# Heterogeneous Treatment Effect with Trained Kernels of the Nadaraya-Watson Regression
## 4 The TNW-CATE Description
For even \(d\), we write the vector of features as \[\mathbf{a}_{i}^{(r)} =\left(\mathbf{x}_{1}^{(r)},\ldots,\mathbf{x}_{n}^{(r)}, \mathbf{x}_{i,j}y_{1}^{(r)},\ldots,y_{n}^{(r)},y_{i}\right),\]8 \[\mathbf{a}_{i}^{(r)} =\left(\mathbf{x}_{1}^{(r)},\ldots,\mathbf{x}_{n}^{(r)}, \mathbf{x}_{i,j}y_{1}^{(r)},\ldots,y_{n}^{(r)},y_{i}\right),\]9 (23) For odd \(d\), there holds \[i =1,\ldots,c,\ r=1,\ldots,N. \tag{11}\]0 (24) The responses are generated as a linear function of \(t\), i.e., they are computed as \(y=at+b\). Values of parameters \(a\), \(b\), and \(t\) for performing numerical experiments with spiral functions areas follows: * The control group: parameters \(a\), \(b\), and \(t\) are uniformly generated from intervals \([1,4]\), \([1,4]\), and \([0,10]\), respectively. * The treatment group: parameters \(a\), \(b\), and \(t\) are uniformly generated from intervals \([8,10]\), \([8,10]\), and \([0,10]\), respectively. 2. **Logarithmic functions:** Features are logarithms of the parameter \(t\), i.e., there holds \[i =1,\ldots,c,\ r=1,\ldots,N. \tag{11}\]1 (25) The responses are generated as a logarithmic function with adding an oscillating term \(b\sin(t)\) to \(y\), i.e., there holds \(y=b(\ln(t)+\sin(t))\). Values of parameters \(a_{1},\ldots,a_{d}\), \(b\) for performing numerical experiments with logarithmic functions are as follows: * Each parameter from the set \(\{a_{1},\ldots,a_{d}\}\) is uniformly generated from intervals \([-4,-1]\cup[1,4]\) for controls as well as for treatments. * Parameter \(b\) is uniformly generated from interval \([1,4]\) for controls and from interval \([-4,-1]\) for treatments. * Values of \(t\) are uniformly generated in interval \([0.5,5]\). 3. **Power functions:** Features are represented as powers of \(t\). For arbitrary \(d\), the vector of features is represented as \[i =1,\ldots,c,\ r=1,\ldots,N. \tag{11}\]2 (26) However, features which are close to linear ones, e.g., \(x_{i}=t^{i/\sqrt{d}}\) for \(0.8<i/\sqrt{d}<1.6\), are replaced with the Gaussian noise having the unit standard deviation and the zero expectation, i.e., \(x_{i}\sim\mathcal{N}(0,1)\). The responses are generated as follows: \[i =1,\ldots,c,\ r=1,\ldots,N. \tag{11}\]3 (27) Values of parameters \(a\), \(b\), \(s\), and \(t\) for performing numerical experiments with power functions are as follows: * The control group: parameters \(a\) and \(b\) are uniformly generated from intervals \([1,2]\) and \([0.25,1]\), respectively; parameter \(s\) is 2.5. The treatment group: parameters \(a\) and \(b\) are uniformly generated from intervals \([2,4]\) and \([1,2]\), respectively; parameter \(s\) is 2.5. Values of \(t\) are uniformly generated in interval \([0,5]\). 4. **Indicator functions**[25]: The functions are expressed through the indicator function \(I\) taking value 1 if its argument is true. * The function for controls is represented as \[i =1,\ldots,c,\ r=1,\ldots,N. \tag{11}\]4 (28) * The function for treatments is represented as \[i =1,\ldots,c,\ r=1,\ldots,N. \tag{11}\]5 \[i =1,\ldots,c,\ r=1,\ldots,N. \tag{11}\]6 (29)
* Vector \(\beta\) is uniformly distributed in interval \([-5;5]^{d}\); values of features \(x_{i}=1,\ldots,d\),are uniformly generated from interval \([-1,1]\). The indicator function differs from other functions considered in numerical examples. It is taken from [25] in order to study TNW-CATE when the assumption of specific and similar domains for the control and treatment feature vectors can be violated. In numerical experiments with the above functions, we take parameter \(\theta\) to be equal to 0.1, 0.5, 0.5, and 0.5 for the spiral, logarithmic, power, and indicator functions, respectively, except for experiments which study how parameter \(\theta\) impacts the MSE.