Chapter map

1. 1. Space-time events
2. 2. Spatial window
3. 3. Time interval
4. 4. Intensity
5. 5. Spatial separation
6. 6. Temporal separation
7. 7. Joint indicator
8. 8. Joint K estimator
9. 9. Spatial marginal K
10. 10. Temporal marginal K
11. 11. Product benchmark
12. 12. Interaction contrast
13. 13. Spatial edge correction
14. 14. Temporal edge correction
15. 15. Simulation envelope
16. 16. Multiscale 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. Space-time events

Here the geometry becomes operational. Each event has verified coordinates and time.

A concrete situation

Four events are mapped over ten days.

What the mathematics does not license

Repeated observations may not be independent. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Define event units. 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 window

This step deserves a careful pause. A bounded region defines geographic opportunity.

A concrete situation

A square window has area 100.

What the mathematics does not license

Holes and barriers change neighbourhoods. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Store exact 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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3. Time interval

The useful question is not only how to calculate, but what the calculation assumes. Observation duration defines temporal opportunity.

A concrete situation

The study covers 10 days.

What the mathematics does not license

Start and end censor partners. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Record boundaries. 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. Intensity

A small example makes the mechanism visible. Events per area-time set the baseline scale.

A concrete situation

Four events occur in 1000 area-days.

What the mathematics does not license

Intensity may vary. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Model first-order trend. 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 separation

This is where modelling choices begin to affect evidence. Compute distance for every distinct pair.

A concrete situation

Six ordered pairs fall within s.

What the mathematics does not license

Network distance may be required. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Choose 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 separation

The central mathematical idea is concrete. Compute absolute lag for every pair.

A concrete situation

Six ordered pairs fall within t.

What the mathematics does not license

Time zones and rounding matter. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Standardize timestamps. 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. Joint indicator

Here the geometry becomes operational. A pair contributes when both conditions hold.

A concrete situation

Four ordered pairs meet both.

What the mathematics does not license

Pair dependence remains. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Use point-process theory. 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. Joint K estimator

This step deserves a careful pause. Scale the corrected joint count by intensity and domain measures.

A concrete situation

Four qualifying pairs contribute to Kst.

What the mathematics does not license

Naive proportions omit edge weights. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Use verified estimators. 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. Spatial marginal K

The useful question is not only how to calculate, but what the calculation assumes. Ignore time and estimate Ks(s).

A concrete situation

Six pairs are spatially close.

What the mathematics does not license

Spatial inhomogeneity distorts interaction. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Use appropriate marginal model. 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. Temporal marginal K

A small example makes the mechanism visible. Ignore space and estimate Kt(t).

A concrete situation

Six pairs are temporally close.

What the mathematics does not license

Seasonality violates stationary time. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Model calendar effects. 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. Product benchmark

This is where modelling choices begin to affect evidence. Multiply marginal functions under separability.

A concrete situation

Normalized marginals .5×.5=.25.

What the mathematics does not license

Separability is an assumption. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Test sensitivity. 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. Interaction contrast

The central mathematical idea is concrete. Subtract product benchmark from joint function.

A concrete situation

4/12−.25≈.083 illustratively.

What the mathematics does not license

This toy contrast is not the scaled estimator. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Keep units and weights. 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. Spatial edge correction

Here the geometry becomes operational. Weight pairs by visible geographic opportunity.

A concrete situation

Boundary pairs receive larger weights.

What the mathematics does not license

Complex geometry challenges formulas. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Compare translation corrections. 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. Temporal edge correction

This step deserves a careful pause. Weight lags by observable overlap in the study interval.

A concrete situation

Long lags have less opportunity.

What the mathematics does not license

Ignoring this biases large t. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Use interval-overlap factors. 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. Simulation envelope

The useful question is not only how to calculate, but what the calculation assumes. Simulate under the fitted no-interaction model.

A concrete situation

999 surfaces form a reference.

What the mathematics does not license

Pointwise cells do not give global control. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Use rank or global envelopes. 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. Multiscale validation

A small example makes the mechanism visible. Interpret coherent regions and confirm elsewhere.

A concrete situation

Positive D persists at 1–2 km and 3–7 days in a later period.

What the mathematics does not license

Pattern still is not mechanism. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Investigate exposure and transmission. 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

Diggle and colleagues extended second-order spatial methods into space and time. A space–time K function counts event pairs within spatial distance s and temporal lag t after scaling for the observation region, duration and intensity. Separate spatial and temporal K functions describe the marginals. Their product provides a no-interaction benchmark under appropriate separability assumptions.

An interaction surface D(s,t)=K_st(s,t)−K_s(s)K_t(t) reveals scales where joint proximity exceeds what the separate spatial and temporal patterns predict. Positive values suggest extra space–time interaction; negative values suggest inhibition. Edge corrections are needed in both dimensions because pairs near geographic or temporal boundaries have missing potential partners.

For an illustrative set of twelve ordered pairs, suppose six are spatially close, six temporally close and four close in both. The normalized joint proportion is 4/12=0.333, while the product of marginal proportions is 0.5×0.5=0.25, giving an illustrative excess of 0.083. A proper estimator restores intensity, window and edge-correction factors.

The method gives a multiscale view that the single-threshold Knox count cannot. Yet intensity trends, seasonality, population movement, reporting delay and repeated measurements can mimic interaction. Strong work models first-order structure, uses global simulation envelopes over the s–t grid, checks edge corrections and validates any scientific explanation with independent events or mechanistic investigation.

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Frequently asked questions

What is a space–time K function?

It is a cumulative second-order statistic counting event pairs within spatial distance s and temporal lag t after intensity and window scaling.

What is the no-interaction benchmark?

Under suitable separability, the joint K equals the product of spatial and temporal marginal K functions.

What is D(s,t)?

D is Kst−KsKt, an interaction surface showing excess or deficit joint proximity by scale.

Why correct edges twice?

Events near geographic and temporal boundaries have unobserved potential partners outside the study domain.

Why use a global envelope?

Many s–t cells are inspected simultaneously, so pointwise bands do not control the whole surface.

Does positive interaction prove transmission?

No. Intensity trends, seasonality, mobility and reporting can create similar surfaces.

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Primary source and fact-check note

P. J. Diggle, A. G. Chetwynd, R. Häggkvist and S. E. Morris, “Second-order analysis of space-time clustering,” Statistical Methods in Medical Research 4(2), 124–136 (1995), DOI 10.1177/096228029500400203. 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
• 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: Space-time events

Begin with this mechanism: Each event has verified coordinates and time. Recreate the concrete case: Four events are mapped over ten days. Write events, area, duration, intensity, spatial distances, time lags, edge weights, joint K, marginal K, interaction surfaces and envelope 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: Repeated observations may not be independent. Complete the transfer step: Define event units. 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 window

Begin with this mechanism: A bounded region defines geographic opportunity. Recreate the concrete case: A square window has area 100. Write events, area, duration, intensity, spatial distances, time lags, edge weights, joint K, marginal K, interaction surfaces and envelope 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: Holes and barriers change neighbourhoods. Complete the transfer step: Store exact 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 3: Time interval

Begin with this mechanism: Observation duration defines temporal opportunity. Recreate the concrete case: The study covers 10 days. Write events, area, duration, intensity, spatial distances, time lags, edge weights, joint K, marginal K, interaction surfaces and envelope 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: Start and end censor partners. Complete the transfer step: Record boundaries. 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: Intensity

Begin with this mechanism: Events per area-time set the baseline scale. Recreate the concrete case: Four events occur in 1000 area-days. Write events, area, duration, intensity, spatial distances, time lags, edge weights, joint K, marginal K, interaction surfaces and envelope 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: Intensity may vary. Complete the transfer step: Model first-order trend. 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 separation

Begin with this mechanism: Compute distance for every distinct pair. Recreate the concrete case: Six ordered pairs fall within s. Write events, area, duration, intensity, spatial distances, time lags, edge weights, joint K, marginal K, interaction surfaces and envelope 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: Network distance may be required. Complete the transfer step: Choose 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 separation

Begin with this mechanism: Compute absolute lag for every pair. Recreate the concrete case: Six ordered pairs fall within t. Write events, area, duration, intensity, spatial distances, time lags, edge weights, joint K, marginal K, interaction surfaces and envelope 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 zones and rounding matter. Complete the transfer step: Standardize timestamps. 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: Joint indicator

Begin with this mechanism: A pair contributes when both conditions hold. Recreate the concrete case: Four ordered pairs meet both. Write events, area, duration, intensity, spatial distances, time lags, edge weights, joint K, marginal K, interaction surfaces and envelope 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: Pair dependence remains. Complete the transfer step: Use point-process theory. 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: Joint K estimator

Begin with this mechanism: Scale the corrected joint count by intensity and domain measures. Recreate the concrete case: Four qualifying pairs contribute to Kst. Write events, area, duration, intensity, spatial distances, time lags, edge weights, joint K, marginal K, interaction surfaces and envelope 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: Naive proportions omit edge weights. Complete the transfer step: Use verified estimators. 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: Spatial marginal K

Begin with this mechanism: Ignore time and estimate Ks(s). Recreate the concrete case: Six pairs are spatially close. Write events, area, duration, intensity, spatial distances, time lags, edge weights, joint K, marginal K, interaction surfaces and envelope 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: Spatial inhomogeneity distorts interaction. Complete the transfer step: Use appropriate marginal model. 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: Temporal marginal K

Begin with this mechanism: Ignore space and estimate Kt(t). Recreate the concrete case: Six pairs are temporally close. Write events, area, duration, intensity, spatial distances, time lags, edge weights, joint K, marginal K, interaction surfaces and envelope 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 violates stationary time. Complete the transfer step: Model calendar effects. 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: Product benchmark

Begin with this mechanism: Multiply marginal functions under separability. Recreate the concrete case: Normalized marginals .5×.5=.25. Write events, area, duration, intensity, spatial distances, time lags, edge weights, joint K, marginal K, interaction surfaces and envelope 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: Separability is an assumption. Complete the transfer step: Test sensitivity. 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: Interaction contrast

Begin with this mechanism: Subtract product benchmark from joint function. Recreate the concrete case: 4/12−.25≈.083 illustratively. Write events, area, duration, intensity, spatial distances, time lags, edge weights, joint K, marginal K, interaction surfaces and envelope 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 toy contrast is not the scaled estimator. Complete the transfer step: Keep units and weights. 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: Spatial edge correction

Begin with this mechanism: Weight pairs by visible geographic opportunity. Recreate the concrete case: Boundary pairs receive larger weights. Write events, area, duration, intensity, spatial distances, time lags, edge weights, joint K, marginal K, interaction surfaces and envelope 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: Complex geometry challenges formulas. Complete the transfer step: Compare translation corrections. 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: Temporal edge correction

Begin with this mechanism: Weight lags by observable overlap in the study interval. Recreate the concrete case: Long lags have less opportunity. Write events, area, duration, intensity, spatial distances, time lags, edge weights, joint K, marginal K, interaction surfaces and envelope 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: Ignoring this biases large t. Complete the transfer step: Use interval-overlap factors. 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: Simulation envelope

Begin with this mechanism: Simulate under the fitted no-interaction model. Recreate the concrete case: 999 surfaces form a reference. Write events, area, duration, intensity, spatial distances, time lags, edge weights, joint K, marginal K, interaction surfaces and envelope 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: Pointwise cells do not give global control. Complete the transfer step: Use rank or global envelopes. 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: Multiscale validation

Begin with this mechanism: Interpret coherent regions and confirm elsewhere. Recreate the concrete case: Positive D persists at 1–2 km and 3–7 days in a later period. Write events, area, duration, intensity, spatial distances, time lags, edge weights, joint K, marginal K, interaction surfaces and envelope 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: Pattern still is not mechanism. Complete the transfer step: Investigate exposure and transmission. 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.