Reinforcement Learning Generalization and Function Approximation
Function approximation - Algorithmic Learning 64-360, Part 3d · 2012-06-27 · University of...
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University of Hamburg
MIN Faculty
Department of Informatics
Function approximation
Function approximationAlgorithmic Learning 64-360, Part 3d
Jianwei Zhang
University of HamburgMIN Faculty, Dept. of Informatics
Vogt-Kolln-Str. 30, D-22527 [email protected]
27/06/2012
Zhang 1
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Approximation
Approximation of the relation between x and y (curve, plane,hyperplane ) with a different function, given a limited number ofdata points D = {xi , yi}li=1.
Zhang 2
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Approximation vs. Interpolation
A special case of approximation is interpolation:the model exactly matches all data points.
If many data points are given or measurement data is affected by noise,approximation is preferably used.
Zhang 3
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Approximation without Overfitting
Zhang 4
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Interpolation with Polynomials
Polynomial interpolation:
I Lagrange polynomial,
I Newton polynomial,
I Bernstein polynomial,
I Basis-Splines.
Zhang 5
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Lagrange interpolation
To match l + 1 data points (xi , yi ) (i = 0, 1, . . . , l) with a polynomial ofdegree l , the following approach of LAGRANGE can be used:
pl(x) =l∑
i=0
yiLi (x)
The interpolation polynomial in the Lagrange form is defined as follows:
Li (x) =(x − x0)(x − x1) · · · (x − xi−1)(x − xi+1) · · · (x − xl)
(xi − x0)(xi − x1) · · · (xi − xi−1)(xi − xi+1) · · · (xi − xl)
=
{1 if x = xi0 if x 6= xi
Zhang 6
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Newton Interpolation
The Newton basis polynomials of degree l are constructed as follows:
pl(x) = a0+a1(x−x0)+a2(x−x0)(x−x1)+· · ·+al(x−x0)(x−x1) · · · (x−xl−1)
This approach enables us to calculate the coefficients easily.For n = 2 the following system of equations is obtained:
p2(x0) = a0 = y0p2(x1) = a0 + a1(x1 − x0) = y1p2(x2) = a0 + a1(x2 − x0) + a2(x2 − x0)(x2 − x1) = y2
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Interpolation with Bernstein polynomials - I
Interpolation of two points with Bernstein polynomials:
y = x0B0,1(t) + x1B1,1(t) = x0(1− t) + x1t
Zhang 8
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Interpolation with Bernstein polynomials - II
Interpolation of three points with Bernstein polynomials:
y = x0B0,2(t) + x1B1,2(t) + x2B2,2(t) = x0(1− t)2 + x12t(1− t) + x2t2
Zhang 9
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Interpolation with Bernstein polynomials - III
Interpolation of four points with Bernstein polynomials:
y = x0B0,3(t) + x1B1,3(t) + x2B2,3(t)x3B3,3(t)
= x0(1− t)3 + x13t(1− t)2 + x23t2(1− t) + x3t3
Zhang 10
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Interpolation with Bernstein polynomials - IV
The Bernstein polynomials of degree k + 1 are defined as follows:
Bi,k(t) =
(k
i
)(1− t)k−i t i , i = 0, 1, . . . , k
Interpolation with Bernstein polynomials Bi,k :
y = x0B0,k(t) + x1B1,k(t) + · · ·+ xkBk,k(t)
Zhang 11
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
B-Splines
A normalized B-Splines Ni,k of degree k is defined as follows: For k = 1,
Ni,k(t) =
{1 : for ti ≤ t < ti+1
0 : else
and for k > 1, the recursive definition:
Ni,k(t) =t − ti
ti+k−1 − tiNi,k−1(t)+
ti+k − t
ti+k − ti+1Ni+1,k−1(t)
with i = 0, . . . ,m.
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
B-Spline-Curve
A B-Spline-Curve of degree k is a composite function built piecewisefrom basis B-Splines resulting in a polynomial of degree (k − 1) that is(k-2)-times continuously differentiable (class C k−2) at the borders of thesegments.The Curve is constructed by polynomials, that are defined by thefollowing parameters:
t = (t0, t1, t2, . . . , tm, tm+1, . . . , tm+k),
where
I m: depending on the number of data-points
I k: the fixed degree of the B-Spline curve
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Examples of B-Splines
B-Splines with degree 1, 2, 3 and 4:
t
t
t
tt j j+1 j+2 j+3 j+4tttt
Nj, 1
Nj, 2
Nj, 3
Nj, 4
Between the interval of parameters k B-Splines are overlapping.
Zhang 14
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Examples of cubic B-Splines
Zhang 15
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
B-Splines of degree k - I
The recursive definition procedure of a B-Spline basis functionNi ,k(t):
Zhang 16
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
B-Splines of degree k - II
Current segments of B-Spline basis functions of degree 2, 3 and 4 for
ti ≤ t < ti+1:
Zhang 17
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Uniform B-spline of degree 1 to 4
k=1 k=2
k=3 k=4
Zhang 18
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Non-uniform B-Splines
Degree 3:
Zhang 19
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Properties of B-splines
Partition of unity:∑k
i=0 Ni ,k(t) = 1.
Positivity: Ni ,k(t) ≥ 0.
Local support: Ni ,k(t) = 0 for t /∈ [ti , ti+k ].
C k−2 continuity: If the knots {ti} different in pairsthen Ni ,k(t) ∈ C k−2,i.e. Ni ,k(t) is (k − 2) times continuouslydifferentiable.
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Construction of B-spline curves
A B-spline curve can be constructed blending a number of predefinedvalues (data-points) with B-splines
r(t) =m∑j=0
vj ·Nj,k(t)
where vj are called control points (de Boor-points).
Let t be a given parameter, then r(t) is a point of the B-spline curve.
If t varies from tk−1 to tm+1, then r(t) is a (k-2)-times continuously
differentiable function (class C k−2).
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Calculation of control points from data points
The points vj are only identical with the data points if k = 2(interpolation/otherwise approximation). The control points form aconvex hull of the interpolation curve. Two methods for thecalculation of control points from data points:
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Calculation of control points from data points
1 Solving the following system of equations (Bohm84):
qj(t) =m∑j=0
vj · Nj ,k(t)
where qj are the data-points for interpolation/approximation,j = 0, · · · ,m.
2 Learning based on gradient descent(Zhang98).
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Lattice - I
Zhang 24
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Lattice - II
Zhang 25
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Tensor 2D-NURBS
Zhang 26
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Real-world Problems
I modeling: learning from examples, self-optimized formation,prediction, ...
I control: perception-action cycle, state control, Identification ofdynamic systems,...
Function approximation as a benchmark for the choice of a model
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Function approximation - 1D example
An example function f (x) = 8sin(10x2 + 5x + 1) with −1 < x < 1and the correctly distributed B-Splines:
Zhang 28
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Lattice
The B-spline model – a two-dimensional illustration.
Zhang 29
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Lattice (cont.)
Every n-dimensional square (n > 1) is covered by the j th
multivariate B-spline N jk(x). N j
k(x) is defined by the tensor of nunivariate B-splines:
N jk(x) =
n∏j=1
N jij ,kj
(xj) (1)
Therefore the shape of each B-spline, and thus the shape ofmultivariate ones (Figure 2), is implicitly set by their order andtheir given knot distribution on each input interval.
Zhang 30
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Lattice (cont.)
(a) Tensor of two, order2 univariate B-splines.
(b) Tensor of one order3 and one order 2 univa-riate B-splines.
(c) Tensor of two univa-riate B-splines of order3.
Bivariate B-splines formed by taking the tensor of two univariateB-splines.
Zhang 31
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
General requirements for an approximator
I Universality: Approximation of arbitrary functions
I Generalization: good approximation without Overfitting
I Adaptivity: on the basis of new data
I Parallelism: Computing based on biological models
I Interpretability: at least “Grey-box” instead of “Black-box”
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Importance of the Interpretability of a Model
Richard P. Feynman: “the way we have to describe nature isgenerally incomprehensible to us”.
Albert Einstein: “it should be possible to explain the laws ofphysics to a barmaid”.
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Importance of the Interpretability of a Model (cont.)
Important reasons for the symbolic interpretability of anapproximator:
I Linguistic modeling is a basis of skill transfer from an expert toa computer or robot .
I Automated learning of a transparent model facilitates theanalysis, validation and monitoring in the development cycle ofa model or a controller.
I Transparen models provide diverse applications inDecision-Support Systems.
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
B-Spline ANFIS
In a B-Spline ANFIS with n inputs x1, x2, . . . , xn, the rules are usedthe following form:{Rule(i1, i2, . . . , in): IF (x1 IS N1
i1,k1) AND (x2 IS N2
i2,k2) AND . . .
AND (xn IS Nnin,kn
) THEN y IS Yi1i2...in},where
I xj : input j (j = 1, . . . , n),
I kj : degree of B-spline basis function for xj ,
I N jij ,kj
: with the i-th linguistic term for the xj -associated B-splinefunction,
I ij = 0, . . . ,mj , partitioning of input j ,
I Yi1i2...in : control points for Rule(i1, i2, . . . , in).
I the “AND”-operator: product
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
B-Spline ANFIS (cont.)
Then the output y of the MISO control system is:
y =
m1∑i1=1
. . .
mn∑in=1
(Yi1,...,in
n∏j=1
N jij ,kj
(xj))
This is a general B-spline model that represents the hyperplane ( itNUBS (nonuniform B-spline)).http://equipe.nce.ufrj.br/adriano/fuzzy/transparencias/anfis/anfis.pdf
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Architecture of B-Spline ANFIS
Zhang 37
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
MF(membership function)-Formulation - Tensor
Tensor of 2D-Splines:
Zhang 38
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
The activation of MF by the inputs
Zhang 39
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
B-Spline ANFIS: example
An example with two input variables (x und y) and one Output z .
The parameters of the THEN-clauses are Z1,Z2,Z3,Z4.
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
B-Spline ANFIS: example (cont.)
The linguistic terms of inputs (IF-clauses):
Zhang 41
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
B-Spline ANFIS: example (cont.)
The parameters of the THEN-clauses:
Zhang 42
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Example: control basis
The sample control basis consists of four rules:
Rule1) IF x is X1 and y is Y1 THEN z is Z1
2) IF x is X1 and y is Y2 THEN z is Z2
3) IF x is X2 and y is Y1 THEN z is Z2
4) IF x is X2 and y is Y2 THEN z is Z4
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Illustration of the fuzzy inference
Zhang 44
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Illustration of the fuzzy inference (2)
Zhang 45
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Illustration of the fuzzy inference (3)
Zhang 46
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Illustration of the fuzzy inference (4)
Zhang 47
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Illustration of the fuzzy inference (5)
Zhang 48
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Algorithms for Supervised Learning - I
Let {(X, yd)} be a set of training data, where
I X = (x1, x2, . . . , xn) : the vector of input data,
I yd : the desired output for X.
The LSE is:
E =1
2(yr − yd)2, (2)
where yr is the current real output value during the training cycle.Goal is to find the parameters Yi1,i2,...,in , that minimize the error in(2)
E =1
2(yr − yd)2 ≡ MIN. (3)
Zhang 49
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Algorithms for Supervised Learning - II
Each control point Yi1,...,in can be improved with the followinggradient descend algorithm:
∆Yi1,...,in = −ε ∂E
∂Yi1,...,in
(4)
= ε(yr − yd)n∏
j=1
N jij ,kj
(xj) (5)
where 0 < ε ≤ 1.
Zhang 50
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Algorithms for Supervised Learning - III
The gradient descend algorithm ensures that the learning algorithmconverges to the global minimum of the LSE-function, because thesecond partial derivative of Y(i1, I2, hdots, in) is constant:
∂2E
∂2Yi1,...,in
= (n∏
j=1
N jij ,kj
(xj))2 ≥ 0. (6)
This means that the LSE-function ( ref (error)) is convexY(i1, I2, dots, in is) and therefore has only one (global) minimum.
Zhang 51
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Symbol Transformation of the Core Functions
Positive, convex core functions can be considered as Fuzzy sets, forexample:
µB(x) =1
1 +(x−5010
)2
Zhang 52
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Membership-functions
0
1
-5 0 5
0
1
-5 0 5
0
1
-5 0 5
0
1
-5 0 5
0
1
-5 0 5
0
1
-5 0 5
-5
0
1
0 5
0
1
-5 0 5
(a) (b)
( c )
(e) (f)
(g) (h)
(d)
Zhang 53
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Introduction to fuzzy sets
I fuzzy natural-language gradations of terms like “big”,“beautiful”, “strong” ...
I human thought and behavior models using the one-step logic:�� ��Driving: “IF-THEN”-clauses�� ��Car parking: With millimeter accuracy?
Zhang 54
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Introduction to Fuzzy sets
I Use of fuzzy language instead of numerical description:�� ��brake 2.52 m before the curve
→ only in machine systems�� ��brake shortly before the curve
→ in natural language
Zhang 55
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Definitions
Fuzzy: indistinctive, vague, unclear.
Fuzzy sets / fuzzy logic as a mechanism for
I fuzzy natural-language gradations of terms like “big”,“beautiful”, “strong” ...
I usage of fuzzy language instead of numerical description:.
I abstraction of unnecessary / too complex details.
I human thought and behavior models using the one-step logic.
Zhang 56
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Characteristic function vs. Membership function
For Fuzzy-sets A we used a generalized characteristic function µA thatassigns a real number from [0, 1] to to each member x ∈ X — the“degree” of membership of x to the fuzzy set A:
µA : X → [0, 1]
µA is called membershop-function.
A = {(x , µA(x)|x ∈ X}
Zhang 57
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Membership function
Characteristic of the continuous membership function
I Positive, convex functions (some important core functions).
I Subjective perception
I no probabilistic functions
Zhang 58
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Membership function types - I
Triangle: trimf (x ; a, b, c) = max
(min
(x − a
b − a,
c − x
c − b
), 0
)
Trapeze: trapmf (x ; a, b, c , d) = max
(min
(x − a
b − a, 1,
d − x
d − c
), 0
)
Zhang 59
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Membership function types - II
Gaussian: gaussmf (x ; c , σ) = e−12( x−c
σ )2
B-Splines: bsplinemf (x , xi , xi+1, · · · , xi+k)
Zhang 60
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Linguistic variables
A numeric variable has numerical values:
age = 25
A linguistic variable has linguistic values (terms):age : young
A linguistic value is a fuzzy set.
Zhang 61
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Fuzzy-Partition
Fuzzy partition of the linguistic values “young”, “average” and“old”:
Zhang 62
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Fuzzy Logic: inference mechanisms
A fuzzy rule is formulated as follows:“IF A THEN B”
with Fuzzy-sets A, B and the universes X , Y .
One of the most important inference mechanisms is the generalizedModus-Ponens (GMP):
Implication: IF x is A THEN y is BPremise: x is A′
Conclusion: y is B ′
Zhang 63
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Fuzzy systems for function approximation
Basic idea:
I Description of the desired control behavior through naturallanguage, qualitative rules.
I Quantification of linguistic values by fuzzy sets.
I Evaluation by methods of fuzzy logic or interpolation.
Zhang 64
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Fuzzy systems for function approximation
Fuzzy-rules:
”IF (a set of conditions is met)THEN(a set of consequences can be determined)“
In the premises (Antecedents) of the IF-part: linguistic variablesfrom the domain of process states;
In the conclusions (Consequences) of the THEN-part:linguistic variables from the system domain.
Zhang 65
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Adaptive networks
Architecture:
Feedforward networks with different node functions
Zhang 66
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Rule Extraction
The Fuzzy-Patches (Kosko):
Zhang 67
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Rule Extraction
A Fuzzy-Rule-Patch:
Zhang 68
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Additive Systeme
An additive fuzzy controller adds the “THEN”-Parts of the firedrules.
Fuzzy-Approximations-Rule:
An additive Fuzzy controller can approximate any continuousfunction f : X → Y if X is compact
Zhang 69
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Optimal Fuzzy-Rule-Patches
Optimal fuzzy rule patches cover the extrema of a function:
Zhang 70
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Optimal Fuzzy-Rule-Patches
Projection of the ellipsoids on the input and output axis:
Zhang 71
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Optimal Fuzzy-Rule-Patches
The size of an ellipsoid depends on the training data.
Zhang 72
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Optimal Fuzzy-Rule-Patches
Visualization of the input-output space:
Zhang 73
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Optimal Fuzzy-Rule-Patches
An example for interpolation:
Zhang 74
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Optimal Fuzzy-Rule-Patches
Data cluster along the function:
Zhang 75
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Optimal Fuzzy-Rule-Patches
Zhang 76
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Optimal Fuzzy-Rule-Patches
Approximation with Fuzzy-sets using the projections of extremes:
Zhang 77
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Approximation of a 2D-function
z =f (x, y)
=3(1 − x)2e−x2−(y+1)2 − 10(x
5− x3 − y5)e−x2−y2 −
1
3e−(x+1)2−y2
Zhang 78
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Approximation of a 2D-function
Derivatives of the function:dz
dx=− 6(1− x)e−x2−(y+1)2 − 6(1− x)2xe−x2−(y+1)2
− 10(1
5− 3x2) ∗ e−x2−y2
+ 20(1
5x − x3 − y5)xe−x2−y2
− 1
3(−2x − 2)e−(x+1)2−y2
dz
dy=3(1− x)2(−2y − 2)e−x2−(y+1)2
+ 50y4e−x2−y2
+ 20(1
5x − x3 − y5)ye−x2−y2
+2
3ye−(x+1)2−y2
Zhang 79
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Approximation of a 2D-function
d dzdx
dx=36xe−x2−(y+1)2 − 18x2e−x2−(y+1)2 − 24x3e−x2−(y+1)2
+ 12x4e−x2−(y+1)2 + 72xe−x2−y2
− 148x3e−x2−y2
− 20y5e−x2−y2
+ 40x5e−x2−y2
+ 40x2e−x2−y2
y5
− 2
3e−(x+1)2−y2
− 4
3e−(x+1)2−y2
x2 − 8
3e−(x+1)2−y2
x
Zhang 80
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Approximation of a 2D-function
d( dzdy )
dy=− 6(1− x)2e−x2−y(+1)2 + 3(1− x)2(−2y − 2)2e−x2−(y+1)2
+ 200y3e−x2−y2
− 200y5e−x2−y2
+ 20(1
5x − x3 − y5)e−x2−y2
− 40(1
5x − x3 − y5)y2e−x2−y2
+2
3e−(x+1)2−y2
− 4
3y2e−(x+1)2−y2
Zhang 81
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Global Overview of the Statistical Learning Theory - I
Let {(xi , yi )}li=1 be a set of data points/examples. We are searching for afunction f , which minimizes the following equation:
H[f ] =1
l
l∑i=1
V (yi , f (xi )) + λ‖f ‖2K
where V (·, ·) is a loss function and ‖f ‖2K is a norm in the Hilbert spaceH , which is defined by a positive kernel K , and λ is the regularizationparameter.
The problems in modeling, data regression and pattern classification are
each based on a kind of V (·, ·).
Zhang 82
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Global Overview of the Statistical Learning Theory - II
1. V (yi , f (xi )) = (yi − f (xi ))2
(yi ∈ R1, V : square error function)⇒ Regularization Networks, RN
2. V (yi , f (xi )) = |yi − f (xi )|ε(yi ∈ R1, | · |ε: eine ε-unempfindliche Norm)
⇒ Support Vector Machines Regression, SVMR
3. V (yi , f (xi )) = |1− yi f (xi )|+(yi ∈ {−1, 1}, |x |+ = x fur x = 0, else |x |+ = 0)
⇒ Support Vector Machines Classification, SVMC
Zhang 83
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Global Overview of the Statistical Learning Theory - II
1. V (yi , f (xi )) = (yi − f (xi ))2
(yi : a real number, V : square error function)⇒ Regularization Networks, RN
2. V (yi , f (xi )) = |yi − f (xi )|ε(yi : a real number, | · |ε: an ε-independent norm)
⇒ Support Vector Machines Regression, SVMR
3. V (yi , f (xi )) = |1− yi f (xi )|+(yi : -1 oder 1, |x |+ = x fur x = 0, sonst |x |+ = 0)
⇒ Support Vector Machines Classification, SVMC
For modeling and control tasks, the first definition is mostimportant.
Zhang 84
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Universelal Function Approximation - I
A control-network can approximate all smooth functions with anarbitrary precision.The general solution of this problem is:
f (x) =l∑
i=1
ciK (x ; xi )
where ci are the coefficients.
Zhang 85
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Universelal Function Approximation - II
Proposition of the approximation:
For any continuous function Y that is defined on the compactsubset Rn and the core function K , there is a functiony∗(x) =
∑li=1 ciK (x ; xi ) that fulfills for all x and any ε:
|Y (x)− y∗(x)| < ε
Zhang 86
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Universal Function Approximation - IIUsing different kernel functions leads to different models:
K (x− xi ) = exp(−‖x− xi‖2) Gaussian RBF
K (x− xi ) = (‖x− xi‖2 + c2)−12 Inverse multiquadratic functions
K (x− xi ) = tanh(x · xi − θ) Multilayer perceptron
K (x− xi ) = (1 + x · xi )d Polynomial of degree d
K (x − xi ) = B2n+1(x − xi ) B-Splines
K (x − xi ) =sin(d + 1
2 )(x − xi )
sin (x−xi )2
Trigonometric polynomial
... ...
Zhang 87
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Problems
1. “Curse of dimensionality”because of the exponentialdependency between the memory requirements and thedimension of input space.
2. Aliasing within the feature extraction
3. Not available target data (y).
4. Not available input factors.
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Learning from DJ-Data
Dow-Jones-Index: can the function be modeled?
Zhang 89
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Example: Image Processing in Local Observation scenarios
A sequence of gray scale images of an object is acquired by the
movement along a fixed location:
Zhang 90
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Example: Extraction of Eigenvectors
The first 6 Eigenvectors:
Zhang 91
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Example: Eigenvectors and Eigenvalues
Zhang 92
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
Combination of Dimension Reduction with B-Spline Model
Eigenvectors can be partitioned by linguistic terms.
Such a combination of PCA and B-spline model can be considered as a
Neuro-Fuzzy model.
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
The Neuro-Fuzzy Model
Zhang 94
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University of Hamburg
MIN Faculty
Department of Informatics
Introduction Function approximation
The Training and Application Phases
Zhang 95