On spline based fuzzy transforms

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On spline based fuzzy transforms
Department of Mathematics, University of Latvia

• Irina Vavilcenkova, Svetlana Asmuss
  

ELEVENTH INTERNATIONAL CONFERENCE ON FUZZY SET
THEORY AND APPLICATIONS
Liptovský Ján, Slovak Republic, January 30 -
February 3, 2012
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•  
• This talk is devoted to F-transform
  (fuzzy transform). The core idea of fuzzy
  transform is inwrought with an interval fuzzy
  partitioning into fuzzy subsets, determined by
  their membership functions. In this work we
  consider polynomial splines of degree m and
  defect 1 with respect to the mesh of an interval
  with some additional nodes. The idea of the
  direct F-transform is transformation from a
  function space to a finite dimensional vector
  space. The inverse F-transform is transformation
  back to the function space.

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Fuzzy partitions
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•  

The fuzzy transform was proposed by
I. Perfilieva in 2003 and studied in several
papers 1 I. Perfilieva, Fuzzy transforms
Theory and applications, Fuzzy Sets and Systems,
157 (2006) 9931023.   2 I. Perfilieva, Fuzzy
transforms, in J.F. Peters, et al. (Eds.),
Transactions on Rough Sets II, Lecture Notes
in Computer Science, vol. 3135, 2004, pp. 6381.
  3 L. Stefanini, F- transform with
parametric generalized fuzzy partitions, Fuzzy
Sets and Systems 180 (2011) 98120. 4 I.
Perfilieva, V. Kreinovich, Fuzzy transforms of
higher order approximate derivatives A theorem ,
Fuzzy Sets and Systems 180 (2011) 5568. 5 G.
Patanè, Fuzzy transform and least-squares
approximation Analogies, differences, and
generalizations, Fuzzy Sets and Systems 180
(2011) 4154.
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Classical fuzzy partition
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An uniform fuzzy partition of the interval 0, 9
by fuzzy sets with triangular shaped membership
funcions n7
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Polynomial splines
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Quadratic spline construction
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Cubic spline construction
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An uniform fuzzy partition of the interval 0, 9
by fuzzy sets with cubic spline membership
funcions n7
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On additional nodes
Generally basic functions are specified by the
mesh
but when we construct those functions with the
help of polynomial splines we use additional
nodes.
where k directly depends on the degree of a
spline used for constructing basic functions and
the number of nodes in the original mesh
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Additional nodes for splines with degree
1,2,3...m (uniform partition)
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We consider additional nodes as parameters of
fuzzy partition the shape of basic functions
depends of the choise of additional nodes
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Direct F transform
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Inverse F-transform
The error of the inverse F-transform
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Inverse F-transforms of the test function
f(x)sin(xp/9)
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Inverse F-transform errors in intervals
for the test function f(x)sin(xp/9) in case of
different basic functions, 3/2 step.
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Inverse F-transform and inverse H-transform of
the test function f(x)sin(1/x) in case of linear
spline basic functions, n10 un n20
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Inverse F-transform and inverse H-transform of
the test function f(x)sin(1/x) in case of
quadratic spline basic functions, n10 un n20
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Inverse F-transform and inverse H-transform of
the test function f(x)sin(1/x) in case of cubic
spline basic functions, n10 un n20
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Generalized fuzzy partition
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An uniform fuzzy partition and a fuzzy
m-partition of the interval 0, 9 based on cubic
spline membership funcions, n7, m3
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Numerical example
 
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Numerical example approximation of derivatives
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Approximation of derrivatives
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F-transform and least-squares approximation
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Direct transform
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Direct transform matrix analysis

• Matrix M1 for linear spline basic functions

The evaluation of inverse matrix norm is true
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• Matrix M2 for quadratic spline basic functions

The evaluation of inverse matrix norm is true
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• Matrix M3 for cubic spline basic functions

The evaluation of inverse matrix norm is true
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Direct transform error bounds
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Inverse transform
The error of the inverse transform
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Inverse transforms of the test function
f(x)sin(xp/9) in case of quadratic spline basic
functions, n7
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Inverse transforms approximation errors of the
test function f(x)sin(xp/9) in case of quadratic
spline basic functions, for n10, n20, n40.
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Inverse transforms of the test function
f(x)sin(xp/9) in case of cubic spline basic
functions, n7
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Inverse transforms approximation errors of the
test function f(x)sin(xp/9) in case of cubic
spline basic functions, for n10, n20, n40.
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Thank you for your attention!