WHY MATHEMATICS?

Why Mathematics? | SpatialDWLS Cell Fractions: PAGE Enrichment, Weighted Least Squares, Damping and Two-Round Deconvolution

How SpatialDWLS screens plausible cell types, fits dampened weighted mixtures and refits after removing negligible components while preserving an auditable error model.

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

Calculate a PAGE enrichment score from marker fold changes
Build a nonnegative reference-mixture regression
Use weights and damping to stabilize relative error
Challenge the rare-cell cutoff before refitting

Chapter map

1. Reference profiles
2. Marker-gene sets
3. Spot fold change
4. Marker-set mean
5. Background mean and spread
6. PAGE enrichment score
7. Eligibility cutoff
8. Mixture design matrix
9. Fraction vector
10. Reconstruction residual
11. Relative-error weights
12. Damping constant
13. Cluster-shared settings
14. First-round solution
15. Low-frequency filter
16. Second-round refit

Visit the How Mathematics Works hub for the wider series and its connections to education, reasoning, science, computing, design and careers.

1. Reference profiles

The central mathematical idea is concrete. Columns describe mean expression for candidate cell types.

A concrete situation

Matrix A has 500 genes by ten types.

What the mathematics does not license

Closely related types are collinear. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

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

2. Marker-gene sets

Here the geometry becomes operational. Each type receives a documented marker list.

A concrete situation

Type A has 25 markers.

What the mathematics does not license

Markers can change by tissue state. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use context-specific evidence. 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. Spot fold change

This step deserves a careful pause. A gene value is divided by its across-spot mean.

A concrete situation

Expression six over mean three gives two.

What the mathematics does not license

Low means create instability. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Set zero and low-count 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.

4. Marker-set mean

The useful question is not only how to calculate, but what the calculation assumes. Fold changes are averaged across m markers.

A concrete situation

Twenty-five markers average 1.6.

What the mathematics does not license

A few outliers can dominate. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Inspect the distribution. 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. Background mean and spread

A small example makes the mechanism visible. All-gene fold changes define mu and delta.

A concrete situation

Mu is one and delta is 0.3.

What the mathematics does not license

Genes are correlated. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Treat PAGE as a screen. 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. PAGE enrichment score

This is where modelling choices begin to affect evidence. ES equals (Sm minus mu) times sqrt(m) over delta.

A concrete situation

(1.6-1)*5/0.3 equals ten.

What the mathematics does not license

Marker count boosts evidence. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Compare balanced marker sets. 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. Eligibility cutoff

The central mathematical idea is concrete. Scores above a rule enter deconvolution.

A concrete situation

ES greater than two is eligible.

What the mathematics does not license

Near-cutoff types can flip. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Sweep thresholds. 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. Mixture design matrix

Here the geometry becomes operational. Eligible reference profiles form matrix A.

A concrete situation

Three columns remain for one spot.

What the mathematics does not license

A missing type forces misallocation. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Inspect structured 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.

9. Fraction vector

This step deserves a careful pause. Nonnegative coefficients describe composition.

A concrete situation

p equals [0.5,0.3,0.2].

What the mathematics does not license

RNA fractions need not equal cell counts. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use careful 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.

10. Reconstruction residual

The useful question is not only how to calculate, but what the calculation assumes. Observed y is compared with A times p.

A concrete situation

Residual gene g is two counts.

What the mathematics does not license

Residuals share technical structure. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Plot by expression and gene class. 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. Relative-error weights

A small example makes the mechanism visible. Weights emphasize proportional fit across genes.

A concrete situation

A low-expression gene receives more weight.

What the mathematics does not license

Tiny denominators explode. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use damping. 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. Damping constant

This is where modelling choices begin to affect evidence. A positive d stabilizes the weight denominator.

A concrete situation

Denominator fitted value plus 0.5.

What the mathematics does not license

Large d changes the objective. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

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

13. Cluster-shared settings

The central mathematical idea is concrete. Similar spots reuse weights and damping.

A concrete situation

One tissue cluster shares d.

What the mathematics does not license

Sharing can oversmooth boundaries. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Compare local fits. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

14. First-round solution

Here the geometry becomes operational. Weighted nonnegative least squares estimates all eligible types.

A concrete situation

Type C receives 0.015.

What the mathematics does not license

A small value may be noise or rarity. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

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

15. Low-frequency filter

This step deserves a careful pause. Components below a threshold are removed.

A concrete situation

Fractions below 0.02 leave the model.

What the mathematics does not license

Rare types can disappear. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Plot recall versus cutoff. 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. Second-round refit

The useful question is not only how to calculate, but what the calculation assumes. The reduced model is fitted again and validated.

A concrete situation

Remaining fractions renormalize to one.

What the mathematics does not license

Refitting changes every coefficient. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Publish both rounds and 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.

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

SpatialDWLS separates a deconvolution problem into screening and estimation. First, Parametric Analysis of Gene Set Enrichment uses marker-gene fold changes to ask which cell types are plausible at each spatial location. The PAGE score standardizes the marker-set mean against the background mean and standard deviation and multiplies by the square root of the marker count. A declared cutoff converts this continuous evidence into an eligible cell-type set.

Second, dampened weighted least squares represents the observed spot profile as a mixture of eligible reference profiles. Residuals are weighted so the fit pays attention to relative rather than only absolute error. A damping constant prevents genes with tiny fitted values from receiving unstable influence, and cross-validation is used to choose that constant. Nonnegative mixture weights are normalized into estimated fractions.

The workflow performs a first deconvolution, removes cell types with very small estimated frequency and fits again. This second round can improve specificity when only a few types occupy a location, but it can also delete a real rare population. Marker selection, enrichment threshold, low-frequency cutoff, reference similarity and platform differences therefore belong in the sensitivity analysis.

SpatialDWLS demonstrates why solving a regression equation is only one part of mathematical modelling. The screen defines the hypothesis space, the weights define what error matters, damping controls numerical stability and the refit changes the candidate set. A trustworthy result reports all four, retains residuals and validates fractions on synthetic mixtures or tissue with independent cell counts.

Frequently asked questions

What is SpatialDWLS?

SpatialDWLS is a two-stage cell-type deconvolution method combining PAGE enrichment screening with dampened weighted least squares.

What does PAGE calculate?

It standardizes the mean fold change of a marker set against all-gene fold-change variation and scales it by the square root of marker count.

Why screen cell types first?

A spatial location usually contains few types. Screening narrows the mixture model and can reduce false positive fractions.

Why use weighted least squares?

Gene-specific weights make the fit reflect relative reconstruction error rather than letting highly expressed genes dominate.

What does damping do?

A positive damping constant prevents tiny fitted values from causing extreme weights and improves numerical stability.

Can the second-round threshold remove a real rare cell type?

Yes. The cutoff trades specificity against rare-population recall and must be tested across plausible values.

Primary source and fact-check note

Rui Dong and Guo-Cheng Yuan, “SpatialDWLS: accurate deconvolution of spatial transcriptomic data,” Genome Biology 22, 145 (2021), DOI 10.1186/s13059-021-02362-7. 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

Marker audit

Version cell-type markers and tissue context.

PAGE audit

Keep fold changes, background mean, spread and cutoff.

Matrix audit

Inspect reference collinearity and missing types.

Weight audit

Publish the relative-error formula.

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.

Damping audit

Show cross-validation and sensitivity for d.

Threshold audit

Measure rare-type recall across cutoffs.

Refit audit

Store first- and second-round fractions and residuals.

Final claim

Regression fractions depend on screening and reference quality.

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: Reference profiles

Begin with this mechanism: Columns describe mean expression for candidate cell types. Recreate the concrete case: Matrix A has 500 genes by ten types. Write marker fold changes, set means, background means, standard deviations, PAGE scores, design entries, weights, damping, fractions and residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Closely related types are collinear. Complete the transfer step: Inspect condition and marker separation. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 2: Marker-gene sets

Begin with this mechanism: Each type receives a documented marker list. Recreate the concrete case: Type A has 25 markers. Write marker fold changes, set means, background means, standard deviations, PAGE scores, design entries, weights, damping, fractions and residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Markers can change by tissue state. Complete the transfer step: Use context-specific evidence. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 3: Spot fold change

Begin with this mechanism: A gene value is divided by its across-spot mean. Recreate the concrete case: Expression six over mean three gives two. Write marker fold changes, set means, background means, standard deviations, PAGE scores, design entries, weights, damping, fractions and residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Low means create instability. Complete the transfer step: Set zero and low-count rules. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 4: Marker-set mean

Begin with this mechanism: Fold changes are averaged across m markers. Recreate the concrete case: Twenty-five markers average 1.6. Write marker fold changes, set means, background means, standard deviations, PAGE scores, design entries, weights, damping, fractions and residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: A few outliers can dominate. Complete the transfer step: Inspect the distribution. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

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: Background mean and spread

Begin with this mechanism: All-gene fold changes define mu and delta. Recreate the concrete case: Mu is one and delta is 0.3. Write marker fold changes, set means, background means, standard deviations, PAGE scores, design entries, weights, damping, fractions and residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Genes are correlated. Complete the transfer step: Treat PAGE as a screen. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 6: PAGE enrichment score

Begin with this mechanism: ES equals (Sm minus mu) times sqrt(m) over delta. Recreate the concrete case: (1.6-1)*5/0.3 equals ten. Write marker fold changes, set means, background means, standard deviations, PAGE scores, design entries, weights, damping, fractions and residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Marker count boosts evidence. Complete the transfer step: Compare balanced marker sets. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 7: Eligibility cutoff

Begin with this mechanism: Scores above a rule enter deconvolution. Recreate the concrete case: ES greater than two is eligible. Write marker fold changes, set means, background means, standard deviations, PAGE scores, design entries, weights, damping, fractions and residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Near-cutoff types can flip. Complete the transfer step: Sweep thresholds. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 8: Mixture design matrix

Begin with this mechanism: Eligible reference profiles form matrix A. Recreate the concrete case: Three columns remain for one spot. Write marker fold changes, set means, background means, standard deviations, PAGE scores, design entries, weights, damping, fractions and residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: A missing type forces misallocation. Complete the transfer step: Inspect structured residuals. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

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: Fraction vector

Begin with this mechanism: Nonnegative coefficients describe composition. Recreate the concrete case: p equals [0.5,0.3,0.2]. Write marker fold changes, set means, background means, standard deviations, PAGE scores, design entries, weights, damping, fractions and residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: RNA fractions need not equal cell counts. Complete the transfer step: Use careful labels. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 10: Reconstruction residual

Begin with this mechanism: Observed y is compared with A times p. Recreate the concrete case: Residual gene g is two counts. Write marker fold changes, set means, background means, standard deviations, PAGE scores, design entries, weights, damping, fractions and residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Residuals share technical structure. Complete the transfer step: Plot by expression and gene class. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 11: Relative-error weights

Begin with this mechanism: Weights emphasize proportional fit across genes. Recreate the concrete case: A low-expression gene receives more weight. Write marker fold changes, set means, background means, standard deviations, PAGE scores, design entries, weights, damping, fractions and residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Tiny denominators explode. Complete the transfer step: Use damping. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 12: Damping constant

Begin with this mechanism: A positive d stabilizes the weight denominator. Recreate the concrete case: Denominator fitted value plus 0.5. Write marker fold changes, set means, background means, standard deviations, PAGE scores, design entries, weights, damping, fractions and residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Large d changes the objective. Complete the transfer step: Choose by cross-validation. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

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: Cluster-shared settings

Begin with this mechanism: Similar spots reuse weights and damping. Recreate the concrete case: One tissue cluster shares d. Write marker fold changes, set means, background means, standard deviations, PAGE scores, design entries, weights, damping, fractions and residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Sharing can oversmooth boundaries. Complete the transfer step: Compare local fits. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 14: First-round solution

Begin with this mechanism: Weighted nonnegative least squares estimates all eligible types. Recreate the concrete case: Type C receives 0.015. Write marker fold changes, set means, background means, standard deviations, PAGE scores, design entries, weights, damping, fractions and residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: A small value may be noise or rarity. Complete the transfer step: Retain uncertainty. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 15: Low-frequency filter

Begin with this mechanism: Components below a threshold are removed. Recreate the concrete case: Fractions below 0.02 leave the model. Write marker fold changes, set means, background means, standard deviations, PAGE scores, design entries, weights, damping, fractions and residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Rare types can disappear. Complete the transfer step: Plot recall versus cutoff. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 16: Second-round refit

Begin with this mechanism: The reduced model is fitted again and validated. Recreate the concrete case: Remaining fractions renormalize to one. Write marker fold changes, set means, background means, standard deviations, PAGE scores, design entries, weights, damping, fractions and residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Refitting changes every coefficient. Complete the transfer step: Publish both rounds and residuals. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.