WHY MATHEMATICS?

Why Mathematics? | Azimuth for Reference Mapping: Supervised PCA, Transfer Anchors, Label Scores and UMAP Projection

How Azimuth maps a new single-cell dataset into an established reference: supervised PCA extracts transcriptomic directions aligned with multimodal reference structure, anchors connect comparable cells, weighted votes transfer hierarchical labels, and the query is projected onto the stored reference map.

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

Understand the reference and supervised-PC geometry
Find anchor pairs between reference and query
Transfer labels with inspectable support scores
Project UMAP carefully and expose reference gaps

Chapter map

1. Why map to a reference
2. Multimodal reference graph
3. Ordinary PCA baseline
4. Supervised PCA objective
5. Projecting query expression
6. Anchor discovery
7. Anchor weights
8. Hierarchical label transfer
9. Prediction-score vector
10. Embedding correction
11. UMAP projection
12. Mapping score
13. Closed-reference boundary
14. Worked anchor vote
15. Independent validation
16. Governance and careers

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

1. Why map to a reference

This step deserves a careful pause. A curated atlas supplies consistent labels and coordinates, reducing repeated manual annotation across query studies.

A concrete situation

A vaccination dataset maps to a healthy PBMC atlas.

What the mathematics does not license

The atlas defines the available vocabulary. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

State the reference scope first. 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. Multimodal reference graph

The useful question is not only how to calculate, but what the calculation assumes. The Cell paper builds a weighted-nearest-neighbour graph that learns the relative usefulness of RNA and protein for each reference cell.

A concrete situation

Protein separates states with similar RNA.

What the mathematics does not license

A query may measure RNA only. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Document modalities used to build the reference. 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. Ordinary PCA baseline

A small example makes the mechanism visible. PCA chooses linear directions that maximise total variance in X.

A concrete situation

A technical depth axis can dominate variance.

What the mathematics does not license

High variance is not always cell identity. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Compare PCs with biological labels 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. Supervised PCA objective

This is where modelling choices begin to affect evidence. Given expression X and reference graph Y, sPCA selects U so the kernel of U-transformed X has high Hilbert–Schmidt dependence with Y.

A concrete situation

Transcript directions align with multimodal neighbours.

What the mathematics does not license

The graph supervises what structure matters. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Record U and dependence settings. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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. Projecting query expression

The central mathematical idea is concrete. The learned sPCA transformation is applied to the query’s shared genes, producing coordinates in reference-informed transcript space.

A concrete situation

An RNA-only query gets sPC coordinates.

What the mathematics does not license

Missing genes and preprocessing mismatch distort projection. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Checksum features and normalisation. 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. Anchor discovery

Here the geometry becomes operational. FindTransferAnchors searches for mutually compatible reference and query neighbours in the reduced space.

A concrete situation

A query memory T cell pairs with several reference neighbours.

What the mathematics does not license

Disease states may lack valid anchors. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Count anchors and distances per cell. 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. Anchor weights

This step deserves a careful pause. Nearby, consistently supported anchors receive greater influence in downstream transfer.

A concrete situation

Five close anchors outweigh one distant pair.

What the mathematics does not license

Many correlated anchors do not equal independent evidence. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Inspect effective support size. 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. Hierarchical label transfer

The useful question is not only how to calculate, but what the calculation assumes. TransferData can move several reference annotation levels, from broad lineage to fine subtype, into each query cell.

A concrete situation

A cell receives L1 immune and L2 memory T labels.

What the mathematics does not license

Fine labels may be less reliable than broad labels. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Calibrate each resolution 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.

9. Prediction-score vector

A small example makes the mechanism visible. Weighted anchor votes produce one score per candidate label and a maximum score for the chosen annotation.

A concrete situation

The top score is 0.82 and runner-up 0.12.

What the mathematics does not license

Scores need empirical calibration by domain. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Preserve top labels and 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.

10. Embedding correction

This is where modelling choices begin to affect evidence. IntegrateEmbeddings uses anchors to adjust the query projection toward the reference coordinate system.

A concrete situation

A batch-shifted query moves near matching reference cells.

What the mathematics does not license

Correction can hide real condition effects. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Compare before and after biology. 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. UMAP projection

The central mathematical idea is concrete. ProjectUMAP applies the stored reference UMAP model to corrected query embeddings without refitting the reference map.

A concrete situation

Query cells appear on a familiar PBMC layout.

What the mathematics does not license

UMAP position is not a direct biological measurement. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Read labels, scores and genes together. 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. Mapping score

Here the geometry becomes operational. Neighbourhood-based mapping diagnostics estimate how well reference geometry represents each query cell.

A concrete situation

A familiar monocyte scores high; an uncovered tumour state scores low.

What the mathematics does not license

A high score does not prove the transferred fine label. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Use mapping and prediction scores jointly. 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. Closed-reference boundary

This step deserves a careful pause. Azimuth normally produces a label for every query cell unless the analyst applies score and coverage rules.

A concrete situation

An absent type receives the least-wrong reference label.

What the mathematics does not license

Automatic labels can conceal novelty. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Run an omitted-type test. 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. Worked anchor vote

The useful question is not only how to calculate, but what the calculation assumes. Use three labels and six anchors with explicit distances and weights; calculate label totals, top score and margin.

A concrete situation

Four coherent anchors support label A.

What the mathematics does not license

Toy weights omit high-dimensional uncertainty. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Perturb one anchor and recalculate. 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. Independent validation

A small example makes the mechanism visible. The Cell study validated RNA-only mapping with query protein measurements held out from mapping.

A concrete situation

Predicted regulatory T cells show expected CD25 protein.

What the mathematics does not license

Concordance is dataset-specific evidence. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Reserve an independent assay or label. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

16. Governance and careers

This is where modelling choices begin to affect evidence. Version reference, sPCA, anchors, scores, corrected embeddings and UMAP. This connects kernels, neighbours and data engineering.

A concrete situation

Students can begin with PCA and weighted votes.

What the mathematics does not license

A projection remains conditional on reference coverage. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Write a reference-mapping report. 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

An Azimuth audit begins with a carefully curated reference and a query that fits the reference’s biological scope. The reference can use a weighted-nearest-neighbour graph built from RNA and protein; supervised principal components learn transcriptomic directions aligned with that graph, transfer anchors link comparable reference and query cells, and weighted votes transfer labels and confidence scores.

Record reference version, assays, genes, SCTransform model, supervised-PC transformation, anchor pairs, anchor filters, weights, hierarchical label levels, prediction scores, mapping scores, corrected query embeddings and the stored UMAP model. A projected point is a placement inside reference geometry, not a newly measured protein profile or proof that every query state exists in the atlas.

Validate with held-out query protein or known labels, absent-type challenges, tissue and disease shifts, anchor consistency, neighbourhood preservation and score calibration. Reference mapping is most useful when low scores, uncovered biology and reference boundaries remain visible.

Frequently asked questions

What is Azimuth?

Azimuth is a reference-mapping workflow and application built around Seurat methods. It maps query cells to curated single-cell references, transfers reference annotations and projects queries onto stored reference visualisations.

Why use supervised PCA?

Ordinary PCA maximises expression variance. The published reference-mapping framework uses supervised PCA to find transcriptomic directions that capture structure in a multimodal weighted-nearest-neighbour graph.

What is a transfer anchor?

An anchor is a reference–query cell pair judged biologically similar in a shared reduced space. Many anchors provide guideposts for label transfer and query correction.

What is a prediction score?

For each transferred annotation level, Azimuth returns a score between zero and one summarising weighted support from anchors. It is a model confidence measure, not a guaranteed probability of truth.

What is a mapping score?

The mapping score assesses how well a query cell is represented by the reference and how well its local neighbourhood structure is preserved during mapping.

Can Azimuth discover a type absent from the reference?

Reference mapping labels cells within the supplied reference vocabulary. Low scores or poor mapping can flag a coverage problem, but novel-type discovery needs separate analysis and validation.

Primary source and fact-check note

Yuhan Hao, Stephanie Hao, Erica Andersen-Nissen et al., “Integrated analysis of multimodal single-cell data,” Cell 184, 3573–3587.e29 (2021), DOI 10.1016/j.cell.2021.04.048. 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

Reference-contract audit

Record reference version, donors, tissues, modalities, label hierarchy and known exclusions before mapping.

Feature-alignment audit

Checksum shared genes, order, SCTransform model and sPCA transformation. Silent mismatch invalidates anchor geometry.

Anchor ledger

Store reference–query pairs, distances, filters and effective weights by query cell and label.

Score calibration

Relate prediction and mapping scores to correctness using independent query labels or held-out assays.

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.

Resolution audit

Compare broad and fine annotation levels. A high lineage score does not validate a fragile subtype.

Omitted-type challenge

Remove a reference type and quantify least-wrong labels, low scores and neighbourhood distortion.

Biology-preservation test

Compare markers, condition effects and neighbourhoods before and after embedding correction.

Final claim

Name reference, sPCA, anchors, scores and UMAP. Mapping is conditional on reference coverage.

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: Why map to a reference

Start with the mechanism: A curated atlas supplies consistent labels and coordinates, reducing repeated manual annotation across query studies. The worked case is concrete: A vaccination dataset maps to a healthy PBMC atlas. Expose the boundary: The atlas defines the available vocabulary. Complete this evidence step: State the reference scope first. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare graph proximities, singular values, anchor weights, token masks, attention matrices, probabilities, embeddings, losses or calibration. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 2: Multimodal reference graph

Start with the mechanism: The Cell paper builds a weighted-nearest-neighbour graph that learns the relative usefulness of RNA and protein for each reference cell. The worked case is concrete: Protein separates states with similar RNA. Expose the boundary: A query may measure RNA only. Complete this evidence step: Document modalities used to build the reference. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare graph proximities, singular values, anchor weights, token masks, attention matrices, probabilities, embeddings, losses or calibration. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 3: Ordinary PCA baseline

Start with the mechanism: PCA chooses linear directions that maximise total variance in X. The worked case is concrete: A technical depth axis can dominate variance. Expose the boundary: High variance is not always cell identity. Complete this evidence step: Compare PCs with biological labels and batches. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare graph proximities, singular values, anchor weights, token masks, attention matrices, probabilities, embeddings, losses or calibration. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 4: Supervised PCA objective

Start with the mechanism: Given expression X and reference graph Y, sPCA selects U so the kernel of U-transformed X has high Hilbert–Schmidt dependence with Y. The worked case is concrete: Transcript directions align with multimodal neighbours. Expose the boundary: The graph supervises what structure matters. Complete this evidence step: Record U and dependence settings. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare graph proximities, singular values, anchor weights, token masks, attention matrices, probabilities, embeddings, losses or calibration. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

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: Projecting query expression

Start with the mechanism: The learned sPCA transformation is applied to the query’s shared genes, producing coordinates in reference-informed transcript space. The worked case is concrete: An RNA-only query gets sPC coordinates. Expose the boundary: Missing genes and preprocessing mismatch distort projection. Complete this evidence step: Checksum features and normalisation. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare graph proximities, singular values, anchor weights, token masks, attention matrices, probabilities, embeddings, losses or calibration. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 6: Anchor discovery

Start with the mechanism: FindTransferAnchors searches for mutually compatible reference and query neighbours in the reduced space. The worked case is concrete: A query memory T cell pairs with several reference neighbours. Expose the boundary: Disease states may lack valid anchors. Complete this evidence step: Count anchors and distances per cell. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare graph proximities, singular values, anchor weights, token masks, attention matrices, probabilities, embeddings, losses or calibration. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 7: Anchor weights

Start with the mechanism: Nearby, consistently supported anchors receive greater influence in downstream transfer. The worked case is concrete: Five close anchors outweigh one distant pair. Expose the boundary: Many correlated anchors do not equal independent evidence. Complete this evidence step: Inspect effective support size. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare graph proximities, singular values, anchor weights, token masks, attention matrices, probabilities, embeddings, losses or calibration. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 8: Hierarchical label transfer

Start with the mechanism: TransferData can move several reference annotation levels, from broad lineage to fine subtype, into each query cell. The worked case is concrete: A cell receives L1 immune and L2 memory T labels. Expose the boundary: Fine labels may be less reliable than broad labels. Complete this evidence step: Calibrate each resolution separately. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare graph proximities, singular values, anchor weights, token masks, attention matrices, probabilities, embeddings, losses or calibration. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

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: Prediction-score vector

Start with the mechanism: Weighted anchor votes produce one score per candidate label and a maximum score for the chosen annotation. The worked case is concrete: The top score is 0.82 and runner-up 0.12. Expose the boundary: Scores need empirical calibration by domain. Complete this evidence step: Preserve top labels and margins. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare graph proximities, singular values, anchor weights, token masks, attention matrices, probabilities, embeddings, losses or calibration. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 10: Embedding correction

Start with the mechanism: IntegrateEmbeddings uses anchors to adjust the query projection toward the reference coordinate system. The worked case is concrete: A batch-shifted query moves near matching reference cells. Expose the boundary: Correction can hide real condition effects. Complete this evidence step: Compare before and after biology. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare graph proximities, singular values, anchor weights, token masks, attention matrices, probabilities, embeddings, losses or calibration. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 11: UMAP projection

Start with the mechanism: ProjectUMAP applies the stored reference UMAP model to corrected query embeddings without refitting the reference map. The worked case is concrete: Query cells appear on a familiar PBMC layout. Expose the boundary: UMAP position is not a direct biological measurement. Complete this evidence step: Read labels, scores and genes together. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare graph proximities, singular values, anchor weights, token masks, attention matrices, probabilities, embeddings, losses or calibration. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 12: Mapping score

Start with the mechanism: Neighbourhood-based mapping diagnostics estimate how well reference geometry represents each query cell. The worked case is concrete: A familiar monocyte scores high; an uncovered tumour state scores low. Expose the boundary: A high score does not prove the transferred fine label. Complete this evidence step: Use mapping and prediction scores jointly. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare graph proximities, singular values, anchor weights, token masks, attention matrices, probabilities, embeddings, losses or calibration. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

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: Closed-reference boundary

Start with the mechanism: Azimuth normally produces a label for every query cell unless the analyst applies score and coverage rules. The worked case is concrete: An absent type receives the least-wrong reference label. Expose the boundary: Automatic labels can conceal novelty. Complete this evidence step: Run an omitted-type test. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare graph proximities, singular values, anchor weights, token masks, attention matrices, probabilities, embeddings, losses or calibration. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 14: Worked anchor vote

Start with the mechanism: Use three labels and six anchors with explicit distances and weights; calculate label totals, top score and margin. The worked case is concrete: Four coherent anchors support label A. Expose the boundary: Toy weights omit high-dimensional uncertainty. Complete this evidence step: Perturb one anchor and recalculate. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare graph proximities, singular values, anchor weights, token masks, attention matrices, probabilities, embeddings, losses or calibration. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 15: Independent validation

Start with the mechanism: The Cell study validated RNA-only mapping with query protein measurements held out from mapping. The worked case is concrete: Predicted regulatory T cells show expected CD25 protein. Expose the boundary: Concordance is dataset-specific evidence. Complete this evidence step: Reserve an independent assay or label. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare graph proximities, singular values, anchor weights, token masks, attention matrices, probabilities, embeddings, losses or calibration. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 16: Governance and careers

Start with the mechanism: Version reference, sPCA, anchors, scores, corrected embeddings and UMAP. This connects kernels, neighbours and data engineering. The worked case is concrete: Students can begin with PCA and weighted votes. Expose the boundary: A projection remains conditional on reference coverage. Complete this evidence step: Write a reference-mapping report. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare graph proximities, singular values, anchor weights, token masks, attention matrices, probabilities, embeddings, losses or calibration. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.