The observation with the smallest residual can have the greatest influence
A survey adjustment fits its observations closely. One especially important observation has almost no residual, so it seems safe. Yet a highly influential observation can pull the fitted model toward itself and leave only a small discrepancy behind. A residual measures what the fit did not absorb, not the full error that may have entered.
This guide develops that distinction with a fictional linear profile. We calculate a fit by hand, introduce one observation error, and track how much appears in the estimated parameters and how much remains in the residuals. Then we change the geometry, the error pattern and the uncertainty assumptions to see which checks can actually reveal a problem.
The profile is a transparent mathematical fixture, not an operational geodetic network or a survey acceptance procedure. Heights, distances and uncertainties are invented. NIST’s regression-diagnostics reference provides the terminology of leverage and influence; the original calculations explain why those ideas matter when interpreting adjusted measurements. The existing free-network guide addresses the separate question of missing datum information.
Geodesy subject guide · Learning that transfers into life · How X Works hub
Choose your reading route
Route 1 · Chapters 1–2: Define the fitted and unfitted parts
Route 2 · Chapters 3–5: Work an influential observation
Build a three-observation profile that can be solved exactly
Route 3 · Chapters 6–8: Understand residual uncertainty
Route 4 · Chapters 9–10: Improve evidence rather than appearances
Route 5 · Chapters 11–12: Transfer and report
Choose independent checks that can challenge the intended claim
Full chapter index · Independent worked case · Questions and practical checks
Chapter index
Define the fitted and unfitted parts
Define a residual before treating it as an error estimate
Split the observations into fitted and residual components
Work an influential observation
Build a three-observation profile that can be solved exactly
Calculate leverage rather than judging it from a plot alone
Inspect where the error went, not just where it started
Understand residual uncertainty
Propagate observation uncertainty into residual uncertainty
Explain why several residuals need not supply several independent checks
Construct an error pattern that residuals cannot see
Improve evidence rather than appearances
Change the geometry to strengthen a weak check
Keep weighting and correlation attached to the diagnostic
CHAPTER 1 OF 12 · Define the fitted and unfitted parts
1. Define a residual before treating it as an error estimate
Let observations satisfy the linear model y = Aβ + e. The vector β contains unknown parameters, A describes their observation sensitivities, and e represents error relative to that model. Least squares produces β_hat and fitted observations Aβ_hat. In this guide define the residual as r = y − Aβ_hat: observed minus fitted.
Some adjustment references use the opposite residual sign. That is not necessarily a disagreement. State the convention before comparing equations or signed corrections. The variance of a residual does not change when every residual sign is reversed, but its interpretation as something added to or subtracted from an observation does.
The actual observation error e is generally unknown. The residual r can be calculated because the fit has supplied β_hat. They are therefore different objects. A model can absorb part of an error into its estimated parameters, leaving a smaller residual. Conversely, a valid observation can receive a residual because errors elsewhere affected the shared fit.
This distinction changes the first review question. Instead of asking only whether a particular residual is small, ask how the observation influences the fitted parameters and which error patterns the complete observation system can distinguish. A residual table without its model, geometry and stochastic assumptions is not a self-contained certificate of measurement quality.
CHAPTER 2 OF 12 · Define the fitted and unfitted parts
2. Split the observations into fitted and residual components
For full-column-rank A and equal independent observation variances, β_hat = (AᵀA)⁻¹Aᵀy. Define H = A(AᵀA)⁻¹Aᵀ. Then the fitted observations are Hy, and residuals are Ry with R = I − H. H is often called the hat matrix because it maps y to y_hat.
Both H and R are symmetric and idempotent in this ordinary least-squares setting: applying either projection twice gives the same result as applying it once. They also satisfy HA = A and RA = 0. Thus an observation pattern that the model can represent lies entirely in the fitted part and contributes nothing to the residual part.
If an error δ is added to observation i only, write its vector as δe_i, where e_i is the corresponding unit vector. The fitted observations change by H e_i δ, while residuals change by R e_i δ. This column-by-column description shows how one error can affect several observations’ residuals simultaneously.
These formulas assume the stated full-rank, equal-variance model. A free network may require an appropriate datum treatment before an ordinary inverse is available; correlated or unequal-variance observations require a compatible weighted formulation. Keep the simplified derivation within its scope. The symbols alone do not establish that a real adjustment satisfies these assumptions.
CHAPTER 3 OF 12 · Work an influential observation
3. Build a three-observation profile that can be solved exactly
Take a fictional straight profile y = a + bx, where y is height in millimetres and x is distance in metres. Observe at x = 0, 1 and 10 m. The design matrix rows are (1,0), (1,1) and (1,10). Both intercept a and slope b are unknown, giving three observations and two fitted parameters.
Assume the true teaching profile is horizontal at zero height. Now insert a one-millimetre error only in the observation at ten metres, so y = (0,0,1). The design cross-product is AᵀA = [[3,11],[11,101]], whose determinant is 182. Its inverse is (1/182)[[101,−11],[−11,3]].
Because Aᵀy = (1,10), the fitted intercept is (101 − 110)/182 = −9/182 mm, and the fitted slope is (−11 + 30)/182 = 19/182 mm per metre. Numerically these are approximately −0.04945 mm and 0.10440 mm/m. The fitted line has tilted substantially to accommodate the distant observation.
Calculate the fitted heights: −9/182, 10/182 and 181/182 mm. Subtracting them from the observations gives residuals (9,−10,1)/182 mm. The observation containing the full one-millimetre error has residual only 1/182 mm, about 0.00549 mm. This is the central counterexample: a tiny residual did not mean a tiny original error.
CHAPTER 4 OF 12 · Work an influential observation
4. Calculate leverage rather than judging it from a plot alone
The leverage of observation i in this setting is h_ii, the corresponding diagonal of H. For a row (1,x), h(x) = [101 − 22x + 3x²]/182. At x = 0, 1 and 10, the leverages are 101/182, 82/182 and 181/182. The last is close to one.
The residual response to an isolated error in its own observation is r_i = (1 − h_ii)δ. Define its residual redundancy number as rnumber_i = 1 − h_ii for this simple ordinary least-squares model. To avoid confusing a scalar number with the residual vector r, we will write ρ_i for that number.
The three ρ values are 81/182, 100/182 and 1/182. They sum to one, matching the residual degrees of freedom n − p = 3 − 2. The lone distant observation has very little of its own independent error left in its own residual. Most of that error has entered the fitted line.
High leverage is a property of the design and weights, not a diagnosis that a measurement is bad. A correct distant observation can be highly valuable. The warning is interpretive: if it is wrong, a residual-only screen may have weak sensitivity to that wrongness. NIST similarly distinguishes leverage from influence; actual influence also depends on the observations and the resulting parameter changes.
CHAPTER 5 OF 12 · Work an influential observation
5. Inspect where the error went, not just where it started
The fitted parameter changes caused by a one-millimetre error at ten metres were −9/182 mm in the intercept and +19/182 mm/m in the slope. Those numbers describe how the error affected the estimated profile. They are not recovered measurements of the true error; they are a response calculation made possible because the teaching truth was stipulated.
At the original two nearby locations the residuals are approximately +0.04945 and −0.05495 mm. They are larger in magnitude than the faulty distant observation’s +0.00549 mm residual. Deleting whichever row has the largest raw residual would therefore point first toward an observation that was originally exact in our fixture.
This does not mean residual examination is useless. The three residuals form a coherent pattern under the shared fit. Their sum is zero and their distance-weighted sum is zero: 9 − 10 + 1 = 0 and 0(9) + 1(−10) + 10(1) = 0 after the common division by 182. These are the least-squares orthogonality conditions.
A leave-one-out calculation provides a different sensitivity view. Remove the ten-metre observation temporarily. The remaining two exact observations define the zero line, which predicts zero at ten metres. Comparing the omitted observation with that prediction gives a one-millimetre discrepancy. For this ordinary least-squares setting, the deleted-prediction residual equals the full-fit residual divided by 1−h_ii, when that denominator is nonzero. Here (1/182)/(1/182) gives exactly one millimetre.
The large amplification is informative, but it is not a new independent field observation. It reuses the existing data under a changed fitting arrangement. Moreover, predicting far beyond the retained points can be uncertain in noisy data. A leave-one-out discrepancy therefore needs its appropriate uncertainty and extrapolation context. It should not be relabelled as a direct measurement of the original error merely because the teaching fixture makes it numerically equal to that error. The exercise reveals influence and weak checking geometry; causal attribution still requires additional evidence.
Use those conditions as implementation checks, then investigate physical causes with appropriate evidence. The residual pattern alone does not prove which instrument or row was wrong. In a real problem several different error patterns can produce the same residual component. Mathematical consistency can narrow an investigation without supplying an unsupported causal diagnosis.
CHAPTER 6 OF 12 · Understand residual uncertainty
6. Propagate observation uncertainty into residual uncertainty
Assume independent equal observation errors with variance σ², so Cov(e) = σ²I. Since r = Re and R is symmetric and idempotent, Cov(r) = σ²R. The variance of residual i is therefore σ²(1 − h_ii) = σ²ρ_i. Different residuals can be correlated even when the original observations were independent.
For the ten-metre observation, the residual standard deviation is σ/√182. Its response to a fixed isolated error δ is δ/182. Dividing that response by its own standard deviation gives δ/(σ√182). This is the standardized response under the stipulated known noise scale, not a universal probability or rejection rule.
Choose a fictional known σ = 0.1 mm and δ = 1 mm. The standardized response is about 0.741. The error is ten times the individual observation noise scale, yet its own standardized residual response is modest because the geometry absorbs most of it. No operational tolerance or significance level is implied by these numbers.
If σ is estimated from the same very small residual sample, interpretation changes again. A variance estimate with one residual degree of freedom is itself highly uncertain, and an error can affect that estimate. Distinguish an externally supplied stochastic scale from a scale inferred after fitting. Do not mix the two while retaining a confidence statement derived for only one.
CHAPTER 7 OF 12 · Understand residual uncertainty
7. Explain why several residuals need not supply several independent checks
Our three-observation fit has one residual degree of freedom. Its residual vector always lies along a single direction proportional to (9,−10,1). Three printed residual entries therefore do not represent three independent tests of the observations. They are three components of one remaining contrast after two parameters have been fitted.
This follows directly from the projection’s rank. R has rank n − p = 1 in this fixture. Once one nonzero residual component is specified, the others are determined by the orthogonality relations. Counting the number of residual rows as though it were the number of independent checks would overstate the evidence.
A global weighted residual sum can test consistency with a stated stochastic model under appropriate distributional assumptions. It does not automatically identify the offending observation, and its interpretation depends on degrees of freedom, weighting and the hypothesis being tested. No distribution-free threshold follows simply from having a residual table.
When reporting a diagnostic, state whether it concerns an individual standardized residual, a global model check or a parameter-sensitivity calculation. These answer different questions. An adjustment can have ample observations yet little useful independent information about a particular error direction. The relevant quantity is the geometry and dependence of the checks, not only their count.
CHAPTER 8 OF 12 · Understand residual uncertainty
8. Construct an error pattern that residuals cannot see
Let an added observation error have the form e = Aγ for some parameter-sized vector γ. Because RA = 0, its residual contribution is exactly zero. The fitted parameters shift by γ and the observations still fit the model perfectly. This is an algebraic blind spot of that observation-model combination, not a numerical failure of least squares.
For the straight-profile fixture, add a common +2 mm offset to every height. This is Aγ with γ = (2,0). The fitted intercept increases by 2 mm and all residuals remain unchanged. If the intended quantity is absolute height relative to an independently established reference, a residual-only check cannot establish that the common offset is absent.
Similarly, add a distance-proportional error of 0.03x mm. It shifts the fitted slope by 0.03 mm/m without adding a residual pattern. This example is deliberately stipulated, not evidence that an actual instrument has that specific error. It shows why a compatible external reference or independent measurement pathway can be necessary for some claims.
The lesson is not that all systematic errors are invisible. Only the component in the model’s fitted column space is absorbed completely in this linear account. Other components remain in the residual space. A general error can be decomposed into both. State which component a test can reveal instead of labelling residuals either an infallible detector or a useless formality.
CHAPTER 9 OF 12 · Improve evidence rather than appearances
9. Change the geometry to strengthen a weak check
A useful redesign asks where another observation would add an independent sensitivity. Repeating a near-identical observation may reduce one noise component while leaving a weak geometry largely intact. Extending or redistributing the design can change leverage and residual redundancy, but the improvement must be calculated under the actual weights and dependence.
Add a fourth independent equal-variance observation at x = −1 m to our fixture. The distances are now −1,0,1,10. The new cross-product is [[4,10],[10,102]], with determinant308. The leverage of the ten-metre observation becomes 302/308 =151/154, leaving residual redundancy3/154 instead of 1/182.
An isolated one-millimetre error at ten metres now produces residual3/154 mm there, approximately0.01948 mm. That is a larger fraction of the error than before, though still far below one millimetre. The improved check has not made the distant observation independent of the fitted slope or supplied a guarantee that every error is detectable.
This calculation also exposes a design question: another observation near the existing cluster may help less than an observation that independently constrains the poorly checked part of the profile. We do not prescribe an optimal site here. The objective and constraints of an actual survey determine what additional measurement is feasible and valuable. Evaluate geometry, independence and physical access together rather than equating more rows with better protection.
CHAPTER 10 OF 12 · Improve evidence rather than appearances
10. Keep weighting and correlation attached to the diagnostic
For unequal or correlated observations, the equal-variance projection formulas need an appropriate weighted treatment. A common route whitens the observations using a factor of their covariance, then performs the corresponding ordinary projection in whitened coordinates. The resulting diagnostics concern that transformed system and must be interpreted through its observation relationships.
A very small assigned variance gives an observation greater authority in the fit. If that variance omits a persistent bias or shared reference uncertainty, the adjustment can become more confident without gaining physical accuracy. Weighting is a statement about an error model; it is not a reward for a preferred measurement.
Suppose two observations share an unknown reference offset. Treating them as independent duplicates can overstate the independent information about an intercept. Their difference may constrain a relative quantity well while their common height reference remains uncertain. The existing datum and covariance guides develop that distinction; residual diagnostics should not silently undo it by using incompatible weights.
Preserve the original covariance, any whitening or differencing transformation, and the diagnostic definition. If an implementation reports a redundancy number, establish which formulation it uses rather than transferring the simple1−h_ii rule indiscriminately to every matrix printed by a weighted adjustment. The transparent fixture is a foundation for understanding, not permission to ignore the specialist model.
CHAPTER 11 OF 12 · Transfer and report
11. Choose independent checks that can challenge the intended claim
An internal check reuses relationships among the observations and fitted model. An external check introduces information with a genuinely different sensitivity or reference. Both can be useful, but agreement between two computations using the same biased input does not test the bias independently.
For our profile, a separately established reference height could challenge a common intercept shift. An independently measured long-baseline height difference could challenge a slope error. Recomputing the same normal equations with another solver mainly checks implementation. The appropriate check depends on whether the claim concerns absolute height, relative change, slope or software correctness.
Changing an observation’s weight or deleting it temporarily can show how sensitive the estimate is to that row. Such a sensitivity analysis is not automatic authority to remove the observation from a final solution. Investigate instrument records, setup changes, field notes and alternative explanations before deciding which data treatment is justified.
Keep unsuccessful checks in the evidence account. A small residual with high leverage, an unstable result after deletion, and an unknown shared reference are distinct findings. Flattening them into a single pass label hides the next useful action. A clear diagnostic report explains what was tested, what the test could have detected and which important error directions remain untested.
CHAPTER 12 OF 12 · Transfer and report
12. Report adjustment consistency without calling it ground truth
A reusable report names the observation model, parameterisation, datum, weights and covariance assumptions. It gives residual conventions and the relevant degrees of freedom. Where an observation has high leverage or low residual redundancy, explain that feature beside the residual rather than leaving a reader to equate small with safe.
Separate observed values, fitted values, residuals and estimated parameter uncertainty. They play different roles. If a sensitivity exercise changes the solution appreciably, report which quantity changed and why that matters for the intended application. Do not convert a conditional mathematical diagnostic into a claim that a real station moved or that an engineering tolerance was met.
The teaching fixture can be summarised precisely: a one-millimetre error in the distant observation tilted the fitted line while leaving only about 0.00549 mm in its own residual. That result follows from the declared design and equal independent weights. It is not a universal attenuation factor for every geodetic network.
The broader habit is to follow information through the fit. Ask which part of an error becomes a parameter change, which part remains testable and which independent evidence can challenge the untested part. That is how a residual table becomes a useful diagnostic instrument rather than a reassuring list of small numbers.
Independent worked case and evidence checks
Work the changed four-observation geometry independently. At distances−1,0,1,10 m, keep the true profile zero and place a one-millimetre error only at ten metres. The fitted intercept is 1/154 mm and slope 15/154 mm/m. The residuals, in that order, are(14,−1,−16,3)/154 mm. Their sum and distance-weighted sum are both zero.
Compare the distant residual3/154 with the earlier1/182. The new geometry makes more of that isolated error visible in its own residual. Yet the fitted ten-metre height still absorbs151/154 mm. Explain both statements together; improved detectability does not mean the full error is recovered directly.
Now add a common2 mm error to all four observations instead. The fitted intercept shifts by 2 mm, slope remains zero and all residuals vanish. The extra observation did not make a common offset distinguishable from the intercept. To test that claim, the observing system needs information whose reference or sensitivity differs appropriately.
Finally assign a fictional known observation standard deviation0.1 mm. The distant residual’s standard deviation is 0.1√(3/154)mm, and its standardized response to the one-millimetre error is 10√(3/154), approximately1.396. These are sensitivity calculations, not automatic rejection criteria. State the stipulated noise model and do not treat the number as professional survey approval.
Worked comparison
| Fictional case | Calculated result | What it shows |
|---|---|---|
| Three-point profile; distant error 1 mm | Distant residual1/182 mm | The fit absorbs most of that error |
| Same observation | Leverage181/182 | Small residual can coexist with high authority |
| All three heights shifted2 mm | Residual change zero | A fitted common offset is internally invisible |
| Add independent observation at −1 m | Distant redundancy3/154 | Geometry changes detectability |
| More residual entries | Not necessarily more independent tests | Use residual rank and covariance |
Questions and practical checks
Does the largest residual identify the bad observation?
Not by itself. Errors propagate through a shared fit and can create residuals in other rows. The deliberately faulty distant observation in our example has the smallest residual. Investigate geometry, covariance and independent evidence before attributing fault.
Is high leverage always undesirable?
No. It describes an observation’s authority under the model and weights. A correct high-leverage observation may be essential. The concern is whether its possible errors are adequately checked and whether the assigned uncertainty is justified.
Why can a perfect fit still be wrong?
An error pattern that lies in the fitted model space changes the estimated parameters without producing residuals. A common height offset is indistinguishable from an intercept change in our fixture. Independent reference information is needed to challenge that particular claim.
Can I use these standardized values as thresholds?
No. They illustrate sensitivity under a stipulated known noise scale. Actual decision rules need their model, distributional assumptions, multiple-testing treatment and application requirements. No operational acceptance threshold is supplied.
What should accompany a residual report?
The model, residual sign, weighting, covariance, degrees of freedom and relevant leverage or redundancy definitions. Add the independent checks and unresolved error directions. This lets the next reader understand what a small residual can and cannot establish.
Continue with the connected subjects
Keep the reference and dependence visible
Separate datum freedoms from measurable relative geometry
Preserve nuisance uncertainty and the provenance of combined information
Distinguish coordinate uncertainty from a residual diagnostic
Connect observation design with temporal evidence
Check temporal coverage and correlated errors in a station trend
Compare a levelling-loop check with its variance-weighted adjustment
Sources and scope
NIST Dataplot: leverage, influence and the limits of ordinary residual analysis
The numerical cases in this guide are original fictional teaching examples, not measurements from a real site. This is an educational explanation, not a field safety instruction, engineering approval or an official certification. For a real project, use qualified practitioners, current product documentation and site-specific evidence. Sources were checked on 3 November 2026.
Connect the observation with the reliability check
An endpoint model defines what was observed, a temporal model connects records made at different moments, and an estimation diagnostic tests only the error directions it can see. Use these complementary guides to keep those questions separate.
Trace the observation
VLBI source switching: test interpolation across unobserved time gaps
Satellite antenna offsets: rotate endpoint connections before projecting range
Test the inference
Integer ambiguity search: separate the best candidate from a validated fix