Let $\xx = (x_1,x_2,x_3,x_4,x_5)\in\X = [0 , 1]^5$. A second toy example is defined as follows:
\begin{eqnarray*}
    G(\xx,N,U_1,U_2,B) = (x_1+ 2x_2 + U_1)\sin(3x_3- 4x_4+N) + U_2+10x_5B + \sum\limits_{i=1}^5 ix_i
\end{eqnarray*}
where
\begin{eqnarray*}
N & \sim & \mathcal{N}(0,1) \\
U_1 & \sim & \mathcal{U}([0,1]) \\
U_2 & \sim & \mathcal{U}([1,2]) \\
B & \sim & \mathcal{B}ern(1/2).
\end{eqnarray*}
Figure \ref{figure:exemple:2:data} represents the output for $N = 100$ points simulated uniformly and independently in \X.

\subsubsection{Kernel regression}

In this section, the  kernel regression is tested on this second toy example with isotropic (Figure \ref{figure:exemple:2:boxplot:uni}) and anisotropic bandwidth (Figure \ref{figure:exemple:2:boxplot:multi}). As in the first example, Figure \ref{figure:exemple:2:1} represents for different values of $x_0$ the true probability density function (blue dashed line) and the estimation obtained from the two estimators of the kernel regression (red plain line and orange dots line). In this example, the two estimators give different results.
In red is represented the relative error for the estimator based on Hellinger distance, in blue for the estimator based on $\xLn{2}$ norm. The Hellinger estimator gives a better approximation of the quantities presented in Figure \ref{figure:exemple:2:boxplot:uni}, except for the mean. Contrary to the case of the first toy example, the errors are quite high.
The variance of the error on the norms, presented in Figure \ref{figure:exemple:2:boxplot:uni}, is low but does not seem to decrease steadily. 
The use of an anisotropic bandwidth does not seem to improve the quality of the estimation, while it is much longer to compute. The errors are approximately the same with isotropic or anisotropic bandwidth. The greater precision brought by the anisotropic bandwidth is compensated by the difficulty to estimate it.

\begin{figure}
\center
		\includegraphics[scale=0.4, bb=0 0 1076 587]{fig_exemple_5_data.pdf}
	\caption{Toy example 2: representations of outputs for $N = 100$ different parameters.}
\label{figure:exemple:2:data}
\end{figure}

\begin{figure}
\center
		\includegraphics[scale=0.4, bb=0 0 1076 587]{fig_exemple_5_1.pdf}
	\caption{Toy example 2: curves in dashed gray lines are probability density functions, red plain line (resp. orange dots line) represents the estimation of a density for a fixed $\xx_0$ from the first (resp. second) method and dashed blue line is the true probability density function. These estimations were made with $h$ isotropic. The orange dotted line and the red line are superposed.}
\label{figure:exemple:2:1}
\end{figure}


\begin{figure}
\center
		
		\includegraphics[scale = 0.4, bb=0 0 720 720]{function5D_huni_boxplot_1.pdf}
	\caption{Toy example 2: boxplot of the errors for different sizes of $N$. Left: estimator given by $\xLn{2}$ norm. Right: estimator given by the Hellinger distance. Results have been averaged with 25 independent experiments. These estimations were made with $h$ isotropic. Blue: estimator given by $\xLn{2}$ norm. Red: estimator given by the Hellinger distance.}
\label{figure:exemple:2:boxplot:uni}
\end{figure}


\begin{figure}
\center
		
		\includegraphics[scale = 0.4, bb=0 0 720 720]{function5D_multi_boxplot_1.pdf}
	\caption{Toy example 2: boxplot of the errors for different sizes of $N$. Left: estimator given by $\xLn{2}$ norm. Right: estimator given by the Hellinger distance. Results have been averaged with 25 independent experiments. These estimations were made with $h$ anisotropic. Blue: estimator given by $\xLn{2}$ norm. Red: estimator given by the Hellinger distance.}
\label{figure:exemple:2:boxplot:multi}
\end{figure}


\subsubsection{Functional decomposition methods}


In this section, the four functional decompositions MMP with $\xLn{2}$ and Hellinger distances, AQM and CPCA are compared. The decompositions are applied to learning samples whose sizes are $N=50$ and 100 and for decomposition basis sizes ranging from 1 to 20. The relative errors between the learning sample functions and their approximations are computed for the 9 quantities of interest. Figure \ref{figure:exemple:2:decomposition} represents, in logarithmic scale, these relative errors in function of the basis size for the three decompositions. 
First, the relative errors are low for all quantities of interest, as in the previous example. 
The relative errors decrease is less even than in the one-dimensional case. The behavior of the errors on 1\% and 99\% quantiles is particularly unsteady. CPCA method gives better results than the other methods for $\xLn{1}$, $\xLn{2}$ norms, 25\% and 75\% quantiles. However, for all studied quantities, the AQM and CPCA method seem to give quite equivalent results. MMP method errors are slightly higher than AQM errors for most of the quantities.


\begin{figure}
\center
		\includegraphics[scale=0.35, bb=0 0 1296 864]{fig_exemple_5_decompositions.pdf}
	\caption{Toy example 2: comparison of the relative error for different quantities in function of the size of the basis $q$ with a design of experiments of size $N = 50$ (circles) and 100 (filled circles). Blue: MMP decomposition given by $\xLn{2}$ norm. Red: MMP decomposition given by the Hellinger distance. Black: AQM decomposition. Green: CPCA decomposition.}
\label{figure:exemple:2:decomposition}
\end{figure}
{"config": "arxiv", "file": "1406.6348/Rapport_CEMRACS_numerical_toy_example_2.tex"}