21  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  0.        ]
 [-3.61017994  6.90781409  2.        ]
 [ 9.91204705  5.09395275  2.        ]
 [-4.4616725   1.3617128   1.        ]
 [-2.40987728  8.05505365  2.        ]]

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   0.          32.09388125  32.09388125]
 [ -3.61017994   6.90781409   2.          43.965223    33.965223  ]
 [  9.91204705   5.09395275   2.           6.25588575  -3.74411425]
 [ -4.4616725    1.3617128    1.         212.41884106 222.41884106]
 [ -2.40987728   8.05505365   2.           9.25981051  -0.74018949]]

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, n_theta=3, method="interpolation", var_type=["num", "num", "num"])
S.fit(X, y)
Sf = Kriging(name='kriging',  seed=123, log_level=50, n_theta=3, method="interpolation", var_type=["num", "num", "factor"])
Sf.fit(X, y)
Kriging(eps=np.float64(1.4901161193847656e-08), method='interpolation',
        model_fun_evals=100,
        model_optimizer=<function differential_evolution at 0x116cf0400>,
        n_theta=3, 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 6.684749 2.830670 2.829951
1 105.865258 105.550815 105.550827
2 29.811774 26.399345 26.399287
3 18.177150 18.916744 18.916944
4 -9.031623 -9.312008 -9.312057 
df.tail()
y Prediction Prediction_factor
9995 73.620503 72.473199 72.473371
9996 76.187178 77.374995 77.375113
9997 29.494401 32.392714 32.392753
9998 35.390268 40.383484 40.383385
9999 36.261264 38.221339 38.221482 
s=np.sum(np.abs(y_pred - y_true))
sf=np.sum(np.abs(y_factor_pred - y_true))
res = (sf - s)
print(res)
1.7980820210359525
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")

21.1 Jupyter Notebook

Note