Chapter map

1. 1. Event identity
2. 2. Coordinate system
3. 3. Time scale
4. 4. Spatial distance matrix
5. 5. Temporal lag matrix
6. 6. Spatial rank list
7. 7. Temporal rank list
8. 8. First-k indicator
9. 9. Joint-rank indicator
10. 10. J k statistic
11. 11. Time-label permutation
12. 12. Permutation p-value
13. 13. Scale sequence
14. 14. Tie sensitivity
15. 15. Influence analysis
16. 16. External validation

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 step deserves a careful pause. Each row denotes one defined event.

A concrete situation

Four events have unique IDs 1 to 4.

What the mathematics does not license

Duplicate records create artificial neighbours. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Deduplicate by a written 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. Coordinate system

The useful question is not only how to calculate, but what the calculation assumes. All locations share a meaningful distance unit.

A concrete situation

Coordinates are projected in metres.

What the mathematics does not license

Latitude-longitude degrees distort local distance. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Document the metric and projection. 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. Time scale

A small example makes the mechanism visible. Timestamps use one origin and resolution.

A concrete situation

Onset dates become elapsed days.

What the mathematics does not license

Reporting dates may not equal event dates. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Separate onset, report and sample time. 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. Spatial distance matrix

This is where modelling choices begin to affect evidence. Compute every nonself geographic separation.

A concrete situation

Event 1 is closest to event 2.

What the mathematics does not license

Barriers can make Euclidean proximity misleading. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Compare a network or travel metric. 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. Temporal lag matrix

The central mathematical idea is concrete. Compute absolute separation between event times.

A concrete situation

Event 1 is closest in time to event 2.

What the mathematics does not license

Rounded dates create ties. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Preserve original 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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6. Spatial rank list

Here the geometry becomes operational. Sort other events by spatial distance for each origin.

A concrete situation

Ranks for event 1 begin 2,3,4.

What the mathematics does not license

Ranks can flip under small coordinate error. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Perform perturbation checks. 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. Temporal rank list

This step deserves a careful pause. Sort other events by temporal lag.

A concrete situation

Ranks for event 1 begin 2,4,3.

What the mathematics does not license

Simultaneous events require a tie rule. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Publish the rule. 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. First-k indicator

The useful question is not only how to calculate, but what the calculation assumes. Mark whether j is among i’s first k neighbours.

A concrete situation

For k=1, S12 equals one.

What the mathematics does not license

The relation is directed. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Do not silently symmetrize. 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-rank indicator

A small example makes the mechanism visible. Multiply spatial and temporal membership indicators.

A concrete situation

Event 1 contributes for neighbour 2.

What the mathematics does not license

Shared origins make products dependent. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Use the full test design. 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. J k statistic

This is where modelling choices begin to affect evidence. Sum all directed joint memberships.

A concrete situation

Three of four origins match at k=1.

What the mathematics does not license

A count alone has no null scale. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Construct 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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11. Time-label permutation

The central mathematical idea is concrete. Shuffle times across fixed locations.

A concrete situation

Each shuffle rebuilds temporal ranks.

What the mathematics does not license

Season and strata may forbid swaps. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Restrict 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 p-value

Here the geometry becomes operational. Rank observed Jk among simulated values.

A concrete situation

Nine of 999 as large gives about .01.

What the mathematics does not license

Few simulations limit resolution. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Use enough replicates and a fixed seed. 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. Scale sequence

This step deserves a careful pause. Repeat a declared set of k values.

A concrete situation

k=1,2,3 reveals local to broader overlap.

What the mathematics does not license

Selecting the best k inflates evidence. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use global or adjusted inference. 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. Tie sensitivity

The useful question is not only how to calculate, but what the calculation assumes. Recompute under justified tie handling.

A concrete situation

Two equal lags share or randomly break rank.

What the mathematics does not license

Favourable tie-breaking biases results. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Report all legitimate rules. 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. Influence analysis

A small example makes the mechanism visible. Find events contributing many overlaps.

A concrete situation

One event is in three mutual pairs.

What the mathematics does not license

Removing it after inspection is invalid. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Report influence and reasons. 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 validation

This is where modelling choices begin to affect evidence. Repeat on a later period or separate region.

A concrete situation

The same k-scale appears in held-out events.

What the mathematics does not license

Replication remains association. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Investigate mechanisms independently. 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

Jacquez’s k-nearest-neighbour test asks whether the same event pairs are unusually close in both space and time. For every event, the analyst ranks all other events by geographic distance and again by temporal separation. The statistic counts directed pairs that appear within the first k positions of both lists. Rank neighbourhoods replace the fixed kilometre and day thresholds used by a Knox test, so the mathematical question is about overlap between local orderings.

The null hypothesis keeps the observed locations and observed times but breaks their pairing, commonly by permuting times among locations. Each permutation rebuilds the temporal nearest-neighbour lists and recounts the overlap. This preserves the separate spatial and temporal patterns while removing their event-level alignment. Exchangeability can fail when seasons, regions, reporting systems or person-level strata impose restrictions, so the permutation design is part of the scientific claim.

Consider four events and k=1. Suppose the nearest spatial neighbours of events 1 to 4 are 2, 1, 4 and 3, while the nearest temporal neighbours are 2, 1, 2 and 3. Events 1, 2 and 4 have the same first neighbour in both rankings, so the illustrative directed overlap count is three. The count is interpreted against its permutation distribution, not as three independent successes.

The method is useful when interaction is expected locally but a distance threshold would be arbitrary. It still requires a declared k or a multiplicity-aware analysis across several k values. Coordinate error, timestamp ties, repeated events, edge effects and mobility can change ranks. A strong analysis publishes the neighbour lists, tie rules, restricted permutation plan, influence checks and independent validation before suggesting transmission or mechanism.

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Frequently asked questions

What does the Jacquez test count?

It counts directed event pairs that belong to the first k nearest neighbours in both geographic distance and temporal separation.

Why use ranks?

Ranks avoid declaring one kilometre and day threshold, while still defining a local scale through k.

What is permuted?

Times are reassigned to fixed locations, subject to any strata required by the design, and the temporal neighbour lists are rebuilt.

Are directed pairs independent?

No. Events appear in many rankings and can be mutual neighbours, so the count needs its permutation reference.

How should k be chosen?

Use a scientifically motivated value or a predeclared set with multiplicity-aware calibration.

Does a large overlap prove transmission?

No. Common exposure, mobility, reporting and shared schedules may also align space and time.

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Primary source and fact-check note

Geoffrey M. Jacquez, “A k nearest neighbour test for space-time interaction,” Statistics in Medicine 15(18), 1935–1949 (1996), DOI 10.1002/(SICI)1097-0258(19960930)15:18<1935::AID-SIM406>3.0.CO;2-I. 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

• Knox space–time interaction tests
• Mantel matrix correlation tests
• Diggle space–time K functions
• 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

Identity audit

Deduplicate events and preserve timestamp provenance.

Rank audit

Publish spatial and temporal neighbour lists.

Tie audit

Repeat every defensible tie rule.

Null audit

Restrict time permutations to valid exchangeability sets.

Scale audit

Treat multiple k values as one search.

Influence audit

Map mutual pairs and high-contribution events.

Transfer audit

Reserve regions or periods outside tuning.

Final claim

Rank overlap is interaction evidence, not transmission proof.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Event identity

Begin with this mechanism: Each row denotes one defined event. Recreate the concrete case: Four events have unique IDs 1 to 4. Write event IDs, coordinates, timestamps, distance matrices, lag matrices, spatial ranks, temporal ranks, first-k indicators, joint indicators, directed overlap counts, permutation counts and tail 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: Duplicate records create artificial neighbours. Complete the transfer step: Deduplicate by a written protocol. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 2: Coordinate system

Begin with this mechanism: All locations share a meaningful distance unit. Recreate the concrete case: Coordinates are projected in metres. Write event IDs, coordinates, timestamps, distance matrices, lag matrices, spatial ranks, temporal ranks, first-k indicators, joint indicators, directed overlap counts, permutation counts and tail 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: Latitude-longitude degrees distort local distance. Complete the transfer step: Document the metric and projection. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 3: Time scale

Begin with this mechanism: Timestamps use one origin and resolution. Recreate the concrete case: Onset dates become elapsed days. Write event IDs, coordinates, timestamps, distance matrices, lag matrices, spatial ranks, temporal ranks, first-k indicators, joint indicators, directed overlap counts, permutation counts and tail 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: Reporting dates may not equal event dates. Complete the transfer step: Separate onset, report and sample time. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 4: Spatial distance matrix

Begin with this mechanism: Compute every nonself geographic separation. Recreate the concrete case: Event 1 is closest to event 2. Write event IDs, coordinates, timestamps, distance matrices, lag matrices, spatial ranks, temporal ranks, first-k indicators, joint indicators, directed overlap counts, permutation counts and tail 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: Barriers can make Euclidean proximity misleading. Complete the transfer step: Compare a network or travel metric. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 5: Temporal lag matrix

Begin with this mechanism: Compute absolute separation between event times. Recreate the concrete case: Event 1 is closest in time to event 2. Write event IDs, coordinates, timestamps, distance matrices, lag matrices, spatial ranks, temporal ranks, first-k indicators, joint indicators, directed overlap counts, permutation counts and tail 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: Rounded dates create ties. Complete the transfer step: Preserve original precision. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 6: Spatial rank list

Begin with this mechanism: Sort other events by spatial distance for each origin. Recreate the concrete case: Ranks for event 1 begin 2,3,4. Write event IDs, coordinates, timestamps, distance matrices, lag matrices, spatial ranks, temporal ranks, first-k indicators, joint indicators, directed overlap counts, permutation counts and tail 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: Ranks can flip under small coordinate error. Complete the transfer step: Perform perturbation checks. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 7: Temporal rank list

Begin with this mechanism: Sort other events by temporal lag. Recreate the concrete case: Ranks for event 1 begin 2,4,3. Write event IDs, coordinates, timestamps, distance matrices, lag matrices, spatial ranks, temporal ranks, first-k indicators, joint indicators, directed overlap counts, permutation counts and tail 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: Simultaneous events require a tie rule. Complete the transfer step: Publish the rule. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 8: First-k indicator

Begin with this mechanism: Mark whether j is among i’s first k neighbours. Recreate the concrete case: For k=1, S12 equals one. Write event IDs, coordinates, timestamps, distance matrices, lag matrices, spatial ranks, temporal ranks, first-k indicators, joint indicators, directed overlap counts, permutation counts and tail 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 relation is directed. Complete the transfer step: Do not silently symmetrize. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 9: Joint-rank indicator

Begin with this mechanism: Multiply spatial and temporal membership indicators. Recreate the concrete case: Event 1 contributes for neighbour 2. Write event IDs, coordinates, timestamps, distance matrices, lag matrices, spatial ranks, temporal ranks, first-k indicators, joint indicators, directed overlap counts, permutation counts and tail 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: Shared origins make products dependent. Complete the transfer step: Use the full test design. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 10: J k statistic

Begin with this mechanism: Sum all directed joint memberships. Recreate the concrete case: Three of four origins match at k=1. Write event IDs, coordinates, timestamps, distance matrices, lag matrices, spatial ranks, temporal ranks, first-k indicators, joint indicators, directed overlap counts, permutation counts and tail 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 has no null scale. Complete the transfer step: Construct permutations. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 11: Time-label permutation

Begin with this mechanism: Shuffle times across fixed locations. Recreate the concrete case: Each shuffle rebuilds temporal ranks. Write event IDs, coordinates, timestamps, distance matrices, lag matrices, spatial ranks, temporal ranks, first-k indicators, joint indicators, directed overlap counts, permutation counts and tail 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: Season and strata may forbid swaps. Complete the transfer step: Restrict permutations. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 12: Permutation p-value

Begin with this mechanism: Rank observed Jk among simulated values. Recreate the concrete case: Nine of 999 as large gives about .01. Write event IDs, coordinates, timestamps, distance matrices, lag matrices, spatial ranks, temporal ranks, first-k indicators, joint indicators, directed overlap counts, permutation counts and tail 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: Few simulations limit resolution. Complete the transfer step: Use enough replicates and a fixed seed. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 13: Scale sequence

Begin with this mechanism: Repeat a declared set of k values. Recreate the concrete case: k=1,2,3 reveals local to broader overlap. Write event IDs, coordinates, timestamps, distance matrices, lag matrices, spatial ranks, temporal ranks, first-k indicators, joint indicators, directed overlap counts, permutation counts and tail 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 best k inflates evidence. Complete the transfer step: Use global or adjusted inference. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 14: Tie sensitivity

Begin with this mechanism: Recompute under justified tie handling. Recreate the concrete case: Two equal lags share or randomly break rank. Write event IDs, coordinates, timestamps, distance matrices, lag matrices, spatial ranks, temporal ranks, first-k indicators, joint indicators, directed overlap counts, permutation counts and tail 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: Favourable tie-breaking biases results. Complete the transfer step: Report all legitimate rules. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 15: Influence analysis

Begin with this mechanism: Find events contributing many overlaps. Recreate the concrete case: One event is in three mutual pairs. Write event IDs, coordinates, timestamps, distance matrices, lag matrices, spatial ranks, temporal ranks, first-k indicators, joint indicators, directed overlap counts, permutation counts and tail 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: Removing it after inspection is invalid. Complete the transfer step: Report influence and reasons. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 16: External validation

Begin with this mechanism: Repeat on a later period or separate region. Recreate the concrete case: The same k-scale appears in held-out events. Write event IDs, coordinates, timestamps, distance matrices, lag matrices, spatial ranks, temporal ranks, first-k indicators, joint indicators, directed overlap counts, permutation counts and tail 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 remains association. Complete the transfer step: Investigate mechanisms independently. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.