WHY MATHEMATICS?

Why Mathematics? | nnSVG for Spatially Variable Genes: Nearest-Neighbor Gaussian Processes, Gene-Specific Length Scales and Likelihood-Ratio Testing

How nnSVG keeps gene-specific spatial ranges while making Gaussian-process testing practical for thousands of genes and many tissue locations.

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

Model each gene with spatial and independent variance
Approximate the dense GP with ordered nearest-neighbor conditionals
Estimate a length scale separately for every gene
Separate statistical evidence, effect size and biological interpretation

Chapter map

1. Transformed expression
2. Design matrix
3. Spatial random effect
4. Exponential covariance
5. Gene-specific length scale
6. Spatial variance
7. Nugget variance
8. Proportion spatial
9. Location ordering
10. Neighbour sets
11. Conditional likelihood
12. Linear scalability
13. Spatial alternative model
14. Nonspatial null
15. Likelihood-ratio and FDR
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.

1. Transformed expression

The central mathematical idea is concrete. Each gene supplies log-normalized values or deviance residuals.

A concrete situation

One vector covers all quality-controlled spots.

What the mathematics does not license

Input transformation changes Gaussian fit. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Document assay values. 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. Design matrix

Here the geometry becomes operational. Covariates enter as fixed effects before spatial covariance.

A concrete situation

Known tissue domains receive indicator columns.

What the mathematics does not license

Post-hoc covariates can remove desired signal. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Choose them prospectively. 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. Spatial random effect

This step deserves a careful pause. A zero-mean Gaussian process represents correlated residual variation.

A concrete situation

Nearby residuals tend to move together.

What the mathematics does not license

Stationarity may fail across tissue. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Map local residuals. 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. Exponential covariance

The useful question is not only how to calculate, but what the calculation assumes. Correlation decays exponentially with distance over a range parameter.

A concrete situation

Half-millimetre structure has a longer scale than a vessel.

What the mathematics does not license

One scale may not fit multiscale genes. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Compare residual semivariograms. 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. Gene-specific length scale

A small example makes the mechanism visible. Every gene estimates its own spatial range.

A concrete situation

Immune and cortical markers occupy different scales.

What the mathematics does not license

Boundary estimates can be unstable. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Report profiles and limits. 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. Spatial variance

This is where modelling choices begin to affect evidence. Sigma squared measures fitted correlated variation.

A concrete situation

A layer marker has substantial spatial variance.

What the mathematics does not license

Magnitude depends on transformation. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use standardized effect summaries. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

7. Nugget variance

The central mathematical idea is concrete. Tau squared captures independent residual variation.

A concrete situation

Measurement noise remains at each spot.

What the mathematics does not license

Nugget mixes biology and technical noise. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Avoid causal labels. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

8. Proportion spatial

Here the geometry becomes operational. Sigma squared is divided by sigma plus tau variance.

A concrete situation

A value near one indicates dominant fitted spatial structure.

What the mathematics does not license

Large effect may be imprecise for weak genes. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Pair effect with uncertainty. 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. Location ordering

This step deserves a careful pause. NNGP factorization conditions each site on earlier sites.

A concrete situation

Coordinates are ordered before neighbour search.

What the mathematics does not license

Different orderings change the approximation. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Fix and record ordering. 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. Neighbour sets

The useful question is not only how to calculate, but what the calculation assumes. Each location uses a small set of nearest predecessors.

A concrete situation

Fifteen neighbours approximate local dependence.

What the mathematics does not license

Too few neighbours lose information. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Sweep m and runtime. 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. Conditional likelihood

A small example makes the mechanism visible. The joint density becomes a product of local Gaussian conditionals.

A concrete situation

One spot is predicted from its neighbour vector.

What the mathematics does not license

Approximation is not exact full GP. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Compare small-data full fits. 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. Linear scalability

This is where modelling choices begin to affect evidence. Fixed neighbour size bounds per-location computation.

A concrete situation

Runtime grows roughly with spot count.

What the mathematics does not license

Large gene counts still multiply work. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Benchmark end to end. 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. Spatial alternative model

The central mathematical idea is concrete. Fixed effects, GP covariance and nugget form the full fit.

A concrete situation

A gene retains fine structure after domain adjustment.

What the mathematics does not license

Optimization can meet local or boundary solutions. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Inspect convergence flags. 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. Nonspatial null

Here the geometry becomes operational. The null removes the spatial variance component.

A concrete situation

Only fixed effects and independent noise remain.

What the mathematics does not license

Null quality depends on covariates. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Check residual mean structure. 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. Likelihood-ratio and FDR

This step deserves a careful pause. Log-likelihood improvement is calibrated and corrected across genes.

A concrete situation

A gene ranks highly with adjusted p below threshold.

What the mathematics does not license

Asymptotic tails and filtering affect calls. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Report ranks, p-values and family 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.

16. Governance and learning

The useful question is not only how to calculate, but what the calculation assumes. Version ordering, m, covariance, covariates, optimizer and correction.

A concrete situation

A student writes six conditional factors.

What the mathematics does not license

Scale and significance answer different questions. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Write an nnSVG 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.

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

nnSVG fits a Gaussian process for each gene, but replaces the full dense likelihood with a nearest-neighbor Gaussian-process approximation. Each location is conditioned on a small set of earlier nearby locations. With a fixed neighbour count, the computation grows approximately linearly with the number of spatial locations.

The fitted covariance has a spatial variance, an independent nugget variance and a gene-specific length scale. A likelihood-ratio statistic compares the spatial model with a nonspatial linear model. The proportion of spatial variance is an effect-size summary, while the length scale describes the distance over which correlation decays.

Covariates can remove broad known domains before testing within-domain spatial variation. This is useful, but conditioning changes the question: a gene significant before adjustment and not after may be a domain marker, while a gene remaining significant may contain finer structure. Covariates must be chosen before inspecting the desired result.

Students can order six locations, choose two nearest predecessors for each one and write the approximate likelihood as conditional factors. Calculate spatial and nugget variance proportions, compare two exponential length scales and form a likelihood-ratio statistic. Reorder the locations to see why neighbour construction and reproducibility matter.

Frequently asked questions

What is nnSVG?

nnSVG identifies spatially variable genes with nearest-neighbor Gaussian-process models fitted gene by gene.

Why use a nearest-neighbor approximation?

A full Gaussian-process likelihood requires dense matrix operations, while conditioning each location on a small neighbour set can scale approximately linearly for fixed neighbour count.

What is the gene-specific length scale?

It is an estimated covariance-range parameter describing how quickly expression correlation decays with physical distance for that gene.

What is proportion of spatial variance?

It is estimated spatial variance divided by spatial plus nugget variance, used as a model-based effect-size summary.

How are genes tested?

A likelihood-ratio statistic compares the spatial Gaussian-process model with a nonspatial linear model, followed by multiple-testing correction.

Can covariates remove broad domains?

Yes, domain indicators may be included to test residual within-domain patterns, but this changes the scientific question and requires pre-specified covariates.

Primary source and fact-check note

Lukas M. Weber, Arkajyoti Saha, Abhirup Datta, Kasper D. Hansen and Stephanie C. Hicks, “nnSVG for the scalable identification of spatially variable genes using nearest-neighbor Gaussian processes,” Nature Communications 14, 4059 (2023), DOI 10.1038/s41467-023-39748-z. 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

Assay audit

Record logcounts or residual construction.

Covariate audit

State the before- versus within-domain question.

Order audit

Fix and record coordinate ordering.

Neighbour audit

Sweep m and compare small full-GP fits.

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.

Range audit

Report gene-specific length-scale limits.

Effect audit

Pair propSV with likelihood evidence.

Convergence audit

Inspect optimizer and boundary flags.

Final claim

Spatial range and significance are distinct.

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: Transformed expression

Begin with this mechanism: Each gene supplies log-normalized values or deviance residuals. Work through the concrete case: One vector covers all quality-controlled spots. Make the limitation visible: Input transformation changes Gaussian fit. Complete the evidence step: Document assay values. Write the coordinates, distances, count offsets, kernel entries, covariance matrices, variances, length scales, likelihoods, test statistics, p-values or correction rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, genes, regions or cell types outside tuning when testing transfer. Keep measurement, fitted model, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 2: Design matrix

Begin with this mechanism: Covariates enter as fixed effects before spatial covariance. Work through the concrete case: Known tissue domains receive indicator columns. Make the limitation visible: Post-hoc covariates can remove desired signal. Complete the evidence step: Choose them prospectively. Write the coordinates, distances, count offsets, kernel entries, covariance matrices, variances, length scales, likelihoods, test statistics, p-values or correction rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, genes, regions or cell types outside tuning when testing transfer. Keep measurement, fitted model, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 3: Spatial random effect

Begin with this mechanism: A zero-mean Gaussian process represents correlated residual variation. Work through the concrete case: Nearby residuals tend to move together. Make the limitation visible: Stationarity may fail across tissue. Complete the evidence step: Map local residuals. Write the coordinates, distances, count offsets, kernel entries, covariance matrices, variances, length scales, likelihoods, test statistics, p-values or correction rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, genes, regions or cell types outside tuning when testing transfer. Keep measurement, fitted model, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 4: Exponential covariance

Begin with this mechanism: Correlation decays exponentially with distance over a range parameter. Work through the concrete case: Half-millimetre structure has a longer scale than a vessel. Make the limitation visible: One scale may not fit multiscale genes. Complete the evidence step: Compare residual semivariograms. Write the coordinates, distances, count offsets, kernel entries, covariance matrices, variances, length scales, likelihoods, test statistics, p-values or correction rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, genes, regions or cell types outside tuning when testing transfer. Keep measurement, fitted model, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

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: Gene-specific length scale

Begin with this mechanism: Every gene estimates its own spatial range. Work through the concrete case: Immune and cortical markers occupy different scales. Make the limitation visible: Boundary estimates can be unstable. Complete the evidence step: Report profiles and limits. Write the coordinates, distances, count offsets, kernel entries, covariance matrices, variances, length scales, likelihoods, test statistics, p-values or correction rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, genes, regions or cell types outside tuning when testing transfer. Keep measurement, fitted model, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 6: Spatial variance

Begin with this mechanism: Sigma squared measures fitted correlated variation. Work through the concrete case: A layer marker has substantial spatial variance. Make the limitation visible: Magnitude depends on transformation. Complete the evidence step: Use standardized effect summaries. Write the coordinates, distances, count offsets, kernel entries, covariance matrices, variances, length scales, likelihoods, test statistics, p-values or correction rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, genes, regions or cell types outside tuning when testing transfer. Keep measurement, fitted model, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 7: Nugget variance

Begin with this mechanism: Tau squared captures independent residual variation. Work through the concrete case: Measurement noise remains at each spot. Make the limitation visible: Nugget mixes biology and technical noise. Complete the evidence step: Avoid causal labels. Write the coordinates, distances, count offsets, kernel entries, covariance matrices, variances, length scales, likelihoods, test statistics, p-values or correction rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, genes, regions or cell types outside tuning when testing transfer. Keep measurement, fitted model, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 8: Proportion spatial

Begin with this mechanism: Sigma squared is divided by sigma plus tau variance. Work through the concrete case: A value near one indicates dominant fitted spatial structure. Make the limitation visible: Large effect may be imprecise for weak genes. Complete the evidence step: Pair effect with uncertainty. Write the coordinates, distances, count offsets, kernel entries, covariance matrices, variances, length scales, likelihoods, test statistics, p-values or correction rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, genes, regions or cell types outside tuning when testing transfer. Keep measurement, fitted model, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

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: Location ordering

Begin with this mechanism: NNGP factorization conditions each site on earlier sites. Work through the concrete case: Coordinates are ordered before neighbour search. Make the limitation visible: Different orderings change the approximation. Complete the evidence step: Fix and record ordering. Write the coordinates, distances, count offsets, kernel entries, covariance matrices, variances, length scales, likelihoods, test statistics, p-values or correction rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, genes, regions or cell types outside tuning when testing transfer. Keep measurement, fitted model, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 10: Neighbour sets

Begin with this mechanism: Each location uses a small set of nearest predecessors. Work through the concrete case: Fifteen neighbours approximate local dependence. Make the limitation visible: Too few neighbours lose information. Complete the evidence step: Sweep m and runtime. Write the coordinates, distances, count offsets, kernel entries, covariance matrices, variances, length scales, likelihoods, test statistics, p-values or correction rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, genes, regions or cell types outside tuning when testing transfer. Keep measurement, fitted model, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 11: Conditional likelihood

Begin with this mechanism: The joint density becomes a product of local Gaussian conditionals. Work through the concrete case: One spot is predicted from its neighbour vector. Make the limitation visible: Approximation is not exact full GP. Complete the evidence step: Compare small-data full fits. Write the coordinates, distances, count offsets, kernel entries, covariance matrices, variances, length scales, likelihoods, test statistics, p-values or correction rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, genes, regions or cell types outside tuning when testing transfer. Keep measurement, fitted model, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 12: Linear scalability

Begin with this mechanism: Fixed neighbour size bounds per-location computation. Work through the concrete case: Runtime grows roughly with spot count. Make the limitation visible: Large gene counts still multiply work. Complete the evidence step: Benchmark end to end. Write the coordinates, distances, count offsets, kernel entries, covariance matrices, variances, length scales, likelihoods, test statistics, p-values or correction rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, genes, regions or cell types outside tuning when testing transfer. Keep measurement, fitted model, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

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: Spatial alternative model

Begin with this mechanism: Fixed effects, GP covariance and nugget form the full fit. Work through the concrete case: A gene retains fine structure after domain adjustment. Make the limitation visible: Optimization can meet local or boundary solutions. Complete the evidence step: Inspect convergence flags. Write the coordinates, distances, count offsets, kernel entries, covariance matrices, variances, length scales, likelihoods, test statistics, p-values or correction rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, genes, regions or cell types outside tuning when testing transfer. Keep measurement, fitted model, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 14: Nonspatial null

Begin with this mechanism: The null removes the spatial variance component. Work through the concrete case: Only fixed effects and independent noise remain. Make the limitation visible: Null quality depends on covariates. Complete the evidence step: Check residual mean structure. Write the coordinates, distances, count offsets, kernel entries, covariance matrices, variances, length scales, likelihoods, test statistics, p-values or correction rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, genes, regions or cell types outside tuning when testing transfer. Keep measurement, fitted model, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 15: Likelihood-ratio and FDR

Begin with this mechanism: Log-likelihood improvement is calibrated and corrected across genes. Work through the concrete case: A gene ranks highly with adjusted p below threshold. Make the limitation visible: Asymptotic tails and filtering affect calls. Complete the evidence step: Report ranks, p-values and family size. Write the coordinates, distances, count offsets, kernel entries, covariance matrices, variances, length scales, likelihoods, test statistics, p-values or correction rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, genes, regions or cell types outside tuning when testing transfer. Keep measurement, fitted model, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 16: Governance and learning

Begin with this mechanism: Version ordering, m, covariance, covariates, optimizer and correction. Work through the concrete case: A student writes six conditional factors. Make the limitation visible: Scale and significance answer different questions. Complete the evidence step: Write an nnSVG model card. Write the coordinates, distances, count offsets, kernel entries, covariance matrices, variances, length scales, likelihoods, test statistics, p-values or correction rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, genes, regions or cell types outside tuning when testing transfer. Keep measurement, fitted model, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.