# Kernel Learning by Spectral Representation and Gaussian Mixtures
## 4 Experiments
### Regression  
For the regression experiments, we use a synthetic data set in our regression attempt. For this, samples are taken from the Gaussian Mixture distribution shown in Equation (12). After that, \(M=250\), and \(\boldsymbol{\beta}\sim\mathcal{N}(0,I_{501})\) are set. In fact, the number 501 is because the extended vector is used, which takes samples from \(y_{i}\sim\mathcal{N}(\boldsymbol{\varphi}_{p}(x_{i})^{T}\boldsymbol{\beta})\), where \(\boldsymbol{\beta}_{p}\) represents the random features from \(\rho\), i.e. \(\omega_{i}\sim\rho(\omega),i=1,2,...,M\). Furthermore, an instance of this problem is shown in Figure 2. Also, 250 samples and vague/non-informative prior to learn the function are used, and the result is shown in Figure 3.  
\[p(\omega)=\pi_{1}p\bigg{(}\omega|0,\frac{1}{2^{2}}\bigg{)}+\pi_{2 }p\bigg{(}\omega|\frac{3\pi}{4},\frac{1}{2^{2}}\bigg{)}+\pi_{3}p\bigg{(}\omega| \frac{11\pi}{8},\frac{1}{4^{2}}\bigg{)} \tag{12}\] \[\text{with }\pi_{1}=\pi_{2}=\pi_{3}=\frac{1}{3} \tag{13}\]  
All the results of the GaBaSR algorithm uses 500 samples (\(M\)) and 5 swaps each one. For regression problems we use some small datasets, Mauna LOA CO\({}_{2}\), California Houses, Boston house-price, and Diabetes. The Mauna LOA CO\({}_{2}\)[29] from the global monitoring laboratory this collects the information of the monthly mean CO\({}_{2}\), as we can see from Figure 4, this data it is stationary, has some repetitions and increments. The California Houses is a set of 20,640 rows with 8 columns. The Boston house price dataset was collected in 1978 from various suburbs in Boston. Diabetes Dataset has ten variables and the progression of the disease one year after.  
In this section we show three tables of results, Table 4 shows the results of our algorithm. Tables 5 and 6 we present the results of the linear regression and the Support Vector Machine with linear kernel, respectively.  
In this subsection, the experiments are performed with simple data and the result of a simple linear regression are shown. The results are shown in Tables 4 and 5.  
As it can be seen, an important result is the given by the _Mauna LOA CO\({}_{2}\)_ dataset. This dataset contains data from the year 1958 to 2001. Thus, for this experiment the algorithm is trained with \(M=250\) and performing five swaps. After the model has been trained, to learn stationary kernels, it is possible to asses the performance of the model. For example, the achieved MSE is 0.6052 which helps at the estimation ot the CO\({}_{2}\) outputs. For example, at the sample 2002.13, we have prediction 376.873 where the real measure for this value is 373.08. The total results of Mauna LOA CO\({}_{2}\) are shown in Figure 4 and Table 4.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Dataset** & **N** & **M** & **Swaps** & **AUC** \\ \hline Breast Cancer [26] & 569 & 500 & 5 & 0.51348 \\ Credit-g [26] & 1000 & 500 & 5 & 0.5142 \\ Blood & 748 & 500 & 5 & 0.54063 \\ Transfusion [27] & 45,312 & 500 & 5 & 0.74991 \\ Egg-eye-state [26] & 14,980 & 500 & 5 & 0.519743 \\ Kr vs Kp [26] & 3196 & 500 & 5 & 0.9045 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Results of GaBaSR Classification.  
Figure 3: The real vs. predicted using GaBaSR.  
Figure 2: An instance of the samples taken from Equation (12).  
We use the following datasets: (1) Mauna LOA CO\({}_{2}\) from [29], (2) California houses from [30], (3) Boston house-price from [30] and Diabetes from [30]