# Subsampling Algorithms for Irregularly Spaced Autoregressive Models
## 4 Subsampling Methods for the IS-AR(1) Model
### Optimal Subsampling Under A-Optimality  
Optimal subsampling is a stochastic subsampling strategy satisfying a certain optimality property. The inclusion of subsampled data points is determined by carefully designed probabilities in order to meet the optimality criterion. In this section, we implement optimal Poisson subsampling, which is suitable for time series and avoids the need to simultaneously access the full data. In comparison to sampling with replacement,Poisson sampling cannot sample the same data point more than once. Since repeatedly sampled data points do not contribute any new information, Poisson sampling preserves more distinct information from the full data and is therefore more efficient than sampling with replacement. See [PERSON] et al. (2019) for a theoretical justification of the superiority of Poisson sampling. Let \(\eta_{j}\) be the sampling probability and \(\delta_{j}\) be the indicator variable for the inclusion of the \(j\)-th data point in the subsample. With \(u_{j}\) randomly sampled from the uniform distribution \(U(0,1)\), we set \(\delta_{j}=1\) when \(u_{j}\leq\eta_{j}\) and \(\delta_{j}=0\) otherwise. The actual subsample size from Poisson sampling, which is denoted by \(r^{*}=\sum_{j=1}^{m}\delta_{j}\), is random. The expected subsample size is \(r=\sum_{j=1}^{m}\eta_{j}\). We obtain the subsample estimator \(\tilde{\mathbf{\theta}}\) by maximizing the following target function:  
\[\mathrm{M}_{r}^{\mathrm{(opt)}}(\mathbf{\theta})=\frac{1}{m}\sum_{j=1}^{m}\delta_{ j}\frac{\ell(\mathbf{\theta};y_{j},y_{j-1},d_{j})}{\eta_{j}}. \tag{6}\]  
The efficacy of \(\tilde{\mathbf{\theta}}\) depends on the subsampling probabilities \(\{\eta_{j}\}_{j=1}^{m}\). We obtain the A-optimal subsampling probabilities which depend on the unknown true parameter and minimize the asymptotic mean squared error of \(\tilde{\mathbf{\theta}}\)([PERSON] and [PERSON], 2019; [PERSON] et al., 2019). In practice, this unknown parameter could be replaced by a pilot estimate \(\mathbf{\theta}^{\mathrm{(plt)}}\) obtained from a pilot subsample of size \(r_{0}\). The A-optimal subsampling probabilities with the pilot estimate \(\mathbf{\theta}^{\mathrm{(plt)}}\) take the general form  
\[\hat{\eta}_{j}^{\mathrm{(opt)}}=\left\|\kappa\Big{\{}\bar{\mathbf{M}}_{r_{0}} ^{\mathrm{(plt)}}(\mathbf{\theta}^{\mathrm{(plt)}})\Big{\}}^{-1}\hat{\ell}(\mathbf{ \theta}^{\mathrm{(plt)}};y_{j},y_{j-1},d_{j})\right\|\wedge 1,\text{ for }j=1, \ldots,m, \tag{7}\]  
where \(\kappa\) is a parameter to ensure that the expected sample size is set around \(r\) and prevent it from being too small, the notation \(a\wedge b=\min(a,b)\),  
\[\bar{\mathbf{M}}_{r_{0}}^{\mathrm{(plt)}}(\mathbf{\theta})=\frac{1}{r_{0}}\sum_{j= 1}^{m}\delta_{j}^{\mathrm{(plt)}}\hat{\ell}(\mathbf{\theta};y_{j},y_{j-1},d_{j}) \tag{8}\]  
is the average Hessian matrix with the pilot subsample, and \(\delta_{j}^{\mathrm{(plt)}}\) is the indicator variable for inclusion of the \(j\)-th data point in the pilot sample. While the exact value of \(\kappa\) requires additional computation, it can be approximated by  
\[\kappa_{a}=\frac{r}{\sum_{j=1}^{m}\left\|\left\{\bar{\mathbf{M}}_{r_{0}}^{ \mathrm{(plt)}}(\mathbf{\theta}^{\mathrm{(plt)}})\right\}^{-1}\hat{\ell}(\mathbf{ \theta}^{\mathrm{(plt)}};y_{j},y_{j-1},d_{j})\right\|}\]  
when the sampling rate \(r/m\) is small. This is usually the case in practice. The approximated subsampling probabilities \(\hat{\eta}_{j}^{\mathrm{(opt)}}\) in (7) are subject to additional disturbance from the pilot estimation, and small values of \(\hat{\eta}_{j}^{\mathrm{(opt)}}\) may inflate the asymptotic variance of the resulting estimator. To address this problem, we mix \(\hat{\eta}_{j}^{\mathrm{(opt)}}\) with the uniform subsampling probabilities to prevent the sampling probabilities from getting too small. Specifically, we use  
\[(1-\alpha_{1})\eta_{j}^{\mathrm{(opt)}}+\alpha_{1}\frac{r}{m},\text{ for }j=1,\ldots,m,\]  
where \(\alpha_{1}\in(0,1)\) is a tuning parameter.