Chapter map

1. 1. Count-aware embedding
2. 2. Delaunay triangulation
3. 3. Graph components
4. 4. Layer zero features
5. 5. One-hop mean
6. 6. Multi-hop expansion
7. 7. Standard deviation summary
8. 8. Concatenated niche vector
9. 9. Gaussian component mean
10. 10. Full covariance
11. 11. Mixing proportion
12. 12. Posterior responsibility
13. 13. Mixture likelihood
14. 14. Fowlkes–Mallows stability
15. 15. Niche enrichment
16. 16. Shape and outcome analysis

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1. Count-aware embedding

The central mathematical idea is concrete. A latent model compresses expression or other omics features.

A concrete situation

scVI supplies a low-dimensional vector.

What the mathematics does not license

The embedding may retain batch. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Validate batch and biology separately. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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2. Delaunay triangulation

Here the geometry becomes operational. Planar triangles connect geometrically nearby observations.

A concrete situation

Three cells form one local triangle.

What the mathematics does not license

Sparse boundaries create long edges. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Remove and report long edges. 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. Graph components

This step deserves a careful pause. Connected components reveal isolated tissue pieces.

A concrete situation

A torn section produces two components.

What the mathematics does not license

Aggregation cannot cross a missing bridge. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Map every component. 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. Layer zero features

The useful question is not only how to calculate, but what the calculation assumes. The focal observation keeps its own latent vector.

A concrete situation

z0 contains twenty values.

What the mathematics does not license

Latent rotation changes coordinates. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Compare stable subspaces. 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. One-hop mean

A small example makes the mechanism visible. Immediate-neighbour vectors are averaged.

A concrete situation

Three neighbours average to a local centre.

What the mathematics does not license

A mean hides bimodality. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Add dispersion summaries. 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. Multi-hop expansion

This is where modelling choices begin to affect evidence. Repeated graph layers enlarge the local field.

A concrete situation

Depth three sees neighbours of neighbours.

What the mathematics does not license

Deep aggregation blends adjacent niches. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Sweep depth. 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. Standard deviation summary

The central mathematical idea is concrete. Local spread describes neighbourhood heterogeneity.

A concrete situation

One feature has local SD 0.8.

What the mathematics does not license

Small degree yields noisy spread. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Condition on degree. 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. Concatenated niche vector

Here the geometry becomes operational. Focal and layer summaries form one fixed-length input.

A concrete situation

Three layers multiply feature blocks.

What the mathematics does not license

Scale differs across blocks. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Standardize before clustering. 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. Gaussian component mean

This step deserves a careful pause. Each niche has a centre in aggregated-feature space.

A concrete situation

Component four centres on immune-rich context.

What the mathematics does not license

A mean may not describe a curved niche. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

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

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10. Full covariance

The useful question is not only how to calculate, but what the calculation assumes. Covariance describes correlated variation within a component.

A concrete situation

Two markers rise together.

What the mathematics does not license

High dimension risks singular estimates. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Regularize and check conditioning. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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11. Mixing proportion

A small example makes the mechanism visible. The model estimates each niche population share.

A concrete situation

A rare niche has weight 0.03.

What the mathematics does not license

Small weights can be unstable. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Bootstrap complete samples. 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. Posterior responsibility

This is where modelling choices begin to affect evidence. Every cell receives probabilities across niches.

A concrete situation

Probabilities 0.6, 0.3 and 0.1 show ambiguity.

What the mathematics does not license

Hard labels hide uncertainty. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Map entropy. 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. Mixture likelihood

The central mathematical idea is concrete. Expectation-maximization improves fit iteratively.

A concrete situation

One update revises means and covariances.

What the mathematics does not license

Local optima depend on starts. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Repeat initializations. 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. Fowlkes–Mallows stability

Here the geometry becomes operational. Pairs of runs compare co-clustered assignments.

A concrete situation

A stable K scores near one.

What the mathematics does not license

Stable partitions can still be wrong. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use held-out anatomy. 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. Niche enrichment

This step deserves a careful pause. Observed cell-type counts are compared with a null or baseline.

A concrete situation

A niche has twofold neutrophil enrichment.

What the mathematics does not license

Composition and abundance can confound. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use patient-stratified tests. 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. Shape and outcome analysis

The useful question is not only how to calculate, but what the calculation assumes. Geometry and patient-level niche proportions support interpretation.

A concrete situation

Elongation describes one connected component.

What the mathematics does not license

Cells within a patient are dependent. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Keep patients as the unit. 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

CellCharter asks a neighbourhood question: which cells or spots occupy comparable local tissue environments? It first learns or receives a low-dimensional feature representation, commonly from a variational model such as scVI when count distributions and batch effects matter. It then constructs a spatial graph, with Delaunay triangulation used in the study, and removes implausibly long edges when appropriate. The graph states which local contexts the algorithm is allowed to see.

For every observation, CellCharter aggregates continuous features over successive graph layers. A one-layer summary describes immediate neighbours; deeper layers enlarge the receptive field. Means describe the local centre, while other summaries such as standard deviations, minima or maxima can describe variation and extremes. Concatenating the focal and neighbourhood summaries turns tissue context into a fixed-length vector.

A Gaussian mixture model clusters the aggregated vectors into cellular niches. Each component has a mean, covariance and mixing proportion, and posterior probabilities preserve uncertainty before a hard label is chosen. Candidate cluster counts can be compared through repeated fits and the Fowlkes–Mallows stability of assignments. Information criteria, negative log-likelihood and biological interpretability provide complementary checks rather than automatic truth.

The resulting niches can be compared through cell-type enrichment, proportions, asymmetric neighbourhood enrichment and shape summaries such as purity, elongation, linearity and curl. These are descriptive quantities. A niche associated with an outcome is not automatically causal, and patient-level analysis must keep donors independent. Robust studies vary graph layers and covariance assumptions, reserve patients for validation and keep rare niches visible rather than tuning them away.

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Frequently asked questions

What is CellCharter?

CellCharter is a framework for identifying, characterizing and comparing spatial cellular niches from continuous cell or spot representations and neighbourhood graphs.

Why use a learned embedding first?

A count-aware latent model can compress high-dimensional measurements and account for nuisance effects before spatial context is aggregated.

What does graph depth mean?

Depth states how many successive spatial-neighbour layers contribute to each observation’s context vector.

Why use a Gaussian mixture?

It models niche vectors as a weighted collection of multivariate Gaussian components and provides posterior membership probabilities.

How is the number of niches chosen?

Repeated fits across candidate counts can be compared with assignment stability, together with information criteria and biological validation.

What is the correct unit for outcome tests?

When the outcome belongs to a patient or sample, inference should use independent patients or samples rather than treating millions of cells as independent replicates.

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Primary source and fact-check note

Marco Varrone, Daniele Tavernari, Albert Santamaria-Martínez et al., “CellCharter reveals spatial cell niches associated with tissue remodeling and cell plasticity,” Nature Genetics 56, 74–84 (2024), DOI 10.1038/s41588-023-01588-4. 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

• BANKSY neighbourhood feature augmentation
• MENDER multiscale tissue context
• NicheCompass interpretable niches
• 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

Embedding audit

Separate biological signal from batch removal.

Graph audit

Plot triangulation, long-edge pruning and components.

Depth audit

Compare local contexts across graph layers.

Summary audit

Test mean and dispersion aggregates.

Mixture audit

Repeat starts, covariance models and candidate K.

Uncertainty audit

Map posterior entropy and rare components.

Patient audit

Use samples, not cells, as independent outcome units.

Final claim

A niche is a probabilistic description of local context.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Count-aware embedding

Begin with this mechanism: A latent model compresses expression or other omics features. Recreate the concrete case: scVI supplies a low-dimensional vector. Write latent features, Delaunay edges, graph degrees, hop layers, local means, standard deviations, concatenated vectors, component means, covariance entries, mixing weights, responsibilities, stability pairs, enrichment counts and patient proportions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: The embedding may retain batch. Complete the transfer step: Validate batch and biology separately. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 2: Delaunay triangulation

Begin with this mechanism: Planar triangles connect geometrically nearby observations. Recreate the concrete case: Three cells form one local triangle. Write latent features, Delaunay edges, graph degrees, hop layers, local means, standard deviations, concatenated vectors, component means, covariance entries, mixing weights, responsibilities, stability pairs, enrichment counts and patient proportions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Sparse boundaries create long edges. Complete the transfer step: Remove and report long edges. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 3: Graph components

Begin with this mechanism: Connected components reveal isolated tissue pieces. Recreate the concrete case: A torn section produces two components. Write latent features, Delaunay edges, graph degrees, hop layers, local means, standard deviations, concatenated vectors, component means, covariance entries, mixing weights, responsibilities, stability pairs, enrichment counts and patient proportions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Aggregation cannot cross a missing bridge. Complete the transfer step: Map every component. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 4: Layer zero features

Begin with this mechanism: The focal observation keeps its own latent vector. Recreate the concrete case: z0 contains twenty values. Write latent features, Delaunay edges, graph degrees, hop layers, local means, standard deviations, concatenated vectors, component means, covariance entries, mixing weights, responsibilities, stability pairs, enrichment counts and patient proportions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Latent rotation changes coordinates. Complete the transfer step: Compare stable subspaces. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 5: One-hop mean

Begin with this mechanism: Immediate-neighbour vectors are averaged. Recreate the concrete case: Three neighbours average to a local centre. Write latent features, Delaunay edges, graph degrees, hop layers, local means, standard deviations, concatenated vectors, component means, covariance entries, mixing weights, responsibilities, stability pairs, enrichment counts and patient proportions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: A mean hides bimodality. Complete the transfer step: Add dispersion summaries. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 6: Multi-hop expansion

Begin with this mechanism: Repeated graph layers enlarge the local field. Recreate the concrete case: Depth three sees neighbours of neighbours. Write latent features, Delaunay edges, graph degrees, hop layers, local means, standard deviations, concatenated vectors, component means, covariance entries, mixing weights, responsibilities, stability pairs, enrichment counts and patient proportions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Deep aggregation blends adjacent niches. Complete the transfer step: Sweep depth. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 7: Standard deviation summary

Begin with this mechanism: Local spread describes neighbourhood heterogeneity. Recreate the concrete case: One feature has local SD 0.8. Write latent features, Delaunay edges, graph degrees, hop layers, local means, standard deviations, concatenated vectors, component means, covariance entries, mixing weights, responsibilities, stability pairs, enrichment counts and patient proportions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Small degree yields noisy spread. Complete the transfer step: Condition on degree. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 8: Concatenated niche vector

Begin with this mechanism: Focal and layer summaries form one fixed-length input. Recreate the concrete case: Three layers multiply feature blocks. Write latent features, Delaunay edges, graph degrees, hop layers, local means, standard deviations, concatenated vectors, component means, covariance entries, mixing weights, responsibilities, stability pairs, enrichment counts and patient proportions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Scale differs across blocks. Complete the transfer step: Standardize before clustering. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 9: Gaussian component mean

Begin with this mechanism: Each niche has a centre in aggregated-feature space. Recreate the concrete case: Component four centres on immune-rich context. Write latent features, Delaunay edges, graph degrees, hop layers, local means, standard deviations, concatenated vectors, component means, covariance entries, mixing weights, responsibilities, stability pairs, enrichment counts and patient proportions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: A mean may not describe a curved niche. Complete the transfer step: Inspect residual geometry. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 10: Full covariance

Begin with this mechanism: Covariance describes correlated variation within a component. Recreate the concrete case: Two markers rise together. Write latent features, Delaunay edges, graph degrees, hop layers, local means, standard deviations, concatenated vectors, component means, covariance entries, mixing weights, responsibilities, stability pairs, enrichment counts and patient proportions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: High dimension risks singular estimates. Complete the transfer step: Regularize and check conditioning. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 11: Mixing proportion

Begin with this mechanism: The model estimates each niche population share. Recreate the concrete case: A rare niche has weight 0.03. Write latent features, Delaunay edges, graph degrees, hop layers, local means, standard deviations, concatenated vectors, component means, covariance entries, mixing weights, responsibilities, stability pairs, enrichment counts and patient proportions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Small weights can be unstable. Complete the transfer step: Bootstrap complete samples. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 12: Posterior responsibility

Begin with this mechanism: Every cell receives probabilities across niches. Recreate the concrete case: Probabilities 0.6, 0.3 and 0.1 show ambiguity. Write latent features, Delaunay edges, graph degrees, hop layers, local means, standard deviations, concatenated vectors, component means, covariance entries, mixing weights, responsibilities, stability pairs, enrichment counts and patient proportions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Hard labels hide uncertainty. Complete the transfer step: Map entropy. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 13: Mixture likelihood

Begin with this mechanism: Expectation-maximization improves fit iteratively. Recreate the concrete case: One update revises means and covariances. Write latent features, Delaunay edges, graph degrees, hop layers, local means, standard deviations, concatenated vectors, component means, covariance entries, mixing weights, responsibilities, stability pairs, enrichment counts and patient proportions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Local optima depend on starts. Complete the transfer step: Repeat initializations. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 14: Fowlkes–Mallows stability

Begin with this mechanism: Pairs of runs compare co-clustered assignments. Recreate the concrete case: A stable K scores near one. Write latent features, Delaunay edges, graph degrees, hop layers, local means, standard deviations, concatenated vectors, component means, covariance entries, mixing weights, responsibilities, stability pairs, enrichment counts and patient proportions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Stable partitions can still be wrong. Complete the transfer step: Use held-out anatomy. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 15: Niche enrichment

Begin with this mechanism: Observed cell-type counts are compared with a null or baseline. Recreate the concrete case: A niche has twofold neutrophil enrichment. Write latent features, Delaunay edges, graph degrees, hop layers, local means, standard deviations, concatenated vectors, component means, covariance entries, mixing weights, responsibilities, stability pairs, enrichment counts and patient proportions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Composition and abundance can confound. Complete the transfer step: Use patient-stratified tests. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 16: Shape and outcome analysis

Begin with this mechanism: Geometry and patient-level niche proportions support interpretation. Recreate the concrete case: Elongation describes one connected component. Write latent features, Delaunay edges, graph degrees, hop layers, local means, standard deviations, concatenated vectors, component means, covariance entries, mixing weights, responsibilities, stability pairs, enrichment counts and patient proportions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Cells within a patient are dependent. Complete the transfer step: Keep patients as the unit. Change exactly one gene filter, normalization, principal-component count, coordinate scale, tensor cell, convolution window, pseudo-label rule, edge penalty, graph radius, feature-neighbour count, attention score, decoder distribution, loss weight, Delaunay edge, graph depth, aggregation statistic, covariance model, mixture count, initialization, latent dimension, GP dimension, kernel parameter, beta weight, inducing-point grid, tissue section, patient group or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, graph or tensor, probability model, objective, optimization, domain or niche label, interpretation and independent validation on separate notebook lines. Reserve complete slides, tissues, patients or platforms outside tuning when testing transfer.