Cumulative Fixed-Time Markout

• Cumulative fixed-time markout is a quantitative framework that segments data into uniform time or position intervals to measure aggregates like semantic scores, profits, or flow rates.
• It provides a systematic method for analyzing time-local fluctuations and trends in areas such as text analysis, auction theory, algorithm benchmarking, temporal logic, and traffic optimization.
• By aligning measurements with natural segmentation scales, this approach enhances structural insight and forecast accuracy compared to normalized or end-point methods.

A cumulative fixed-time markout is a quantitative construct indexing some aggregate or difference in values—such as semantic scores, cumulative profits, arrival/share functions, or treatment effects—over a fixed interval of time or position (such as word count, minutes, or cycles). This paradigm appears in diverse domains, including computational text analysis, auction design, epidemiological statistics, time-series forecasting, and cyber-physical specification, with a unifying principle: tracking or segmenting the cumulative quantity or effect accrued within prescribed, uniform, absolute time intervals, rather than only employing normalized, relative, or end-point comparisons. The cumulative fixed-time markout thus elucidates not just total or average outcomes, but also the structure, predictability, time-local fluctuations, or efficiency associated with the temporal or positional distribution of the modeled phenomenon.

1. Cumulative Fixed-Time Markout in Computational Text Analysis

The cumulative fixed-time markout framework is exemplified by the cumulative word-time methodology introduced in the ousiometric paper of book structure (Fudolig et al., 2022). Here, the "time" variable is defined as the running word count: for a text with tokens $w_1, \ldots, w_N$, the $i$th token is assigned $t_i = i$. To analyze semantic evolution, the text is windowed into non-overlapping segments of fixed size ($N_w = 50$ words), producing a time series $S_j$ of averaged semantic scores (power, danger) for each window. This uniform absolute scale is critical for capturing real structural fluctuations:

• Empirical Mode Decomposition (EMD) is then applied to the windowed time series, splitting it into a non-oscillatory trend plus a set of oscillatory intrinsic mode functions (IMFs).
• Comparisons to shuffled, order-randomized baselines allow for order-sensitive fluctuation detection: short texts exhibit only a trend, while longer texts manifest oscillatory structure with characteristic periods of $10^3$–$10^4$ words, closely tracking editorial units such as chapters.
• Unlike fraction-of-book time or wide-window smoothing, fixed-time markout enables alignment with natural narrative segmentation and avoids erasing mid-scale phenomena.

This approach empirically substantiates that longer books are not mere expansions of shorter ones, but instead comprise concatenations of segment-scale units, a structure rendered visible only through the cumulative fixed-time markout lens (Fudolig et al., 2022).

2. Auction Theory and Financial Markout: Minute-Interval Analysis

In quantitative finance and blockchain auction research, cumulative fixed-time markout quantifies the total realized value (e.g., profit, arbitrage) extracted within fixed time windows. The TimeBoost auction analysis formalizes this concept for high-frequency trading in discrete rounds (Mamageishvili et al., 23 Nov 2025):

• Each round $i$ covers a fixed interval of $\Delta=1$ minute, during which the auction winner may execute multiple trades.
• The instantaneous markout of trade $t$ is

$$\Pi(t) = p_A(t_T + m) t_A - p_B(t_T + m) t_B - \mathrm{fee}_t$$

where $p_A$ and $p_B$ are mid-prices at a fixed lag, and the fee is the execution cost.

• The cumulative fixed-time markout for round $i$ is the sum over all fast-lane trades in that minute:

$M_i = \sum_{t\in\mathcal T_i} \Pi(t)$

with $\mathcal T_i$ denoting all relevant trades in $[(i-1)\Delta, i \Delta)$.

• Analyses aggregate $M_i$ over longer windows, e.g., $M_i^{(T)} = \sum_{j=0}^{T-1} M_{i+j}$ for $T$ consecutive minutes.

This markout is compared to "winning" and "paid" bids to paper forecastability and auction efficiency. Results show minute-by-minute markouts are only weakly predicted by bids (e.g., Wintermute's Pearson correlation with $M_i$ was only 0.32–0.33), but correlations rise markedly when markouts are aggregated over longer intervals (up to 0.88 at 60-minute windows), demonstrating that trend detection, rather than precise per-interval prediction, dominates bidder behavior (Mamageishvili et al., 23 Nov 2025).

Interval	Pearson corr. (Wintermute)
1 minute ($M_i$)	0.32–0.33
30 minutes	0.83–0.82
60 minutes	0.88–0.88

3. Benchmarking and Protocols: Cumulative Makeout Curves in Metaheuristics

In the benchmarking of optimization algorithms, cumulative fixed-time markout is operationalized as the empirical cumulative distribution function (ECDF) of the times at which performance targets are first reached, under a strict wall-clock constraint (Lian, 10 Sep 2025). Key constructs:

• For algorithm $A$ and total budget $T$, define $t_i$ as the time-to-target for run $i$; if the target is not reached, $t_i = T$.
• The cumulative fixed-time markout curve is $F_A(t; q) = (1/R)|\{i: t_i \le t\}|$, the fraction of runs attaining the target $q$ by $t$.
• The expected running time (ERT) at target $q$ and budget $T$ is $$\operatorname{ERT}_A(q; T) = \frac{\sum_{i=1}^R \min\{t_i, T\}}{s}$$ where $s$ is the number of successful runs before $T$.
• Performance profiles aggregate per-instance performance: $$P_A(\tau) = \frac{1}{|I|} |\{i \in I : c_A(i) \le \tau \min_B c_B(i)\}|$$ with $c_A(i)$ the cost (e.g., ERT) on instance $i$.

The standardized restart-fair protocol ensures all algorithms are compared under equal fixed-time constraints, with restart usage and reporting practices strictly documented. This protocol exposes not just means or best values but the full cumulative time-to-target distributions, facilitating robust, time-fair performance assessment (Lian, 10 Sep 2025).

4. Cumulative Fixed-Time Markout in Signal Temporal Logic and Monitoring

The extension of temporal logic with cumulative-time operators in Cumulative-Time Signal Temporal Logic (CT-STL) provides a syntactic and semantic formalism directly capturing fixed-window cumulative duration properties (Chen et al., 14 Apr 2025):

• The cumulative operator $C_{I}^\tau \varphi$ asserts that "within interval $I$, the number of time steps where $\varphi$ is true is at least $\tau$".
• Boolean semantics: $$\llbracket C_{[a,b]}^\tau \varphi \rrbracket_{\xi, t} = 1 \quad \text{iff} \quad \sum_{d=a}^b \llbracket \varphi \rrbracket_{\xi, t+d} \geq \tau$$
• Robustness semantics assesses the $\tau$-th largest value of the quantitative score of $\varphi$ over the interval.
• Online monitoring employs an efficient sliding "τ-th largest" heap algorithm with $O(\log w)$ complexity per window, enabling real-time evaluation of sliding cumulative requirements.

CT-STL enables explicit specification and real-time verification of cumulative markout-type requirements, as in "at least $k$ time units safe in every interval of length $T$"—a level of expressivity not afforded by classical STL (Chen et al., 14 Apr 2025).

5. Statistical Inference of Cumulative Markouts in Time-to-Event Data

In survival analysis and competing risks, cumulative fixed-time markout refers to differences in cumulative incidence functions (CIF) at prespecified time points (Chen et al., 2018):

• For group-specific CIFs $\operatorname{CIF}_{ri}(t)$, the fixed-time markout at time $t_0$ is $$\Delta_i(t_0) = \operatorname{CIF}_{1i}(t_0) - \operatorname{CIF}_{2i}(t_0)$$, indicating the additional cumulative risk by $t_0$ attributable to treatment or exposure group 1.
• Pseudo-value regression enables model-free pointwise inference for $\Delta_i(t_0)$, with transformation functions (log–log recommended) and variance estimation methods (Gaynor’s estimator for precision in small samples).
• The statistical test statistic for the fixed-time markout is

$$X^2 = \frac{\left[\varphi(\hat{C}_1) - \varphi(\hat{C}_2)\right]^2}{V_1[\varphi'(\hat{C}_1)]^2 + V_2[\varphi'(\hat{C}_2)]^2}$$

which is compared to $\chi^2_1$ under the null.

This methodology allows statistically robust and interpretable comparison of cumulative effect “marked out” by a particular time, which is critical in the presence of non-proportional hazards or crossing CIF curves (Chen et al., 2018).

6. Modeling and Forecasting Cumulative Markout in Irregular Time Series

The monotonic neural ODE (MODE) approach models and forecasts cumulative series with strictly monotonic behavior, predicting markout values at fixed future horizons (Chen et al., 2023):

• Given observed cumulative sequence $s(t_1), \ldots, s(t_n)$ at (possibly uneven) times $t_1 < \dots < t_n$, the task is to forecast $s(t_n + \Delta t)$.
• The ODE model is $d\ell(t)/dt = f(\ell(t); \theta)$ with monotonicity enforced via $$f(\ell; \theta) = \operatorname{ReLU}(\operatorname{MLP}_\theta(\ell))$$.
• No explicit monotonicity regularizer is needed—guaranteed by architecture; irregular sampling is handled natively by ODE solvers.
• The loss is sum-of-squares at observed points; inference at the fixed time is by ODE integration forward by $\Delta t$.

MODE demonstrates superior performance and robustness in real-world forecasting scenarios, particularly as it naturally handles monotonicity and irregularly spaced data, directly yielding cumulative fixed-time markout predictions at arbitrary horizons (Chen et al., 2023).

7. Queueing and Flow Optimization: Cumulative Markout in Fixed-Cycle Control

Cumulative fixed-time markout methods appear in transportation systems as the primary criteria for optimizing fixed-time traffic signal plans via cumulative flow diagrams (CFDs) (Tan et al., 2022):

• For each intersection phase, the CFD plots cumulative arrivals $A(t)$ and departures $D(t)$ over a cycle of length $C$.
• The queue at $t$ is $Q(t) = A(t) - D(t)$. The queue dissipation time $t_d$ solves $D(t_d) = A(t_d)$.
• The fixed-time markout $M = t_d - t_g^e$ (with $t_g^e$ the green-phase end) measures how much longer than the green interval a queue persists.
• The multi-objective optimization minimizes first the maximum markout across phases (requiring $M \leq 0$ for all, preventing queue spillovers), then the average stopped delay.

This framework is solved with a bi-level particle swarm optimization, leveraging connected vehicle trajectory data to drive the CFD and adapt timing plans. The fixed-time markout here is the key metric for feasibility and efficiency in real-world, time-driven signal optimization (Tan et al., 2022).

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Cumulative fixed-time markout thus emerges as a fundamental cross-domain concept, enabling robust analysis, segmentation, forecasting, optimization, and inference by imposing absolute, interpretable time windows on cumulative or difference measures. Its applications range from neuro-linguistic segmentation and clinical trial statistics to online auctions, vehicular queue control, and dynamical forecasting, providing both methodological rigor and practical interpretability across the sciences and engineering.