# BENK: The Beran Estimator with Neural Kernels for Estimating the Heterogeneous Treatment Effect
## 7 Numerical Experiments
### Generating Synthetic Datasets  
As has been described above, we consider generating the artificial complex feature spaces and outcomes in the numerical experiments. All the vectors of features, including controls \(\mathbf{x}\) and treatments \(\mathbf{y}\), are generated by means of three functions: the spiral function, the bell-shaped function and the circular function. The idea to use these functions stems from the goal to obtain complex structures of data, which are poorly processed by many standard methods. The above functions are defined through a parameter \(\xi\) as follows:  
1. Spiral functions: The feature vectors, having dimensionality \(d\) and being located on the Archimedean spirals, are defined for even \(d\) as \[\mathbf{x}=(\xi\sin(\xi),\xi\cos(\xi),\ldots,\xi\sin(\xi\cdot d/2),\xi\cos( \xi\cdot d/2)),\] (24) and for odd \(d\) as \[\mathbf{x}=(\xi\sin(\xi),\xi\cos(\xi),\ldots,\xi\sin(\xi\cdot\lceil d/2\rceil )).\] (25) Values of \(\xi\) are uniformly generated from the interval \([0,10]\) for all numerical experiments;
2. Bell-shaped functions: Features are represented as a set of almost non-overlapping Gaussians. As \(\xi\) is uniformly generated in the numerical experiments, we can define \(\xi_{\min}\) and \(\xi_{\max}\) as corresponding bounds of the uniform distribution. Therefore, the feature vector of dimensionality \(d\) is represented as \[\mathbf{x} =(x_{0},x_{1},\ldots,x_{d-1}),\] \[\sigma =\frac{\xi_{\max}-\xi_{\min}}{6d},\;\mu=\frac{\xi_{\max}-\xi_{ \min}}{d-1},\] \[x_{i} =\frac{1}{\sigma\sqrt{2\pi}}\cdot\exp\left(\frac{-(\xi-i\cdot\mu )^{2}}{2\sigma^{2}}\right),\;i=1,\ldots,d-1.\] (26) Therefore, each feature \(x_{i}\) corresponds to its own region in the \(\xi\) distribution; \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Meta-Model**} \\ \hline Survival regression algorithms & T-learner & S-learner & X-learner \\ \hline Beran estimator & T-Beran & S-Beran & X-Beran \\ \hline Cox model & T-Cox & S-Cox & X-Cox \\ \hline RSF & T-SF & S-SF & X-SF \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations of the models, depending on meta-learners and base models. 3. Circular functions: The corresponding feature space is generated by using only the even numbers of features. The feature vectors are located on the two-dimensional circles as follows: \[c_{num} = \frac{d}{2},\ c_{range}=\frac{\xi_{\max}-\xi_{\min}}{c_{num}},\] \[\mathbf{x} = (x_{1}^{1},x_{1}^{2},x_{2}^{1},x_{2}^{2},\ldots,x_{c_{num}}^{1},x_ {c_{num}}^{2}),\] \[x_{i}^{1} = \sin\biggl{(}\frac{2\pi(\xi-(i-1)\cdot c_{range})}{c_{range}} \biggr{)}\cdot\mathbf{I}_{i},\] \[x_{i}^{2} = \cos\biggl{(}\frac{2\pi(\xi-(i-1)\cdot c_{range})}{c_{range}} \biggr{)}\cdot\mathbf{I}_{i},\] \[\mathbf{I}_{i} = \mathbf{I}\{(i-1)\cdot c_{range}\leq\xi<i\cdot c_{range}\},\ i=1, \ldots,c_{num},\] (27) where \(\mathbf{I}_{i}\) is an indicator function. Each pair of features \((x_{i}^{(1)},x_{i}^{(2)})\) corresponds to their own two-dimensional circle and to their own region in the \(\xi\) distribution. In all experiments, feature vectors \(\mathbf{y}\) are generated in the same way as vectors \(\mathbf{x}\). However, for feature vectors \(\mathbf{x}\) and \(\mathbf{y}\), from the control and treatment groups, the corresponding times to events \(f\) and \(h\) are different and are generated by using the Weibull distribution, as follows:  
\[f(\xi)=-\biggl{(}\frac{\log(u)}{0.0005\cdot\exp{(1.6\cdot\xi)}} \biggr{)}^{1/2}, \tag{28}\] \[h(\xi)=-\biggl{(}\frac{\log(u)}{0.005\cdot\exp{(0.8\cdot\xi)}} \biggr{)}^{1/2}, \tag{29}\]  
where \(u\) is the random variable, uniformly distributed on the interval \((0,1)\); values \(f\) and \(h\) larger than 2000 are clipped to this value. This way for generating \(f\) and \(h\) is in agreement with the Cox model. Hence, we can use the Cox model as a base model among RSFs and the Beran estimator with Gaussian kernels in the numerical experiments. The proportion of censored data, denoted as \(p\), is taken as 33% of all observations in the experiments.