Chapter map

1. 1. Reference cells
2. 2. Cell-type averages
3. 3. Spatial library size
4. 4. Poisson count law
5. 5. Mixture mean
6. 6. Platform-effect vector
7. 7. Weight optimization
8. 8. Log likelihood
9. 9. Singlet score
10. 10. Doublet comparison
11. 11. Full mixture mode
12. 12. Worked two-gene example
13. 13. Residual map
14. 14. Reference-swap test
15. 15. Spatial expression after decomposition
16. 16. Governance and learning

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

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1. Reference cells

This is where modelling choices begin to affect evidence. Annotated single-cell counts define the profiles available to explain spatial measurements.

A concrete situation

Three donors contribute neuronal, glial and vascular labels.

What the mathematics does not license

Wrong or missing labels redirect weights. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Audit donors, labels and condition coverage. 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. Cell-type averages

The central mathematical idea is concrete. For every gene and type, the reference estimates an average expression rate after filtering.

A concrete situation

A vascular marker has a high endothelial rate and low neuronal rates.

What the mathematics does not license

Averages hide within-type states. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Compare donor- and state-specific profiles. 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. Spatial library size

Here the geometry becomes operational. The total count at a pixel scales all expected gene counts for that location.

A concrete situation

A pixel with twice the reads has twice the expected counts under equal composition.

What the mathematics does not license

Depth can correlate with tissue density. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Plot depth against histology and fitted weights. 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. Poisson count law

This step deserves a careful pause. Each gene count is modeled as Poisson with variance tied to its expected mean.

A concrete situation

A mean of 6 makes counts near 6 most plausible.

What the mathematics does not license

Extra-Poisson variation can weaken fit. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

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

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5. Mixture mean

The useful question is not only how to calculate, but what the calculation assumes. The expected expression is a nonnegative weighted sum of reference type profiles.

A concrete situation

Weights 0.7 and 0.3 mix two gene-rate vectors.

What the mathematics does not license

Similar profiles are hard to separate. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Measure profile correlations. 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. Platform-effect vector

A small example makes the mechanism visible. A positive gene-specific factor aligns relative expression between reference and spatial technologies.

A concrete situation

A ribosomal gene needs a 0.6 correction while a marker stays near one.

What the mathematics does not license

Biology can be mistaken for technology. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Estimate effects across diverse locations. 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. Weight optimization

This is where modelling choices begin to affect evidence. Numerical optimization finds the nonnegative weights that maximize the spatial count likelihood.

A concrete situation

A grid search on two types shows one clear likelihood peak.

What the mathematics does not license

Local choices and constraints matter. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Repeat with diagnostics and perturbations. 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. Log likelihood

The central mathematical idea is concrete. Products of Poisson probabilities become sums of count times log mean minus mean terms.

A concrete situation

One implausible high marker count changes the score.

What the mathematics does not license

Large genes can dominate totals. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Inspect gene contributions. 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. Singlet score

Here the geometry becomes operational. The best one-type model supplies a simple explanation and baseline likelihood.

A concrete situation

An oligodendrocyte-only fit explains nearly every marker.

What the mathematics does not license

Simple does not mean true. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Compare residual markers. 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. Doublet comparison

This step deserves a careful pause. Two-type candidates are compared with penalties or thresholds that discourage gratuitous complexity.

A concrete situation

Adding microglia sharply improves a mixed pixel.

What the mathematics does not license

Thresholds create borderline cases. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Publish score margins. 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. Full mixture mode

The useful question is not only how to calculate, but what the calculation assumes. A location may be represented by more than two types when resolution and tissue biology require it.

A concrete situation

A Visium spot contains immune, stromal and epithelial RNA.

What the mathematics does not license

More flexibility increases ambiguity. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Test sparsity and stability. 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. Worked two-gene example

A small example makes the mechanism visible. Multiply two type profiles by weights, apply depth and platform factors, then compute Poisson probabilities.

A concrete situation

Observed counts 9 and 3 favour a 75:25 mixture.

What the mathematics does not license

Two genes are not identifiable in practice. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Add genes and correlated 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.

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13. Residual map

This is where modelling choices begin to affect evidence. Observed minus expected counts reveal genes and regions the reference mixture does not explain.

A concrete situation

A disease programme leaves positive residuals along a lesion.

What the mathematics does not license

Residuals mix biology and technical failure. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Follow with condition-matched data. 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. Reference-swap test

The central mathematical idea is concrete. Fit plausible atlases from different donors or technologies and compare weights.

A concrete situation

A rare subtype disappears with a shallow atlas.

What the mathematics does not license

Agreement may reflect shared bias. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Use orthogonal staining. 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. Spatial expression after decomposition

Here the geometry becomes operational. Cell-type localization can support tests of within-type spatial expression when uncertainty is respected.

A concrete situation

A neuronal gene changes near a boundary.

What the mathematics does not license

Localization error propagates downstream. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Carry weights and uncertainty forward. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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16. Governance and learning

This step deserves a careful pause. Version counts, labels, effects, thresholds, modes and residual checks.

A concrete situation

A student traces one pixel from counts to a doublet decision.

What the mathematics does not license

A colour is not an explanation. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Write a complete evidence ledger. 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

An RCTD analysis begins with annotated single-cell RNA sequencing and spatial transcript counts. It estimates a reference expression profile for each declared cell type, models spatial counts with a Poisson distribution, and corrects gene-level platform effects between technologies. The nonnegative weights at each pixel or spot describe how strongly each reference type contributes to that location.

The mathematics separates three questions that are often blended in casual language: which reference profiles are plausible, how much of each profile is supported at a location, and whether one or two cell types explain the signal better than a more complex mixture. Doublet mode is therefore a model-comparison procedure under explicit penalties and thresholds, not a visual guess from a coloured map.

Validate with simulations, reference-donor swaps, absent-type tests, known anatomical markers, pixels held out by section and sensitivity to cell-type similarity. A fitted weight is conditional on the reference and platform correction. It is not direct observation of a whole cell, and a doublet label does not prove that exactly two intact cells occupied the capture region.

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Frequently asked questions

What is RCTD?

RCTD is a supervised spatial transcriptomics decomposition method that uses cell-type profiles learned from single-cell data to estimate which types contribute to each spatial location while correcting cross-platform differences.

Why a Poisson model?

RCTD treats spatial transcript counts as discrete events whose expected value is built from cell-type weights, reference profiles, library size and gene platform effects. Model checks are needed when variability exceeds the assumptions.

What is a platform effect?

It is a systematic gene-specific difference between the reference and spatial technologies. RCTD estimates correction factors so a gene that is relatively easier to detect on one platform does not automatically distort cell-type weights.

What are the weights?

They are nonnegative fitted contributions of reference cell types at a location. Their interpretation depends on normalization, reference quality and the count model.

What is doublet mode?

Doublet mode compares simpler one-type and two-type explanations and penalizes unnecessary complexity. It produces a statistical decision under thresholds, not a direct count of physical cells.

Can RCTD discover a missing cell type?

It may reveal poor fit or unusual residuals, but a supervised decomposition cannot name a type absent from its reference. Novelty requires additional evidence and revised references.

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Primary source and fact-check note

Dylan M. Cable, Evan Murray, Luli S. Zou et al., “Robust decomposition of cell type mixtures in spatial transcriptomics,” Nature Biotechnology 40, 517–526 (2022), DOI 10.1038/s41587-021-00830-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

• cell2location abundance maps
• Stereoscope mixture weights
• Tangram spatial mapping
• 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

Reference audit

Check labels, donors, platform and absent states.

Poisson audit

Inspect means, zeros, tails and spatial residuals.

Platform audit

Review gene corrections for biology-technology confounding.

Mode audit

State singlet, doublet or full mode and every threshold.

Similarity test

Challenge closely related reference types.

Section holdout

Reserve complete tissue sections.

External check

Use histology or targeted spatial markers.

Final claim

A mixture weight and doublet decision are model-based evidence.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Reference cells

Begin with this mechanism: Annotated single-cell counts define the profiles available to explain spatial measurements. Work through the concrete case: Three donors contribute neuronal, glial and vascular labels. Make the limitation visible: Wrong or missing labels redirect weights. Complete the evidence step: Audit donors, labels and condition coverage. Write the relevant vectors, matrices, probability distribution, graph, loss or posterior update with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Test transfer on complete donors, tissue sections, capture locations or gene sets outside tuning. Keep measurement, fitted parameter, prediction, interpretation and decision on separate notebook lines so the mathematical claim remains auditable.

Workshop 2: Cell-type averages

Begin with this mechanism: For every gene and type, the reference estimates an average expression rate after filtering. Work through the concrete case: A vascular marker has a high endothelial rate and low neuronal rates. Make the limitation visible: Averages hide within-type states. Complete the evidence step: Compare donor- and state-specific profiles. Write the relevant vectors, matrices, probability distribution, graph, loss or posterior update with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Test transfer on complete donors, tissue sections, capture locations or gene sets outside tuning. Keep measurement, fitted parameter, prediction, interpretation and decision on separate notebook lines so the mathematical claim remains auditable.

Workshop 3: Spatial library size

Begin with this mechanism: The total count at a pixel scales all expected gene counts for that location. Work through the concrete case: A pixel with twice the reads has twice the expected counts under equal composition. Make the limitation visible: Depth can correlate with tissue density. Complete the evidence step: Plot depth against histology and fitted weights. Write the relevant vectors, matrices, probability distribution, graph, loss or posterior update with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Test transfer on complete donors, tissue sections, capture locations or gene sets outside tuning. Keep measurement, fitted parameter, prediction, interpretation and decision on separate notebook lines so the mathematical claim remains auditable.

Workshop 4: Poisson count law

Begin with this mechanism: Each gene count is modeled as Poisson with variance tied to its expected mean. Work through the concrete case: A mean of 6 makes counts near 6 most plausible. Make the limitation visible: Extra-Poisson variation can weaken fit. Complete the evidence step: Use residual and dispersion diagnostics. Write the relevant vectors, matrices, probability distribution, graph, loss or posterior update with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Test transfer on complete donors, tissue sections, capture locations or gene sets outside tuning. Keep measurement, fitted parameter, prediction, interpretation and decision on separate notebook lines so the mathematical claim remains auditable.

Workshop 5: Mixture mean

Begin with this mechanism: The expected expression is a nonnegative weighted sum of reference type profiles. Work through the concrete case: Weights 0.7 and 0.3 mix two gene-rate vectors. Make the limitation visible: Similar profiles are hard to separate. Complete the evidence step: Measure profile correlations. Write the relevant vectors, matrices, probability distribution, graph, loss or posterior update with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Test transfer on complete donors, tissue sections, capture locations or gene sets outside tuning. Keep measurement, fitted parameter, prediction, interpretation and decision on separate notebook lines so the mathematical claim remains auditable.

Workshop 6: Platform-effect vector

Begin with this mechanism: A positive gene-specific factor aligns relative expression between reference and spatial technologies. Work through the concrete case: A ribosomal gene needs a 0.6 correction while a marker stays near one. Make the limitation visible: Biology can be mistaken for technology. Complete the evidence step: Estimate effects across diverse locations. Write the relevant vectors, matrices, probability distribution, graph, loss or posterior update with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Test transfer on complete donors, tissue sections, capture locations or gene sets outside tuning. Keep measurement, fitted parameter, prediction, interpretation and decision on separate notebook lines so the mathematical claim remains auditable.

Workshop 7: Weight optimization

Begin with this mechanism: Numerical optimization finds the nonnegative weights that maximize the spatial count likelihood. Work through the concrete case: A grid search on two types shows one clear likelihood peak. Make the limitation visible: Local choices and constraints matter. Complete the evidence step: Repeat with diagnostics and perturbations. Write the relevant vectors, matrices, probability distribution, graph, loss or posterior update with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Test transfer on complete donors, tissue sections, capture locations or gene sets outside tuning. Keep measurement, fitted parameter, prediction, interpretation and decision on separate notebook lines so the mathematical claim remains auditable.

Workshop 8: Log likelihood

Begin with this mechanism: Products of Poisson probabilities become sums of count times log mean minus mean terms. Work through the concrete case: One implausible high marker count changes the score. Make the limitation visible: Large genes can dominate totals. Complete the evidence step: Inspect gene contributions. Write the relevant vectors, matrices, probability distribution, graph, loss or posterior update with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Test transfer on complete donors, tissue sections, capture locations or gene sets outside tuning. Keep measurement, fitted parameter, prediction, interpretation and decision on separate notebook lines so the mathematical claim remains auditable.

Workshop 9: Singlet score

Begin with this mechanism: The best one-type model supplies a simple explanation and baseline likelihood. Work through the concrete case: An oligodendrocyte-only fit explains nearly every marker. Make the limitation visible: Simple does not mean true. Complete the evidence step: Compare residual markers. Write the relevant vectors, matrices, probability distribution, graph, loss or posterior update with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Test transfer on complete donors, tissue sections, capture locations or gene sets outside tuning. Keep measurement, fitted parameter, prediction, interpretation and decision on separate notebook lines so the mathematical claim remains auditable.

Workshop 10: Doublet comparison

Begin with this mechanism: Two-type candidates are compared with penalties or thresholds that discourage gratuitous complexity. Work through the concrete case: Adding microglia sharply improves a mixed pixel. Make the limitation visible: Thresholds create borderline cases. Complete the evidence step: Publish score margins. Write the relevant vectors, matrices, probability distribution, graph, loss or posterior update with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Test transfer on complete donors, tissue sections, capture locations or gene sets outside tuning. Keep measurement, fitted parameter, prediction, interpretation and decision on separate notebook lines so the mathematical claim remains auditable.

Workshop 11: Full mixture mode

Begin with this mechanism: A location may be represented by more than two types when resolution and tissue biology require it. Work through the concrete case: A Visium spot contains immune, stromal and epithelial RNA. Make the limitation visible: More flexibility increases ambiguity. Complete the evidence step: Test sparsity and stability. Write the relevant vectors, matrices, probability distribution, graph, loss or posterior update with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Test transfer on complete donors, tissue sections, capture locations or gene sets outside tuning. Keep measurement, fitted parameter, prediction, interpretation and decision on separate notebook lines so the mathematical claim remains auditable.

Workshop 12: Worked two-gene example

Begin with this mechanism: Multiply two type profiles by weights, apply depth and platform factors, then compute Poisson probabilities. Work through the concrete case: Observed counts 9 and 3 favour a 75:25 mixture. Make the limitation visible: Two genes are not identifiable in practice. Complete the evidence step: Add genes and correlated types. Write the relevant vectors, matrices, probability distribution, graph, loss or posterior update with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Test transfer on complete donors, tissue sections, capture locations or gene sets outside tuning. Keep measurement, fitted parameter, prediction, interpretation and decision on separate notebook lines so the mathematical claim remains auditable.

Workshop 13: Residual map

Begin with this mechanism: Observed minus expected counts reveal genes and regions the reference mixture does not explain. Work through the concrete case: A disease programme leaves positive residuals along a lesion. Make the limitation visible: Residuals mix biology and technical failure. Complete the evidence step: Follow with condition-matched data. Write the relevant vectors, matrices, probability distribution, graph, loss or posterior update with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Test transfer on complete donors, tissue sections, capture locations or gene sets outside tuning. Keep measurement, fitted parameter, prediction, interpretation and decision on separate notebook lines so the mathematical claim remains auditable.

Workshop 14: Reference-swap test

Begin with this mechanism: Fit plausible atlases from different donors or technologies and compare weights. Work through the concrete case: A rare subtype disappears with a shallow atlas. Make the limitation visible: Agreement may reflect shared bias. Complete the evidence step: Use orthogonal staining. Write the relevant vectors, matrices, probability distribution, graph, loss or posterior update with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Test transfer on complete donors, tissue sections, capture locations or gene sets outside tuning. Keep measurement, fitted parameter, prediction, interpretation and decision on separate notebook lines so the mathematical claim remains auditable.

Workshop 15: Spatial expression after decomposition

Begin with this mechanism: Cell-type localization can support tests of within-type spatial expression when uncertainty is respected. Work through the concrete case: A neuronal gene changes near a boundary. Make the limitation visible: Localization error propagates downstream. Complete the evidence step: Carry weights and uncertainty forward. Write the relevant vectors, matrices, probability distribution, graph, loss or posterior update with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Test transfer on complete donors, tissue sections, capture locations or gene sets outside tuning. Keep measurement, fitted parameter, prediction, interpretation and decision on separate notebook lines so the mathematical claim remains auditable.

Workshop 16: Governance and learning

Begin with this mechanism: Version counts, labels, effects, thresholds, modes and residual checks. Work through the concrete case: A student traces one pixel from counts to a doublet decision. Make the limitation visible: A colour is not an explanation. Complete the evidence step: Write a complete evidence ledger. Write the relevant vectors, matrices, probability distribution, graph, loss or posterior update with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Test transfer on complete donors, tissue sections, capture locations or gene sets outside tuning. Keep measurement, fitted parameter, prediction, interpretation and decision on separate notebook lines so the mathematical claim remains auditable.