Chapter map

1. 1. Event identity
2. 2. Spatial coordinates
3. 3. Event times
4. 4. Unordered pairs
5. 5. Spatial distance
6. 6. Temporal lag
7. 7. Spatial indicator
8. 8. Temporal indicator
9. 9. Joint indicator
10. 10. Knox count
11. 11. Rough benchmark
12. 12. Permutation scheme
13. 13. Pseudo p-value
14. 14. Threshold grid
15. 15. Influential events
16. 16. External investigation

Visit the How Mathematics Works hub for the wider series and its connections to education, reasoning, science, computing, design and careers.

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1. Event identity

This is where modelling choices begin to affect evidence. Each record represents one defined event.

A concrete situation

Five cases have unique IDs.

What the mathematics does not license

Duplicates inflate pair counts. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Deduplicate by protocol. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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2. Spatial coordinates

The central mathematical idea is concrete. Locations share a coordinate reference and unit.

A concrete situation

Addresses are geocoded in metres.

What the mathematics does not license

Geocoding error blurs short distances. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Audit precision. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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3. Event times

Here the geometry becomes operational. Times use one scale and resolution.

A concrete situation

Dates are converted to days.

What the mathematics does not license

Interval-censored onset is not exact. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Record uncertainty. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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4. Unordered pairs

This step deserves a careful pause. n events create n(n−1)/2 pairs.

A concrete situation

Five events create ten.

What the mathematics does not license

Ordered counting doubles totals. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Use one convention. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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5. Spatial distance

The useful question is not only how to calculate, but what the calculation assumes. Calculate separation for every pair.

A concrete situation

Four pairs fall within 2 km.

What the mathematics does not license

Road or network distance may be more relevant. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Choose domain geometry. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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6. Temporal lag

A small example makes the mechanism visible. Use absolute time difference.

A concrete situation

Three pairs occur within seven days.

What the mathematics does not license

Seasonality changes the baseline. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Model calendar structure. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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7. Spatial indicator

This is where modelling choices begin to affect evidence. Aij equals one inside the spatial threshold.

A concrete situation

Four A values equal one.

What the mathematics does not license

Threshold ties need a rule. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Declare ≤ or <. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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8. Temporal indicator

The central mathematical idea is concrete. Bij equals one inside the time threshold.

A concrete situation

Three B values equal one.

What the mathematics does not license

Time rounding creates ties. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Preserve source precision. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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9. Joint indicator

Here the geometry becomes operational. Multiply Aij and Bij.

A concrete situation

Two pairs meet both rules.

What the mathematics does not license

The same event appears in many products. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Respect dependence. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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10. Knox count

This step deserves a careful pause. Sum all joint indicators.

A concrete situation

X equals 2.

What the mathematics does not license

A count alone is not evidence. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Construct the null. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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11. Rough benchmark

The useful question is not only how to calculate, but what the calculation assumes. S×T divided by pair count gives 1.2.

A concrete situation

Observed 2 exceeds 1.2.

What the mathematics does not license

This ignores exact dependence. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Use permutations. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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12. Permutation scheme

A small example makes the mechanism visible. Shuffle times across fixed places.

A concrete situation

Every shuffle recomputes X.

What the mathematics does not license

Strata may forbid some swaps. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Use design-valid restrictions. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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13. Pseudo p-value

This is where modelling choices begin to affect evidence. Rank observed X among simulated counts.

A concrete situation

Nine of 999 are as large gives about .01.

What the mathematics does not license

Replicate count limits resolution. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Set seed and alternative. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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14. Threshold grid

The central mathematical idea is concrete. Repeat declared space and time scales.

A concrete situation

Results are shown for 1,2,5 km and 3,7,14 days.

What the mathematics does not license

Selecting the minimum p-value is biased. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Use global correction or descriptive grid. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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15. Influential events

Here the geometry becomes operational. Remove-one-event diagnostics show dominance.

A concrete situation

One case participates in both close pairs.

What the mathematics does not license

Deletion after seeing results is invalid. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Report influence transparently. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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16. External investigation

This step deserves a careful pause. Validate with new periods and exposure evidence.

A concrete situation

A later wave shows similar interaction.

What the mathematics does not license

Replication still does not prove cause. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Investigate mechanisms. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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A seven-day practice lab

1. Day 1: define the decision, observation unit, X variables and Y or reference condition.
2. Day 2: reproduce the smallest worked example and check every dimension, unit and sign.
3. Day 3: build a transparent baseline and write what it assumes.
4. Day 4: fit the main method inside a training-only pipeline; record every parameter.
5. Day 5: challenge one assumption with a deliberately difficult dataset or operating regime.
6. Day 6: compare validation metrics with the practical workload created by errors.
7. Day 7: present one supported conclusion, one unresolved uncertainty and one safe next experiment.

Questions for a student, parent or mentor

• Which part of the model is a mathematical definition, and which part is a choice?
• What small calculation would catch a sign, scale or dimension error?
• What evidence comes from training, validation and the untouched test?
• What benign condition could look like a fault or prediction failure?
• How would the answer change for another response, asset, school project or time period?
• What should a human review before acting on the score?

A method-specific audit before deployment

The Knox test asks whether pairs of events that are close in space are also close in time more often than expected. Every unordered pair receives two indicators: one for spatial closeness and one for temporal closeness. The statistic counts pairs for which both indicators equal one. The thresholds are part of the hypothesis, not harmless display choices.

Under a common null, the observed places and times are preserved but times are permuted among places. This breaks space–time interaction while retaining the marginal spatial and temporal patterns. The entire count is recomputed for each permutation. A small p-value describes unusual overlap under that exchangeability rule; it does not identify a source of transmission.

With five events there are ten unordered pairs. Suppose four are spatially close, three are temporally close and two meet both criteria. A rough independence benchmark is 4×3/10=1.2 overlapping pairs, while the valid finite-sample reference comes from the permutation distribution because pair indicators share events and are dependent.

Knox-style analysis is useful for disease surveillance, faults, crime and ecological events, but results can move sharply with thresholds, location error and event-time precision. Responsible work predeclares meaningful distances and lags, repeats a sensitivity grid without selecting only the best result, preserves strata when needed, controls repeated analyses and follows statistical interaction with domain evidence.

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Frequently asked questions

What does the Knox test measure?

It counts event pairs that meet both a chosen spatial-distance threshold and a chosen temporal-lag threshold.

What is the usual null?

Times are exchangeable among fixed locations, preserving the marginal spatial and temporal patterns while removing their pairing.

Why are ordinary pairwise tests wrong?

Pairs share events, so their indicators are dependent rather than independent Bernoulli observations.

How should thresholds be chosen?

Use domain-relevant scales declared before outcome inspection and report a sensitivity grid.

What does a small p-value mean?

The observed joint-close count is extreme under the stated permutation scheme and alternative.

Does interaction prove transmission?

No. Common exposure, reporting processes and population movement can produce the same pattern.

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Primary source and fact-check note

E. G. Knox, “The Detection of Space-Time Interactions,” Journal of the Royal Statistical Society Series C 13(1), 25–29 (1964), DOI 10.2307/2985220. Definitions and core claims in this article were checked against this primary publication. Calculations shown here were independently recomputed. Examples are educational illustrations; they are not reported industrial performance results.

Useful next reading

• Kulldorff spatial scan statistics
• Ripley’s K point patterns
• How Mathematics Works

Keep the options open. Mathematics education supports clearer questions, transferable problem-solving skills and informed choices across science, engineering, computing, AI, statistics, design and many other careers. It does not guarantee intelligence, income, admission or employment; those outcomes depend on far more than one subject or model.

Extended laboratory: topic-specific validation and responsible use

Data audit

Verify IDs, units, precision, missingness and duplicates.

Geometry audit

Publish windows, metrics, thresholds and edge rules.

Calculation audit

Reproduce a hand example and software output.

Null audit

State exchangeability, conditioning and simulations.

Multiplicity audit

Control threshold grids, matrix tests or surfaces.

Sensitivity audit

Repeat scientifically plausible scales and designs.

Transfer audit

Reserve independent regions, periods or samples.

Final claim

Report interaction or association without automatic causal claims.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Event identity

Begin with this mechanism: Each record represents one defined event. Recreate the concrete case: Five cases have unique IDs. Write events, coordinates, times, pair distances, lags, spatial indicators, temporal indicators, joint counts, permutation counts and p-value ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Duplicates inflate pair counts. Complete the transfer step: Deduplicate by protocol. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or samples outside tuning when testing transfer.

Workshop 2: Spatial coordinates

Begin with this mechanism: Locations share a coordinate reference and unit. Recreate the concrete case: Addresses are geocoded in metres. Write events, coordinates, times, pair distances, lags, spatial indicators, temporal indicators, joint counts, permutation counts and p-value ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Geocoding error blurs short distances. Complete the transfer step: Audit precision. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or samples outside tuning when testing transfer.

Workshop 3: Event times

Begin with this mechanism: Times use one scale and resolution. Recreate the concrete case: Dates are converted to days. Write events, coordinates, times, pair distances, lags, spatial indicators, temporal indicators, joint counts, permutation counts and p-value ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Interval-censored onset is not exact. Complete the transfer step: Record uncertainty. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or samples outside tuning when testing transfer.

Workshop 4: Unordered pairs

Begin with this mechanism: n events create n(n−1)/2 pairs. Recreate the concrete case: Five events create ten. Write events, coordinates, times, pair distances, lags, spatial indicators, temporal indicators, joint counts, permutation counts and p-value ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Ordered counting doubles totals. Complete the transfer step: Use one convention. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or samples outside tuning when testing transfer.

Workshop 5: Spatial distance

Begin with this mechanism: Calculate separation for every pair. Recreate the concrete case: Four pairs fall within 2 km. Write events, coordinates, times, pair distances, lags, spatial indicators, temporal indicators, joint counts, permutation counts and p-value ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Road or network distance may be more relevant. Complete the transfer step: Choose domain geometry. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or samples outside tuning when testing transfer.

Workshop 6: Temporal lag

Begin with this mechanism: Use absolute time difference. Recreate the concrete case: Three pairs occur within seven days. Write events, coordinates, times, pair distances, lags, spatial indicators, temporal indicators, joint counts, permutation counts and p-value ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Seasonality changes the baseline. Complete the transfer step: Model calendar structure. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or samples outside tuning when testing transfer.

Workshop 7: Spatial indicator

Begin with this mechanism: Aij equals one inside the spatial threshold. Recreate the concrete case: Four A values equal one. Write events, coordinates, times, pair distances, lags, spatial indicators, temporal indicators, joint counts, permutation counts and p-value ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Threshold ties need a rule. Complete the transfer step: Declare ≤ or <. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or samples outside tuning when testing transfer.

Workshop 8: Temporal indicator

Begin with this mechanism: Bij equals one inside the time threshold. Recreate the concrete case: Three B values equal one. Write events, coordinates, times, pair distances, lags, spatial indicators, temporal indicators, joint counts, permutation counts and p-value ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Time rounding creates ties. Complete the transfer step: Preserve source precision. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or samples outside tuning when testing transfer.

Workshop 9: Joint indicator

Begin with this mechanism: Multiply Aij and Bij. Recreate the concrete case: Two pairs meet both rules. Write events, coordinates, times, pair distances, lags, spatial indicators, temporal indicators, joint counts, permutation counts and p-value ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: The same event appears in many products. Complete the transfer step: Respect dependence. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or samples outside tuning when testing transfer.

Workshop 10: Knox count

Begin with this mechanism: Sum all joint indicators. Recreate the concrete case: X equals 2. Write events, coordinates, times, pair distances, lags, spatial indicators, temporal indicators, joint counts, permutation counts and p-value ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: A count alone is not evidence. Complete the transfer step: Construct the null. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or samples outside tuning when testing transfer.

Workshop 11: Rough benchmark

Begin with this mechanism: S×T divided by pair count gives 1.2. Recreate the concrete case: Observed 2 exceeds 1.2. Write events, coordinates, times, pair distances, lags, spatial indicators, temporal indicators, joint counts, permutation counts and p-value ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: This ignores exact dependence. Complete the transfer step: Use permutations. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or samples outside tuning when testing transfer.

Workshop 12: Permutation scheme

Begin with this mechanism: Shuffle times across fixed places. Recreate the concrete case: Every shuffle recomputes X. Write events, coordinates, times, pair distances, lags, spatial indicators, temporal indicators, joint counts, permutation counts and p-value ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Strata may forbid some swaps. Complete the transfer step: Use design-valid restrictions. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or samples outside tuning when testing transfer.

Workshop 13: Pseudo p-value

Begin with this mechanism: Rank observed X among simulated counts. Recreate the concrete case: Nine of 999 are as large gives about .01. Write events, coordinates, times, pair distances, lags, spatial indicators, temporal indicators, joint counts, permutation counts and p-value ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Replicate count limits resolution. Complete the transfer step: Set seed and alternative. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or samples outside tuning when testing transfer.

Workshop 14: Threshold grid

Begin with this mechanism: Repeat declared space and time scales. Recreate the concrete case: Results are shown for 1,2,5 km and 3,7,14 days. Write events, coordinates, times, pair distances, lags, spatial indicators, temporal indicators, joint counts, permutation counts and p-value ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Selecting the minimum p-value is biased. Complete the transfer step: Use global correction or descriptive grid. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or samples outside tuning when testing transfer.

Workshop 15: Influential events

Begin with this mechanism: Remove-one-event diagnostics show dominance. Recreate the concrete case: One case participates in both close pairs. Write events, coordinates, times, pair distances, lags, spatial indicators, temporal indicators, joint counts, permutation counts and p-value ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Deletion after seeing results is invalid. Complete the transfer step: Report influence transparently. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or samples outside tuning when testing transfer.

Workshop 16: External investigation

Begin with this mechanism: Validate with new periods and exposure evidence. Recreate the concrete case: A later wave shows similar interaction. Write events, coordinates, times, pair distances, lags, spatial indicators, temporal indicators, joint counts, permutation counts and p-value ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Replication still does not prove cause. Complete the transfer step: Investigate mechanisms. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or samples outside tuning when testing transfer.