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  1.        ]
 [ 9.91204705  5.09395275  0.        ]
 [-4.4616725   1.3617128   0.        ]
 [-2.40987728  8.05505365  0.        ]]

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   1.          43.965223    53.965223  ]
 [  9.91204705   5.09395275   0.           6.25588575   6.25588575]
 [ -4.4616725    1.3617128    0.         212.41884106 212.41884106]
 [ -2.40987728   8.05505365   0.           9.25981051   9.25981051]]

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, 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 0x10d84c680>,
        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 -3.315251 -8.529671 -8.531265
1 105.865258 105.831670 105.831675
2 49.811774 49.435283 49.435284
3 -1.822850 6.199391 6.199415
4 0.968377 -5.417527 -5.417969 
df.tail()
y Prediction Prediction_factor
9995 73.620503 73.929808 73.929766
9996 96.187178 95.269931 95.270115
9997 49.494401 51.023448 51.023323
9998 35.390268 22.258425 22.258813
9999 26.261264 25.092604 25.092367 
s=np.sum(np.abs(y_pred - y_true))
sf=np.sum(np.abs(y_factor_pred - y_true))
res = (sf - s)
print(res)
0.43224212594213895
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