Chapter map

1. 1. Binary coding
2. 2. Spatial graph
3. 3. Undirected convention
4. 4. BB count
5. 5. WW count
6. 6. BW count
7. 7. Join total
8. 8. Class margins
9. 9. Expected BB
10. 10. Expected WW
11. 11. Expected BW
12. 12. Observed contrast
13. 13. Permutation test
14. 14. Join table test
15. 15. Local join counts
16. 16. Threshold and transfer

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. Binary coding

This is where modelling choices begin to affect evidence. Every location is assigned a documented zero or one.

A concrete situation

Six areas are 1,1,1,0,0,0.

What the mathematics does not license

Arbitrary thresholds discard information. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Repeat plausible cut points. 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 graph

The central mathematical idea is concrete. Edges define admissible joins.

A concrete situation

A six-node cycle has six undirected edges.

What the mathematics does not license

Rook and queen contiguity differ. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Publish the adjacency list. 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. Undirected convention

Here the geometry becomes operational. Each edge is counted once.

A concrete situation

Edge 1–2 is one join, not two.

What the mathematics does not license

Directed storage can double counts. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Canonicalize edge pairs. 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. BB count

This step deserves a careful pause. Count edges joining two ones.

A concrete situation

Edges 1–2 and 2–3 give BB=2.

What the mathematics does not license

Class labels may be reversed. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Define black substantively. 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. WW count

The useful question is not only how to calculate, but what the calculation assumes. Count edges joining two zeros.

A concrete situation

Edges 4–5 and 5–6 give WW=2.

What the mathematics does not license

Missing values can masquerade as zero. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Keep absence separate from missingness. 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. BW count

A small example makes the mechanism visible. Count mixed-label edges.

A concrete situation

Edges 3–4 and 6–1 give BW=2.

What the mathematics does not license

Orientation should not split BW and WB in an undirected graph. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Audit the table. 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. Join total

This is where modelling choices begin to affect evidence. BB+WW+BW must equal edge count.

A concrete situation

2+2+2 equals 6.

What the mathematics does not license

A mismatch signals duplicated or lost edges. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Use this invariant. 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. Class margins

The central mathematical idea is concrete. Record nB and nW.

A concrete situation

Both equal 3.

What the mathematics does not license

Rare classes limit attainable counts. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Report margins with results. 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. Expected BB

Here the geometry becomes operational. Under free randomization E[BB]=m nB(nB−1)/(n(n−1)).

A concrete situation

6×3×2/(6×5)=1.2.

What the mathematics does not license

Other nulls yield other expectations. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

State the conditioning. 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. Expected WW

This step deserves a careful pause. Replace nB with nW.

A concrete situation

Expected WW is also 1.2.

What the mathematics does not license

Symmetry here comes from equal margins. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Do not generalize it. 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. Expected BW

The useful question is not only how to calculate, but what the calculation assumes. E[BW]=m 2nBnW/(n(n−1)).

A concrete situation

Expected BW is 3.6.

What the mathematics does not license

Graph dependence affects variance. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Use exact formulas or 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. Observed contrast

A small example makes the mechanism visible. Compare counts with their expected centres.

A concrete situation

BB=2 exceeds 1.2 while BW=2 is below 3.6.

What the mathematics does not license

Differences alone are not significance tests. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Build a reference distribution. 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. Permutation test

This is where modelling choices begin to affect evidence. Shuffle labels while preserving margins.

A concrete situation

Enumerate or sample binary allocations.

What the mathematics does not license

Few unique allocations limit resolution. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Use exact enumeration when feasible. 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. Join table test

The central mathematical idea is concrete. A contingency summary can combine pair types.

A concrete situation

Observed and expected counts form a table.

What the mathematics does not license

Edges are dependent, so naive chi-square can mislead. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Use spatially valid calibration. 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. Local join counts

Here the geometry becomes operational. Node-level counts identify specific clusters.

A concrete situation

Area two touches two black neighbours.

What the mathematics does not license

Many nodes create multiple tests. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Control false discoveries. 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. Threshold and transfer

This step deserves a careful pause. Check other cut points and independent maps.

A concrete situation

BB excess persists at two thresholds and a later date.

What the mathematics does not license

Stability remains association. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Investigate processes separately. 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

Join-count statistics study a binary map by counting the labels on connected pairs. A black–black join links two locations coded one, a white–white join links two coded zero, and a black–white join crosses the classes. The graph decides which pairs exist, while the coding decides what the classes mean. Both are part of the hypothesis.

Under free randomization, the observed number of black and white locations is preserved while labels are assigned to the fixed graph. Expected BB, WW and BW counts follow from the number of undirected edges and class counts. Excess same-colour joins suggest positive spatial association; excess mixed joins suggest negative association or alternation. Directed adjacency must be converted or counted consistently.

A permutation test provides a transparent reference for irregular graphs, islands and small maps. Every permutation preserves the chosen margins, recomputes the counts and ranks the observation. A chi-square summary of the join table can be useful, but separate one-sided questions about BB or BW require declared alternatives and multiplicity control.

Join counts are useful for land-cover classes, presence–absence data, electoral categories and thresholded biological states. Yet thresholding a continuous variable discards magnitude and can manufacture instability near the cut point. Strong work repeats reasonable cut points, reports ties and class balance, maps influential edges and distinguishes spatial association from contagion, preference or cause.

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Frequently asked questions

What is a join count?

It counts the class labels at the two ends of each spatial edge in a binary map.

What are BB, WW and BW?

They are black–black, white–white and mixed black–white neighbour pairs.

What does excess BB mean?

More BB joins than expected under the chosen randomization suggests positive association among black locations.

Why preserve class totals?

Free randomization conditions on the observed numbers of black and white locations while reallocating them over the fixed graph.

Can directed weights be used?

Yes, but ordered-pair counting and formulas must be consistent; many textbook join counts use undirected edges.

Does clustering prove contagion?

No. Shared environment, thresholding, boundaries and measurement processes can create the same pattern.

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Primary source and fact-check note

Andrew D. Cliff and J. K. Ord, Spatial Autocorrelation, Pion (1973), ISBN 0-85086-036-9. 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

• BinSpect binary spatial states
• Moran’s I spatial autocorrelation
• 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 definitions, alignment, missingness and units.

Structure audit

Publish the graph, window, weights or metric.

Calculation audit

Reproduce a hand-worked example and software output.

Null audit

State exactly what is randomized or simulated.

Multiplicity audit

Account for local tests or searched windows.

Sensitivity audit

Repeat plausible scales, thresholds and boundaries.

Transfer audit

Reserve independent regions, periods or samples.

Final claim

Report association under the model without claiming automatic cause.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Binary coding

Begin with this mechanism: Every location is assigned a documented zero or one. Recreate the concrete case: Six areas are 1,1,1,0,0,0. Write binary labels, graph edges, BB, WW and BW counts, class margins, expected joins, permutation ranks, thresholds and local contributions 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: Arbitrary thresholds discard information. Complete the transfer step: Repeat plausible cut points. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 2: Spatial graph

Begin with this mechanism: Edges define admissible joins. Recreate the concrete case: A six-node cycle has six undirected edges. Write binary labels, graph edges, BB, WW and BW counts, class margins, expected joins, permutation ranks, thresholds and local contributions 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: Rook and queen contiguity differ. Complete the transfer step: Publish the adjacency list. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 3: Undirected convention

Begin with this mechanism: Each edge is counted once. Recreate the concrete case: Edge 1–2 is one join, not two. Write binary labels, graph edges, BB, WW and BW counts, class margins, expected joins, permutation ranks, thresholds and local contributions 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: Directed storage can double counts. Complete the transfer step: Canonicalize edge pairs. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 4: BB count

Begin with this mechanism: Count edges joining two ones. Recreate the concrete case: Edges 1–2 and 2–3 give BB=2. Write binary labels, graph edges, BB, WW and BW counts, class margins, expected joins, permutation ranks, thresholds and local contributions 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: Class labels may be reversed. Complete the transfer step: Define black substantively. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 5: WW count

Begin with this mechanism: Count edges joining two zeros. Recreate the concrete case: Edges 4–5 and 5–6 give WW=2. Write binary labels, graph edges, BB, WW and BW counts, class margins, expected joins, permutation ranks, thresholds and local contributions 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: Missing values can masquerade as zero. Complete the transfer step: Keep absence separate from missingness. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 6: BW count

Begin with this mechanism: Count mixed-label edges. Recreate the concrete case: Edges 3–4 and 6–1 give BW=2. Write binary labels, graph edges, BB, WW and BW counts, class margins, expected joins, permutation ranks, thresholds and local contributions 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: Orientation should not split BW and WB in an undirected graph. Complete the transfer step: Audit the table. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 7: Join total

Begin with this mechanism: BB+WW+BW must equal edge count. Recreate the concrete case: 2+2+2 equals 6. Write binary labels, graph edges, BB, WW and BW counts, class margins, expected joins, permutation ranks, thresholds and local contributions 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 mismatch signals duplicated or lost edges. Complete the transfer step: Use this invariant. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 8: Class margins

Begin with this mechanism: Record nB and nW. Recreate the concrete case: Both equal 3. Write binary labels, graph edges, BB, WW and BW counts, class margins, expected joins, permutation ranks, thresholds and local contributions 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: Rare classes limit attainable counts. Complete the transfer step: Report margins with results. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 9: Expected BB

Begin with this mechanism: Under free randomization E[BB]=m nB(nB−1)/(n(n−1)). Recreate the concrete case: 6×3×2/(6×5)=1.2. Write binary labels, graph edges, BB, WW and BW counts, class margins, expected joins, permutation ranks, thresholds and local contributions 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: Other nulls yield other expectations. Complete the transfer step: State the conditioning. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 10: Expected WW

Begin with this mechanism: Replace nB with nW. Recreate the concrete case: Expected WW is also 1.2. Write binary labels, graph edges, BB, WW and BW counts, class margins, expected joins, permutation ranks, thresholds and local contributions 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: Symmetry here comes from equal margins. Complete the transfer step: Do not generalize it. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 11: Expected BW

Begin with this mechanism: E[BW]=m 2nBnW/(n(n−1)). Recreate the concrete case: Expected BW is 3.6. Write binary labels, graph edges, BB, WW and BW counts, class margins, expected joins, permutation ranks, thresholds and local contributions 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: Graph dependence affects variance. Complete the transfer step: Use exact formulas or permutations. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 12: Observed contrast

Begin with this mechanism: Compare counts with their expected centres. Recreate the concrete case: BB=2 exceeds 1.2 while BW=2 is below 3.6. Write binary labels, graph edges, BB, WW and BW counts, class margins, expected joins, permutation ranks, thresholds and local contributions 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: Differences alone are not significance tests. Complete the transfer step: Build a reference distribution. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 13: Permutation test

Begin with this mechanism: Shuffle labels while preserving margins. Recreate the concrete case: Enumerate or sample binary allocations. Write binary labels, graph edges, BB, WW and BW counts, class margins, expected joins, permutation ranks, thresholds and local contributions 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 unique allocations limit resolution. Complete the transfer step: Use exact enumeration when feasible. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 14: Join table test

Begin with this mechanism: A contingency summary can combine pair types. Recreate the concrete case: Observed and expected counts form a table. Write binary labels, graph edges, BB, WW and BW counts, class margins, expected joins, permutation ranks, thresholds and local contributions 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: Edges are dependent, so naive chi-square can mislead. Complete the transfer step: Use spatially valid calibration. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 15: Local join counts

Begin with this mechanism: Node-level counts identify specific clusters. Recreate the concrete case: Area two touches two black neighbours. Write binary labels, graph edges, BB, WW and BW counts, class margins, expected joins, permutation ranks, thresholds and local contributions 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: Many nodes create multiple tests. Complete the transfer step: Control false discoveries. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 16: Threshold and transfer

Begin with this mechanism: Check other cut points and independent maps. Recreate the concrete case: BB excess persists at two thresholds and a later date. Write binary labels, graph edges, BB, WW and BW counts, class margins, expected joins, permutation ranks, thresholds and local contributions 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: Stability remains association. Complete the transfer step: Investigate processes separately. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.