Chapter map

1. 1. Why reduce dimensions
2. 2. Training-only preprocessing
3. 3. Singular value decomposition
4. 4. Principal-component variance
5. 5. Class-informative components
6. 6. One-versus-rest models
7. 7. Support-vector margins
8. 8. Radial kernel geometry
9. 9. Cross-validation
10. 10. Class probabilities
11. 11. The 0.9 acceptance rule
12. 12. Projecting a new query
13. 13. Alignment and domain shift
14. 14. Novel-type challenge
15. 15. A student SVM project
16. 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.

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1. Why reduce dimensions

Here the geometry becomes operational. Gene matrices contain thousands of correlated features, noise and far fewer independent biological samples.

A concrete situation

Ten thousand genes become tens of components.

What the mathematics does not license

Compression can discard rare signals. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Plot class performance versus component count. 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. Training-only preprocessing

This step deserves a careful pause. Normalisation, gene filtering, centring and scaling parameters must be learned without the final test data.

A concrete situation

Counts become a declared comparable matrix.

What the mathematics does not license

Leakage makes transfer look easier. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Fit every transformation inside folds. 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. Singular value decomposition

The useful question is not only how to calculate, but what the calculation assumes. SVD factors the centred matrix into cell scores, singular values and gene loading vectors.

A concrete situation

A two-dimensional toy matrix yields orthogonal axes.

What the mathematics does not license

Sign choices are arbitrary but projections must be consistent. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Reconstruct a small matrix by hand. 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. Principal-component variance

A small example makes the mechanism visible. Squared singular values quantify how much training variation each component carries.

A concrete situation

A component above the documented variance screen enters testing.

What the mathematics does not license

High variance can be batch, not biology. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Relate scores to donor and 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.

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5. Class-informative components

This is where modelling choices begin to affect evidence. Component-score distributions are tested for association with each class, reducing noisy axes before training.

A concrete situation

One component separates beta cells from others.

What the mathematics does not license

Multiple testing and small classes affect selection. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Repeat selection inside cross-validation. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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6. One-versus-rest models

The central mathematical idea is concrete. A separate classifier can learn each class against all remaining cells.

A concrete situation

The beta model distinguishes beta from non-beta.

What the mathematics does not license

Negative classes can be heterogeneous. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Report support and imbalance per model. 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. Support-vector margins

Here the geometry becomes operational. An SVM finds a boundary with a large geometric margin between labelled examples.

A concrete situation

Support vectors lie nearest the boundary.

What the mathematics does not license

Outliers and cost settings alter the boundary. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Calculate a linear margin on toy points. 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. Radial kernel geometry

This step deserves a careful pause. The radial basis kernel turns Euclidean component distance into local similarity and permits curved boundaries.

A concrete situation

Two clusters are separated by a nonlinear contour.

What the mathematics does not license

Gamma that is too large memorises points. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Tune cost and gamma within folds. 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. Cross-validation

The useful question is not only how to calculate, but what the calculation assumes. Training folds tune the model while held-out folds estimate performance without seeing their labels.

A concrete situation

Each biological donor becomes a test fold.

What the mathematics does not license

Random cells share donor signatures. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Use grouped folds and save splits. 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. Class probabilities

A small example makes the mechanism visible. The fitted model converts decision values into class probability estimates used for acceptance.

A concrete situation

A query receives scores across several labels.

What the mathematics does not license

Probability calibration can shift across studies. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Plot reliability by class and domain. 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. The 0.9 acceptance rule

This is where modelling choices begin to affect evidence. The documented default requires a class probability above 0.9; otherwise the cell is unassigned.

A concrete situation

A 0.87 winner is withheld.

What the mathematics does not license

One threshold has different effects for rare classes. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Plot coverage, error and rejection. 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. Projecting a new query

The central mathematical idea is concrete. Query expression is aligned to training genes and multiplied by the saved loading basis before classification.

A concrete situation

A new cell enters the original coordinate system.

What the mathematics does not license

Recomputing PCA creates incompatible axes. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Unit-test projection on known cells. 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. Alignment and domain shift

Here the geometry becomes operational. When reference and query differ systematically, any alignment step must be fitted without using final labels and audited for distortion.

A concrete situation

Studies mix more after declared alignment.

What the mathematics does not license

Alignment can erase real condition biology. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Compare before and after class separation. 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. Novel-type challenge

This step deserves a careful pause. Withhold an entire type and test whether the threshold rejects it rather than assigning a nearby class.

A concrete situation

Epsilon cells are absent from training.

What the mathematics does not license

High confidence can persist outside domain. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Report false-known and unassigned rates. 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. A student SVM project

The useful question is not only how to calculate, but what the calculation assumes. Compute PCA on training points, select label-associated axes, fit a radial SVM and add a probability threshold.

A concrete situation

Known truth exposes leakage and rejection.

What the mathematics does not license

Real data add thousands of genes and batch effects. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Keep one group untouched. 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 careers

A small example makes the mechanism visible. Version processing, genes, means, loadings, tests, kernel, hyperparameters, calibration and thresholds. This joins linear algebra and machine learning.

A concrete situation

Students can begin with SVD and margins.

What the mathematics does not license

A predicted class proves no function or diagnosis. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

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

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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

A scPred audit starts with labelled cells, a carefully processed expression matrix and a strict split by biological replicate. Singular value decomposition defines orthogonal principal components. Class-informative components are selected, and a probability-producing support-vector machine learns one-versus-rest class boundaries in that reduced space.

Record normalisation, genes, training means and loadings, component variance, the informative-component tests, kernel, cost, gamma, cross-validation folds, class probabilities, acceptance threshold and unassigned results. The query must be projected through the training basis; recalculating a new basis changes the coordinate system and invalidates the trained model.

Validate on untouched studies and types, inspect component stability, class balance, probability calibration, threshold coverage and rejected cells. A 0.9 default threshold is a policy for high-confidence assignment, not proof that every reported probability is calibrated in a new domain.

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Frequently asked questions

What is scPred?

scPred is a supervised single-cell classifier that selects informative principal components and trains probability-producing machine-learning models, with a radial support-vector machine as the documented default.

Why use principal components?

They provide orthogonal combinations of genes that can concentrate shared biological signal and reduce dimensionality before classification.

How are informative components chosen?

Components are screened for association with cell classes; the original paper describes class-wise statistical testing of component-score distributions.

What does the SVM do?

It learns boundaries in the selected component space; a radial kernel can create nonlinear boundaries from distances between cells.

When is a cell unassigned?

If no class probability exceeds the declared threshold, such as the documented default of 0.9, the query can remain unassigned.

How should scPred be validated?

Use donor- and study-held-out folds, locked preprocessing and loadings, class-wise metrics, calibration, threshold curves and withheld-type challenges.

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Primary source and fact-check note

Jose Alquicira-Hernandez, Anuja Sathe, Hanlee P. Ji, Quan Nguyen and Joseph E. Powell, “scPred: accurate supervised method for cell-type classification from single-cell RNA-seq data,” Genome Biology 20, 264 (2019), DOI 10.1186/s13059-019-1862-5. 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

• scmap projection and unassigned queries
• CellTypist decision scores
• SingleR reference scoring

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

Projection unit test

Save training means, scales and loadings. Require held-out reference cells to reproduce expected component scores before any biological interpretation.

Component-stability audit

Repeat SVD and informative-component selection across biological folds. Compare subspaces and class performance, not component signs alone.

Kernel grid

Tune cost and gamma inside grouped folds. Plot margin complexity, support-vector counts and external error; the most flexible boundary is rarely the most trustworthy.

Probability calibration

Build reliability diagrams by class, study and tissue. Domain shift can make a confident SVM wrong even when internal validation looked strong.

Threshold curve

Vary the acceptance threshold in validation data and report coverage, selective risk and unassigned rate. Lock the chosen rule before final testing.

Withheld-type challenge

Omit one type during training and measure how often it is rejected versus confidently assigned to a neighbour. Closed-set confidence needs explicit stress tests.

Leakage audit

Confirm that gene filters, PCA, feature tests, alignment, tuning and calibration never saw final test labels or cells. Record grouped split identifiers.

Final claim

Name preprocessing, basis, selected components, SVM settings, calibration and threshold. A high probability is model evidence within a tested domain, not proof of biology.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Why reduce dimensions

Start with the mechanism: Gene matrices contain thousands of correlated features, noise and far fewer independent biological samples. The worked case is concrete: Ten thousand genes become tens of components. Expose the boundary: Compression can discard rare signals. Complete this evidence step: Plot class performance versus component count. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare correlations, binary pairs, margins, probabilities, paths or labels. 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: Training-only preprocessing

Start with the mechanism: Normalisation, gene filtering, centring and scaling parameters must be learned without the final test data. The worked case is concrete: Counts become a declared comparable matrix. Expose the boundary: Leakage makes transfer look easier. Complete this evidence step: Fit every transformation inside folds. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare correlations, binary pairs, margins, probabilities, paths or labels. 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: Singular value decomposition

Start with the mechanism: SVD factors the centred matrix into cell scores, singular values and gene loading vectors. The worked case is concrete: A two-dimensional toy matrix yields orthogonal axes. Expose the boundary: Sign choices are arbitrary but projections must be consistent. Complete this evidence step: Reconstruct a small matrix by hand. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare correlations, binary pairs, margins, probabilities, paths or labels. 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: Principal-component variance

Start with the mechanism: Squared singular values quantify how much training variation each component carries. The worked case is concrete: A component above the documented variance screen enters testing. Expose the boundary: High variance can be batch, not biology. Complete this evidence step: Relate scores to donor and 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 correlations, binary pairs, margins, probabilities, paths or labels. 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 5: Class-informative components

Start with the mechanism: Component-score distributions are tested for association with each class, reducing noisy axes before training. The worked case is concrete: One component separates beta cells from others. Expose the boundary: Multiple testing and small classes affect selection. Complete this evidence step: Repeat selection inside cross-validation. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare correlations, binary pairs, margins, probabilities, paths or labels. 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: One-versus-rest models

Start with the mechanism: A separate classifier can learn each class against all remaining cells. The worked case is concrete: The beta model distinguishes beta from non-beta. Expose the boundary: Negative classes can be heterogeneous. Complete this evidence step: Report support and imbalance per model. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare correlations, binary pairs, margins, probabilities, paths or labels. 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: Support-vector margins

Start with the mechanism: An SVM finds a boundary with a large geometric margin between labelled examples. The worked case is concrete: Support vectors lie nearest the boundary. Expose the boundary: Outliers and cost settings alter the boundary. Complete this evidence step: Calculate a linear margin on toy points. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare correlations, binary pairs, margins, probabilities, paths or labels. 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: Radial kernel geometry

Start with the mechanism: The radial basis kernel turns Euclidean component distance into local similarity and permits curved boundaries. The worked case is concrete: Two clusters are separated by a nonlinear contour. Expose the boundary: Gamma that is too large memorises points. Complete this evidence step: Tune cost and gamma within folds. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare correlations, binary pairs, margins, probabilities, paths or labels. 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 9: Cross-validation

Start with the mechanism: Training folds tune the model while held-out folds estimate performance without seeing their labels. The worked case is concrete: Each biological donor becomes a test fold. Expose the boundary: Random cells share donor signatures. Complete this evidence step: Use grouped folds and save splits. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare correlations, binary pairs, margins, probabilities, paths or labels. 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: Class probabilities

Start with the mechanism: The fitted model converts decision values into class probability estimates used for acceptance. The worked case is concrete: A query receives scores across several labels. Expose the boundary: Probability calibration can shift across studies. Complete this evidence step: Plot reliability by class and domain. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare correlations, binary pairs, margins, probabilities, paths or labels. 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: The 0.9 acceptance rule

Start with the mechanism: The documented default requires a class probability above 0.9; otherwise the cell is unassigned. The worked case is concrete: A 0.87 winner is withheld. Expose the boundary: One threshold has different effects for rare classes. Complete this evidence step: Plot coverage, error and rejection. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare correlations, binary pairs, margins, probabilities, paths or labels. 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: Projecting a new query

Start with the mechanism: Query expression is aligned to training genes and multiplied by the saved loading basis before classification. The worked case is concrete: A new cell enters the original coordinate system. Expose the boundary: Recomputing PCA creates incompatible axes. Complete this evidence step: Unit-test projection on known cells. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare correlations, binary pairs, margins, probabilities, paths or labels. 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 13: Alignment and domain shift

Start with the mechanism: When reference and query differ systematically, any alignment step must be fitted without using final labels and audited for distortion. The worked case is concrete: Studies mix more after declared alignment. Expose the boundary: Alignment can erase real condition biology. Complete this evidence step: Compare before and after class separation. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare correlations, binary pairs, margins, probabilities, paths or labels. 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: Novel-type challenge

Start with the mechanism: Withhold an entire type and test whether the threshold rejects it rather than assigning a nearby class. The worked case is concrete: Epsilon cells are absent from training. Expose the boundary: High confidence can persist outside domain. Complete this evidence step: Report false-known and unassigned rates. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare correlations, binary pairs, margins, probabilities, paths or labels. 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: A student SVM project

Start with the mechanism: Compute PCA on training points, select label-associated axes, fit a radial SVM and add a probability threshold. The worked case is concrete: Known truth exposes leakage and rejection. Expose the boundary: Real data add thousands of genes and batch effects. Complete this evidence step: Keep one group untouched. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare correlations, binary pairs, margins, probabilities, paths or labels. 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 processing, genes, means, loadings, tests, kernel, hyperparameters, calibration and thresholds. This joins linear algebra and machine learning. The worked case is concrete: Students can begin with SVD and margins. Expose the boundary: A predicted class proves no function or diagnosis. Complete this evidence step: Write a projection model card. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare correlations, binary pairs, margins, probabilities, paths or labels. 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.