# Fast Minimum Error Entropy for Linear Regression
## 4 Experiments  
In this section, a comprehensive array of experiments is undertaken, encompassing both numerical simulations and real-world scenarios. The numerical simulations serve to validate the efficacy of the FMEE approach and to scrutinize the time efficiency of both FMEE and MEE methodologies. Concurrently, practical experiments are conducted aimed at forecasting power outages within a city located in northwest China. Regarding the numerical simulations, we consider a scenario where \(\mathbf{x}\in\mathbb{R}^{10}\), with the model defined as \(y=\mathbf{w}*^{\mathsf{T}}\mathbf{x}+e\), where \(\mathbf{w}*=[1,-1,1,-1,1,-1,1,-1,1,-1]^{\mathsf{T}}\) and \(\mathbf{x}\) follows a normal distribution \(\mathcal{N}(0,I_{10})\). Two distinct types of noise are examined in our experiments. Firstly, Gaussian noise \(e\sim\mathcal{N}(0,1)\) is employed, while the second type comprises generalized Gaussian noise characterized by a probability density function \(f(e)\propto\exp(-\alpha|e|)^{0.3}\). This heavy-tailed distribution, utilized herein, aligns with previous works, such as references [27; 35]. Although FMEE is derived under the assumption of near-Gaussian noise, it is anticipated that FMEE remains effective in the presence of non-Gaussian, heavy-tailed noises. Here, \(\alpha\) denotes a constant adjusted to maintain a variance of 1 for \(e\). For MEE, the step size is set to \(\sqrt{0.005\pi}\), and the scale parameter for the Gaussian kernel is 10, mirroring the settings adopted in Reference [27]. Across the experiments, sample sizes range from 100 to 500, with each combination of sample size and noise type repeated 100 times. The ensuing experimental results are detailed below. Table 1 provides a comparative analysis of the time consumption between MEE and FMEE in the presence of Gaussian noise. FMEE demonstrates a notable acceleration in computational efficiency compared to MEE. Specifically, while the computational time of MEE exhibits quadratic growth relative to the sample size, FMEE's computational overhead scales linearly with the number of samples. Notably, FMEE achieves remarkable speed, processing 500 objects in approximately 0.1 s, thereby positioning it as a highly promising candidate for time-sensitive applications. Table 2 compares the mean squared error between MEE and FMEE in the context of Gaussian noise. Overall, MEE exhibits superior performance compared to FMEE in terms of MSE. Particularly noteworthy is MEE's substantial advantage over FMEE, especially evident when the sample size is 100. However, as the sample size increases, the discrepancy in MSE between the two algorithms diminishes. By the time the sample size reaches 500, the difference dwindles to a mere 2%. This observation suggests that while FMEE may lag slightly behind MEE, particularly in larger-scale datasets, such disparity remains acceptable in practical applications, given FMEE's significantly reduced computational time compared to MEE. \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Sample Size**} & \multicolumn{2}{c}{**Average Running Time**} & \multicolumn{2}{c}{**Optimal Running Time**} \\ \cline{2-5}  & **MEE** & **FMEE** & **MEE** & **FMEE** \\ \hline
100 & 103.89 & 0.0401 & _77.77_ & 0.0156 \\
200 & 384.31 & 0.0580 & 310.05 & 0.0312 \\
300 & 801.52 & 0.0769 & 654.21 & 0.0468 \\
400 & 1430.61 & 0.1000 & 1254.12 & 0.0468 \\
500 & 2306.24 & 0.1104 & 2072.50 & 0.0624 \\ \hline \end{tabular}
\end{table} 
Table 1: The running time of MEE and FMEE for Gaussian noise. Table 3 analyses the iteration counts required for convergence between MEE and FMEE in the context of Gaussian noise. Notably, FMEE achieves convergence with significantly fewer iterations compared to MEE. Furthermore, it is observed that the iteration count of MEE gradually decreases as the sample size increases. Conversely, the iteration count of FMEE remains relatively constant, irrespective of variations in sample size. Table 4 provides a comparative assessment of the time consumption between MEE and FMEE in the context of non-Gaussian noise.