Model support: marginal ranges can conceal joint extrapolation
Checking each input against its observed minimum and maximum is not enough to establish that a new combination lies among the combinations studied. A point can pass every one-variable range check and still require joint extrapolation. Preserve the observed input pairs and use a stated support diagnostic before describing a model prediction as interpolation.
The deliverable is a support map, reproducible combination weights and a record of what additional observations would test the proposed use. It evaluates the relationship between a model query and the studied domain. It is not a certificate that predictions inside a geometric boundary are accurate.
Define the domain in the model’s actual inputs
The NIST discussion of use beyond the data domain distinguishes prediction inside and outside the data domain and urges caution about extrapolation. This article adds its own explicit geometric worksheet for a hypothetical two-input example; it does not claim that NIST prescribes this worksheet as a universal validation rule.
Suppose X1 and X2 are two continuous task features measured on documented standardized scales. Coordinate zero is a chosen reference, not an assertion that a document has zero content. Keep the transformations, original units, version and feasible combinations. A query transformed differently from the training rows cannot be located meaningfully in their domain.
The following three observations represent compatible verification tasks. The response is invented mean completion time in minutes, with values one, three and three. They make the geometry visible rather than reporting a real operating result.
A consistent invertible affine change of units preserves the convex-combination relationship. For example, replacing X1 with twice X1 plus one moves the target’s first coordinate to five but leaves its combination weights unchanged. Changing only the query’s preprocessing would instead make the comparison inconsistent.
| Observed task | X1 | X2 | Response |
|---|---|---|---|
| A | 0 | 0 | 1 |
| B | 2 | 0 | 3 |
| C | 0 | 2 | 3 |
Both input columns range from zero to two. Their observed combinations form a triangle, not the entire square defined by the separate ranges. The missing upper-right part of that square is the issue to investigate.
Check identification before blaming the geometry
Include an intercept in the design matrix, giving rows (1,0,0), (1,2,0) and (1,0,2). Its determinant is four, so the three columns are linearly independent. A plane can be fitted to these three response means with separately determined coefficients.
That matters because this is not an example of two predictors lying on one line and preventing separate slope identification. The design is full rank. Nevertheless, full rank does not establish that every combination of individually familiar values has been observed or is well supported.
Three cell means also do not provide a residual-variance estimate for a saturated three-parameter plane. Keep the ability to solve coefficients separate from precision and validation. A matrix operation returning a unique prediction does not establish the prediction’s reliability.
Test the apparently in-range query
Consider target T = (2,2). Each coordinate equals an observed maximum, so both simple range checks pass. To see whether it lies in the triangle, solve for weights on A, B and C whose sum is one and whose weighted coordinates equal T.
The X1 coordinate requires two times B’s weight to equal two, so B’s weight is one. The X2 coordinate similarly requires C’s weight to be one. The sum-to-one condition then gives A’s weight minus one.
| Target | Weight on A | Weight on B | Weight on C | Geometric result |
|---|---|---|---|---|
| T = (2,2) | −1 | 1 | 1 | Outside the convex hull |
| S = (0.5,0.5) | 0.5 | 0.25 | 0.25 | Inside the convex hull |
Convex combinations require nonnegative weights summing to one. T’s negative weight therefore puts it outside the observed triangle, despite passing the separate ranges. For this worksheet, that is a joint-extrapolation flag. It does not mean the query is impossible or its prediction necessarily wrong; it means the fitted use extends beyond the observed combinations under this geometric diagnostic.
For comparison, S has weights one-half, one-quarter and one-quarter. They reproduce both coordinates and sum to one without a negative component. It is geometrically inside the triangle. This is a different statement from having many nearby observations or an independently validated prediction there.
Show how the prediction relies on model structure
The fitted plane is response-hat = 1 + X1 + X2. At T it predicts five. The same result follows from the affine weights: minus one times one, plus three, plus three. The negative weight is a visible reminder that this is not averaging nearby observed responses with nonnegative proportions.
Now consider another mean surface, 1 + X1 + X2 + X1×X2. It fits the three observed means equally well because the product term is zero at every observed point. At T it predicts nine rather than five. The existing three observations cannot distinguish those two surfaces.
This is an example of model-class uncertainty, not an arithmetic contradiction. The plane assumes away the interaction. That assumption may be justified by other knowledge, but a perfect fit at three vertices does not test it. Save the assumed response form instead of presenting the plane’s result as directly measured at T.
An observation at T, if feasible and measured comparably, could help distinguish the two predictions. One new result would not validate every future point. It would address a specified gap, and its measurement uncertainty and task comparability would still need attention.
Suppose a new compatible measurement at T centers near seven. If its uncertainty comfortably includes five and nine, it may not distinguish the models. If repeated precise measurements exclude both, the evidence challenges both response forms rather than automatically selecting the closer one. Adding T also expands the observed hull, but that geometric change does not certify the rest of the new interior. Save the measurement and model checks separately.
Test another boundary: inside the hull is insufficient
At S = (0.5,0.5), the plane predicts two while the alternative surface predicts 2.25. Even the geometric interior has untested curvature in this sparse example. Thus an inside-hull label cannot establish model correctness, local information density or adequate error calibration.
A high-dimensional convex hull can also contain large regions with few or no nearby observations. Preserve distances or neighborhood information relevant to the task alongside any hull label. Their interpretation depends on feature scaling and measurement meaning; do not combine kilograms and arbitrary category codes into an unexplained distance.
Categorical inputs require their own treatment. A numerical code halfway between two jurisdictions or document formats does not define a meaningful intermediate category. A convex combination of codes can be mathematically available while the corresponding task is not feasible. Keep the feasible domain distinct from a numerical envelope.
Likewise, a bounded model interval does not certify coverage outside the relevant domain. A prediction routine can compute a standard error under the assumed plane even where omitted curvature dominates. Report what uncertainty sources it includes rather than treating a narrow interval as evidence that extrapolation risk has disappeared.
Build a support-review workflow
First retain the exact training observations and preprocessing recipe. Then locate the query in individual ranges and joint combinations. For a simple continuous two-input case, reconstruct the weights or solve the nonnegative-combination feasibility problem. For a larger design, document the chosen diagnostic and its computational limitations instead of claiming that a scatterplot of two selected columns covers every feature.
If a query is outside, identify whether the combination is physically or operationally feasible. When suitable, collect a comparable confirmatory observation near that combination. If new data cannot be obtained, narrow the use claim or label it an assumption-dependent extrapolation. Do not add the model’s own prediction to the training set and call that new evidence.
| Field | Evidence to retain | Review question |
|---|---|---|
| Inputs | Units, transformations and versions | Same measurements at training and query? |
| Joint sample | Actual combinations | Has the combination been studied? |
| Geometry | Weights or stated diagnostic | Is the flag reproducible? |
| Model class | Plane, interaction or other terms | Which shapes were assumed away? |
| Local evidence | Neighbors, repeats and validation | Is the interior actually informative? |
| Follow-up | Feasible new observations | Which competing prediction would it test? |
Revisit the classification after adding observations, revising features or changing the intended task population. Save the old domain and the reason the new evidence changes it. The useful conclusion distinguishes a calculable prediction from a well-supported one, while keeping both geometric and substantive limits visible.
FAQ
Do familiar values in every column guarantee interpolation?
No. The (2,2) query passes both ranges but lies outside the triangle of observed combinations.
Is a point inside the convex hull guaranteed to be predicted correctly?
No. Sparse local data, incorrect model structure and incompatible measurements can still invalidate the prediction.