DEMANDE Dataset

<p>This dataset contains the features and probabilites of ten different functions. Each dataset is saved using numpy arrays. &nbsp;\item The data set \textit{Arc} corresponds to a two-dimensional random sample drawn from a random vector $$X=(X_1,X_2)$$ with probability density function given by $$f(x_1,x_2)=\mathcal{N}(x_2|0,4)\mathcal{N}(x_1|0.25x_2^2,1)$$ where $$\mathcal{N}(u|\mu,\sigma^2)$$ denotes the density function of a normal distribution with mean $$\mu$$ and variance $$\sigma^2$$. \cite{Papamakarios2017} used this data set to evaluate his neural density estimation methods. \item The data set \textit{Potential 1} corresponds to a two-dimensional random sample drawn from a random vector $$X=(X_1,X_2)$$ with probability density function given by $$f(x_1,x_2)=\frac{1}{2}\left(\frac{||x||-2}{0.4}\right)^2 - \ln{\left(\exp\left\{-\frac{1}{2}\left[\frac{x_1-2}{0.6}\right]^2\right\}+\exp\left\{-\frac{1}{2}\left[\frac{x_1+2}{0.6}\right]^2\right\}\right)}$$ with a normalizing constant of approximately 6.52 calculated by Monte Carlo integration. \item The data set \textit{Potential 2} corresponds to a two-dimensional random sample drawn from a random vector $$X=(X_1,X_2)$$ with probability density function given by $$f(x_1,x_2)=\frac{1}{2}\left[ \frac{x_2-w_1(x)}{0.4}\right]^2$$ where $$w_1(x)=\sin{(\frac{2\pi x_1}{4})}$$ with a normalizing constant of approximately 8 calculated by Monte Carlo integration. \item The data set \textit{Potential 3} corresponds to a two-dimensional random sample drawn from a random vector $$x=(X_1,X_2)$$ with probability density function given by $$f(x_1,x_2)= - \ln{\left(\exp\left\{-\frac{1}{2}\left[\frac{x_2-w_1(x)}{0.35}\right]^2\right\}+\exp\left\{-\frac{1}{2}\left[\frac{x_2-w_1(x)+w_2(x)}{0.35}^2\right]\right\}\right)}$$ where $$w_1(x)=\sin{(\frac{2\pi x_1}{4})}$$ and $$w_2(x)=3 \exp \left\{-\frac{1}{2}\left[ \frac{x_1-1}{0.6}\right]^2\right\}$$ with a normalizing constant of approximately 13.9 calculated by Monte Carlo integration. \item The data set \textit{Potential 4} corresponds to a two-dimensional random sample drawn from a random vector $$x=(X_1,X_2)$$ with probability density function given by $$f(x_1,x_2)= - \ln{\left(\exp\left\{-\frac{1}{2}\left[\frac{x_2-w_1(x)}{0.4}\right]^2\right\}+\exp\left\{-\frac{1}{2}\left[\frac{x_2-w_1(x)+w_3(x)}{0.35}^2\right]\right\}\right)}$$ where $$w_1(x)=\sin{(\frac{2\pi x_1}{4})}$$, $$w_3(x)=3 \sigma \left(\left[ \frac{x_1-1}{0.3}\right]^2\right)$$, and $$\sigma(x)= \frac{1}{1+\exp(x)}$$ with a normalizing constant of approximately 13.9 calculated by Monte Carlo integration. \item The data set \textit{2D mixture} corresponds to a two-dimensional random sample drawn from the random vector $$x=(X_1, X_2)$$ with a probability density function given by $$f(x) = \frac{1}{2}\mathcal{N}(x|\mu_1,\Sigma_1) + \frac{1}{2}\mathcal{N}(x|\mu_2,\Sigma_2)$$ with means and covariance matrices $$\mu_1 = [1, -1]^T$$, $$\mu_2 = [-2, 2]^T$$, $$\Sigma_1=\left[\begin{array}{cc} 1 &amp; 0 \\ 0 &amp; 2 \end{array}\right]$$, and $$\Sigma_1=\left[\begin{array}{cc} 2 &amp; 0 \\ 0 &amp; 1 \end{array}\right]$$ \item The data set \textit{10D-mixture} corresponds to a 10-dimensional random sample drawn from the random vector $$x=(X_1,\cdots,X_{10})$$ with a mixture of four diagonal normal probability density functions $$\mathcal{N}(X_i|\mu_i, \sigma_i)$$, where each $$\mu_i$$ is drawn uniformly in the interval $$[-0.5,0.5]$$, and the $$\sigma_i$$ is drawn uniformly in the interval $$[-0.01, 0.5]$$. Each diagonal normal probability density has the same probability of being drawn $$1/4$$.</p>