WHY MATHEMATICS?

Why Mathematics? | CellTrek Spatial Charting: Random Forest Distances, Mutual Nearest Neighbours, Sparse Graphs and Coordinate Transfer

How CellTrek learns a spatially supervised distance between single cells and tissue spots, builds a sparse matching graph and transfers coordinates with clear limits.

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

Create a shared feature space for cells and spots
Learn two spatial coordinates with a random forest
Keep only mutual nearest-neighbour graph edges
Distinguish transferred coordinates from directly observed locations

Chapter map

1. Single-cell expression
2. Spatial transcriptomics
3. Reference-based co-embedding
4. Principal-component features
5. Multivariate coordinate response
6. Random-forest ensemble
7. Tree separation depth
8. Forest distance matrix
9. Spatial interpolation
10. Distance threshold
11. Mutual nearest neighbours
12. Sparse bipartite graph
13. Coordinate transfer
14. Point repulsion
15. SColoc abstraction
16. External validation

Visit the How Mathematics Works hub for the wider series and its connections to education, reasoning, science, computing, design and careers.

1. Single-cell expression

The useful question is not only how to calculate, but what the calculation assumes. Dissociated cells provide high-dimensional profiles.

A concrete situation

Cell i has 15,000 genes.

What the mathematics does not license

Dissociation loses location. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Keep donor and batch labels. 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. Spatial transcriptomics

A small example makes the mechanism visible. Spots provide expression with X and Y coordinates.

A concrete situation

Spot j is at (120,80).

What the mathematics does not license

Spots may contain mixtures. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Record platform 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.

3. Reference-based co-embedding

This is where modelling choices begin to affect evidence. Cells and spots share a corrected latent space.

A concrete situation

Both map into 30 PCs.

What the mathematics does not license

Integration can erase real differences. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Inspect anchors and batches. 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. Principal-component features

The central mathematical idea is concrete. A reduced vector becomes the predictor input.

A concrete situation

PC1 through PC30 describe spot j.

What the mathematics does not license

Variance is not spatial relevance. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Tune dimension on held-out spots. 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. Multivariate coordinate response

Here the geometry becomes operational. The forest predicts X and Y together.

A concrete situation

Outcome is [120,80].

What the mathematics does not license

Axes can reflect tissue orientation. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

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

6. Random-forest ensemble

This step deserves a careful pause. Many trees partition feature space.

A concrete situation

One thousand trees are trained.

What the mathematics does not license

Trees can learn batch artefacts. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Use donor-held-out forests. 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. Tree separation depth

The useful question is not only how to calculate, but what the calculation assumes. Shared ancestry within a tree defines proximity.

A concrete situation

Two observations split three edges below root.

What the mathematics does not license

Tree depth varies. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Normalize the distance. 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. Forest distance matrix

A small example makes the mechanism visible. Distances are averaged across trees for cell-spot pairs.

A concrete situation

Dij equals 0.18.

What the mathematics does not license

A learned metric inherits training bias. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Compare alternative metrics. 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. Spatial interpolation

This is where modelling choices begin to affect evidence. Pseudo-spots fill gaps between measured locations.

A concrete situation

A grid point is interpolated between four spots.

What the mathematics does not license

Interpolation is not measurement. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Mark synthetic nodes. 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. Distance threshold

The central mathematical idea is concrete. Weak cell-spot matches are removed.

A concrete situation

Edges above 0.5 are discarded.

What the mathematics does not license

Thresholds alter graph coverage. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Plot coverage versus precision. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

11. Mutual nearest neighbours

Here the geometry becomes operational. A pair must be close from both cell and spot views.

A concrete situation

Cell i and spot j choose each other.

What the mathematics does not license

Rare cells may lack mutual matches. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Audit unmatched groups. 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. Sparse bipartite graph

This step deserves a careful pause. Cells connect only to retained spot nodes.

A concrete situation

Ten thousand possible pairs become 600 edges.

What the mathematics does not license

Sparsity may fragment the graph. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Report component sizes. 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. Coordinate transfer

The useful question is not only how to calculate, but what the calculation assumes. A cell inherits or averages neighbour spot coordinates.

A concrete situation

Cell i maps near (118,82).

What the mathematics does not license

Multiple neighbours create ambiguity. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Keep edge weights and spread. 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. Point repulsion

A small example makes the mechanism visible. A layout separates overlapping plotted cells.

A concrete situation

Two markers move a few pixels apart.

What the mathematics does not license

Display movement is not biological. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Retain original inferred coordinates. 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. SColoc abstraction

This is where modelling choices begin to affect evidence. Mapped cells form type-level colocalization graphs.

A concrete situation

Types A and B share many neighbourhoods.

What the mathematics does not license

Abundance can drive co-location. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Use permutation or density controls. 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. External validation

The central mathematical idea is concrete. Known anatomy and independent assays test the chart.

A concrete situation

Mapped B cells overlap a measured marker region.

What the mathematics does not license

Marker agreement may be circular. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Use held-out genes and tissue. 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

CellTrek begins by co-embedding single-cell and spatial-transcriptomics observations in a shared feature space. Principal components or another integrated representation place cells and spatial spots on comparable axes. Using the spatial spots, a multivariate random forest learns to predict the two spatial coordinates from those features. The forest is supervised by location, so its notion of similarity is oriented toward spatially useful expression patterns.

A random-forest distance is calculated from tree structure: observations that remain together deeper in many trees are treated as more similar than observations separated near the root. CellTrek uses this distance between reference cells and spatial spots, including interpolated pseudo-spots that increase the map’s apparent resolution. Interpolation supplies candidate anchors; it does not create new measured molecules or independently observed cells.

The distance matrix is thresholded and filtered with mutual nearest neighbours. A cell and a spot are connected only when each considers the other among its closest partners under the declared rules. This produces a sparse bipartite graph rather than a forced dense map. Coordinates are transferred from connected spots to cells, and a point-repulsion layout reduces visual clumping without adding spatial evidence.

Downstream SColoc summaries can turn the mapped cells into cell-type neighbourhood and colocalization graphs. Every conclusion depends on the co-embedding, forest, interpolation, distance threshold, neighbour number and reference coverage. CellTrek therefore offers a valuable lesson in metric learning: a learned distance can organize evidence for mapping, but the displayed coordinate remains an inference that needs held-out tissue, known anatomy and orthogonal spatial measurements.

Frequently asked questions

What is CellTrek?

CellTrek maps single-cell transcriptomes to tissue coordinates through co-embedding, random-forest metric learning and a sparse cell-spot graph.

Why train a random forest on coordinates?

The forest learns expression features and interactions that distinguish spatial X and Y locations among measured spots.

What is random-forest distance?

It measures how early observations separate inside trained trees, aggregated across the forest.

Why use mutual nearest neighbours?

They keep cell-spot pairs that are locally preferred in both directions, reducing one-sided matches.

What does point repulsion change?

It spreads plotted cell symbols to reduce overlap; it does not add experimental location evidence.

Does a transferred coordinate equal a measured coordinate?

No. It is an inferred location conditional on integration, the learned metric, graph thresholds and reference coverage.

Primary source and fact-check note

Runmin Wei, Siyuan He, Shanshan Bai, Emi Sei, Min Hu, Alastair Thompson, Ken Chen, Savitri Krishnamurthy and Nicholas E. Navin, “Spatial charting of single-cell transcriptomes in tissues,” Nature Biotechnology 40, 1190–1199 (2022), DOI 10.1038/s41587-022-01233-1. 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

Integration audit

Inspect anchors, batches and lost biological variation.

Feature audit

Publish PCs, scaling and held-out selection.

Forest audit

Record trees, depth, seeds and coordinate error.

Distance audit

State the tree-distance formula and aggregation.

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.

Graph audit

Publish thresholds, k and unmatched nodes.

Transfer audit

Keep every cell-spot edge and coordinate spread.

Display audit

Separate point repulsion from inferred coordinates.

Final claim

A mapped coordinate is inferred, not directly observed.

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: Single-cell expression

Begin with this mechanism: Dissociated cells provide high-dimensional profiles. Recreate the concrete case: Cell i has 15,000 genes. Write integrated features, coordinates, tree depths, forest distances, interpolated nodes, neighbour ranks, graph edges, transferred coordinates and colocation counts with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Dissociation loses location. Complete the transfer step: Keep donor and batch labels. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 2: Spatial transcriptomics

Begin with this mechanism: Spots provide expression with X and Y coordinates. Recreate the concrete case: Spot j is at (120,80). Write integrated features, coordinates, tree depths, forest distances, interpolated nodes, neighbour ranks, graph edges, transferred coordinates and colocation counts with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Spots may contain mixtures. Complete the transfer step: Record platform resolution. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 3: Reference-based co-embedding

Begin with this mechanism: Cells and spots share a corrected latent space. Recreate the concrete case: Both map into 30 PCs. Write integrated features, coordinates, tree depths, forest distances, interpolated nodes, neighbour ranks, graph edges, transferred coordinates and colocation counts with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Integration can erase real differences. Complete the transfer step: Inspect anchors and batches. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 4: Principal-component features

Begin with this mechanism: A reduced vector becomes the predictor input. Recreate the concrete case: PC1 through PC30 describe spot j. Write integrated features, coordinates, tree depths, forest distances, interpolated nodes, neighbour ranks, graph edges, transferred coordinates and colocation counts with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Variance is not spatial relevance. Complete the transfer step: Tune dimension on held-out spots. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

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: Multivariate coordinate response

Begin with this mechanism: The forest predicts X and Y together. Recreate the concrete case: Outcome is [120,80]. Write integrated features, coordinates, tree depths, forest distances, interpolated nodes, neighbour ranks, graph edges, transferred coordinates and colocation counts with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Axes can reflect tissue orientation. Complete the transfer step: Preserve coordinate units. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 6: Random-forest ensemble

Begin with this mechanism: Many trees partition feature space. Recreate the concrete case: One thousand trees are trained. Write integrated features, coordinates, tree depths, forest distances, interpolated nodes, neighbour ranks, graph edges, transferred coordinates and colocation counts with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Trees can learn batch artefacts. Complete the transfer step: Use donor-held-out forests. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 7: Tree separation depth

Begin with this mechanism: Shared ancestry within a tree defines proximity. Recreate the concrete case: Two observations split three edges below root. Write integrated features, coordinates, tree depths, forest distances, interpolated nodes, neighbour ranks, graph edges, transferred coordinates and colocation counts with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Tree depth varies. Complete the transfer step: Normalize the distance. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 8: Forest distance matrix

Begin with this mechanism: Distances are averaged across trees for cell-spot pairs. Recreate the concrete case: Dij equals 0.18. Write integrated features, coordinates, tree depths, forest distances, interpolated nodes, neighbour ranks, graph edges, transferred coordinates and colocation counts with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: A learned metric inherits training bias. Complete the transfer step: Compare alternative metrics. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

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: Spatial interpolation

Begin with this mechanism: Pseudo-spots fill gaps between measured locations. Recreate the concrete case: A grid point is interpolated between four spots. Write integrated features, coordinates, tree depths, forest distances, interpolated nodes, neighbour ranks, graph edges, transferred coordinates and colocation counts with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Interpolation is not measurement. Complete the transfer step: Mark synthetic nodes. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 10: Distance threshold

Begin with this mechanism: Weak cell-spot matches are removed. Recreate the concrete case: Edges above 0.5 are discarded. Write integrated features, coordinates, tree depths, forest distances, interpolated nodes, neighbour ranks, graph edges, transferred coordinates and colocation counts with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Thresholds alter graph coverage. Complete the transfer step: Plot coverage versus precision. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 11: Mutual nearest neighbours

Begin with this mechanism: A pair must be close from both cell and spot views. Recreate the concrete case: Cell i and spot j choose each other. Write integrated features, coordinates, tree depths, forest distances, interpolated nodes, neighbour ranks, graph edges, transferred coordinates and colocation counts with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Rare cells may lack mutual matches. Complete the transfer step: Audit unmatched groups. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 12: Sparse bipartite graph

Begin with this mechanism: Cells connect only to retained spot nodes. Recreate the concrete case: Ten thousand possible pairs become 600 edges. Write integrated features, coordinates, tree depths, forest distances, interpolated nodes, neighbour ranks, graph edges, transferred coordinates and colocation counts with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Sparsity may fragment the graph. Complete the transfer step: Report component sizes. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

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: Coordinate transfer

Begin with this mechanism: A cell inherits or averages neighbour spot coordinates. Recreate the concrete case: Cell i maps near (118,82). Write integrated features, coordinates, tree depths, forest distances, interpolated nodes, neighbour ranks, graph edges, transferred coordinates and colocation counts with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Multiple neighbours create ambiguity. Complete the transfer step: Keep edge weights and spread. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 14: Point repulsion

Begin with this mechanism: A layout separates overlapping plotted cells. Recreate the concrete case: Two markers move a few pixels apart. Write integrated features, coordinates, tree depths, forest distances, interpolated nodes, neighbour ranks, graph edges, transferred coordinates and colocation counts with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Display movement is not biological. Complete the transfer step: Retain original inferred coordinates. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 15: SColoc abstraction

Begin with this mechanism: Mapped cells form type-level colocalization graphs. Recreate the concrete case: Types A and B share many neighbourhoods. Write integrated features, coordinates, tree depths, forest distances, interpolated nodes, neighbour ranks, graph edges, transferred coordinates and colocation counts with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Abundance can drive co-location. Complete the transfer step: Use permutation or density controls. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 16: External validation

Begin with this mechanism: Known anatomy and independent assays test the chart. Recreate the concrete case: Mapped B cells overlap a measured marker region. Write integrated features, coordinates, tree depths, forest distances, interpolated nodes, neighbour ranks, graph edges, transferred coordinates and colocation counts with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Marker agreement may be circular. Complete the transfer step: Use held-out genes and tissue. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.