In this third tutorial we consider an example with two dimensional input data and we approach its solution using a multi-objective approach where, aside the loss, we consider the formula complexity as a second objective.

We will use a memetic approach to learn the model parameters while evolution will shape the model itself.

Eventually you will learn:

How to instantiate a multi-objective symbolic regression problem.

How to use a memetic multi-objective approach to find suitable models for your data

```
In [1]:
```# Some necessary imports.
import dcgpy
import pygmo as pg
# Sympy is nice to have for basic symbolic manipulation.
from sympy import init_printing
from sympy.parsing.sympy_parser import *
init_printing()
# Fundamental for plotting.
from matplotlib import pyplot as plt
%matplotlib inline

```
In [2]:
```# We load our data from some available ones shipped with dcgpy.
# In this particular case we use the problem sinecosine from the paper:
# Vladislavleva, Ekaterina J., Guido F. Smits, and Dick Den Hertog.
# "Order of nonlinearity as a complexity measure for models generated by symbolic regression via pareto genetic
# programming." IEEE Transactions on Evolutionary Computation 13.2 (2008): 333-349.
X, Y = dcgpy.generate_sinecosine()

```
In [3]:
```from mpl_toolkits.mplot3d import Axes3D
# And we plot them as to visualize the problem.
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
_ = ax.scatter(X[:,0], X[:,1], Y[:,0])

```
```

```
In [4]:
```# We define our kernel set, that is the mathematical operators we will
# want our final model to possibly contain. What to choose in here is left
# to the competence and knowledge of the user. A list of kernels shipped with dcgpy
# can be found on the online docs. The user can also define its own kernels (see the corresponding tutorial).
ss = dcgpy.kernel_set_double(["sum", "diff", "mul", "sin", "cos"])

```
In [5]:
```# We instantiate the symbolic regression optimization problem
# Note how we specify to consider one ephemeral constant via
# the kwarg n_eph. We also request 100 kernels with a linear
# layout (this allows for the construction of longer expressions) and
# we set the level back to 101 (in an attempt to skew the search towards
# simple expressions)
udp = dcgpy.symbolic_regression(
points = X, labels = Y, kernels=ss(),
rows = 1,
cols = 100,
n_eph = 1,
levels_back = 101,
multi_objective=True)
prob = pg.problem(udp)
print(udp)

```
```

```
In [6]:
```# We instantiate here the evolutionary strategy we want to use to
# search for models. Note we specify we want the evolutionary operators
# to be applied also to the constants via the kwarg *learn_constants*
uda = dcgpy.momes4cgp(gen = 250, max_mut = 4)
algo = pg.algorithm(uda)
algo.set_verbosity(10)

```
In [7]:
```# We use a population of 100 individuals
pop = pg.population(prob, 100)

```
In [11]:
```# Here is where we run the actual evolution. Note that the screen output
# will show in the terminal (not on your Jupyter notebook in case
# you are using it). Note you will have to run this a few times before
# solving the problem entirely.
pop = algo.evolve(pop)

```
In [12]:
```# Compute here the non dominated front.
ndf = pg.non_dominated_front_2d(pop.get_f())

```
In [13]:
```# Inspect the front and print the proposed expressions.
print("{: >20} {: >30}".format("Loss:", "Model:"), "\n")
for idx in ndf:
x = pop.get_x()[idx]
f = pop.get_f()[idx]
a = parse_expr(udp.prettier(x))[0]
print("{: >20} | {: >30}".format(str(f[0]), str(a)), "|")

```
```

```
In [14]:
```# Lets have a look to the non dominated fronts in the final population.
ax = pg.plot_non_dominated_fronts(pop.get_f())
_ = plt.xlabel("loss")
_ = plt.ylabel("complexity")
_ = plt.title("Non dominate fronts")

```
```

```
In [15]:
```# Here we get the log of the latest call to the evolve
log = algo.extract(dcgpy.momes4cgp).get_log()
gen = [it[0] for it in log]
loss = [it[2] for it in log]
compl = [it[4] for it in log]

```
In [19]:
```# And here we plot, for example, the generations against the best loss
_ = plt.plot(gen, loss)
_ = plt.title('last call to evolve')
_ = plt.xlabel('generations')
_ = plt.ylabel('loss')

```
```