constraint on $N_{gal}$ is coming from $R_{1/2}$ in the Millennium simulations. The selection of $R_{1/2}$ should not be surprising. It contains information about the halo mass and, as discussed above, can be related to halo environment through $M_{200}$.

  Parameter       First Round   Second Round
  -------------- ------------- --------------
  $N_p$              0.216         0.170
  $M_{200}$          0.189         0.163
  $\sigma_{v}$       0.220         0.163
  $v_{max}$          0.229         0.163
  $R_{1/2}$          0.163           –
  spin               0.275         0.167

  : \[tab:fs\]MSE values obtained by performing a forward feature selection using SVM. The values quoted are the median MSE from 10 different randomizations of the training, validation and test sets. The halo parameters we use are listed in column 1 and the median MSE values obtained by using only the listed parameter are shown in column 2 (First Round). One can see that $R_{1/2}$ has the smallest median MSE and hence it should be the best predictor of $N_{gal}$ in the context of the Millennium simulations. Adding in additional parameters does not significantly change the median MSE as indicated in the third column (Second Round) which lists the median MSE values obtained using $R_{1/2}$ plus the parameter listed in column 1.

Colour-dependent Mocks {#sec:colour}
----------------------

The Millennium semi-analytic galaxies come with $b,v,r,i,z$ absolute magnitudes which we 