WHY MATHEMATICS?

Why Mathematics? | SpatialDM for Cell Communication: Bivariate Moran Statistics, Analytical Nulls, Local Interaction Maps and Ligand-Receptor Patterns

How SpatialDM prioritizes ligand-receptor pairs whose expression aligns across neighbouring locations, maps local contributions and keeps spatial association separate from proof of signalling.

Why mathematics is important becomes easier to see when a real decision depends on it. This guide follows the mechanism from definitions and dimensions to a worked calculation, validation, practical learning and responsible use. It keeps factual evidence separate from interpretation and treats every illustrative scenario as an example rather than a promise of results.

Choose a reading route

Represent physical proximity with a documented weight matrix
Measure global ligand-receptor spatial co-expression
Use analytical or permutation nulls for scalable evidence
Decompose global scores into local maps and interaction patterns

Chapter map

1. Ligand vector
2. Receptor vector
3. Candidate database
4. Spatial coordinates
5. Weight matrix
6. Row normalization
7. Standardization
8. Bivariate Moran numerator
9. Global scaling
10. Analytical null mean
11. Analytical null variance
12. Permutation mode
13. Multiple pair testing
14. Local interaction map
15. Pattern and pathway grouping
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.

1. Ligand vector

A small example makes the mechanism visible. One expression value is recorded per location for a candidate ligand.

A concrete situation

Six spots contain a ligand gradient.

What the mathematics does not license

mRNA may not equal secreted protein. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

State measurement type. 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.

2. Receptor vector

This is where modelling choices begin to affect evidence. A matched expression vector records the receptor.

A concrete situation

Receptor peaks one spot away.

What the mathematics does not license

Subunits and complexes can matter. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Use curated pair definitions. 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.

3. Candidate database

The central mathematical idea is concrete. Prior ligand-receptor knowledge limits tested pairs.

A concrete situation

One pathway provides twenty pairs.

What the mathematics does not license

Databases can be incomplete or context-free. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Version the resource. 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.

4. Spatial coordinates

Here the geometry becomes operational. Every observation has physical position.

A concrete situation

A Visium grid defines spot centres.

What the mathematics does not license

Spot mixtures blur cell contact. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Match claim to resolution. 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.

Did You Know? In this chapter, the most important number is useful only together with its construction rule. Changing scaling, sample membership or validation design can change the numerical answer without changing the arithmetic.

5. Weight matrix

This step deserves a careful pause. W encodes neighbour strength across locations.

A concrete situation

Adjacent spots receive positive weights.

What the mathematics does not license

Graph choices set communication range. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Plot every edge. 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.

6. Row normalization

The useful question is not only how to calculate, but what the calculation assumes. Weights can sum to one per source spot.

A concrete situation

Three neighbours share influence.

What the mathematics does not license

Normalization changes dense-versus-sparse regions. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

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

7. Standardization

A small example makes the mechanism visible. Ligand and receptor values are centred and scaled.

A concrete situation

Both vectors have mean zero.

What the mathematics does not license

Outliers dominate standard scores. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use influence 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.

8. Bivariate Moran numerator

This is where modelling choices begin to affect evidence. Weighted ligand deviations multiply neighbouring receptor deviations.

A concrete situation

High ligand near high receptor adds positive evidence.

What the mathematics does not license

High-high can reflect one tissue domain. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Adjust or stratify types. 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.

9. Global scaling

The central mathematical idea is concrete. Norms and weight totals make pairs more comparable.

A concrete situation

One pair obtains a positive R.

What the mathematics does not license

Comparability depends on preprocessing. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Preserve raw components. 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.

10. Analytical null mean

Here the geometry becomes operational. A random-association model gives the expected statistic.

A concrete situation

The null mean is near its derived value.

What the mathematics does not license

Exchangeability can fail across sections. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Check design blocks. 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.

11. Analytical null variance

This step deserves a careful pause. Weight-matrix moments determine the z-score denominator.

A concrete situation

Irregular degrees change variance.

What the mathematics does not license

Numerical cancellation can occur. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Validate with 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.

12. Permutation mode

The useful question is not only how to calculate, but what the calculation assumes. Expression or assignments are shuffled under a valid scheme.

A concrete situation

One thousand shuffles form an empirical tail.

What the mathematics does not license

Global shuffling breaks batch structure. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Permute within blocks. 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.

13. Multiple pair testing

A small example makes the mechanism visible. Many ligand-receptor candidates require correction.

A concrete situation

FDR selects a global set.

What the mathematics does not license

Database size changes the family. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

State all tested 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.

14. Local interaction map

This is where modelling choices begin to affect evidence. Each spot receives a contribution from its ligand value and neighbouring receptor values.

A concrete situation

A compact interface lights up.

What the mathematics does not license

Local hits inherit global selection. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Avoid double-dipping claims. 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.

15. Pattern and pathway grouping

The central mathematical idea is concrete. Interactions with similar local maps can be clustered and summarized.

A concrete situation

Several pairs share an interface pattern.

What the mathematics does not license

Co-patterning does not prove one pathway. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use independent validation. 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.

16. Governance and learning

Here the geometry becomes operational. Version database, graph, scaling, null, correction and contrasts.

A concrete situation

A student computes a six-spot cross-product.

What the mathematics does not license

Spatial co-expression is evidence, not messaging proof. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Write a SpatialDM 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.

A seven-day practice lab

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

SpatialDM evaluates a ligand in one location together with its receptor in neighbouring locations. A spatial weight matrix turns physical proximity into the meaning of “near.” After standardization, a bivariate extension of Moran’s statistic measures whether high ligand deviations align with high receptor deviations across weighted edges.

A global statistic ranks ligand-receptor pairs whose spatial co-expression is stronger than expected. SpatialDM provides a permutation mode and a fast analytical z-score mode based on moments of the null distribution. The latter avoids millions of shuffles, but its calibration still depends on the graph, normalization and validity of the null assumptions.

Local interaction statistics decompose the global association into spot-level contributions. Groups of ligand-receptor interactions can then be organized into spatial patterns and pathways, and conditions can be compared. A bright local hit is a weighted expression association, not direct observation of a molecular message travelling between cells.

A six-spot example can place ligand values on one vector and receptor values on another, centre and scale them, row-normalize a neighbour matrix and calculate the weighted cross-product. Mapping each spot contribution reveals why the same global score can arise from one compact hotspot or many moderate interactions.

Frequently asked questions

What is SpatialDM?

SpatialDM is a statistical toolbox that uses a bivariate Moran extension to identify spatially co-expressed ligand-receptor pairs and the locations contributing to them.

What does bivariate Moran association compare?

It compares standardized ligand expression at one location with standardized receptor expression at spatially weighted neighbouring locations.

Why is the weight matrix important?

It defines which locations can contribute to an interaction and how strongly, so radius, k, distance decay and normalization determine the biological scale of the test.

How is significance assessed?

SpatialDM offers a permutation mode and an analytical z-score mode derived from null moments, enabling large analyses when assumptions are suitable.

What is a local interaction score?

It is a spot-level contribution to a ligand-receptor spatial association, used to map where a globally selected pair is active under the model.

Does a significant pair prove communication?

No. It supports spatial co-expression of a known ligand-receptor pair. Protein presence, direction, contact range, receptor activity, perturbation and replication remain separate evidence.

Primary source and fact-check note

Zhuoxuan Li, Tianjie Wang, Pentao Liu and Yuanhua Huang, “SpatialDM for rapid identification of spatially co-expressed ligand–receptor and revealing cell–cell communication patterns,” Nature Communications 14, 3995 (2023), DOI 10.1038/s41467-023-39608-w. 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

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

Pair audit

Version ligand-receptor knowledge.

Graph audit

Match neighbour range to biology and resolution.

Scaling audit

Inspect standardization and outliers.

Null audit

Compare analytical and blocked permutation results.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Global audit

Correct across every tested pair.

Local audit

Map contributions without re-testing selectively.

Validation audit

Use proteins, perturbation or replicate tissue.

Final claim

Spatial co-expression does not prove signalling.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Ligand vector

Begin with this mechanism: One expression value is recorded per location for a candidate ligand. Work through the concrete case: Six spots contain a ligand gradient. Make the limitation visible: mRNA may not equal secreted protein. Complete the evidence step: State measurement type. 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: Receptor vector

Begin with this mechanism: A matched expression vector records the receptor. Work through the concrete case: Receptor peaks one spot away. Make the limitation visible: Subunits and complexes can matter. Complete the evidence step: Use curated pair definitions. 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: Candidate database

Begin with this mechanism: Prior ligand-receptor knowledge limits tested pairs. Work through the concrete case: One pathway provides twenty pairs. Make the limitation visible: Databases can be incomplete or context-free. Complete the evidence step: Version the resource. 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: Spatial coordinates

Begin with this mechanism: Every observation has physical position. Work through the concrete case: A Visium grid defines spot centres. Make the limitation visible: Spot mixtures blur cell contact. Complete the evidence step: Match claim to resolution. 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.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Workshop 5: Weight matrix

Begin with this mechanism: W encodes neighbour strength across locations. Work through the concrete case: Adjacent spots receive positive weights. Make the limitation visible: Graph choices set communication range. Complete the evidence step: Plot every edge. 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: Row normalization

Begin with this mechanism: Weights can sum to one per source spot. Work through the concrete case: Three neighbours share influence. Make the limitation visible: Normalization changes dense-versus-sparse regions. Complete the evidence step: State the convention. 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: Standardization

Begin with this mechanism: Ligand and receptor values are centred and scaled. Work through the concrete case: Both vectors have mean zero. Make the limitation visible: Outliers dominate standard scores. Complete the evidence step: Use influence 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 8: Bivariate Moran numerator

Begin with this mechanism: Weighted ligand deviations multiply neighbouring receptor deviations. Work through the concrete case: High ligand near high receptor adds positive evidence. Make the limitation visible: High-high can reflect one tissue domain. Complete the evidence step: Adjust or stratify types. 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.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Workshop 9: Global scaling

Begin with this mechanism: Norms and weight totals make pairs more comparable. Work through the concrete case: One pair obtains a positive R. Make the limitation visible: Comparability depends on preprocessing. Complete the evidence step: Preserve raw components. 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: Analytical null mean

Begin with this mechanism: A random-association model gives the expected statistic. Work through the concrete case: The null mean is near its derived value. Make the limitation visible: Exchangeability can fail across sections. Complete the evidence step: Check design blocks. 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: Analytical null variance

Begin with this mechanism: Weight-matrix moments determine the z-score denominator. Work through the concrete case: Irregular degrees change variance. Make the limitation visible: Numerical cancellation can occur. Complete the evidence step: Validate with 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 12: Permutation mode

Begin with this mechanism: Expression or assignments are shuffled under a valid scheme. Work through the concrete case: One thousand shuffles form an empirical tail. Make the limitation visible: Global shuffling breaks batch structure. Complete the evidence step: Permute within blocks. 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.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Workshop 13: Multiple pair testing

Begin with this mechanism: Many ligand-receptor candidates require correction. Work through the concrete case: FDR selects a global set. Make the limitation visible: Database size changes the family. Complete the evidence step: State all tested pairs. 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: Local interaction map

Begin with this mechanism: Each spot receives a contribution from its ligand value and neighbouring receptor values. Work through the concrete case: A compact interface lights up. Make the limitation visible: Local hits inherit global selection. Complete the evidence step: Avoid double-dipping claims. 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 and pathway grouping

Begin with this mechanism: Interactions with similar local maps can be clustered and summarized. Work through the concrete case: Several pairs share an interface pattern. Make the limitation visible: Co-patterning does not prove one pathway. Complete the evidence step: Use independent validation. 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 database, graph, scaling, null, correction and contrasts. Work through the concrete case: A student computes a six-spot cross-product. Make the limitation visible: Spatial co-expression is evidence, not messaging proof. Complete the evidence step: Write a SpatialDM 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.