Gas Records, Event Time, and Storage Decisions

Conceptual map

IV-16.01Physical and recorded events
IV-16.02Observed responses
IV-16.03Comparison designs
IV-16.04Decision loss
IV-16.05Timing uncertainty

1. Physical events and recorded events occupy different clocks

A timestamp is an observation generated by a reporting process. Its value for estimation or storage control depends on the relation between recorded time and the physical event.

Let physical event time be $$T$$ and recorded time be $$R=T+U$$ , where the timing error $$U$$ may depend on facility, load, sign, or event magnitude. If a physical response profile is $$g(s)$$ , the mean profile aligned on recorded time is a mixture $\mathbb{E}[g(s+U)\mid\text{record inclusion}]$ . Under independent additive error with density $f_{U}$ , this is a convolution. Dependence or window selection changes the mixing law.

Definition 1 · Decision information set

The information set $\mathcal{W}_{t}$ contains records available when action $a_{t}$ is chosen, including their release times and revisions. A field observed later cannot enter an operational rule at date $$t$$ .

2. The loss function determines the useful timing summary

Suppose the controller predicts a scalar gas need $$G$$ using record $$W$$ and incurs squared loss $(G-a)^{2}$ . The optimal unconstrained action is the conditional mean. The minimum risk is $\mathbb{E}[\operatorname{Var}(G\mid W)]$ . Under absolute loss a conditional median is optimal, and asymmetric inventory costs lead to a conditional quantile.

Proposition 1 · Capacity-constrained squared-loss action

If the feasible storage action is the interval [0, $$K$$ ] with $K\ge0$ , every squared-loss Bayes action is the Euclidean projection of $\mathbb{E}[G\mid W]$ onto $$[0,K]$$ .

Proof. Conditional on $$W$$ , decompose $\mathbb{E}[(G-a)^{2}\mid W]=\operatorname{Var}(G\mid W)+(\mathbb{E}[G\mid W]-a)^{2}$ . The first term is action-invariant. Minimizing the second over a closed interval gives the closest feasible point to the conditional mean. ∎

3. A noisy record can be valuable without revealing the event

Let $$G$$ be zero or two with equal probability. Observe $$W=G+U$$ , where $$U$$ is independently $$-1$$ or 1 with equal probability. Records $$-1$$ and 3 reveal $$G$$ . Record 1 arises from either physical value with equal posterior probability, so its posterior mean is one and posterior variance is one.

The conditional-mean rule is 0 at $$W=-1$$ , 1 at $$W=1$$ , and 2 at $$W=3$$ . Since $$P(W=1)=0.5$$ , its Bayes risk is 0.5. Using the raw record as the action yields E[ $(G-W)^{2}$ ]=E[ $U^{2}$ ]=1. The posterior rule halves expected squared loss even though one record value remains ambiguous.

An event-study coefficient aligned on $$W$$ describes the record-time mixture. It is not itself the Bayes action and need not recover the physical-time profile. The estimand, record process, and decision problem should therefore be reported separately.

4. Windows, censoring, and endogenous records alter the calculation

Failure case · Symmetric timing error inferred from a centered histogram

A centered observed distribution can result from asymmetric errors combined with sample selection or asymmetric physical events. Error symmetry requires information beyond the marginal recorded-time histogram.

A finite observation window censors events shifted across its edges. Storage capacity $$K=0$$ fixes the action at zero; infinite capacity returns the conditional mean. At the equality boundaries $\mathbb{E}[G\mid W]=0$ or $\mathbb{E}[G\mid W]=K$ , the projection is continuous but its derivative has a kink. If record delay depends on $$G$$ , the independent-convolution formula fails. Multiple records for one event create dependence and require event-level clustering or a deduplication rule. A missing timestamp can represent no event, failed recording, or delayed release; those states imply different likelihoods. Causal response claims additionally require an assignment or identification design beyond the clock model.

5. Implementation, exercises, and sources

Keep physical quantity, event timestamp, recording timestamp, publication timestamp, revision timestamp, sensor identifier, unit, and quality flag as separate fields. Define the information vintage available at each decision date. Estimate or bound the record-error law using validation events, propagate it through the declared loss, and compare decisions by out-of-sample loss with chronological splits.

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Exercises

Reproduce the posterior rule and both risks in Section 3.
Find the action when $$K=1.5$$ and compute its risk.
Show how an event at the right edge can disappear from a fixed recorded-time window when $$U>0$$ .

Partial solutions

1. Bayes’ rule gives posterior means 0, 1, and 2; only the middle record has conditional variance one. 2. Projection replaces the action two at $$W=3$$ by 1.5. This state has probability one quarter and adds $0.25\times 0.5^{2}=0.0625$ , giving risk 0.5625. 3. If inclusion requires $R\le b$ , then $T\le b$ can still be excluded whenever $$U>b-T$$ .

James O. Berger, Statistical Decision Theory and Bayesian Analysis.Bayes actions under specified losses.
Raymond J. Carroll, David Ruppert, Leonard A. Stefanski, and Ciprian Crainiceanu, Measurement Error in Nonlinear Models.Measurement-error structure and identification.
Chae-Yeon Xon (2026), “Fuzzy Clock: Decision-Specific Timing Information.”Related event-time and storage application.

6. Audit checkpoint

Physical event, recorded event, release and revision times, additive-error support, dependence, duplicate records, window edges, left and right censoring, missing-state meanings, information vintage, target profile, loss function, action support, capacity kink, posterior calibration, experimental design, clustering level, and chronological out-of-sample evaluation.

7. Scope boundary

The chapter covers discrete timing-error examples and one-period Bayesian storage actions. Continuous-time filtering, dynamic inventory control, and causal event-study identification require additional models.