Chapter map

1. 1. Series definition
2. 2. Rank values
3. 3. Candidate splits
4. 4. Rank sum
5. 5. Ut formula
6. 6. Early split
7. 7. Second split
8. 8. Maximum K
9. 9. Candidate location
10. 10. Pairwise signs
11. 11. Permutation null
12. 12. Approximate p-value
13. 13. Effect direction
14. 14. Multiple changes
15. 15. Trend and seasonality
16. 16. External validation

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. Series definition

This step deserves a careful pause. Fix ordered observations and analysis period.

A concrete situation

Values are 2,3,4,9,10.

What the mathematics does not license

Choosing dates after viewing data biases results. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Predefine the window. 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. Rank values

The useful question is not only how to calculate, but what the calculation assumes. Assign ranks across the complete series.

A concrete situation

Ranks are 1,2,3,4,5.

What the mathematics does not license

Ties need midranks or another rule. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

State the convention. 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. Candidate splits

A small example makes the mechanism visible. Consider t from 1 to n−1.

A concrete situation

Four splits are possible.

What the mathematics does not license

Endpoints can be unstable. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Consider minimum segment size. 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. Rank sum

This is where modelling choices begin to affect evidence. Compute Σi≤t ri.

A concrete situation

At t=3 the sum is 6.

What the mathematics does not license

A coding offset changes every value. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Check by hand. 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. Ut formula

The central mathematical idea is concrete. Compute 2 rank sum−t(n+1).

A concrete situation

U3=12−18=−6.

What the mathematics does not license

Sign depends on ordering convention. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Interpret magnitude 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.

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6. Early split

Here the geometry becomes operational. Calculate U1.

A concrete situation

U1=2−6=−4.

What the mathematics does not license

Tiny segments give noisy locations. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Report segment sizes. 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. Second split

This step deserves a careful pause. Calculate U2.

A concrete situation

U2=2(3)−12=−6.

What the mathematics does not license

Maximum can be tied. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

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

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8. Maximum K

The useful question is not only how to calculate, but what the calculation assumes. Take max absolute Ut.

A concrete situation

K=6 in the example.

What the mathematics does not license

K alone does not give significance. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Calibrate the null. 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. Candidate location

A small example makes the mechanism visible. Find every maximizing t.

A concrete situation

t=2 and t=3 tie.

What the mathematics does not license

Reporting one silently is misleading. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Show all candidates. 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. Pairwise signs

This is where modelling choices begin to affect evidence. Cross-check the rank form.

A concrete situation

Before-after sign sums reproduce Ut.

What the mathematics does not license

Ties contribute zero. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Verify equivalence. 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. Permutation null

The central mathematical idea is concrete. Shuffle values when exchangeability is defensible.

A concrete situation

Each replicate recomputes max |Ut|.

What the mathematics does not license

Serial dependence breaks exchangeability. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use appropriate blocks or models. 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. Approximate p-value

Here the geometry becomes operational. Use the stated large-sample approximation if valid.

A concrete situation

An approximation summarizes tail probability.

What the mathematics does not license

Small n and ties distort it. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Prefer exact or simulation 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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13. Effect direction

This step deserves a careful pause. Compare segment medians and distributions.

A concrete situation

Later values are larger.

What the mathematics does not license

The test is not only a mean test. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Report robust summaries. 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. Multiple changes

The useful question is not only how to calculate, but what the calculation assumes. Inspect residual structure after one split.

A concrete situation

A second shift may remain.

What the mathematics does not license

Recursive testing inflates error. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use a planned multi-change method. 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. Trend and seasonality

A small example makes the mechanism visible. Test whether gradual or cyclic structure explains ranks.

A concrete situation

A trend can mimic one jump.

What the mathematics does not license

A significant split need not be abrupt. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Plot and model alternatives. 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. External validation

This is where modelling choices begin to affect evidence. Check the candidate against independent records.

A concrete situation

A documented process change occurs nearby.

What the mathematics does not license

Temporal proximity is not causation. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Keep evidence separate. 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

Pettitt’s test searches a sequence for one unknown change point without requiring a normal distribution. For every possible split t, it compares observations before and after the split through pairwise signs or an equivalent rank-sum statistic. The split with the largest absolute contrast becomes the candidate change location.

For n observations with ranks ri, a convenient form is Ut=2Σi≤t ri−t(n+1). The test statistic is K=max|Ut|, and the maximizing t estimates the change point. Exact, approximate or permutation calibration is then used under the no-change null, with ties and small samples handled explicitly.

Take observations 2, 3, 4, 9, 10 with ranks 1, 2, 3, 4, 5. At t=3, the rank sum is 6, so U3=2(6)−3(6)=−6 and |U3|=6. Earlier values are U1=−4 and U2=−6, so the maximum is tied at t=2 and t=3 in this small example. A tie rule and null calibration are necessary.

The test is designed around a single abrupt distributional shift. Trend, seasonality, serial dependence, multiple changes and selective choice of the analysed period can distort significance and location. Responsible work predefines the series, uses an appropriate null, reports ties and uncertainty, and validates the interpretation with independent context.

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Frequently asked questions

What does Pettitt’s test find?

It tests for one unknown abrupt distributional change and estimates the split with the largest rank contrast.

Why is it non-parametric?

Its statistic is built from ranks or pairwise signs rather than a specified normal likelihood.

What is Ut?

It is the rank-based contrast between observations before and after candidate split t.

What is K?

It is the maximum absolute Ut over all candidate splits.

How are ties handled?

They require an explicit ranking and null-calibration convention because ties affect the statistic’s distribution.

Can it find several changes?

The basic test targets one change; repeated testing after data-driven segmentation needs separate multiplicity control and validation.

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Primary source and fact-check note

A. N. Pettitt, “A Non-Parametric Approach to the Change-Point Problem,” Journal of the Royal Statistical Society Series C: Applied Statistics 28(2), 126–135 (1979), DOI 10.2307/2346729. 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

• Farrington outbreak detection
• EWMA surveillance charts
• 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

Window audit

Predefine series dates and exclusions.

Rank audit

Publish ranks and tie handling.

Split audit

Calculate Ut for every eligible t.

Maximum audit

Report all tied maximizing locations.

Null audit

Justify exact, approximate or permutation calibration.

Structure audit

Test trend, seasonality and dependence.

Transfer audit

Validate the candidate with independent context.

Final claim

Describe one rank-based shift candidate and limits.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Series definition

Begin with this mechanism: Fix ordered observations and analysis period. Recreate the concrete case: Values are 2,3,4,9,10. Write observations, ranks, tie groups, candidate splits, segment sizes, rank sums, pairwise signs, Ut statistics, absolute contrasts, maximizing locations, null maxima and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Choosing dates after viewing data biases results. Complete the transfer step: Predefine the window. Change exactly one observation, timestamp, gap, rank, tie rule, candidate split, reference mean, drift allowance, running minimum, likelihood model, change size, likelihood ratio, initial value, threshold, state multiplier, transition penalty, state ceiling, null model, simulation seed, dependence pattern, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, model, recursion or search, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 2: Rank values

Begin with this mechanism: Assign ranks across the complete series. Recreate the concrete case: Ranks are 1,2,3,4,5. Write observations, ranks, tie groups, candidate splits, segment sizes, rank sums, pairwise signs, Ut statistics, absolute contrasts, maximizing locations, null maxima and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Ties need midranks or another rule. Complete the transfer step: State the convention. Change exactly one observation, timestamp, gap, rank, tie rule, candidate split, reference mean, drift allowance, running minimum, likelihood model, change size, likelihood ratio, initial value, threshold, state multiplier, transition penalty, state ceiling, null model, simulation seed, dependence pattern, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, model, recursion or search, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 3: Candidate splits

Begin with this mechanism: Consider t from 1 to n−1. Recreate the concrete case: Four splits are possible. Write observations, ranks, tie groups, candidate splits, segment sizes, rank sums, pairwise signs, Ut statistics, absolute contrasts, maximizing locations, null maxima and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Endpoints can be unstable. Complete the transfer step: Consider minimum segment size. Change exactly one observation, timestamp, gap, rank, tie rule, candidate split, reference mean, drift allowance, running minimum, likelihood model, change size, likelihood ratio, initial value, threshold, state multiplier, transition penalty, state ceiling, null model, simulation seed, dependence pattern, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, model, recursion or search, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 4: Rank sum

Begin with this mechanism: Compute Σi≤t ri. Recreate the concrete case: At t=3 the sum is 6. Write observations, ranks, tie groups, candidate splits, segment sizes, rank sums, pairwise signs, Ut statistics, absolute contrasts, maximizing locations, null maxima and calibrated probabilities with 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 coding offset changes every value. Complete the transfer step: Check by hand. Change exactly one observation, timestamp, gap, rank, tie rule, candidate split, reference mean, drift allowance, running minimum, likelihood model, change size, likelihood ratio, initial value, threshold, state multiplier, transition penalty, state ceiling, null model, simulation seed, dependence pattern, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, model, recursion or search, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 5: Ut formula

Begin with this mechanism: Compute 2 rank sum−t(n+1). Recreate the concrete case: U3=12−18=−6. Write observations, ranks, tie groups, candidate splits, segment sizes, rank sums, pairwise signs, Ut statistics, absolute contrasts, maximizing locations, null maxima and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Sign depends on ordering convention. Complete the transfer step: Interpret magnitude first. Change exactly one observation, timestamp, gap, rank, tie rule, candidate split, reference mean, drift allowance, running minimum, likelihood model, change size, likelihood ratio, initial value, threshold, state multiplier, transition penalty, state ceiling, null model, simulation seed, dependence pattern, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, model, recursion or search, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 6: Early split

Begin with this mechanism: Calculate U1. Recreate the concrete case: U1=2−6=−4. Write observations, ranks, tie groups, candidate splits, segment sizes, rank sums, pairwise signs, Ut statistics, absolute contrasts, maximizing locations, null maxima and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Tiny segments give noisy locations. Complete the transfer step: Report segment sizes. Change exactly one observation, timestamp, gap, rank, tie rule, candidate split, reference mean, drift allowance, running minimum, likelihood model, change size, likelihood ratio, initial value, threshold, state multiplier, transition penalty, state ceiling, null model, simulation seed, dependence pattern, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, model, recursion or search, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 7: Second split

Begin with this mechanism: Calculate U2. Recreate the concrete case: U2=2(3)−12=−6. Write observations, ranks, tie groups, candidate splits, segment sizes, rank sums, pairwise signs, Ut statistics, absolute contrasts, maximizing locations, null maxima and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Maximum can be tied. Complete the transfer step: Declare a tie rule. Change exactly one observation, timestamp, gap, rank, tie rule, candidate split, reference mean, drift allowance, running minimum, likelihood model, change size, likelihood ratio, initial value, threshold, state multiplier, transition penalty, state ceiling, null model, simulation seed, dependence pattern, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, model, recursion or search, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 8: Maximum K

Begin with this mechanism: Take max absolute Ut. Recreate the concrete case: K=6 in the example. Write observations, ranks, tie groups, candidate splits, segment sizes, rank sums, pairwise signs, Ut statistics, absolute contrasts, maximizing locations, null maxima and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: K alone does not give significance. Complete the transfer step: Calibrate the null. Change exactly one observation, timestamp, gap, rank, tie rule, candidate split, reference mean, drift allowance, running minimum, likelihood model, change size, likelihood ratio, initial value, threshold, state multiplier, transition penalty, state ceiling, null model, simulation seed, dependence pattern, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, model, recursion or search, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 9: Candidate location

Begin with this mechanism: Find every maximizing t. Recreate the concrete case: t=2 and t=3 tie. Write observations, ranks, tie groups, candidate splits, segment sizes, rank sums, pairwise signs, Ut statistics, absolute contrasts, maximizing locations, null maxima and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Reporting one silently is misleading. Complete the transfer step: Show all candidates. Change exactly one observation, timestamp, gap, rank, tie rule, candidate split, reference mean, drift allowance, running minimum, likelihood model, change size, likelihood ratio, initial value, threshold, state multiplier, transition penalty, state ceiling, null model, simulation seed, dependence pattern, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, model, recursion or search, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 10: Pairwise signs

Begin with this mechanism: Cross-check the rank form. Recreate the concrete case: Before-after sign sums reproduce Ut. Write observations, ranks, tie groups, candidate splits, segment sizes, rank sums, pairwise signs, Ut statistics, absolute contrasts, maximizing locations, null maxima and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Ties contribute zero. Complete the transfer step: Verify equivalence. Change exactly one observation, timestamp, gap, rank, tie rule, candidate split, reference mean, drift allowance, running minimum, likelihood model, change size, likelihood ratio, initial value, threshold, state multiplier, transition penalty, state ceiling, null model, simulation seed, dependence pattern, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, model, recursion or search, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 11: Permutation null

Begin with this mechanism: Shuffle values when exchangeability is defensible. Recreate the concrete case: Each replicate recomputes max |Ut|. Write observations, ranks, tie groups, candidate splits, segment sizes, rank sums, pairwise signs, Ut statistics, absolute contrasts, maximizing locations, null maxima and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Serial dependence breaks exchangeability. Complete the transfer step: Use appropriate blocks or models. Change exactly one observation, timestamp, gap, rank, tie rule, candidate split, reference mean, drift allowance, running minimum, likelihood model, change size, likelihood ratio, initial value, threshold, state multiplier, transition penalty, state ceiling, null model, simulation seed, dependence pattern, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, model, recursion or search, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 12: Approximate p-value

Begin with this mechanism: Use the stated large-sample approximation if valid. Recreate the concrete case: An approximation summarizes tail probability. Write observations, ranks, tie groups, candidate splits, segment sizes, rank sums, pairwise signs, Ut statistics, absolute contrasts, maximizing locations, null maxima and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Small n and ties distort it. Complete the transfer step: Prefer exact or simulation checks. Change exactly one observation, timestamp, gap, rank, tie rule, candidate split, reference mean, drift allowance, running minimum, likelihood model, change size, likelihood ratio, initial value, threshold, state multiplier, transition penalty, state ceiling, null model, simulation seed, dependence pattern, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, model, recursion or search, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 13: Effect direction

Begin with this mechanism: Compare segment medians and distributions. Recreate the concrete case: Later values are larger. Write observations, ranks, tie groups, candidate splits, segment sizes, rank sums, pairwise signs, Ut statistics, absolute contrasts, maximizing locations, null maxima and calibrated probabilities with 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 test is not only a mean test. Complete the transfer step: Report robust summaries. Change exactly one observation, timestamp, gap, rank, tie rule, candidate split, reference mean, drift allowance, running minimum, likelihood model, change size, likelihood ratio, initial value, threshold, state multiplier, transition penalty, state ceiling, null model, simulation seed, dependence pattern, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, model, recursion or search, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 14: Multiple changes

Begin with this mechanism: Inspect residual structure after one split. Recreate the concrete case: A second shift may remain. Write observations, ranks, tie groups, candidate splits, segment sizes, rank sums, pairwise signs, Ut statistics, absolute contrasts, maximizing locations, null maxima and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Recursive testing inflates error. Complete the transfer step: Use a planned multi-change method. Change exactly one observation, timestamp, gap, rank, tie rule, candidate split, reference mean, drift allowance, running minimum, likelihood model, change size, likelihood ratio, initial value, threshold, state multiplier, transition penalty, state ceiling, null model, simulation seed, dependence pattern, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, model, recursion or search, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 15: Trend and seasonality

Begin with this mechanism: Test whether gradual or cyclic structure explains ranks. Recreate the concrete case: A trend can mimic one jump. Write observations, ranks, tie groups, candidate splits, segment sizes, rank sums, pairwise signs, Ut statistics, absolute contrasts, maximizing locations, null maxima and calibrated probabilities with 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 significant split need not be abrupt. Complete the transfer step: Plot and model alternatives. Change exactly one observation, timestamp, gap, rank, tie rule, candidate split, reference mean, drift allowance, running minimum, likelihood model, change size, likelihood ratio, initial value, threshold, state multiplier, transition penalty, state ceiling, null model, simulation seed, dependence pattern, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, model, recursion or search, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 16: External validation

Begin with this mechanism: Check the candidate against independent records. Recreate the concrete case: A documented process change occurs nearby. Write observations, ranks, tie groups, candidate splits, segment sizes, rank sums, pairwise signs, Ut statistics, absolute contrasts, maximizing locations, null maxima and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Temporal proximity is not causation. Complete the transfer step: Keep evidence separate. Change exactly one observation, timestamp, gap, rank, tie rule, candidate split, reference mean, drift allowance, running minimum, likelihood model, change size, likelihood ratio, initial value, threshold, state multiplier, transition penalty, state ceiling, null model, simulation seed, dependence pattern, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, model, recursion or search, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.