Controlled straight-line extrapolation · non-normative worksheet

Documented Linear Threshold-Projection Worksheet

Fit one documented indicator history with ordinary linear least squares and calculate its analytical crossing of a user-sourced threshold inside a declared extrapolation horizon.

no invented confidenceno universal vibration limitanalytical crossingcurrent ISO status

This is not an ISO RUL calculator or a failure-probability model. ISO 13381-1:2025 provides current guidance and requirements for prognosis processes; its public scope does not establish this worksheet’s regression formula, a universal alarm threshold, or a rule that converts R² into confidence. The withdrawn ISO 13381-1:2015 edition is not presented as current.

Controlled projection inputs

Use one indicator, one operating regime and one time coordinate. The last row is treated as the current observation.

Threshold direction

Time-coordinate unit label

Indicator name

Indicator unit label

Documented threshold LNo default is supplied. The threshold must use the same quantity and unit as every y value.

Maximum allowed future projection spanSame time-coordinate unit. A crossing beyond this user-controlled horizon is not reported.

Threshold source, revision and applicability

Measurement/data source and revision

Operating regime and measurement-condition record

Calculation ID and intended decision boundary

Point	Time coordinate t	Indicator y
1
2
3
4
5
6
7
8

Enter at least three complete pairs. Times must be finite, non-negative and distinct; order may be arbitrary. Blank rows are ignored. This three-point minimum is a mathematical UI rule, not an ISO adequacy criterion.

Required projection confirmation The indicator, unit, threshold and all measurements represent the same quantity, processing convention and measurement location. The operating regime and degradation mechanism are sufficiently stable for a straight-line screening projection over the declared horizon. I will not treat R² as confidence, probability of failure, prediction uncertainty or proof that extrapolation is valid.

Reference projection result

Projected remaining span Δt

—

Straight-line threshold projection, not certified RUL.

Projected crossing coordinate tL

—

Fitted slope b

—

Descriptive R²

—

In-sample fit statistic only; not confidence.

Residual standard error s

—

√[Σ(yi−ŷi)²/(n−2)] in indicator units.

Extrapolation/baseline ratio

—

Directional-step conflicts

—

Observed adjacent steps opposing the selected direction.

No confidence level, prediction interval, failure probability, ISO conformity, universal threshold or maintenance/safety verdict is produced.

Equations, status and model boundary

For n ≥ 3 complete observations and the straight-line model ŷ(t)=a+bt, ordinary unweighted least squares gives:

b = Σ[(ti−t̄)(yi−ȳ)] / Σ[(ti−t̄)²] · a = ȳ−bt̄

tL = (L−a)/b · Δt = tL−tlast · s = √{Σ[yi−ŷ(ti)]²/(n−2)}

The calculation uses an analytical crossing, not a stepped search. It reports a result only when the fitted slope points toward a threshold beyond the last observation, the crossing lies in the future, and the remaining span does not exceed the user-declared horizon.

R² is descriptive. It summarizes in-sample lack of fit relative to variation around the sample mean. It is not a confidence percentage, probability of correct prediction, prediction interval, model-validation result or evidence that the mechanism will remain linear outside the data.

Exact numerical checks

Data and control	Fit	Projection
(0,1),(1,3),(2,5); upper L=9; horizon=10	ŷ=1+2t; R²=1; s=0	tL=4; Δt=2
(0,10),(2,8),(4,6); lower L=2; horizon=10	ŷ=10−t; R²=1; s=0	tL=8; Δt=4

Standard and source classification

Claim	Classification	Inspected source
Prognosis-process guidance and requirements	Current ISO framework; no regression equation inferred from public scope	ISO 13381-1:2025, edition 3, published 2025-09
ISO 13381-1:2015 status	Withdrawn 2025-09-02 and revised by the 2025 edition	Official ISO lifecycle card
Straight-line parameter fitting by minimizing squared residuals	General statistical method, not an ISO 13381 formula	NIST/SEMATECH e-Handbook §4.1.4.1 and §4.4.3.1
R² interpretation	Descriptive in-sample statistic; no confidence mapping	NIST/SEMATECH glossary and process-model limitations

ISO 13381-1:2025 official card — current edition, status and public scope.

ISO 13381-1:2015 official lifecycle card — withdrawn edition and replacement.

NIST/SEMATECH linear least-squares guidance and R-squared glossary — method and interpretation.

Accessed: 15 July 2026. Only public ISO scope/status information was used. No inaccessible ISO clause, formula, threshold or acceptance rule was invented.

Screening projection only. A defensible prognosis requires a validated degradation mechanism, representative data, measurement uncertainty, operating-regime controls, threshold provenance, model validation, uncertainty treatment and consequences-based engineering review. Outliers, autocorrelation, heteroscedasticity, interventions and regime changes can invalidate ordinary least-squares extrapolation.