Chapter map

1. 1. Spatial points
2. 2. Gene marks
3. 3. Point-pair distances
4. 4. Conditional mark law
5. 5. Mean-mark function
6. 6. Variance-mark function
7. 7. Stoyan mark correlation
8. 8. Mark variogram
9. 9. Permutation null
10. 10. Global radius test
11. 11. Two-sided evidence
12. 12. Gene-level adjustment
13. 13. Edge correction
14. 14. Weighted KDE
15. 15. Low-dimensional maps
16. 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.

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1. Spatial points

This is where modelling choices begin to affect evidence. Measured cells or regions provide fixed coordinates.

A concrete situation

Five cells occupy an irregular tissue patch.

What the mathematics does not license

Sampling density affects pair counts. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

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

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2. Gene marks

The central mathematical idea is concrete. One scalar expression value attaches to each point.

A concrete situation

A receptor count is the mark for every cell.

What the mathematics does not license

Zeros may be technical or biological. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Inspect detection rate. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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3. Point-pair distances

Here the geometry becomes operational. All cell pairs are grouped by separation radius.

A concrete situation

Pairs within 20 micrometres enter one bin.

What the mathematics does not license

Bin width changes resolution. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Repeat sensible radii. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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4. Conditional mark law

This step deserves a careful pause. Expression pairs are considered given their distance.

A concrete situation

Close pairs may share large marks.

What the mathematics does not license

Dependence can reflect cell composition. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Stratify or validate types. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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5. Mean-mark function

The useful question is not only how to calculate, but what the calculation assumes. Average marks among pairs are traced over distance.

A concrete situation

High local expression raises the short-range curve.

What the mathematics does not license

One intense point influences many pairs. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Run influence 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.

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6. Variance-mark function

A small example makes the mechanism visible. Conditional mark variance changes with distance.

A concrete situation

A boundary creates high short-range variance.

What the mathematics does not license

Variance alone does not show direction. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Plot contributing pairs. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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7. Stoyan mark correlation

This is where modelling choices begin to affect evidence. Mean mark products are normalized by the overall mean squared.

A concrete situation

Neighbouring high marks produce a ratio above one.

What the mathematics does not license

It is not an ordinary Pearson correlation. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

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

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8. Mark variogram

The central mathematical idea is concrete. Half the squared mark difference is averaged over pairs.

A concrete situation

Smooth fields have small short-range variograms.

What the mathematics does not license

Noise and boundaries both raise values. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Compare residual 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.

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9. Permutation null

Here the geometry becomes operational. Marks are reassigned while coordinates remain fixed.

A concrete situation

One thousand shuffles create null curves.

What the mathematics does not license

Exchangeability can fail under batches. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Permute within valid blocks. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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10. Global radius test

This step deserves a careful pause. The largest standardized deviation across radii enters the null comparison.

A concrete situation

One radius drives gene significance.

What the mathematics does not license

The winning radius is selected data-dependently. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Report the full curve. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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11. Two-sided evidence

The useful question is not only how to calculate, but what the calculation assumes. Absolute deviations detect unusually high or low summaries.

A concrete situation

A segregated pattern departs in either direction.

What the mathematics does not license

Direction needs the observed statistic. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Publish signed plots. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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12. Gene-level adjustment

A small example makes the mechanism visible. P-values are corrected across all tested genes.

A concrete situation

BH controls the expected discovery proportion.

What the mathematics does not license

Filtering changes the test family. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

State 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.

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13. Edge correction

This is where modelling choices begin to affect evidence. Boundary points receive correction for missing outside pairs.

A concrete situation

Ripley-style weights adjust exposed cells.

What the mathematics does not license

Irregular masks complicate edges. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

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

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14. Weighted KDE

The central mathematical idea is concrete. Expression weights produce a smoothed spatial intensity map.

A concrete situation

A local high-expression region appears.

What the mathematics does not license

Bandwidth controls hotspot size. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Show several bandwidths. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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15. Low-dimensional maps

Here the geometry becomes operational. The method can test marks on an embedding as well as physical tissue.

A concrete situation

A t-SNE neighbourhood shows a trend.

What the mathematics does not license

Embedding distance is not physical distance. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Name the geometry honestly. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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16. Governance and learning

This step deserves a careful pause. Version radii, permutations, edge correction, bandwidth and FDR.

A concrete situation

A student computes two variogram bins.

What the mathematics does not license

A spatial trend is an association. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

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

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A seven-day practice lab

1. Day 1: define the decision, observation unit, X variables and Y or reference condition.
2. Day 2: reproduce the smallest worked example and check every dimension, unit and sign.
3. Day 3: build a transparent baseline and write what it assumes.
4. Day 4: fit the main method inside a training-only pipeline; record every parameter.
5. Day 5: challenge one assumption with a deliberately difficult dataset or operating regime.
6. Day 6: compare validation metrics with the practical workload created by errors.
7. Day 7: present one supported conclusion, one unresolved uncertainty and one safe next experiment.

Questions for a student, parent or mentor

• Which part of the model is a mathematical definition, and which part is a choice?
• What small calculation would catch a sign, scale or dimension error?
• What evidence comes from training, validation and the untouched test?
• What benign condition could look like a fault or prediction failure?
• How would the answer change for another response, asset, school project or time period?
• What should a human review before acting on the score?

A method-specific audit before deployment

Trendsceek treats each measured cell or tissue region as a point and the expression of one gene as a numerical mark attached to that point. The question is whether the distribution of pairs of marks changes with the distance between their points. This marked-point-process view can detect nonlinear spatial trends without naming a region or pattern in advance.

Four summaries look at different pairwise features: conditional mean, conditional variance, Stoyan’s mark correlation and the mark variogram. Under spatial independence, these summaries should stay within what would be expected after expression marks are reassigned to fixed locations. Permutation supplies the empirical null rather than a strong parametric distribution.

Because many radii and genes are examined, the method uses maximum deviations across radii and then adjusts across genes. A weighted kernel-density step can locate unusually high-expression regions after a gene passes the global test. The localized map is exploratory evidence and inherits bandwidth, edge and multiplicity choices.

A hand calculation can use five points, one gene and two distance bins. List all eligible pairs, compute their average mark product and half squared difference, then permute the five marks. Comparing the observed curve with permutation maxima makes global spatial significance tangible.

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Frequently asked questions

What is Trendsceek?

Trendsceek is a nonparametric marked-point-process method that tests whether gene-expression marks depend on the distances between fixed spatial locations.

What is a mark?

For one gene, the mark is the expression value attached to a measured cell, spot or region represented as a point.

Which statistics are used?

The method evaluates conditional mean, conditional variance, Stoyan’s mark correlation and the mark variogram across pair-separation distances.

How is the null model created?

Expression marks are permuted without replacement over the fixed spatial points, breaking mark-location dependence while preserving the observed values.

Why adjust across radii?

Searching many distances creates multiple opportunities for an extreme deviation, so permutation maxima provide a global gene-level comparison.

Does a high-expression hotspot prove regulation there?

No. It is a smoothed statistical region affected by bandwidth, sampling, edge correction and confounding and needs biological validation.

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Primary source and fact-check note

Daniel Edsgärd, Per Johnsson and Rickard Sandberg, “Identification of spatial expression trends in single-cell gene expression data,” Nature Methods 15, 339–342 (2018), DOI 10.1038/nmeth.4634. 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

• SpatialDE Gaussian-process testing
• SPARK count and kernel tests
• SPARK-X scalable covariance testing
• How Mathematics Works

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

Point audit

Map density, mask and edge exposure.

Mark audit

Check detection and outliers.

Radius audit

Publish bins and pair counts.

Permutation audit

Use valid exchangeability blocks.

Multiplicity audit

Adjust radii and genes.

KDE audit

Sweep bandwidth and edge correction.

Validation audit

Compare cell types and independent tissue.

Final claim

A mark trend is spatial association.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Spatial points

Begin with this mechanism: Measured cells or regions provide fixed coordinates. Work through the concrete case: Five cells occupy an irregular tissue patch. Make the limitation visible: Sampling density affects pair counts. Complete the evidence step: Map density before testing. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 2: Gene marks

Begin with this mechanism: One scalar expression value attaches to each point. Work through the concrete case: A receptor count is the mark for every cell. Make the limitation visible: Zeros may be technical or biological. Complete the evidence step: Inspect detection rate. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 3: Point-pair distances

Begin with this mechanism: All cell pairs are grouped by separation radius. Work through the concrete case: Pairs within 20 micrometres enter one bin. Make the limitation visible: Bin width changes resolution. Complete the evidence step: Repeat sensible radii. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 4: Conditional mark law

Begin with this mechanism: Expression pairs are considered given their distance. Work through the concrete case: Close pairs may share large marks. Make the limitation visible: Dependence can reflect cell composition. Complete the evidence step: Stratify or validate types. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 5: Mean-mark function

Begin with this mechanism: Average marks among pairs are traced over distance. Work through the concrete case: High local expression raises the short-range curve. Make the limitation visible: One intense point influences many pairs. Complete the evidence step: Run influence checks. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 6: Variance-mark function

Begin with this mechanism: Conditional mark variance changes with distance. Work through the concrete case: A boundary creates high short-range variance. Make the limitation visible: Variance alone does not show direction. Complete the evidence step: Plot contributing pairs. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 7: Stoyan mark correlation

Begin with this mechanism: Mean mark products are normalized by the overall mean squared. Work through the concrete case: Neighbouring high marks produce a ratio above one. Make the limitation visible: It is not an ordinary Pearson correlation. Complete the evidence step: Use the correct interpretation. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 8: Mark variogram

Begin with this mechanism: Half the squared mark difference is averaged over pairs. Work through the concrete case: Smooth fields have small short-range variograms. Make the limitation visible: Noise and boundaries both raise values. Complete the evidence step: Compare residual maps. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 9: Permutation null

Begin with this mechanism: Marks are reassigned while coordinates remain fixed. Work through the concrete case: One thousand shuffles create null curves. Make the limitation visible: Exchangeability can fail under batches. Complete the evidence step: Permute within valid blocks. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 10: Global radius test

Begin with this mechanism: The largest standardized deviation across radii enters the null comparison. Work through the concrete case: One radius drives gene significance. Make the limitation visible: The winning radius is selected data-dependently. Complete the evidence step: Report the full curve. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 11: Two-sided evidence

Begin with this mechanism: Absolute deviations detect unusually high or low summaries. Work through the concrete case: A segregated pattern departs in either direction. Make the limitation visible: Direction needs the observed statistic. Complete the evidence step: Publish signed plots. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 12: Gene-level adjustment

Begin with this mechanism: P-values are corrected across all tested genes. Work through the concrete case: BH controls the expected discovery proportion. Make the limitation visible: Filtering changes the test family. Complete the evidence step: State inclusion rules. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 13: Edge correction

Begin with this mechanism: Boundary points receive correction for missing outside pairs. Work through the concrete case: Ripley-style weights adjust exposed cells. Make the limitation visible: Irregular masks complicate edges. Complete the evidence step: Use the real tissue outline. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 14: Weighted KDE

Begin with this mechanism: Expression weights produce a smoothed spatial intensity map. Work through the concrete case: A local high-expression region appears. Make the limitation visible: Bandwidth controls hotspot size. Complete the evidence step: Show several bandwidths. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 15: Low-dimensional maps

Begin with this mechanism: The method can test marks on an embedding as well as physical tissue. Work through the concrete case: A t-SNE neighbourhood shows a trend. Make the limitation visible: Embedding distance is not physical distance. Complete the evidence step: Name the geometry honestly. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, 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 radii, permutations, edge correction, bandwidth and FDR. Work through the concrete case: A student computes two variogram bins. Make the limitation visible: A spatial trend is an association. Complete the evidence step: Write a Trendsceek audit card. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.