WHY MATHEMATICS?

Why Mathematics? | Moran’s I Spatial Autocorrelation: Weight Matrices, Expected Values, Permutation Tests and Local Clusters

How Moran’s I turns a map and a neighbourhood rule into a testable measure of spatial association without confusing autocorrelation with cause.

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

Define locations, values and a defensible weights matrix
Centre the data and calculate weighted cross-products
Compare I with its randomization expectation and permutations
Use local follow-up with multiplicity and validation controls

Chapter map

1. Observation vector
2. Global mean
3. Centred deviations
4. Neighbour definition
5. Weight matrix
6. Weight transformation
7. Total weight S0
8. Weighted cross-product
9. Total variation
10. Global I calculation
11. Expected value
12. Permutation distribution
13. Pseudo p-value
14. Moran scatterplot
15. Local indicators
16. Transfer and causality

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

1. Observation vector

The central mathematical idea is concrete. One numeric value is aligned to each spatial unit.

A concrete situation

Four areas have values 1, 2, 4 and 5.

What the mathematics does not license

Misaligned rows create false pairs. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Verify IDs before calculation. 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. Global mean

Here the geometry becomes operational. The arithmetic mean sets the centre for cross-products.

A concrete situation

The four values have mean 3.

What the mathematics does not license

An outlier moves the centre. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Inspect robust summaries as diagnostics. 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. Centred deviations

This step deserves a careful pause. Subtracting the mean yields z values.

A concrete situation

z equals −2, −1, 1 and 2.

What the mathematics does not license

Centred values are not standardized units. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Record the transformation. 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. Neighbour definition

The useful question is not only how to calculate, but what the calculation assumes. Contiguity, distance or kNN determines admissible pairs.

A concrete situation

A chain connects 1–2, 2–3 and 3–4.

What the mathematics does not license

A graph can encode a desired answer. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Choose rules before outcome inspection. 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. Weight matrix

A small example makes the mechanism visible. wij quantifies the contribution of j to i.

A concrete situation

A binary edge has weight one.

What the mathematics does not license

Directed and symmetric graphs count pairs differently. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Publish W and symmetry. 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. Weight transformation

This is where modelling choices begin to affect evidence. Row standardization makes each non-island row sum to one.

A concrete situation

Two neighbours receive one half each.

What the mathematics does not license

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

Try it, check it, transfer it

Compare binary and row-standardized results. 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. Total weight S0

The central mathematical idea is concrete. All wij values are summed.

A concrete situation

The symmetric chain has S0=6.

What the mathematics does not license

Self-weights alter the null formulas. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Set and report the diagonal. 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. Weighted cross-product

Here the geometry becomes operational. Every edge contributes wij zi zj.

A concrete situation

Directed chain products sum to 6.

What the mathematics does not license

A few hubs can dominate. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Map node contributions. 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. Total variation

This step deserves a careful pause. The denominator sums zi squared.

A concrete situation

4+1+1+4 equals 10.

What the mathematics does not license

Zero variance makes I undefined. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Check variance 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.

10. Global I calculation

The useful question is not only how to calculate, but what the calculation assumes. I=(n/S0) times cross-product over variation.

A concrete situation

(4/6)(6/10)=0.4.

What the mathematics does not license

The number has no meaning without W and null. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Report both. 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. Expected value

A small example makes the mechanism visible. Under randomization E[I]=−1/(n−1).

A concrete situation

For n=4, expectation is −1/3.

What the mathematics does not license

Alternative nulls have different variance assumptions. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

State the reference model. 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. Permutation distribution

This is where modelling choices begin to affect evidence. Values are shuffled while W stays fixed.

A concrete situation

999 shuffles produce 999 simulated I values.

What the mathematics does not license

Few permutations limit p-value resolution. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use a seed and enough replicates. 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. Pseudo p-value

The central mathematical idea is concrete. A rank count includes the observed arrangement.

A concrete situation

Nine as-or-more-extreme values among 999 give about 0.01.

What the mathematics does not license

One- and two-sided alternatives differ. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Declare direction in advance. 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. Moran scatterplot

Here the geometry becomes operational. Standardized z is plotted against its spatial lag.

A concrete situation

The fitted slope summarizes global association.

What the mathematics does not license

Leverage points distort the slope. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Label influential locations. 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. Local indicators

This step deserves a careful pause. Local products reveal high–high, low–low and spatial outliers.

A concrete situation

One high value surrounded by lows is high–low.

What the mathematics does not license

Many local tests inflate false positives. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use multiplicity control and 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.

16. Transfer and causality

The useful question is not only how to calculate, but what the calculation assumes. Independent regions or dates test stability.

A concrete situation

A held-out year retains positive I.

What the mathematics does not license

Stable association still is not mechanism. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Combine design, covariates and domain 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.

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

Moran’s I asks whether nearby places tend to carry similar values. It begins with a variable x, its mean, and a spatial weights matrix W. Each weight wij states how strongly location j counts as a neighbour of location i. The statistic multiplies mean-centred values across weighted pairs, sums those products, and scales the result by the total variation and the total weight. The answer therefore depends on both the data and the neighbourhood definition.

For n observations, centred values zi and S0 equal to the sum of all weights, global Moran’s I is n divided by S0, multiplied by the weighted cross-product sum, divided by the sum of squared z values. Positive products come from high–high or low–low pairs; negative products come from high–low pairs. Under the usual randomization reference with no self-neighbours, the expected value is minus one divided by n minus one, not exactly zero.

Permutation inference keeps W fixed and repeatedly shuffles observed values among locations. This creates a reference distribution for the exact map and attribute values. A pseudo p-value is a rank-based statement about extremeness under that shuffling scheme. It is not the probability that the null is true, and it does not repair a weights matrix chosen after examining the desired result.

Moran’s I is useful in geography, ecology, epidemiology, environmental monitoring and spatial omics, but one global number can hide opposing local patterns. A strong analysis maps the data, publishes W and its units, checks islands and degree, compares plausible neighbourhoods, examines a Moran scatterplot, controls multiplicity for local follow-up and preserves independent regions or time periods for validation.

Frequently asked questions

What is Moran’s I?

Moran’s I is a global statistic that compares weighted cross-products of mean-centred values with total variation across a spatial network.

What does a positive value mean?

Relative to the stated null expectation, positive spatial autocorrelation means similar high or low values tend to be connected by the chosen weights.

Why is the null expectation not zero?

For the common randomization reference with no self-neighbours, finite-sample dependence gives E[I]=−1/(n−1).

What is a spatial weights matrix?

It is an explicit numeric rule describing which locations are neighbours and how much each pair contributes.

What does a permutation p-value mean?

It ranks the observed statistic among values produced by shuffling attributes over fixed locations under a specified alternative.

Does Moran’s I identify a cause?

No. It detects association under a chosen graph; common exposures, trends, boundaries and measurement processes can all create spatial pattern.

Primary source and fact-check note

P. A. P. Moran, “Notes on Continuous Stochastic Phenomena,” Biometrika 37(1–2), 17–23 (1950), DOI 10.1093/biomet/37.1-2.17. 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

Alignment audit

Verify row IDs, geometry and value definitions.

Weights audit

Publish adjacency, units, symmetry, diagonal and transformation.

Influence audit

Map node contributions and leverage.

Null audit

State randomization, normality or design-based assumptions.

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.

Permutation audit

Record alternative, seed, restrictions and replicate count.

Local audit

Correct multiplicity and label cluster quadrants.

Transfer audit

Reserve independent regions or time periods.

Final claim

Spatial autocorrelation is association under W, not cause.

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: Observation vector

Begin with this mechanism: One numeric value is aligned to each spatial unit. Recreate the concrete case: Four areas have values 1, 2, 4 and 5. Write locations, values, means, centred deviations, weights, row sums, S0, cross-products, squared deviations, I values, expectations, permutation ranks, spatial lags and local contributions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Misaligned rows create false pairs. Complete the transfer step: Verify IDs before calculation. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.

Workshop 2: Global mean

Begin with this mechanism: The arithmetic mean sets the centre for cross-products. Recreate the concrete case: The four values have mean 3. Write locations, values, means, centred deviations, weights, row sums, S0, cross-products, squared deviations, I values, expectations, permutation ranks, spatial lags and local contributions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: An outlier moves the centre. Complete the transfer step: Inspect robust summaries as diagnostics. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.

Workshop 3: Centred deviations

Begin with this mechanism: Subtracting the mean yields z values. Recreate the concrete case: z equals −2, −1, 1 and 2. Write locations, values, means, centred deviations, weights, row sums, S0, cross-products, squared deviations, I values, expectations, permutation ranks, spatial lags and local contributions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Centred values are not standardized units. Complete the transfer step: Record the transformation. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.

Workshop 4: Neighbour definition

Begin with this mechanism: Contiguity, distance or kNN determines admissible pairs. Recreate the concrete case: A chain connects 1–2, 2–3 and 3–4. Write locations, values, means, centred deviations, weights, row sums, S0, cross-products, squared deviations, I values, expectations, permutation ranks, spatial lags and local contributions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: A graph can encode a desired answer. Complete the transfer step: Choose rules before outcome inspection. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients 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: Weight matrix

Begin with this mechanism: wij quantifies the contribution of j to i. Recreate the concrete case: A binary edge has weight one. Write locations, values, means, centred deviations, weights, row sums, S0, cross-products, squared deviations, I values, expectations, permutation ranks, spatial lags and local contributions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Directed and symmetric graphs count pairs differently. Complete the transfer step: Publish W and symmetry. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.

Workshop 6: Weight transformation

Begin with this mechanism: Row standardization makes each non-island row sum to one. Recreate the concrete case: Two neighbours receive one half each. Write locations, values, means, centred deviations, weights, row sums, S0, cross-products, squared deviations, I values, expectations, permutation ranks, spatial lags and local contributions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Transformation changes the estimand. Complete the transfer step: Compare binary and row-standardized results. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.

Workshop 7: Total weight S0

Begin with this mechanism: All wij values are summed. Recreate the concrete case: The symmetric chain has S0=6. Write locations, values, means, centred deviations, weights, row sums, S0, cross-products, squared deviations, I values, expectations, permutation ranks, spatial lags and local contributions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Self-weights alter the null formulas. Complete the transfer step: Set and report the diagonal. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.

Workshop 8: Weighted cross-product

Begin with this mechanism: Every edge contributes wij zi zj. Recreate the concrete case: Directed chain products sum to 6. Write locations, values, means, centred deviations, weights, row sums, S0, cross-products, squared deviations, I values, expectations, permutation ranks, spatial lags and local contributions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: A few hubs can dominate. Complete the transfer step: Map node contributions. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients 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: Total variation

Begin with this mechanism: The denominator sums zi squared. Recreate the concrete case: 4+1+1+4 equals 10. Write locations, values, means, centred deviations, weights, row sums, S0, cross-products, squared deviations, I values, expectations, permutation ranks, spatial lags and local contributions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Zero variance makes I undefined. Complete the transfer step: Check variance first. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.

Workshop 10: Global I calculation

Begin with this mechanism: I=(n/S0) times cross-product over variation. Recreate the concrete case: (4/6)(6/10)=0.4. Write locations, values, means, centred deviations, weights, row sums, S0, cross-products, squared deviations, I values, expectations, permutation ranks, spatial lags and local contributions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: The number has no meaning without W and null. Complete the transfer step: Report both. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.

Workshop 11: Expected value

Begin with this mechanism: Under randomization E[I]=−1/(n−1). Recreate the concrete case: For n=4, expectation is −1/3. Write locations, values, means, centred deviations, weights, row sums, S0, cross-products, squared deviations, I values, expectations, permutation ranks, spatial lags and local contributions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Alternative nulls have different variance assumptions. Complete the transfer step: State the reference model. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.

Workshop 12: Permutation distribution

Begin with this mechanism: Values are shuffled while W stays fixed. Recreate the concrete case: 999 shuffles produce 999 simulated I values. Write locations, values, means, centred deviations, weights, row sums, S0, cross-products, squared deviations, I values, expectations, permutation ranks, spatial lags and local contributions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Few permutations limit p-value resolution. Complete the transfer step: Use a seed and enough replicates. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients 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: Pseudo p-value

Begin with this mechanism: A rank count includes the observed arrangement. Recreate the concrete case: Nine as-or-more-extreme values among 999 give about 0.01. Write locations, values, means, centred deviations, weights, row sums, S0, cross-products, squared deviations, I values, expectations, permutation ranks, spatial lags and local contributions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: One- and two-sided alternatives differ. Complete the transfer step: Declare direction in advance. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.

Workshop 14: Moran scatterplot

Begin with this mechanism: Standardized z is plotted against its spatial lag. Recreate the concrete case: The fitted slope summarizes global association. Write locations, values, means, centred deviations, weights, row sums, S0, cross-products, squared deviations, I values, expectations, permutation ranks, spatial lags and local contributions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Leverage points distort the slope. Complete the transfer step: Label influential locations. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.

Workshop 15: Local indicators

Begin with this mechanism: Local products reveal high–high, low–low and spatial outliers. Recreate the concrete case: One high value surrounded by lows is high–low. Write locations, values, means, centred deviations, weights, row sums, S0, cross-products, squared deviations, I values, expectations, permutation ranks, spatial lags and local contributions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Many local tests inflate false positives. Complete the transfer step: Use multiplicity control and maps. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.

Workshop 16: Transfer and causality

Begin with this mechanism: Independent regions or dates test stability. Recreate the concrete case: A held-out year retains positive I. Write locations, values, means, centred deviations, weights, row sums, S0, cross-products, squared deviations, I values, expectations, permutation ranks, spatial lags and local contributions with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Stable association still is not mechanism. Complete the transfer step: Combine design, covariates and domain evidence. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.