Chapter map

1. 1. Gene expression vector
2. 2. Two-state reduction
3. 3. K-means binarization
4. 4. Rank binarization
5. 5. Spatial nodes
6. 6. Delaunay network
7. 7. KNN or distance network
8. 8. Undirected edges
9. 9. Endpoint-state pairs
10. 10. Contingency table
11. 11. Odds ratio
12. 12. Fisher exact probability
13. 13. Multiple genes
14. 14. Threshold sensitivity
15. 15. Pattern limitation
16. 16. Governance and learning

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. Gene expression vector

The useful question is not only how to calculate, but what the calculation assumes. One gene provides a value at every tissue node.

A concrete situation

Seven cells have counts from zero to twelve.

What the mathematics does not license

Depth and transformation change the vector. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Record preprocessing. 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. Two-state reduction

A small example makes the mechanism visible. Each value becomes low or high.

A concrete situation

Four nodes are labelled one.

What the mathematics does not license

Magnitude within a state is lost. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Plot raw values too. 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. K-means binarization

This is where modelling choices begin to affect evidence. Two centroids minimize within-group squared distances.

A concrete situation

Low and high modes separate.

What the mathematics does not license

Outliers can pull a centroid. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Repeat robust 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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4. Rank binarization

The central mathematical idea is concrete. A percentile threshold marks the upper tail high.

A concrete situation

Top thirty percent become one.

What the mathematics does not license

Ties and detection limits matter. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

State tie handling. 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 nodes

Here the geometry becomes operational. Cells or spots become graph vertices.

A concrete situation

Coordinates define seven vertices.

What the mathematics does not license

Filtering changes topology. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Version node inclusion. 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. Delaunay network

This step deserves a careful pause. Triangulation connects geometric neighbours.

A concrete situation

A planar mesh covers the sample.

What the mathematics does not license

Long edges cross holes. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Trim and inspect. 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. KNN or distance network

The useful question is not only how to calculate, but what the calculation assumes. Alternative rules control degree and scale.

A concrete situation

Each node receives six neighbours.

What the mathematics does not license

Large k crosses compartments. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Sweep graph settings. 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. Undirected edges

A small example makes the mechanism visible. Each neighbour pair is counted once.

A concrete situation

Ten unique edges remain after deduplication.

What the mathematics does not license

Orientation conventions affect mixed cells. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Document counting. 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. Endpoint-state pairs

This is where modelling choices begin to affect evidence. Edges receive 00, 01, 10 or 11 labels.

A concrete situation

Five high-high edges support clustering.

What the mathematics does not license

Shared nodes create dependence. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Use graph-aware diagnostics. 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. Contingency table

The central mathematical idea is concrete. The four counts enter a two-by-two table.

A concrete situation

Rows describe one endpoint and columns the other.

What the mathematics does not license

Symmetry can double-count mixed states. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Check implementation 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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11. Odds ratio

Here the geometry becomes operational. Cross-products compare like-state and mixed-state odds.

A concrete situation

An odds ratio above one signals enrichment.

What the mathematics does not license

Zero cells create infinite estimates. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Report intervals or 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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12. Fisher exact probability

This step deserves a careful pause. The hypergeometric distribution conditions on margins.

A concrete situation

One table gets a small one-sided p-value.

What the mathematics does not license

Independent-edge assumptions are imperfect. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Compare 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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13. Multiple genes

The useful question is not only how to calculate, but what the calculation assumes. Thousands of tables create multiplicity.

A concrete situation

Adjusted ranks select candidates.

What the mathematics does not license

Top-k lists do not control FDR. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

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

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14. Threshold sensitivity

A small example makes the mechanism visible. Changing k-means seed or rank cutoff can reorder genes.

A concrete situation

A borderline gene flips labels.

What the mathematics does not license

Stable anatomy may appear fragile. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Publish overlap curves. 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. Pattern limitation

This is where modelling choices begin to affect evidence. A coherent low-expression domain may be informative too.

A concrete situation

Low-low enrichment can form a region.

What the mathematics does not license

Focusing only on high-high narrows the question. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Inspect all four cells. 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. Governance and learning

The central mathematical idea is concrete. Version bin rule, graph, table, tail and adjustment.

A concrete situation

A student counts edges by hand.

What the mathematics does not license

Fast screening still needs validation. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Write a BinSpect audit card. 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

BinSpect is a fast network-enrichment method inside the Giotto spatial-analysis framework. Each gene is reduced to two states. BinSpect-kmeans uses a two-group k-means split; BinSpect-rank uses a chosen expression percentile. This deliberate simplification asks whether high expression occurs next to high expression more often than expected.

Physical coordinates become a spatial network through Delaunay triangulation, k-nearest neighbours or a distance rule. Every undirected edge contributes an endpoint-state pair: 0-0, 0-1, 1-0 or 1-1. These counts form a two-by-two table, and an odds ratio summarizes enrichment of like-state neighbours.

Fisher’s exact test evaluates the contingency table without a large-sample approximation. Because one node participates in several edges, edge observations are not truly independent; the score is best treated as a practical spatial-enrichment screen and should be checked with coordinate-aware permutations or external replication when calibrated inference matters.

Students can draw a seven-node network, label four nodes high and three low, count the four edge types and compute the odds ratio. Moving a single edge across a boundary changes the table. Repeating with a rank threshold and a k-means threshold shows that binarization is part of the scientific question, not a neutral formatting step.

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Frequently asked questions

What is BinSpect?

BinSpect, or Binary Spatial Extraction, is a fast Giotto method that tests whether binary gene-expression states are enriched among spatial neighbours.

How are expression values binarized?

BinSpect-kmeans uses k-means with two clusters; BinSpect-rank marks observations above a chosen expression percentile as high.

How is the spatial network built?

Giotto supports Delaunay triangulation, k-nearest-neighbour connections and fixed-distance networks based on physical coordinates.

What goes into the contingency table?

Every edge contributes the binary states at its endpoints, producing counts for low-low, low-high, high-low and high-high neighbour combinations.

What does Fisher’s exact test answer?

Conditioning on table margins, it evaluates whether the observed association between endpoint states is more extreme than expected under independence.

What can BinSpect miss?

Binarization discards magnitude, thresholds can alter results, and edge dependence complicates textbook test calibration; sensitivity analysis and external validation remain essential.

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Primary source and fact-check note

Ruben Dries, Qian Zhu, Rui Dong et al., “Giotto: a toolbox for integrative analysis and visualization of spatial expression data,” Genome Biology 22, 78 (2021), DOI 10.1186/s13059-021-02286-2. 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

• scGCO graph-cut spatial states
• MERINGUE Moran’s I
• Trendsceek pair-distance statistics
• 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

Binarization audit

Compare k-means and rank thresholds.

Graph audit

Sweep Delaunay, kNN and distance rules.

Edge audit

Verify deduplication and mixed-state counting.

Table audit

Show all four contingency cells.

Exact-test audit

Check odds ratios, tails and zero cells.

Dependence audit

Compare coordinate-aware permutations.

Replication audit

Repeat candidate genes in independent tissue.

Final claim

Neighbour enrichment is a screen, not mechanism.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Gene expression vector

Begin with this mechanism: One gene provides a value at every tissue node. Work through the concrete case: Seven cells have counts from zero to twelve. Make the limitation visible: Depth and transformation change the vector. Complete the evidence step: Record preprocessing. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 2: Two-state reduction

Begin with this mechanism: Each value becomes low or high. Work through the concrete case: Four nodes are labelled one. Make the limitation visible: Magnitude within a state is lost. Complete the evidence step: Plot raw values too. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 3: K-means binarization

Begin with this mechanism: Two centroids minimize within-group squared distances. Work through the concrete case: Low and high modes separate. Make the limitation visible: Outliers can pull a centroid. Complete the evidence step: Repeat robust checks. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 4: Rank binarization

Begin with this mechanism: A percentile threshold marks the upper tail high. Work through the concrete case: Top thirty percent become one. Make the limitation visible: Ties and detection limits matter. Complete the evidence step: State tie handling. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 5: Spatial nodes

Begin with this mechanism: Cells or spots become graph vertices. Work through the concrete case: Coordinates define seven vertices. Make the limitation visible: Filtering changes topology. Complete the evidence step: Version node inclusion. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 6: Delaunay network

Begin with this mechanism: Triangulation connects geometric neighbours. Work through the concrete case: A planar mesh covers the sample. Make the limitation visible: Long edges cross holes. Complete the evidence step: Trim and inspect. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 7: KNN or distance network

Begin with this mechanism: Alternative rules control degree and scale. Work through the concrete case: Each node receives six neighbours. Make the limitation visible: Large k crosses compartments. Complete the evidence step: Sweep graph settings. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 8: Undirected edges

Begin with this mechanism: Each neighbour pair is counted once. Work through the concrete case: Ten unique edges remain after deduplication. Make the limitation visible: Orientation conventions affect mixed cells. Complete the evidence step: Document counting. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 9: Endpoint-state pairs

Begin with this mechanism: Edges receive 00, 01, 10 or 11 labels. Work through the concrete case: Five high-high edges support clustering. Make the limitation visible: Shared nodes create dependence. Complete the evidence step: Use graph-aware diagnostics. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 10: Contingency table

Begin with this mechanism: The four counts enter a two-by-two table. Work through the concrete case: Rows describe one endpoint and columns the other. Make the limitation visible: Symmetry can double-count mixed states. Complete the evidence step: Check implementation rules. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 11: Odds ratio

Begin with this mechanism: Cross-products compare like-state and mixed-state odds. Work through the concrete case: An odds ratio above one signals enrichment. Make the limitation visible: Zero cells create infinite estimates. Complete the evidence step: Report intervals or corrections. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 12: Fisher exact probability

Begin with this mechanism: The hypergeometric distribution conditions on margins. Work through the concrete case: One table gets a small one-sided p-value. Make the limitation visible: Independent-edge assumptions are imperfect. Complete the evidence step: Compare permutations. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 13: Multiple genes

Begin with this mechanism: Thousands of tables create multiplicity. Work through the concrete case: Adjusted ranks select candidates. Make the limitation visible: Top-k lists do not control FDR. Complete the evidence step: State selection logic. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 14: Threshold sensitivity

Begin with this mechanism: Changing k-means seed or rank cutoff can reorder genes. Work through the concrete case: A borderline gene flips labels. Make the limitation visible: Stable anatomy may appear fragile. Complete the evidence step: Publish overlap curves. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 15: Pattern limitation

Begin with this mechanism: A coherent low-expression domain may be informative too. Work through the concrete case: Low-low enrichment can form a region. Make the limitation visible: Focusing only on high-high narrows the question. Complete the evidence step: Inspect all four cells. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 16: Governance and learning

Begin with this mechanism: Version bin rule, graph, table, tail and adjustment. Work through the concrete case: A student counts edges by hand. Make the limitation visible: Fast screening still needs validation. Complete the evidence step: Write a BinSpect audit card. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.