# Subsampling Algorithms for Irregularly Spaced Autoregressive Models
## 2 Background
### Subsampling Methods  
Constructing parameter estimates based on subsamples from the full data is a popular technique to speed up computations. While the simplest approach of uniform sampling may not be effective for extracting useful information from a large dataset, optimized subsampling methods do provide a better trade-off between estimation efficiency and computational efficiency. Such methods have attracted significant attention in recent years because they are designed to (a) give higher preference to more informative data points and (b) be subject to less information loss. Typical practices include stochastic subsampling and deterministic subdata selection.  
Stochastic subsampling methods are successful because they specify inclusion probabilities that allow more informative data points to have a higher chance of being included in the subsample. In early attempts, [PERSON] et al. [19] advocated the use of normalized statistical leverage scores as subsampling probabilities in least squares estimation problems, while [PERSON] et al. [20] showed that using the normalized square roots of the statistical leverage scores provides a tighter error bound. [PERSON] et al. [21] examined the statistical properties of estimators resulting from subsampling methods based on statistical leverage scores, they and termed it _algorithmic leverage_. [PERSON] et al. [22] applied the statistical leverage scores for subsampling under a vector autoregressive model.  
This approach has been used in several statistical modeling frameworks. [PERSON] [23] proposed a subsampling method using the gradients of the objective function for linear models. [PERSON] et al. [24] proposed optimal subsampling probabilities under A-optimality and L-optimality criteria for logistic regression. [PERSON] et al. [25] examined the asymptotic properties of subsampling estimators for generalized linear models under unbounded design. The optimal subsampling framework has been extended to other modeling scenarios such as multiclass logistic regression, generalized linear models, and quantile regressions.  
While the aforementioned studies are based on sampling with replacement (in which the same data point may be included more than once in the subsample), further research derived optimal subsampling probabilities and a distributed sampling strategy under Poisson sampling (sampling without replacement). [PERSON] et al. (2018) further showed that Poisson sampling can be superior to sampling with replacement in terms of estimation efficiency. Poisson sampling for irregular time series analysis is described in Section 4.1.  
Deterministic subdata selection uses a particular criterion to determine a subsample without involving additional randomness. [PERSON] et al. (2018) introduced a novel method, termed _information-based optimal subsampling_ (IBOSS), designed to select data points with extreme values to approximate the D-optimality criterion in designed experiments. [PERSON] and [PERSON] (2018) later proposed an online selection strategy that leverages directional derivatives to decide whether to include a data point in a subsample, aiming to achieve optimality according to a given criterion. This approach processes only the data points encountered up to the current time, determining whether the current point should be included in the subsample. As a result, it is especially well suited for applications involving streaming data.  
Despite the rapid recent developments mentioned above, applications of the stochastic subsampling and deterministic subdata selection methods remain unexplored for irregularly spaced time series.