35.8 Linear Function Verification

Linear Function Verification is the collection of checks used to confirm that a constructed or identified linear function rule is correct, spanning direct substitution checks against known values, rechecks of the rate and initial value individually, and, for contextual models, additional checks on units and real-world reasonableness. Because a linear rule depends on exactly two correctly identified components, the rate and the initial value, verification focuses heavily on confirming each of these independently before accepting the rule as a whole, complementing the general candidate function rule verification approach with checks specific to the linear structure.

This verification work draws together checking techniques already introduced throughout numerical function evaluation, function classification verification, and linear rule construction, applying them specifically and systematically to the finished linear model as a final confirming step.

Known Pair Substitution

Substituting a Known Input-Output Pair Into the Rule

Verification begins by substituting the input of a known pair into the candidate linear rule and comparing the resulting predicted output against the known output, following the same substitution and comparison process established under candidate function rule verification.

Selecting an Independent Pair for This Check

Where possible, selecting a pair that was not directly used during the rule's original construction provides a more independent test, since checking only the exact pair used to build the rule risks simply confirming a calculation that was already assumed correct from the start.

Confirming Agreement Across Multiple Pairs

Checking more than one known pair, rather than relying on a single substitution, provides stronger confirmation that the rule is genuinely correct across its domain rather than only at one specific, possibly coincidental point.

Table Row Agreement

Checking the Rule Against Every Available Row

Where a table of values is available, the constructed rule is checked against every row rather than only a convenient sample, following the same every table row requirement already emphasized throughout candidate rule verification and linear table rule construction.

Confirming Complete Row-by-Row Agreement

Full confirmation requires that every row's recorded output matches the rule's predicted output for that row's input, with even a single disagreement sufficient to indicate the rule requires correction.

Using Row Agreement as the Most Comprehensive Available Check

Because a table typically contains more data points than a single known pair, checking full table row agreement, when a table is available, generally provides the most thorough and comprehensive verification achievable for a constructed linear rule.

Constant Rate Recheck

Recalculating the Rate Independently

The rate coefficient used in the constructed rule is rechecked by recalculating it independently from the original data, using two-point slope determination or rate of change from a table, and comparing the recalculated value against the rate actually placed into the rule.

Confirming the Recalculated Rate Matches

Agreement between the recalculated rate and the rate already present in the rule confirms that this specific component of the construction process was performed correctly.

Responding to a Rate Mismatch

If the recalculated rate does not match the rate in the constructed rule, the original rate identification step is revisited, checking specifically whether the rate coefficient placement process introduced an error during the rule's original assembly.

Initial Value Recheck

Recalculating the Initial Value Independently

The initial value used in the constructed rule is rechecked by recalculating it independently, either by directly reading a known zero-input output or by working backward from a known pair using the confirmed rate, following the missing initial value determination procedure.

Confirming the Recalculated Initial Value Matches

Agreement between the recalculated initial value and the value already present in the rule confirms that this specific component was also identified and placed correctly.

Responding to an Initial Value Mismatch

If the recalculated initial value does not match, the original identification and placement steps are revisited, checking specifically whether the initial value placement process introduced an error separate from any issue with the rate coefficient.

Variable Unit Agreement

Confirming Units Match the Original Declaration

For a contextual linear model, verification includes confirming that the units attached to the rate and initial value, as discussed under model unit interpretation, agree with the units originally declared for the input and output variables under linear model variable declaration.

Detecting a Unit Mismatch

A unit mismatch, such as a rate expressed in the wrong combination of units relative to the declared input and output, signals that an error occurred either during the original extraction of the rate from the context or during the variable declaration itself.

Correcting a Detected Unit Mismatch

Correcting a detected mismatch requires returning to the original contextual description, re-extracting the per-unit context change and its associated units, and confirming the corrected units now align consistently throughout the model.

Contextual Output Reasonableness

Checking Whether Predicted Outputs Make Sense

For a contextual model, verification includes checking whether outputs predicted at reasonable, in-context inputs produce results that make sense given the described situation, such as confirming a predicted cost is a plausible, positive dollar amount rather than an implausible or nonsensical value.

Recognizing an Unreasonable Output

An output that is wildly out of proportion to the described situation, or that violates an obvious practical expectation, such as a negative count of physical items, signals a likely error somewhere in the model's construction even if the underlying arithmetic was performed correctly.

Using Reasonableness Checks as a Complement to Numerical Verification

Because a reasonableness check relies on judgment about the described situation rather than a strict numerical comparison, it serves as a useful complement to, rather than a replacement for, the more precise numerical verification techniques described earlier in this topic.

Domain Compliance Check

Confirming Predictions Respect Any Domain Restriction

Verification includes confirming that any evaluation or prediction made using the model respects whatever domain restriction was established, whether the explicit domain restriction priority discussed for elementary rule domains or the contextual domain restriction established when the model was originally built from a real situation.

Detecting a Domain Compliance Violation

A violation is detected when a model is evaluated at, or a prediction is reported for, an input value that falls outside its established domain, regardless of whether the underlying rule itself would still compute a technically valid output for that value.

Correcting a Detected Domain Violation

Correcting a detected violation requires discarding any prediction made at a disallowed input and, if a valid result is still needed, selecting a different input that actually falls within the model's established domain before repeating the evaluation.