Analytic Programming
Comparative Study
Ivan Zelinka, Zuzana Oplatková
http://www.ft.utb.cz/people/zelinka
Email {zelinka,oplatkova}@ft.utb.cz
Tomas Bata University in Zlin
Faculty of Technology
Institut of Information Technologies
Mostni 5139
Zlin 760 01
Czech Republic
Main Principles of AP I
C2
• Set of functions and its possible arguments (terminals)
Sin
e
Round
C1
Example: {Sin, Tan, e, Tanh, t,...}
Tanh
Catalan
• Rule for construction of analytic solution from given individual
Glaischer
User function
t
FractionalPart
• Rule for critical situations treating:
• pathological functions (without arguments, self-looped...)
• functions with imaginary or real part (if not expected))
• infinity in functions
• „frozen“ functions (extremely long time to get its cost value - hrs...)
• Rule for cost function evaluation
GFS0arg=
GFS1arg=
GFS2arg=
GFS3arg=
GFSall =
{t, x, y, z, C1, C2, Kinchin, ...}
{Sin, Cos, Tan, Abs, Re, Im, ...}
{+, -, /, ^, Log, Mod, GammaRegularized...}
{BetaRegularized, ...}
{+, -, /, ^, d/dt, Sin, Cos, Tan, t, C1, Mod, ...}
Fcost = |DataSet – FAP(t )|
Main Principles of AP - II
^/GFS0arg=
GFS1arg=
GFS2arg=
GFS3arg=
GFSall =
{t, x, y, z, C1, C2, Kinchin, ...}
{Sin, Cos, Tan, Abs, Re, Im, ...}
{+, -, /, ^, Log, Mod, GammaRegularized...}
{BetaRegularized, ...}
{+, -, /, ^, d/dt, Sin, Cos, Tan, t, C1, Mod, ...}
z/x
Functions generated by AP
sin
tan
1
coth
1
log csc
1
log C
cosh
log cos 1 t
cosh
csc 1 sin
sinh 1 cos
cosh 1 t
csc 1 sin
sinh 1 cos
cosh 1 t
t
log
t
csc 1 sin
sinh 1 cos
cosh 1 t
1
cosh cot
1
sec
1
Glaisher
1
1
t
coth sec
1
C
cos
t Khinchin 1
t Khinchin 1
log
log coth
1 4
log
csch
1
Khinchin
log Glaisher
log sinh cosh
log t
sec 1 Glaisher
log t
logsec
1 Glaisher
sinh sec cos 1
t Khinchin 1
cosh
1
2
1
tan tan
1
1
sin
2
t
1
1
log sin Glaisher
1
cosh cot
csc 1 sin
sinh 1 cos
cosh 1 t
log tanh 1 tan t
coth
1
1
csch t
1
1
sin t
1
t
2t
Variants of AP
Analytic programming and structure
of set of functions and terminals
Standard (Koza)
+, -, *, /, x, rand1…rand100
=> 105 elements
Metaevolution
+, -, *, /, x, K => only 6 elements
EA is used to find general solution like is
demonstrated below. Then is called other
EA (can be the same) to estimate
numerical values of constants K[[…]].
Result then look like
Nonlinear fitting
The same like in
metaevolution. Only
nonlinear fitting method
is used instead of EA to
estimate constants
K[[…]]
Sinus “Four and Three”
SOMA
S3 DE S3 GA S3 SA S3 SOMA
S4 DE S4
GA S4 SA S4
S3 DE S3 GA S3 SA S3 SOMA
S4 DE S4
GA S4 SA S4
20
DE
Three
Four
15
20
15
12.5
10
7.5
5
2.5
15
CV
10
10
5
7
8
9
10 11 12
8
13
10
SOMA Three
12
14
16
Four
0
SOMA
12
25
5
10
20
8
15
6
10
Sinus Three
4
5
2
2
3
4
GA
5
6
7
4
Three
6
8
10
Four
25
20
20
15
15
10
10
5
5
5
10
5
15
10
15
20
3
Sinus Four
2
SA
Three
14
12
10
8
6
4
2
Four
1
12
10
8
-3
4
2
5
7.5
10 12.5 15
-1
1
-1
6
2.5
-2
-2
10
12
14
16
-3
2
3
Quintic and Sextic
DE
Quintic
40
30
30
20
20
10
10
-11
-11
10 2.
SOMA
-11
10 3.
10 4.
10
-11
Quintic
30
30
20
20
10
10
-12 -12 -12 -12 -11
10
6.
GA
10
8.
-10
10 1.2
1.
10 1.4
10
-11 -11
1.5
10
1
10
5
10
QDE Q GA Q SA Q
SOMA
SDE S GA S SA SSOMA
QDE Q GA Q SA Q
-7
101.5
-9
102.
-9
102.5
10
-9
-7
-7
-8
0
Sextic
0.3
Quintic
0.2
0.1
-9
2.
10
-9
10 1.
50
40
10
4.
CV
5.
40
2.
10
SDE S GA S SA SSOMA
Sextic
40
1.
2
SOMA
-9
10 4.
Quintic
106.
-9
-9
108.
10 1.
10
-8
-1
-0.5
0.5
1
-0.1
Sextic
-0.2
50
40
40
-0.3
30
30
20
20
10
10
2.
-11
10
4.
-11
10
6.
SA
-11
10
8.
-11
10
1.
-10
10
1.2
10
-10
0.15
1.
-11
10 2.
Quintic
-11
-11
10 3.
10 4.
10
0.125
-11
0.1
Sextic
0.075
40
40
30
30
20
20
10
10
2.
10
4.
-6
10
6.
-6
10
8.
-6
-6
0.00001
0.000014
10 0.000012
Sextic
0.05
0.025
5.
-7
101.
-6
101.5
-6
10
2.
-6
102.5
10
-6
-1
-0.5
0.5
1
Conclusions
1.
On two types of set of functions and terminals three methods were used to estimate solution:
•
Standard set like in Koza’s GP
•
Modified set with only one universal constant K, indexed later in evolutionary process
2.
Three methods were used to estimate solution:
•
Standard AP i.e. EA=>exact programm estimation
•
AP based on metaevolution i.e. EA=>general programm=>EA=>coefficients estimation
•
AP based on nonlinear fitting i.e.
EA=>general programm=>nonlinear fitting=>coefficients estimation
3.
Numerical difficulty:
•
Koza : 4000 individuals, 1 400 000 – 3 000 000 cost function evaluations
•
AP : 60 – 150 individuals, 500 – 18 000 cost functions evaluations
1.
Future ressearch:
•
Use AP for solution of different problems (mathematical physics, cybernetics,…)
•
Use AP for new EAs construction in metaevolutionary way