Algorithm 2.

Estimation of covariate relationships via sparse boosting

Step 1: Initialization. Set m = 0 and ${\mathit{F}}^{(0)}={\left(F_1^{(0)}\cdots F_{n_c}^{(0)}\right)}^{\prime }=\mathbf{0}$.

Step 2: Fit and update. m = m + 1.

Compute

$$\left({\widehat{\eta }}_{k0},{\widehat{\mathit{\eta }}}_k\right)=\text{argmin}\sum _{i=1}^{n_c}{\left(r_i^{(m)}-{\eta }_{k0}-{\mathit{u}}_{ik}{\mathit{\eta }}_k^{\prime }\right)}^2,k=2,\cdots ,q,$$ (12)

where $r_i^{(m)}=z_{i1}-F_i^{(m-1)}$.

For variable selection, compute

$$S_m=\underset{2\le k\le q}{\text{argmin}}\left\{\log \left(\sum _{i=1}^{n_c}{\left(r_i^{(m)}-{\eta }_{k0}-{\mathit{u}}_{ik}{\mathit{\eta }}_k^{\prime }\right)}^2\right)+d{f}_m(k)\log \left(n_c\right)/n_c\right\}.$$ (13)

Update ${\mathit{F}}^{(m)}={\mathit{F}}^{(m-1)}+\nu {\widehat{\mathit{r}}}^{(m)}$, where ν = 0.1, and ${\widehat{\mathit{r}}}^{(m)}={\left({\widehat{\eta }}_{S_m,0}+{\mathit{u}}_{1,S_m}{\widehat{\mathit{\eta }}}_{S_m}^{\prime }\cdots {\widehat{\eta }}_{S_m,0}+{\mathit{u}}_{n_c,S_m}{\widehat{\mathit{\eta }}}_{S_m}^{\prime }\right)}^{\prime }$.

Step 3: Iteration and stopping. Repeat Step 2 for M iterations. Estimate the stopping iteration by

$$m_{\text{stop}}=\underset{1\le m\le M}{\text{argmin}}\left\{\log \left(\sum _{i=1}^{n_c}{\left(z_{i1}-F_i^{(m)}\right)}^2\right)+d{f}_m\log \left(n_c\right)/n_c\right\}.$$ (14)

Variables selected and their estimates at iteration mstop are the final results.