WHY MATHEMATICS?

Why Mathematics? | SpatialDE for Spatially Variable Genes: Gaussian Processes, Covariance Kernels, Length Scales and Variance Decomposition

How SpatialDE turns distances between tissue locations into gene-by-gene covariance models, tests spatial variation and distinguishes effect size, pattern scale and statistical significance.

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

Turn pairwise distances into a spatial covariance matrix
Separate spatial variance from independent noise
Compare a spatial model with a no-spatial null
Interpret length scale and false-discovery-controlled results

Chapter map

1. Expression vector
2. Coordinate matrix
3. Pairwise distances
4. Squared-exponential kernel
5. Length scale
6. Spatial variance
7. Independent noise
8. Fraction of spatial variance
9. Marginal likelihood
10. Null model
11. Likelihood-ratio test
12. Multiple testing
13. Alternative kernels
14. BIC pattern choice
15. Automatic expression histology
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. Expression vector

A small example makes the mechanism visible. One gene supplies a normalized value at every measured location.

A concrete situation

A marker is measured at fifty tissue spots.

What the mathematics does not license

Transformation changes covariance. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Record counts, normalization and filtering. 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. Coordinate matrix

This is where modelling choices begin to affect evidence. Each location has two- or three-dimensional physical coordinates.

A concrete situation

Spot i is at 120 and 80 micrometres.

What the mathematics does not license

Pixel indices are not physical units. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Verify scale and axes. 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. Pairwise distances

The central mathematical idea is concrete. Euclidean distances summarize separation between locations.

A concrete situation

Two adjacent spots are 100 micrometres apart.

What the mathematics does not license

Barriers and anisotropy can violate distance similarity. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Test alternative metrics. 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. Squared-exponential kernel

Here the geometry becomes operational. Covariance decays exponentially with squared distance.

A concrete situation

Close spots receive a kernel value near one.

What the mathematics does not license

Smooth kernels blur abrupt boundaries. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Plot kernel versus distance. 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. Length scale

This step deserves a careful pause. A parameter controls the distance of correlation decay.

A concrete situation

A small value captures a compact hotspot.

What the mathematics does not license

Grid limits constrain estimates. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Report the search grid. 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

The useful question is not only how to calculate, but what the calculation assumes. A scale multiplies the spatial kernel.

A concrete situation

One gene has strong location-dependent covariance.

What the mathematics does not license

Variance depends on preprocessing. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Compare robust transformations. 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. Independent noise

A small example makes the mechanism visible. A diagonal term captures nonspatial variation.

A concrete situation

Replicate-like fluctuations remain after the spatial fit.

What the mathematics does not license

Noise may be heteroscedastic. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Inspect residual variance by mean. 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. Fraction of spatial variance

This is where modelling choices begin to affect evidence. Spatial variance is divided by total variance.

A concrete situation

A fitted FSV of 0.7 indicates a large model share.

What the mathematics does not license

It is not variance caused by anatomy. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Use effect and uncertainty 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.

9. Marginal likelihood

The central mathematical idea is concrete. Mean, variance and covariance determine joint Gaussian plausibility.

A concrete situation

A kernel explains a smooth expression gradient.

What the mathematics does not license

Gaussian tails may misfit normalized counts. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use predictive 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.

10. Null model

Here the geometry becomes operational. The no-spatial model retains mean and independent variance only.

A concrete situation

Shuffling coordinates should remove the advantage.

What the mathematics does not license

A null can omit confounders. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Add planned covariates when supported. 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. Likelihood-ratio test

This step deserves a careful pause. Twice the log-likelihood difference measures evidence for spatial covariance.

A concrete situation

A patterned gene improves the full fit.

What the mathematics does not license

Boundary parameters complicate asymptotics. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Check calibration and permutations. 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. Multiple testing

The useful question is not only how to calculate, but what the calculation assumes. Thousands of gene tests require false-discovery control.

A concrete situation

Q-values rank evidence across the transcriptome.

What the mathematics does not license

FDR is an expected proportion, not a guarantee per gene. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Report tested gene 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.

13. Alternative kernels

A small example makes the mechanism visible. Linear and periodic covariance functions represent different pattern families.

A concrete situation

A repeating stripe fits a periodic kernel.

What the mathematics does not license

The library cannot express every geometry. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Inspect maps, not only 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.

14. BIC pattern choice

This is where modelling choices begin to affect evidence. Penalized likelihood compares covariance families.

A concrete situation

A periodic model beats a linear trend after penalty.

What the mathematics does not license

Selection uncertainty is often hidden. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Show competing BIC 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.

15. Automatic expression histology

The central mathematical idea is concrete. Spatially variable genes can be grouped into shared patterns.

A concrete situation

A gene cluster outlines one tissue region.

What the mathematics does not license

Co-patterning is not co-regulation. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Validate modules independently. 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

Here the geometry becomes operational. Version expression, coordinates, kernels, grids, null and correction method.

A concrete situation

A student builds one five-by-five kernel.

What the mathematics does not license

A p-value is not a mechanism. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Write a SpatialDE evidence 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

SpatialDE asks whether variation in one gene is partly explained by distance between measured locations. It represents the expression vector with a multivariate Gaussian model whose covariance is the sum of a spatial kernel and independent noise. A likelihood comparison against a no-spatial-covariance model turns this geometric idea into a statistical test.

The squared-exponential kernel makes covariance decay with squared distance. Its characteristic length scale tells us how quickly a fitted pattern changes across tissue. The fraction of spatial variance compares the fitted spatial component with total variance. Neither number is an anatomical label; both are model-dependent summaries of one gene.

After significance testing, alternative linear, periodic and squared-exponential covariance patterns can be compared with BIC. Multiple genes require false-discovery-rate control. Normalization, uneven sampling, duplicated tissue regions and unmodeled cell-type composition can all create apparent patterns, so the p-value is the beginning of interpretation rather than its end.

A student can place five spots on a line, compute their squared distances, build two Gaussian kernels and add a diagonal noise matrix. Calculate one quadratic-form contribution to the marginal likelihood, compare the spatial and null models and see how the preferred length scale changes after one outlying expression value.

Frequently asked questions

What is SpatialDE?

SpatialDE is a gene-by-gene Gaussian-process method that tests whether expression covariance contains a spatial component related to measurement coordinates.

What is a covariance kernel?

It is a rule that converts two coordinates into expected covariance; the squared-exponential kernel gives nearby locations larger covariance than distant ones.

What does length scale mean?

It controls how quickly fitted spatial correlation decays with distance, so smaller values describe finer-scale change under the model.

What is fraction of spatial variance?

It is the fitted spatial variance divided by total spatial plus independent variance, a model-based effect-size summary.

How is significance tested?

The fitted spatial model is compared with a null model containing only independent covariance, followed by multiple-testing correction across genes.

Does a spatially variable gene prove a tissue mechanism?

No. Spatial association can reflect anatomy, cell composition, technical gradients or other covariates and needs independent biological validation.

Primary source and fact-check note

Valentine Svensson, Sarah A. Teichmann and Oliver Stegle, “SpatialDE: identification of spatially variable genes,” Nature Methods 15, 343–346 (2018), DOI 10.1038/nmeth.4636. 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

Input audit

Repeat normalization and filtering.

Distance audit

Verify coordinates, units and anisotropy.

Kernel audit

Plot several length scales and families.

Likelihood audit

Inspect residuals and null calibration.

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.

Effect audit

Report FSV with significance.

Multiplicity audit

State tested genes and FDR method.

Pattern audit

Show competing BIC values and maps.

Final claim

Spatial covariance is evidence, not mechanism.

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: Expression vector

Begin with this mechanism: One gene supplies a normalized value at every measured location. Work through the concrete case: A marker is measured at fifty tissue spots. Make the limitation visible: Transformation changes covariance. Complete the evidence step: Record counts, normalization and filtering. 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: Coordinate matrix

Begin with this mechanism: Each location has two- or three-dimensional physical coordinates. Work through the concrete case: Spot i is at 120 and 80 micrometres. Make the limitation visible: Pixel indices are not physical units. Complete the evidence step: Verify scale and axes. 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: Pairwise distances

Begin with this mechanism: Euclidean distances summarize separation between locations. Work through the concrete case: Two adjacent spots are 100 micrometres apart. Make the limitation visible: Barriers and anisotropy can violate distance similarity. Complete the evidence step: Test alternative metrics. 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: Squared-exponential kernel

Begin with this mechanism: Covariance decays exponentially with squared distance. Work through the concrete case: Close spots receive a kernel value near one. Make the limitation visible: Smooth kernels blur abrupt boundaries. Complete the evidence step: Plot kernel versus distance. 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: Length scale

Begin with this mechanism: A parameter controls the distance of correlation decay. Work through the concrete case: A small value captures a compact hotspot. Make the limitation visible: Grid limits constrain estimates. Complete the evidence step: Report the search grid. 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: A scale multiplies the spatial kernel. Work through the concrete case: One gene has strong location-dependent covariance. Make the limitation visible: Variance depends on preprocessing. Complete the evidence step: Compare robust transformations. 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: Independent noise

Begin with this mechanism: A diagonal term captures nonspatial variation. Work through the concrete case: Replicate-like fluctuations remain after the spatial fit. Make the limitation visible: Noise may be heteroscedastic. Complete the evidence step: Inspect residual variance by mean. 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: Fraction of spatial variance

Begin with this mechanism: Spatial variance is divided by total variance. Work through the concrete case: A fitted FSV of 0.7 indicates a large model share. Make the limitation visible: It is not variance caused by anatomy. Complete the evidence step: Use effect and uncertainty together. 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: Marginal likelihood

Begin with this mechanism: Mean, variance and covariance determine joint Gaussian plausibility. Work through the concrete case: A kernel explains a smooth expression gradient. Make the limitation visible: Gaussian tails may misfit normalized counts. Complete the evidence step: Use predictive 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 10: Null model

Begin with this mechanism: The no-spatial model retains mean and independent variance only. Work through the concrete case: Shuffling coordinates should remove the advantage. Make the limitation visible: A null can omit confounders. Complete the evidence step: Add planned covariates when supported. 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: Likelihood-ratio test

Begin with this mechanism: Twice the log-likelihood difference measures evidence for spatial covariance. Work through the concrete case: A patterned gene improves the full fit. Make the limitation visible: Boundary parameters complicate asymptotics. Complete the evidence step: Check calibration and permutations. 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: Multiple testing

Begin with this mechanism: Thousands of gene tests require false-discovery control. Work through the concrete case: Q-values rank evidence across the transcriptome. Make the limitation visible: FDR is an expected proportion, not a guarantee per gene. Complete the evidence step: Report tested gene count. 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: Alternative kernels

Begin with this mechanism: Linear and periodic covariance functions represent different pattern families. Work through the concrete case: A repeating stripe fits a periodic kernel. Make the limitation visible: The library cannot express every geometry. Complete the evidence step: Inspect maps, not only 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 14: BIC pattern choice

Begin with this mechanism: Penalized likelihood compares covariance families. Work through the concrete case: A periodic model beats a linear trend after penalty. Make the limitation visible: Selection uncertainty is often hidden. Complete the evidence step: Show competing BIC 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 15: Automatic expression histology

Begin with this mechanism: Spatially variable genes can be grouped into shared patterns. Work through the concrete case: A gene cluster outlines one tissue region. Make the limitation visible: Co-patterning is not co-regulation. Complete the evidence step: Validate modules independently. 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 expression, coordinates, kernels, grids, null and correction method. Work through the concrete case: A student builds one five-by-five kernel. Make the limitation visible: A p-value is not a mechanism. Complete the evidence step: Write a SpatialDE evidence 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.