# Subsampling Algorithms for Irregularly Spaced Autoregressive Models
## 4 Subsampling Methods for the IS-AR(1) Model
### Sequential Thinning on Streaming Data  
The **thin** method proposed in [PERSON] and [PERSON] [28] is another deterministic subdata selection approach. Unlike **iboss**, which requires simultaneous access to the full data, **thin** only uses the data points that are already included in the subsample to determine whether the next data point should be included in the subsample or not. This online decision nature of the **thin** algorithm makes it suitable for time series data. We present the main idea of the **thin** method based on D-optimality below. Let the average information matrix of a subsample indexed by \(S=\{\delta_{j}^{S}\}_{j=1}^{m}\) be  
\[\mathbf{\mathcal{I}}_{S}(\mathbf{\theta})=\frac{1}{r_{S}}\sum_{j=1}^{m}\delta_{j}^{S} \mathbf{I}_{j}(\mathbf{\theta}|y_{j-1},d_{j}), \tag{12}\]  
where \(r_{S}=\sum_{j=1}^{m}\delta_{j}^{S}\) and \(\mathbf{I}_{j}(\mathbf{\theta}|y_{j-1},d_{j})\) is defined in (10). The contribution of a data point, say \((y_{j^{\prime}},y_{j^{\prime}-1},d_{j^{\prime}})\), to the average information matrix, if included in the subsample, can be measured by the directional derivative, which is defined as  
\[\zeta_{\mathbf{\mathcal{I}}_{S}(\mathbf{\theta}),j^{\prime}}=\text{tr}\Big{(}\mathbf{ \mathcal{I}}_{S}(\mathbf{\theta})^{-1}\mathbf{I}_{j^{\prime}}(\mathbf{\theta}|y_{j^{\prime }-1},d_{j^{\prime}})\Big{)}-2. \tag{13}\]  
The **thin** method aims to include \((y_{j^{\prime}},y_{j^{\prime}-1},d_{j^{\prime}})\) in the subsample if \(\zeta_{\mathbf{\mathcal{I}}_{S}(\mathbf{\theta}),j^{\prime}}\) is large enough. This is motivated by the key result in optimal experimental design theory that the optimal design consists of design points with the largest directional derivatives in the design space [31; 32]. In the context of subsampling, this means that we need to find the subsample with the \(r\) largest directional derivatives under the unknown optimal average information matrix. In order to achieve this in an online manner, we need to sequentially estimate the upper \(r/m\) quantile for the distribution of the directional derivatives. Unlike the linear models discussed in [PERSON] and [PERSON] [28] for which the information matrix is completely known, the information matrix for our model depends on the unknown parameter \(\mathbf{\theta}\). We therefore need a pilot step to obtain a pilot estimator. We present the outline of the **thin** method for our problem in the following steps:  
1. Reserve the first \(r_{0}\) data points up to time \(t_{r_{0}}\) as a pilot sample, and use it to obtain a pilot estimate \(\mathbf{\theta}^{\text{(ptl)}}\) as in Section 4.1. 2. Calculate \(\{\zeta_{\mathbf{\mathcal{I}}_{T_{0}}(\mathbf{\theta}^{\text{(ptl)}}),j^{\prime}}\}_{ j=1}^{r_{0}}\) and obtain its sample upper \(\alpha\)-quantile \(\hat{C}_{r_{0}}\), where \(\alpha=r/(m-r_{0})\) and \(\mathbf{\mathcal{I}}_{r_{0}}(\mathbf{\theta}^{\text{(ptl)}})\) is the average information matrix for the pilot sample. 3. For \(j=r_{0}+1,\ldots,m\), let \(\mathbf{\mathcal{I}}_{j-1}(\mathbf{\theta}^{\text{(ptl)}})\) be the average information matrix and \(\hat{C}_{j-1}\) be the estimated quantile from the subsample collected up to time \(t_{j-1}\). If \(\zeta_{\mathbf{\mathcal{I}}_{j-1}(\mathbf{\theta}^{\text{(ptl)}}),j}>\hat{C}_{j-1}\), include the data point \((y_{j},y_{j-1},d_{j})\) in the subsample and calculate the updated \(\mathbf{\mathcal{I}}_{j}(\mathbf{\theta}^{\text{(ptl)}})\); otherwise, \(\mathbf{\mathcal{I}}_{j}(\mathbf{\theta}^{\text{(ptl)}})=\mathbf{\mathcal{I}}_{j-1}(\mathbf{ \theta}^{\text{(ptl)}})\). Calculate the updated \(\hat{C}_{j}\). 4. Let \(\delta_{j}^{\text{(thin)}}\) be the indicators for the **thin** subsample collected in Step 3. Obtain the subsample estimator \(\tilde{\mathbf{\theta}}^{\text{(thin)}}\) by maximizing \[\text{M}_{r}^{\text{(thin)}}(\mathbf{\theta})=\frac{1}{r}\sum_{j=r_{0}+1}^{m} \delta_{j}^{\text{(thin)}}\ell(\mathbf{\theta};y_{j},y_{j-1},d_{j}).\]
5. Obtain the final subsample estimator \(\tilde{\mathbf{\theta}}^{\text{(thin)}}\) by \[