Time-Weighted Averaging (T-WA) Methods

• Time-Weighted Averaging (T-WA) is a dynamic framework that assigns temporal weights to data points, enhancing estimates in evolving environments.
• It utilizes various schemes such as fixed decay kernels and adaptive eigenanalysis-based methods to adjust for time-dependent data quality.
• T-WA improves uncertainty quantification and performance in applications like frequency analysis, online learning, and time-varying optimization.

Time-Weighted Averaging (T-WA) is a rigorous statistical framework for combining measurements or model outputs where the relative contribution of each datum is explicitly modulated by temporal considerations. T-WA arises naturally in fields such as time-resolved metrology, frequency analysis, time series modeling, and online learning, serving as a principled extension of classic weighted averaging by introducing dynamic, time-dependent, or adaptive weights. Its generalization encompasses fixed temporal decay kernels, kernel-aligned elastic averaging, and quadratic-form adaptive schemes derived from eigenvalue problems. T-WA procedures allow for robust estimation of means, frequency stability, optimization tracking, or domain adaptation under nonstationary or temporally evolving conditions, with uncertainty quantification that accounts for both formal errors and observed sample scatter.

1. Mathematical Foundations of Time-Weighted Averaging

The foundational T-WA estimator for a set of measurements $x_1, x_2, \ldots, x_n$ with associated uncertainties $s_i$ modifies the classical weighted mean by introducing time-dependent weights:

• For static error-based weights: $p_i = 1 / s_i^2$
• For time-weighted schemes: $p_i = w_{\text{time}}(t_i) / s_i^2$, where $w_{\text{time}}(t_i)$ is a kernel reflecting recency, temporal proximity, or adaptive factors from the underlying physical process (Malkin, 2011).

The general form for the T-WA mean is:

$$\bar{x}_{\text{TWA}} = \frac{\sum_{i=1}^n p_i x_i}{\sum_{i=1}^n p_i}$$

Analogous constructions appear in streaming optimization (Abrar et al., 15 Oct 2025) with the instantaneous objective:

$F_t(w) = \sum_{i=1}^t a_i(t) f_i(w)$

where $a_i(t)$ is the temporal weight. Choices include:

• Uniform: $a_i(t) = 1/t$
• Discounted: $a_i(t) = (1-\gamma)/(1-\gamma^t)\,\gamma^{t-i}$

In moving average analysis (Boudjemila et al., 2022), the temporal weight may be a product of fixed kernel (e.g., exponential decay) and a polynomial determined from eigenanalysis of system observables, yielding adaptive and state-dependent weighting.

2. Uncertainty Quantification and Error Propagation in T-WA

Uncertainty estimation in T-WA extends the standard approaches for weighted means. Three principal strategies are identified (Malkin, 2011):

Estimator	Formula	Context
$\sigma_1$ (formal)	$\sigma_1 = 1/\sqrt{p}$	Only reported errors, low scatter
$\sigma_2$ (LS-based)	$\sigma_2 = \sigma_1 \sqrt{H / (n - 1)}$<br>$H = \sum p_i (x_i - \bar{x}_w)^2$	Accounts for the sample scatter
$\sigma_c$ (combined)	$\sigma_c = \sqrt{\sigma_1^2 + \sigma_2^2}$	Robust to both consistent/discrepant data

The combined estimator $\sigma_c$ is particularly useful for T-WA where measurement discrepancies may arise due to temporal trends or nonstationarity. This quadrature-based combination provides resilience to both underestimated reported errors and excess scatter, especially in time-dependent analyses.

In frequency analysis, uncertainty for a time-weighted average is expressed in the Fourier domain (Benkler et al., 2015):

$u^2 = \int_0^\infty S_y(f) |W(f)|^2 df$

where $S_y(f)$ is the power spectral density and $W(f)$ is the Fourier transform of the time-dependent weighting kernel.

In time-varying optimization, the "tracking error" (TE) quantifies deviation between the iteratively computed parameter and the time-evolving optimum (Abrar et al., 15 Oct 2025). With uniform weighting, TE asymptotically diminishes as $\mathcal{O}(1/t)$; with discounted weighting, TE approaches a nonzero error floor determined by the weight decay $\gamma$ and algorithmic parameters.

3. Temporal Weighting Schemes: Kernels and Adaptive Methods

T-WA implementation spans a variety of weighting function types:

• Fixed kernels: Rectangular, exponential, or polynomial windowing (Boudjemila et al., 2022). These encode explicit time scales but do not adapt to data properties.
• Kernel-based elastic alignment: Probabilistic methods use time-elastic kernels (e.g., KDTW) and alignment matrices to compute centroids across time series, yielding robust averaging by integrating contributions from all alignment paths (Marteau, 2015).
• Eigenproblem-derived weighting: Internal degrees of freedom within the weight function are set via eigenanalysis of an observable (e.g., execution flow), producing adaptive weights that instantaneously "switch" in response to dynamic changes (Boudjemila et al., 2022).
• Frequency domain weights: Averaging functions for frequency measurements include rectangular ("$\Pi$"-weighting), linear ("$\Lambda$"-weighting), and parabolic regression weighting. Each optimally suppresses noise of different types (Benkler et al., 2015).
• Discounted/decaying weights: Consistently used in online learning and streaming optimization to emphasize recent data while discounting memory (Abrar et al., 15 Oct 2025).

Temporal weighting design is commonly motivated by the need to balance responsiveness to change against statistical stability, as evidenced by adaptive averaging strategies that switch between kernels depending on noise regime and measurement context.

4. Robust T-WA Algorithms and Performance Evaluation

Time-elastic centroid (TEC) computation via kernel methods reframes centroid averaging as a preimage problem in RKHS (Marteau, 2015). Algorithms developed for probabilistic interpretation of kernel alignment matrices include:

• Iterative KDBA: Uses kernel alignment information to perform barycenter averaging along optimal alignment paths iteratively.
• Progressive KDTW-PWA: Averages both sample values and time indices; averages along the time axis confer improved noise reduction and robustness, especially where time-shifts and irregular sampling are present.

Empirical results on diverse datasets demonstrate that centroid-based T-WA methods outperform medoid-based techniques (lower error rates in 1-NC classification tasks), with kernel-based progressive averaging delivering highest robustness against noise and misalignment.

For frequency analysis, the use of modADEV and parADEV two-sample deviations, linked to $\Lambda$- and parabolic-weighted averaging, yields steeper uncertainty reductions in white phase noise environments (Benkler et al., 2015).

5. Applications of T-WA in Time-Varying Optimization and Temporal Generalization

In online learning from streams, T-WA is formalized as minimization of a weighted average objective, allowing the learning agent to "track" the moving optimum. Choice of weighting (uniform vs. discounted) dictates asymptotic behavior:

• Uniform weighting: For stationary environments, the tracking error TE vanishes as $\mathcal{O}(1/t)$, ensuring eventual convergence to the optimal parameter (Abrar et al., 15 Oct 2025).
• Discounted weighting: For nonstationary or fast-changing environments, recent observations are prioritized, leading to a persistent error floor determined by $\gamma$.

Design guidelines for streaming applications prescribe the selection of weight decay and number of gradient updates to meet specified error tolerances.

In temporal domain generalization, Temporal Experts Averaging (TEA) uses adaptive averaging across temporally fine-tuned expert models. TEA projects expert weight trajectories into principal component space, forecasts future positions, and adaptively computes averaging coefficients based on proximity to the anticipated future domain (Liu et al., 30 Sep 2025). The bias-variance decomposition guides this process, emphasizing low-bias and functionally similar experts, yielding superior generalization to future domains while minimizing computational overhead.

6. Limitations, Variability, and Practical Implementation Concerns

A central concern in T-WA is the proper accounting of both systematic error and sample variability. Classical estimators may underestimate uncertainty in the presence of measurement inconsistencies or nonstationarity. The quadrature sum approach—$\sigma_c = \sqrt{\sigma_1^2 + \sigma_2^2}$—ameliorates this but presumes suitable characterizations of both formal error and scatter.

In adaptive moving averages with internal degrees of freedom, the complexity of eigenvalue decompositions and polynomial fitting may introduce computational overhead and demand higher-quality data (Boudjemila et al., 2022).

Preimage computations in kernel-based T-WA are non-convex and susceptible to ill-posedness or overfitting, especially for long or high-dimensional series (Marteau, 2015). In weight-based optimization, discounted weighting trades off sensitivity for error stability, and explicit formulas provide bounds that must be reconciled with resource limitations and the stationarity or drift rate of the underlying data source (Abrar et al., 15 Oct 2025).

A plausible implication is that robust practical implementation of T-WA requires flexible adjustment of weighting schemes, uncertainty estimation methods, and computational resources based on the statistical properties and operating regime of the application.

7. T-WA Across Measurement, Analysis, and Learning Domains

Time-Weighted Averaging is a unifying concept for dynamic estimation processes where the temporal, statistical, or model-based context evolves. Its application ranges from precision metrology (robust mean and uncertainty estimation (Malkin, 2011)), frequency measurement and stability analysis (adaptive kernel averaging (Benkler et al., 2015)), time-series centroiding (probabilistic kernel alignment (Marteau, 2015)), market and physical system analysis (eigenproblem-based adaptive moving averages (Boudjemila et al., 2022)), time-varying online optimization (streaming data handling with tracked error bounds (Abrar et al., 15 Oct 2025)), and machine learning for domain adaptation under temporal shift (TEA (Liu et al., 30 Sep 2025)).

The essential feature throughout is the rigorous design of temporal weighting so as to balance recency, robustness, and uncertainty. The adaptive and theoretically justified schemes described in these works collectively provide practitioners with versatile frameworks for incorporating time into weighted averages, delivering estimators and predictors that are both statistically sound and resilient to evolving data properties.