Sequential Leveraging Sampling (SLS)

• SLS is an online, leverage-based block sampling method that selects a single, contiguous block from streaming autoregressive time series based on a random start and sequential stopping rule.
• The method uses pilot estimates to compute surrogate leverage scores, allowing for efficient, per-observation computations without full matrix inversions even in high-speed, unbounded streams.
• SLS preserves temporal dependence and adapts block lengths via an information threshold, making it particularly useful in applications such as seismic monitoring and other dynamic data environments.

Searching arXiv for the cited papers to ground the article and check for any closely related usage. Sequential Leveraging Sampling (SLS) is an online, leverage-based block sampling method for streaming autoregressive time series. In the formulation introduced in "Online Sequential Leveraging Sampling Method for Streaming Autoregressive Time Series with Application to Seismic Data" (Xie et al., 25 Sep 2025), SLS performs statistically valid inference from a single, carefully chosen consecutive block of a data stream rather than from the entire series. Its defining mechanism is a leverage-based random choice of the block starting time together with a sequential stopping rule that expands the block until a prescribed information threshold is reached. The method is intended for high-speed, potentially unbounded streams in which temporal dependence must be preserved, and it is developed for both linear and nonlinear autoregressive settings.

1. Conceptual basis and motivation

SLS addresses a specific streaming regime: data arrive continuously, the total stream length is unknown or effectively unbounded, working memory is limited, and the inferential target is least-squares estimation of autoregressive parameters. In seismic monitoring, the motivating application, sensors can generate data at tens of Hz on the macroseismic scale up to MHz in laboratory or exploration geophysics. Under such conditions, storing and processing the full stream for least squares or maximum likelihood estimation is infeasible (Xie et al., 25 Sep 2025).

The method is motivated by three structural difficulties. First, streaming time series are temporally dependent, so pointwise i.i.d. subsampling destroys lag relationships and invalidates standard autoregressive inference. Second, informative segments are not uniformly distributed: strong events and regime changes are typically rare relative to quiescent periods, so uniform subsampling expends budget on low-information segments. Third, any online procedure must avoid repeated inversion of a growing information matrix and must maintain per-observation cost independent of accumulated sample size.

The leverage component of SLS generalizes the regression notion of leverage score to an autoregressive stream. In classical linear regression with design matrix $X$, the hat matrix is

$H = X (X^T X)^{\dagger} X^T,$

and the leverage score of observation $i$ is

$h_{ii} = x_i^T (X^T X)^{\dagger} x_i.$

In SLS, each autoregressive observation is associated with a lag-vector regressor $z_i$, and the full-sample inverse is replaced by a streaming surrogate estimated from a small pilot dataset. This allows leverage-guided online sampling without recomputing a full inverse as the stream grows.

2. Statistical model and streaming setting

The principal model is a univariate autoregressive process of order $p$, observed as a stream $\{X_i\}_{i=1}^\infty$: $$X_i = \beta_1 X_{i-1} + \beta_2 X_{i-2} + \cdots + \beta_p X_{i-p} + \varepsilon_i,$$ with parameter vector $\boldsymbol{\beta} = (\beta_1,\ldots,\beta_p)^T$. The innovations $\{\varepsilon_i\}$ are i.i.d., mean zero, variance $H = X (X^T X)^{\dagger} X^T,$0, and independent of the initial values. For data up to time $H = X (X^T X)^{\dagger} X^T,$1, the regressor vector is $H = X (X^T X)^{\dagger} X^T,$2, the design matrix is

$H = X (X^T X)^{\dagger} X^T,$3

and the response vector is $H = X (X^T X)^{\dagger} X^T,$4. The full-sample least-squares estimator is

$H = X (X^T X)^{\dagger} X^T,$5

which becomes impractical for very large or effectively infinite $H = X (X^T X)^{\dagger} X^T,$6 (Xie et al., 25 Sep 2025).

The supplementary development extends SLS to nonlinear autoregressive and nonlinear additive ARX-type models of the forms

$H = X (X^T X)^{\dagger} X^T,$7

and

$H = X (X^T X)^{\dagger} X^T,$8

with smooth $H = X (X^T X)^{\dagger} X^T,$9 or $i$0. In these settings, the same block-selection mechanism is retained, while least-squares or analogous $i$1-estimators on the selected block are analyzed under dependence, mixing, and smoothness conditions.

The stated assumptions are tailored to streaming inference. Innovations are i.i.d. with finite variance and independent of the block starting state. The parameter set for AR($i$2) includes both stable and nearly or fully unstable cases, including unit-root regimes. At the same time, the process is treated as locally stationary over the SLS block, a condition emphasized as crucial when seismic events induce nonstationarity over longer horizons.

3. Algorithmic construction

SLS proceeds through a fixed pilot phase and an online block-construction phase. The pilot data $i$3 are used to determine the autoregressive order $i$4 via BIC, or alternatively AIC or MDL, and to estimate the expected information or precision matrix

$i$5

through a plug-in estimator $i$6. The computational cost of this step is $i$7, and it is performed once (Xie et al., 25 Sep 2025).

For each subsequent time $i$8, the streaming leverage score is defined by

$i$9

This serves as an online surrogate for the classical leverage score

$h_{ii} = x_i^T (X^T X)^{\dagger} x_i.$0

Because $h_{ii} = x_i^T (X^T X)^{\dagger} x_i.$1 is fixed after the pilot stage, no large matrix inversion is required per time step, and the per-observation cost is $h_{ii} = x_i^T (X^T X)^{\dagger} x_i.$2, independent of $h_{ii} = x_i^T (X^T X)^{\dagger} x_i.$3.

The starting time is chosen by independent Bernoulli trials,

$h_{ii} = x_i^T (X^T X)^{\dagger} x_i.$4

If $h_{ii} = x_i^T (X^T X)^{\dagger} x_i.$5, the algorithm advances to $h_{ii} = x_i^T (X^T X)^{\dagger} x_i.$6. If $h_{ii} = x_i^T (X^T X)^{\dagger} x_i.$7, then $h_{ii} = x_i^T (X^T X)^{\dagger} x_i.$8 becomes the start of the SLS block. This biases selection toward observations with larger $h_{ii} = x_i^T (X^T X)^{\dagger} x_i.$9, interpreted in the paper as a larger Mahalanobis norm and hence greater influence under the assumed autoregressive structure.

After a starting time is chosen, the block expands consecutively until cumulative information exceeds a threshold $z_i$0: $z_i$1 If innovations are Gaussian, $z_i$2 equals the trace of the observed Fisher information matrix for the block. Larger $z_i$3 implies more collected data and smaller asymptotic variance; smaller $z_i$4 yields shorter blocks and faster computation.

Inference is then performed only on the selected block. With block design matrix

$z_i$5

and block response vector $z_i$6, the least-squares estimator is

$z_i$7

The use of a single consecutive block preserves autoregressive dependence, limits storage to the block and fixed-size summaries, and yields memory complexity $z_i$8, independent of total stream length.

4. Asymptotic theory and inferential targets

For AR(1),

$z_i$9

the SLS least-squares estimator takes the form

$p$0

The central theoretical statement is a uniform asymptotic normality result: $p$1 The uniformity over $p$2 covers both stable and unit-root regimes. The stated interpretation is that, under information-based stopping, the normalized estimation error converges to a normal law with variance $p$3 regardless of proximity to instability (Xie et al., 25 Sep 2025).

The paper interprets the information threshold $p$4 as a proxy for effective sample size. For stable AR(1), the asymptotic relative efficiency of $p$5 versus the full-sample least-squares estimator $p$6 is approximately

$p$7

and the variance is controlled by

$p$8

For AR($p$9), with

$\{X_i\}_{i=1}^\infty$0

the main result asserts that, under regularity conditions on $\{X_i\}_{i=1}^\infty$1 covering stable and some unstable cases,

$\{X_i\}_{i=1}^\infty$2

for any compact $\{X_i\}_{i=1}^\infty$3, where $\{X_i\}_{i=1}^\infty$4 is the multivariate normal CDF with covariance $\{X_i\}_{i=1}^\infty$5. Correspondingly, the quadratic pivot

$\{X_i\}_{i=1}^\infty$6

converges in distribution to $\{X_i\}_{i=1}^\infty$7 uniformly on $\{X_i\}_{i=1}^\infty$8.

These results support fixed-width confidence regions. For significance level $\{X_i\}_{i=1}^\infty$9 and target width $$X_i = \beta_1 X_{i-1} + \beta_2 X_{i-2} + \cdots + \beta_p X_{i-p} + \varepsilon_i,$$0, the ellipsoid

$$X_i = \beta_1 X_{i-1} + \beta_2 X_{i-2} + \cdots + \beta_p X_{i-p} + \varepsilon_i,$$1

satisfies

$$X_i = \beta_1 X_{i-1} + \beta_2 X_{i-2} + \cdots + \beta_p X_{i-p} + \varepsilon_i,$$2

for an appropriate choice of $$X_i = \beta_1 X_{i-1} + \beta_2 X_{i-2} + \cdots + \beta_p X_{i-p} + \varepsilon_i,$$3. In the authors’ presentation, this is the statistical rationale for coupling leverage-guided starts with information-based stopping rather than using fixed-length blocks.

5. Seismic applications and empirical behavior

The principal empirical application is the 2023 Turkey–Syria earthquake doublet. The dataset is a seismogram from station NL.TERZ, covering 12 hours from 00:00 to 12:00 UTC on 6 February 2023, sampled at 40 Hz for approximately 1.726 million records. It contains two major earthquakes, $$X_i = \beta_1 X_{i-1} + \beta_2 X_{i-2} + \cdots + \beta_p X_{i-p} + \varepsilon_i,$$4 at 01:17 UTC and $$X_i = \beta_1 X_{i-1} + \beta_2 X_{i-2} + \cdots + \beta_p X_{i-p} + \varepsilon_i,$$5 at 10:24 UTC, together with numerous aftershocks. The first 10 minutes, or 24,000 records, were used as pilot data; BIC selected autoregressive order $$X_i = \beta_1 X_{i-1} + \beta_2 X_{i-2} + \cdots + \beta_p X_{i-p} + \varepsilon_i,$$6, and the information threshold $$X_i = \beta_1 X_{i-1} + \beta_2 X_{i-2} + \cdots + \beta_p X_{i-p} + \varepsilon_i,$$7 was chosen to correspond roughly to the Fisher information of 1 minute of stationary seismic activity. Under this setup, SLS produced 3,159 blocks scattered over the 12-hour stream (Xie et al., 25 Sep 2025).

Several findings are reported for this macroseismic case. When the first two lag coefficients are compared across representative blocks, the coefficients change markedly during major events: before events they are similar to pilot estimates; during large earthquakes the magnitude of lag-1 increases while lag-2 decreases; after shocks both lag-1 and lag-2 become positive. Event detection is implemented through the block-based statistic

$$X_i = \beta_1 X_{i-1} + \beta_2 X_{i-2} + \cdots + \beta_p X_{i-p} + \varepsilon_i,$$8

where $$X_i = \beta_1 X_{i-1} + \beta_2 X_{i-2} + \cdots + \beta_p X_{i-p} + \varepsilon_i,$$9 and $\boldsymbol{\beta} = (\beta_1,\ldots,\beta_p)^T$0 are pilot estimates. The reported spikes in $\boldsymbol{\beta} = (\beta_1,\ldots,\beta_p)^T$1 align with the onset of the two major earthquakes and with significant aftershocks, even though only the SLS blocks are used for inference. The paper also reports that streaming leverage scores themselves display sharp spikes coinciding with major events, suggesting a role as an alert indicator.

A second application uses a microseismic sequence from Oklahoma on 26 October 2014 consisting of 16,000 points and including one distinct event lasting seconds. In this case, the pilot size is $\boldsymbol{\beta} = (\beta_1,\ldots,\beta_p)^T$2, the selected autoregressive order is $\boldsymbol{\beta} = (\beta_1,\ldots,\beta_p)^T$3, the first 14,000 points are used for training, and the last 2,000 for testing. The study considers 11 values of the information threshold $\boldsymbol{\beta} = (\beta_1,\ldots,\beta_p)^T$4 between $\boldsymbol{\beta} = (\beta_1,\ldots,\beta_p)^T$5 and $\boldsymbol{\beta} = (\beta_1,\ldots,\beta_p)^T$6, each with 100 replicates. Streaming leverage scores again highlight the main event as a spike, and across all $\boldsymbol{\beta} = (\beta_1,\ldots,\beta_p)^T$7 values SLS yields lower prediction MSE than uniform sequential sampling while using smaller block sizes and shorter running times.

The simulation studies extend these observations beyond seismic examples. The authors examine AR(1), AR(2), and nonlinear AR or additive AR models. In the AR(1) experiments, $\boldsymbol{\beta} = (\beta_1,\ldots,\beta_p)^T$8, the innovations follow a $\boldsymbol{\beta} = (\beta_1,\ldots,\beta_p)^T$9 distribution, the pilot size is $\{\varepsilon_i\}$0, and the main evaluation metrics are MSE, total running time, and block size. Across these designs, SLS typically achieves lower MSE than uniform sampling for the same $\{\varepsilon_i\}$1, comparable or better MSE than fixed-length block sampling, shorter blocks than uniform sampling to reach the same information threshold, and faster computation, especially when $\{\varepsilon_i\}$2. Sensitivity analyses in the $\{\varepsilon_i\}$3 case indicate that MSE is relatively insensitive to pilot size once $\{\varepsilon_i\}$4 is large enough to estimate $\{\varepsilon_i\}$5 and $\{\varepsilon_i\}$6 reliably.

6. Limitations, extensions, and terminological scope

The stated theoretical guarantees are tied to autoregressive models with i.i.d. innovations and local stationarity over the selected block. Strong long-range dependence, non-Markovian dynamics, or heavy nonlinearity beyond the considered nonlinear extensions may violate these guarantees. Global nonstationarity, including slow drift and structural breaks, may require re-estimation of the pilot-based precision surrogate over time. Model misspecification is another explicit limitation: if AR($\{\varepsilon_i\}$7) is a poor approximation, then leverage scores based on $\{\varepsilon_i\}$8 may misidentify informative segments and inference may be biased. The paper emphasizes that seismic signals are complex and that autoregressive models are used as first-order approximations (Xie et al., 25 Sep 2025).

Several extensions are suggested or implied. These include ARMA and state-space models, multivariate time series such as VAR processes and sensor streams, spatial and spatiotemporal data in which blocks would be defined jointly over time and space, and non-seismic high-speed streams such as network traffic, medical imaging signals, acoustic monitoring, and IoT sensor data. A plausible implication is that the central design principle of SLS is not restricted to seismology: leverage-guided block initiation together with information-based stopping constitutes a general data-reduction strategy for dependent streams whenever contiguous local structure is indispensable for inference.

A separate terminological point arises from contemporaneous work in LLM reinforcement learning. "The Road Less Traveled: Enhancing Exploration in LLMs via Sequential Sampling" (Kang et al., 17 Oct 2025) presents SESA, a two-stage sequential sampling framework in which short solution sketches are generated sequentially and then expanded into full reasoning paths. That paper explicitly notes that "Sequential Leveraging Sampling (SLS) is not a term used explicitly in the SESA paper." This suggests a useful distinction: in current arXiv usage, SLS refers to the leverage-score-based online block-sampling method for streaming autoregressive time series, whereas SESA concerns history-conditioned exploration in LLM rollouts.