WHY MATHEMATICS?

Why Mathematics? | SPARK for Spatial Expression Patterns: Count Models, Spatial Kernels, Penalized Quasi-Likelihood and Cauchy P-Value Combination

How SPARK tests gene-expression counts across tissue with several spatial geometries while separating count scale, spatial scale, significance and biological importance.

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 sparse counts with a generalized linear spatial model
Represent several pattern scales with spatial kernels
Fit efficiently with penalized quasi-likelihood
Combine kernel evidence while controlling transcriptome-wide error

Chapter map

1. Raw count response
2. Library-size offset
3. Link function
4. Fixed covariates
5. Spatial random effect
6. Gaussian kernels
7. Periodic kernels
8. Generalized spatial model
9. Penalized quasi-likelihood
10. Kernel score test
11. Mixture chi-square law
12. Kernel-specific p-values
13. Cauchy combination
14. Transcriptome-wide FDR
15. Power and effect
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. Raw count response

The central mathematical idea is concrete. Each gene supplies nonnegative counts across spatial locations.

A concrete situation

One gene has many zeros and a few large counts.

What the mathematics does not license

Zero inflation can reflect depth or biology. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Plot count distributions. 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. Library-size offset

Here the geometry becomes operational. An offset accounts for different total counts per location.

A concrete situation

A high-depth spot has a larger baseline mean.

What the mathematics does not license

Depth can correlate with tissue quality. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Map offsets spatially. 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. Link function

This step deserves a careful pause. A logarithmic link keeps fitted means positive.

A concrete situation

Linear predictors become multiplicative count effects.

What the mathematics does not license

Interpretation differs from additive expression. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Translate coefficients carefully. 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. Fixed covariates

The useful question is not only how to calculate, but what the calculation assumes. Measured nuisance or design variables enter the mean model.

A concrete situation

Batch or section is included before spatial testing.

What the mathematics does not license

Unmeasured confounding remains. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Pre-register covariates. 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. Spatial random effect

A small example makes the mechanism visible. A kernel covariance represents location-dependent latent variation.

A concrete situation

Nearby spots share a smooth random effect.

What the mathematics does not license

The effect is a model, not a measured field. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Inspect fitted maps. 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. Gaussian kernels

This is where modelling choices begin to affect evidence. Several bandwidths capture smooth hotspots at different scales.

A concrete situation

A compact immune niche fits a narrow kernel.

What the mathematics does not license

Bandwidth grids can miss unusual scales. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Show power across widths. 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. Periodic kernels

The central mathematical idea is concrete. Cosine-like kernels represent repeating spatial patterns.

A concrete situation

Alternating tissue bands create periodic covariance.

What the mathematics does not license

Periodic fit can be mathematically plausible but biologically odd. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Validate morphology. 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. Generalized spatial model

Here the geometry becomes operational. Counts, link, offsets, covariates and random effects combine.

A concrete situation

One gene has count overdispersion and a broad gradient.

What the mathematics does not license

Misspecified mean creates residual spatial signal. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Audit residuals first. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

9. Penalized quasi-likelihood

This step deserves a careful pause. Approximate weighted mixed-model updates fit each gene.

A concrete situation

Weights change with the current fitted mean.

What the mathematics does not license

PQL can be biased for low counts. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use calibration checks. 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. Kernel score test

The useful question is not only how to calculate, but what the calculation assumes. Evidence asks whether the spatial variance component is zero.

A concrete situation

A smooth gene yields a large quadratic score.

What the mathematics does not license

A score does not name the pattern cause. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Pair scores with maps. 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. Mixture chi-square law

A small example makes the mechanism visible. Kernel eigenvalues determine the null distribution of a quadratic statistic.

A concrete situation

Several weighted chi-square components form one p-value.

What the mathematics does not license

Numerical tails require care. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Check extreme-p calibration. 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. Kernel-specific p-values

This is where modelling choices begin to affect evidence. Each geometry supplies a separate test.

A concrete situation

A broad kernel detects one gene while a narrow one does not.

What the mathematics does not license

Trying many tests inflates opportunity. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use the planned omnibus rule. 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. Cauchy combination

The central mathematical idea is concrete. Transformed p-values are averaged and mapped back through arctangent.

A concrete situation

One strong kernel can influence the combined result.

What the mathematics does not license

Tiny numerical p-values need stable computation. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

State weights and clipping. 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. Transcriptome-wide FDR

Here the geometry becomes operational. Combined p-values are corrected across tested genes.

A concrete situation

A five-percent threshold yields a ranked candidate set.

What the mathematics does not license

Low-expression filtering changes the family. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Report all inclusion rules. 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. Power and effect

This step deserves a careful pause. Count modeling and multiple kernels can improve detection.

A concrete situation

A weak broad pattern becomes detectable with depth.

What the mathematics does not license

Power is not biological importance. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Add effect and replication criteria. 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 counts, offsets, covariates, kernels, weights and corrections.

A concrete situation

A student combines two p-values by hand.

What the mathematics does not license

Significance remains conditional. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

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

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

SPARK models spatial count data directly with a generalized linear spatial model. For sequencing data, a count distribution and link function connect the mean to observed library size, covariates and a spatial random effect. This avoids pretending that sparse counts are already continuous Gaussian measurements.

A single kernel can miss a pattern at the wrong scale. SPARK evaluates several Gaussian and periodic kernels, derives calibrated kernel-specific p-values from a mixture of chi-square distributions and combines them with a Cauchy rule. The combined test searches several plausible geometries while retaining one gene-level result.

Penalized quasi-likelihood makes the fitting procedure computationally practical, but it remains an approximation. Type-I-error control depends on the statistical construction, preprocessing and model assumptions. A small p-value indicates evidence against the no-spatial-effect null; it does not measure biological importance or causal regulation.

A worked example uses six locations, Poisson means proportional to library size and two spatial kernels. Calculate fitted mean counts, Pearson-style residuals, one kernel score and a Cauchy combination from two illustrative p-values. Then change the offset or bandwidth to see how count scale and spatial scale affect evidence differently.

Frequently asked questions

What is SPARK?

SPARK is Spatial PAttern Recognition via Kernels, a generalized linear spatial model for detecting spatially expressed genes from count data.

Why model counts directly?

Poisson-type mean and variance relationships, library-size offsets and sparse observations are not preserved by treating every transformed value as Gaussian.

Why use several kernels?

Genes can show broad gradients, local hotspots or periodic patterns at different scales, so one fixed covariance geometry may have low power.

What is penalized quasi-likelihood?

It is an approximate fitting method that turns a generalized spatial mixed model into computationally manageable updates.

How are kernel results combined?

Kernel-specific p-values are combined with a Cauchy rule to produce one omnibus gene-level p-value while accommodating dependence.

Does a smaller p-value mean a larger biological effect?

No. Significance also depends on count depth, sample size, model fit and noise; expression maps and effect summaries remain essential.

Primary source and fact-check note

Shiquan Sun, Jiaqiang Zhu and Xiang Zhou, “Statistical analysis of spatial expression patterns for spatially resolved transcriptomic studies,” Nature Methods 17, 193–200 (2020), DOI 10.1038/s41592-019-0701-7. 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

Count audit

Map depth, zeros and mean-variance structure.

Offset audit

Check library-size gradients.

Covariate audit

Pre-specify design variables.

Kernel audit

Publish scale-specific evidence.

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.

PQL audit

Test low-count calibration.

Combination audit

State Cauchy weights and numeric handling.

Replication audit

Require effect maps and independent tissue.

Final claim

A small p-value is not a large biological effect.

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: Raw count response

Begin with this mechanism: Each gene supplies nonnegative counts across spatial locations. Work through the concrete case: One gene has many zeros and a few large counts. Make the limitation visible: Zero inflation can reflect depth or biology. Complete the evidence step: Plot count distributions. 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: Library-size offset

Begin with this mechanism: An offset accounts for different total counts per location. Work through the concrete case: A high-depth spot has a larger baseline mean. Make the limitation visible: Depth can correlate with tissue quality. Complete the evidence step: Map offsets spatially. 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: Link function

Begin with this mechanism: A logarithmic link keeps fitted means positive. Work through the concrete case: Linear predictors become multiplicative count effects. Make the limitation visible: Interpretation differs from additive expression. Complete the evidence step: Translate coefficients carefully. 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: Fixed covariates

Begin with this mechanism: Measured nuisance or design variables enter the mean model. Work through the concrete case: Batch or section is included before spatial testing. Make the limitation visible: Unmeasured confounding remains. Complete the evidence step: Pre-register covariates. 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: Spatial random effect

Begin with this mechanism: A kernel covariance represents location-dependent latent variation. Work through the concrete case: Nearby spots share a smooth random effect. Make the limitation visible: The effect is a model, not a measured field. Complete the evidence step: Inspect fitted maps. 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: Gaussian kernels

Begin with this mechanism: Several bandwidths capture smooth hotspots at different scales. Work through the concrete case: A compact immune niche fits a narrow kernel. Make the limitation visible: Bandwidth grids can miss unusual scales. Complete the evidence step: Show power across widths. 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: Periodic kernels

Begin with this mechanism: Cosine-like kernels represent repeating spatial patterns. Work through the concrete case: Alternating tissue bands create periodic covariance. Make the limitation visible: Periodic fit can be mathematically plausible but biologically odd. Complete the evidence step: Validate morphology. 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: Generalized spatial model

Begin with this mechanism: Counts, link, offsets, covariates and random effects combine. Work through the concrete case: One gene has count overdispersion and a broad gradient. Make the limitation visible: Misspecified mean creates residual spatial signal. Complete the evidence step: Audit residuals first. 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: Penalized quasi-likelihood

Begin with this mechanism: Approximate weighted mixed-model updates fit each gene. Work through the concrete case: Weights change with the current fitted mean. Make the limitation visible: PQL can be biased for low counts. Complete the evidence step: Use calibration checks. 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: Kernel score test

Begin with this mechanism: Evidence asks whether the spatial variance component is zero. Work through the concrete case: A smooth gene yields a large quadratic score. Make the limitation visible: A score does not name the pattern cause. Complete the evidence step: Pair scores with maps. 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: Mixture chi-square law

Begin with this mechanism: Kernel eigenvalues determine the null distribution of a quadratic statistic. Work through the concrete case: Several weighted chi-square components form one p-value. Make the limitation visible: Numerical tails require care. Complete the evidence step: Check extreme-p calibration. 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: Kernel-specific p-values

Begin with this mechanism: Each geometry supplies a separate test. Work through the concrete case: A broad kernel detects one gene while a narrow one does not. Make the limitation visible: Trying many tests inflates opportunity. Complete the evidence step: Use the planned omnibus rule. 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: Cauchy combination

Begin with this mechanism: Transformed p-values are averaged and mapped back through arctangent. Work through the concrete case: One strong kernel can influence the combined result. Make the limitation visible: Tiny numerical p-values need stable computation. Complete the evidence step: State weights and clipping. 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: Transcriptome-wide FDR

Begin with this mechanism: Combined p-values are corrected across tested genes. Work through the concrete case: A five-percent threshold yields a ranked candidate set. Make the limitation visible: Low-expression filtering changes the family. Complete the evidence step: Report all inclusion rules. 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: Power and effect

Begin with this mechanism: Count modeling and multiple kernels can improve detection. Work through the concrete case: A weak broad pattern becomes detectable with depth. Make the limitation visible: Power is not biological importance. Complete the evidence step: Add effect and replication criteria. 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 counts, offsets, covariates, kernels, weights and corrections. Work through the concrete case: A student combines two p-values by hand. Make the limitation visible: Significance remains conditional. Complete the evidence step: Write a SPARK testing ledger. 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.