connected components. We test two instances of this example: one with noise and signal variables in both components, and another with a component containing only noise variables. The fourth example is taken from the elastic net paper [@enet]. All signal variables in that example belong to three connected components, and the remaining noise variables are independent. The elastic net is known to perform well under such conditions, and is shown in [@enet] to be better than the lasso at picking out the relevant correlated variables.

Our examples were generated as follows:

-   Example 1: $p=8$, $\sigma=3$ and $\beta=(3,1.5,0,0,2,0,0,0)$. We simulate 100 20/20/200 sets of observations with pairwise correlation ${\rm corr}(i,j)=0.5^{|i-j|}$. This gave an average signal to noise ratio (SNR) of 2.38.

-   Example 2: $p=8$, $\sigma=5$ and $\beta=(3,1.5,0,0,2,3,0,0)$ or $\beta=(3,1.5,2,3,0,0,0,0)$. We simulate 100 20/20/200 sets of observations in the following way: $$x_{i} = Z_1 + \epsilon_i \text{ if } i \in 1,\ldots,4$$ $$x_{i} = Z_2 + \epsilon_i \text{ if } i \in 5,\ldots,8$$ where $Z_1$ and $Z_2$ $\sim \mathcal{N}(0,2)$ and $\epsilon_i \sim \mathcal{N}(0,0.5)$. The main point of this example is to compare the performance of the component lasso depending on whether the signal variables are in separate connected components (signal 