Highly Accurate Symbolic Regression with Noisy Training Data 95
F.x/Dinv abs sqroot square cube quart exp ln cos sin tan tanh) in reasonable
computation times, of a maximum 20 h (on an advanced laptop built in Dec 2012)
and a maximum 40 h (on an advanced laptop built in Jan 2008). Most noiseless
problems finish far quicker than these maximum time horizons.
Pushing things to the extreme, the enhanced algorithm will achieve extremely
accurate champions for all of the problems inU 2 (1)[50]throughU 1 (5)[50]in a
maximum of 160 h (on an advanced laptop built in Dec 2012). Most noiseless
problems finish far quicker than these maximum time horizons.
Obviously a cloud configuration will greatly speed up the enhanced EA algo-
rithm, and we will address cloud configurations and extreme accuracy in a later
paper. For this chapter, we will develop an extremely accurate SR algorithm which
any scientist can use on their personal laptop.
1.1 Example Test Problems
In this section we list the example test problems which we will address. All of
these test problems lie in the domain of eitherU 2 (1)[25],U 1 (25)[25],U 1 (5)[150],or
F.x/(5)[3000], where the function set F.x/D(inv abs sqroot square cube quart
exp ln cos sin tan tanh), and the terminal set is the featuresx 0 thruxM 1 plus
the real number constantcwithcbitD 18. Our training data sets will contain 25
features, 150, and 3000 features as specified. Our core assertion is that the enhanced
algorithm will find extremely accurate champions for all of these problems and for
all similar problemsin practical time on a laptop computer.
Similar problemsare easily obtained by substituting all other possibilities within
U 2 (1)[25],U 1 (25)[25],U 1 (5)[150],orF.x/(5)[3000]. For instance one problem in
U 2 (1)[25]isy D 1 : 687 C. 94 : 183 .x 3 x 2 //. By substitution,y D 1 : 687 C
. 94 : 183 .x 3 =x 2 //andy D 1 : 687 C. 94 : 183 .x 23 x 12 //are also inU 2 (1)[25].
Another problem inU 2 (1)[25]isyD 2 : 36 C. 28 : 413 ln.x 2 /=x 3 /. By substitution,
yD 2 : 36 C. 28 : 413 cos.x 12 /x 6 /andyD 2 : 36 C. 28 : 413 sqroot.x 21 /x 10 /
are also inU 2 (1)[25]. Our core assertion is that the EA algorithm not only finds
accurate solutions to the 45 test problems listed below, but also toall other possible
test problemsinU 2 (1)[25],U 1 (25)[25],U 1 (5)[150],orF.x/(5)[3000].
- Deep problems in U 2 (1)[25]
- ..Note: these problems trained on 10,000 examples with 25 features each
•(T1): yD1:57C.14:3x 3 /
•(T2): yD3:57C.24:33=x 3 /
•(T3): yD1:687C.94:183.x 3 x 2 //
•(T4): yD21:37C.41:13.x 3 =x 2 //
•(T5): yD1:57C.2:3..x 3 x 0 /x 2 //
•(T6): yD9:00C.24:983..x 3 x 0 /.x 2 x 4 ///
•(T7): yD71:57C.64:3..x 3 x 0 /=x 2 //
•(T8): yD5:127C.21:3..x 3 x 0 /=.x 2 x 4 ///