x) f F,n (x) < ε. Uniform convergence : f F,n (x) n f(x).

9 UNIFORM CONVERGENCE n = 5-1 n = 20 Inverse Transforms of sin(1/x) Based on Triangular Shaped Basic Functions n = 4-1 n = 20 Inverse Transforms of sin(1/x) Based on Sinusoidal Shaped Basic Functions

10 CONVINCING UNIFORM CONVERGENCE sin(x) and its Inverse Transform Based on Triangular Shaped Basic Functions sin(x) and its Inverse Transform Based on Sinusoidal Shaped Basic Functions

11 INDEPENDENCE ON SHAPES OF BASIC FUNCTIONS Lemma 2 Let f F,n (x) and f F,n (x) be two inverse F-transforms of f(x) w.r.t. n-tuples of different basic functions. Then f F,n (x) f F,n (x) 2ω(f,2h) where ω is the modulus of continuity of f(x): ω(f,2h) = max max f(x + δ) f(x). δ 2h x [a,b]

12 INDEPENDENCE ON SHAPES. ILLUSTRATION f F,n (x) and f F,n (x) for Triangular and Sinusoidal Basic Functions

13 3. REMOVING NOISE BY F-TRANSFORM Theorem 1 Let f(x) = sin ω(x + ϕ) and x [a, b], basic functions have triangular or sinusoidal shapes, ω = 2πk h for some k N, ϕ R, 2h the length of supports of basic functions. Then for both F-transforms of f F i =0, i = 2,..., n 1, F 1 O( 1 πk ), F n O( 1 πk ).

14 Removing Noise Illustration Sinusoidal Noise Random Noise

15 5. NUMERICAL INTEGRATION Theorem 2 Let f(x) continuous function on [a, b], f F,n (x) the inverse F-transform w.r.t. some n-tuple of basic functions. Then f(x) and f F,n (x) have equal integral average values 1 b a b a f(x)dx = 1 b a b a f F,n(x)dx = = 1 n 1 (F F F n 1 + F n 2 ).

16 6. NUMERICAL SOLUTION TO CAUCHY PROBLEM The Cauchy problem y (x) = f(x, y), y(x 1 ) = y 1 can be approximately solved if an F-transform is applied to both sides of the differential equation. The following scheme Y 1 = y 1, Y k+1 = Y k + hˆf k, k = 1,..., n 1. where ˆF k = ba f(x, Y k )A k (x)dx ba. A k (x)dx generalizes the Euler method.

17 The inverse F-transform y Y,n (x) = n k=1 Y k A k (x) approximates the solution y(x). The local approximation error has the order h 2.

18 Euler Method - Illustration The precise solution (grey line) and the two approximate ones obtained by the classical (numeric) Euler method and the generalized (F-transform) Euler method for the Cauchy problem with f(x, y) = x 2 y The number of nodes n = 7, the order of precision 10 1.

19 THE GENERALIZED EULER-CAUCHY METHOD The following scheme computes the components of F-transform of y(x) w.r.t. some basic functions A 1,..., A n according to the generalized Euler-Cauchy method. Y 1 = y 1, Y k+1 = Y k + hˆf k, Y k+1 = Y k + h 2 (ˆF k + ˆF k+1 ), k = 1,..., n 1 where ˆF k = ba f(x, Ŷ k )A k (x)dx ba, A k (x)dx ba ˆF k+1 = f(x, Ŷk+1 )A k+1(x)dx ba. A k+1 (x)dx Local approximation error has the order h 3.

20 EULER-CAUCHY Method - Illustration The precise solution (grey line) and the approximate one obtained by the generalized (F-transform) Euler-Cauchy method for the Cauchy problem with f(x, y) = x 2 y The number of nodes n = 10.

21 7. NUMERICAL SOLUTION TO CAUCHY PROBLEM WITH NOISE Consider the Cauchy problem with noisy right-hand side y (x) = f(x, y) + sin(ωx + ϕ), y(x 1 ) = y 1. The classical numeric methods are unstable: Number of nodes n = 7 and n = 10.

22 The number of nodes n = 10 and n = 13.

23 F-Transform Methods for the Noisy Cauchy Problem F-transform methods are stable Number of nodes n = 7. Comparison of classical numerical and F-transform methods Number of nodes n = 7.

24 F-Transform Methods for the Normal and Noisy Cauchy Problem The number of nodes n = 7.

25 8. REEF GROWTH MODEL AND THE INVERSE PROBLEM The following differential equation characterizes the reef growth process: dh(t) dt = G m tanh ( I0 I k exp( k[h 0 + h(t) (s 0 + s(t))]) ) where h(t) is the growth increment, h 0 - the initial height, G m - maximal growth rate, I 0 - surface light intensity, I k - saturating light intensity, k - extinction coefficient, s 0 - the initial sea level position and s(t) - sea level variation. As light decreases, so does reef growth.

26 Sea Level Model The sea level curve s(t) of the past years was reconstructed from the numeric data related to the pattern by the use of F-transforms

27 Reef Growth Picture The carbonate production versus depth and distance determined by solving differential equation using the generalized Euler method.

28 Sea Level History Problem: Using a stratigraphic measured section with the growth increment h(t), find the sea level history. h(t) can be taken from: a sequence of thicknesses of various types of rocks divided into respective cycles plus the information about the correspondence between a rock type and a water depth. Rock Rock Cycle Cycle type thickness number thickness

29 Rock Rock Cycle Cycle type thickness number thickness Sea Level Data We solve equation dh(t) dt = G m tanh ( I0 I k exp( k[h 0 + h(t) (s 0 + s(t))]) with respect to unknown s(t) for each cycle and obtain the data which characterizes the sea level history. )

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31 Sea Level Curve Obtained Using F-transform

32 Sea Level Data+Curve

33 Conclusions A new approach to the approximate representation of continuous functions is developed. It includes: - Generalization of points by basic functions - Representation of a function by the vector of its average values (direct F-transform: f F n [f]) - Approximation of a function by inverse F-transform: f F n [f] f F,n. Advantages of this kind of representation (removing noise) are demonstrated

34 Importance of F-transforms to practical applications is demonstrated on examples of the generalized numerical methods: integration and solution to the ordinary differential equations.