22  Factorial Variables

Until now, we have considered continuous variables. However, in many applications, the variables are not continuous, but rather discrete or categorical. For example, the number of layers in a neural network, the number of trees in a random forest, or the type of kernel in a support vector machine are all discrete variables. In the following, we will consider a simple example with two numerical variables and one categorical variable.

from spotpython.design.spacefilling import SpaceFilling
from spotpython.surrogate.kriging import Kriging
from spotpython.fun.objectivefunctions import Analytical
import numpy as np

First, we generate the test data set for fitting the Kriging model. We use the SpaceFilling class to generate the first two diemnsion of \(n=30\) design points. The third dimension is a categorical variable, which can take the values \(0\), \(1\), or \(2\).

gen = SpaceFilling(2)
n = 30
rng = np.random.RandomState(1)
lower = np.array([-5,-0])
upper = np.array([10,15])
fun_orig = Analytical().fun_branin
fun = Analytical().fun_branin_factor

X0 = gen.scipy_lhd(n, lower=lower, upper = upper)
X1 = np.random.randint(low=0, high=3, size=(n,))
X = np.c_[X0, X1]
print(X[:5,:])
[[-2.84117593  5.97308949  2.        ]
 [-3.61017994  6.90781409  0.        ]
 [ 9.91204705  5.09395275  0.        ]
 [-4.4616725   1.3617128   2.        ]
 [-2.40987728  8.05505365  1.        ]]

The objective function is the fun_branin_factor in the analytical class [SOURCE]. It calculates the Branin function of \((x_1, x_2)\) with an additional factor based on the value of \(x_3\). If \(x_3 = 1\), the value of the Branin function is increased by 10. If \(x_3 = 2\), the value of the Branin function is decreased by 10. Otherwise, the value of the Branin function is not changed.

y = fun(X)
y_orig = fun_orig(X0)
data = np.c_[X, y_orig, y]
print(data[:5,:])
[[ -2.84117593   5.97308949   2.          32.09388125  22.09388125]
 [ -3.61017994   6.90781409   0.          43.965223    43.965223  ]
 [  9.91204705   5.09395275   0.           6.25588575   6.25588575]
 [ -4.4616725    1.3617128    2.         212.41884106 202.41884106]
 [ -2.40987728   8.05505365   1.           9.25981051  19.25981051]]

We fit two Kriging models, one with three numerical variables and one with two numerical variables and one categorical variable. We then compare the predictions of the two models.

S = Kriging(name='kriging',  seed=123, log_level=50, method="interpolation", var_type=["num", "num", "num"])
S.fit(X, y)
Sf = Kriging(name='kriging',  seed=123, log_level=50, method="interpolation", var_type=["num", "num", "factor"])
Sf.fit(X, y)
Anisotropic model: n_theta set to 3
Anisotropic model: n_theta set to 3
Kriging(eps=np.float64(1.4901161193847656e-08), method='interpolation',
        model_fun_evals=100,
        model_optimizer=<function differential_evolution at 0x1109263e0>,
        name='kriging', seed=123, var_type=['num', 'num', 'factor'])
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We can now compare the predictions of the two models. We generate a new test data set and calculate the sum of the absolute differences between the predictions of the two models and the true values of the objective function. If the categorical variable is important, the sum of the absolute differences should be smaller than if the categorical variable is not important.

n = 100
k = 100
y_true = np.zeros(n*k)
y_pred= np.zeros(n*k)
y_factor_pred= np.zeros(n*k)
for i in range(k):
  X0 = gen.scipy_lhd(n, lower=lower, upper = upper)
  X1 = np.random.randint(low=0, high=3, size=(n,))
  X = np.c_[X0, X1]
  a = i*n
  b = (i+1)*n
  y_true[a:b] = fun(X)
  y_pred[a:b] = S.predict(X)
  y_factor_pred[a:b] = Sf.predict(X)
import pandas as pd
df = pd.DataFrame({"y":y_true, "Prediction":y_pred, "Prediction_factor":y_factor_pred})
df.head()
y Prediction Prediction_factor
0 16.684749 14.065908 14.857172
1 105.865258 105.792426 105.710852
2 49.811774 49.251145 49.937606
3 18.177150 18.152566 18.621294
4 10.968377 -2.918720 2.995571 
df.tail()
y Prediction Prediction_factor
9995 93.620503 93.935619 93.764760
9996 86.187178 96.735830 87.559558
9997 29.494401 30.541060 30.166253
9998 25.390268 26.009600 28.823019
9999 16.261264 15.555043 16.906403 
s=np.sum(np.abs(y_pred - y_true))
sf=np.sum(np.abs(y_factor_pred - y_true))
res = (sf - s)
print(res)
-25450.289482965745
from spotpython.plot.validation import plot_actual_vs_predicted
plot_actual_vs_predicted(y_test=df["y"], y_pred=df["Prediction"], title="Default")
plot_actual_vs_predicted(y_test=df["y"], y_pred=df["Prediction_factor"], title="Factor")

22.1 Jupyter Notebook

Note