Chapter map

1. 1. Counting unit
2. 2. Exposure
3. 3. Baseline λ0
4. 4. Target λ1
5. 5. Likelihood ratio
6. 6. Score formula
7. 7. Recursive sum
8. 8. Reset rule
9. 9. Decision limit
10. 10. Run length
11. 11. False-alarm rate
12. 12. Detection delay
13. 13. Fast initial response
14. 14. Overdispersion
15. 15. Dependence
16. 16. Alarm review

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. Counting unit

Here the geometry becomes operational. Fix the event and interval.

A concrete situation

Daily incidents are counted once.

What the mathematics does not license

Duplicate records inflate evidence. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Deduplicate. 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. Exposure

This step deserves a careful pause. Record opportunities at risk.

A concrete situation

Each day has comparable coverage.

What the mathematics does not license

Varying denominators alter the rate. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Use an offset or rate 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.

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3. Baseline λ0

The useful question is not only how to calculate, but what the calculation assumes. Estimate the in-control mean count.

A concrete situation

λ0=4 per day.

What the mathematics does not license

Contaminated baselines bias design. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Use stable history. 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. Target λ1

A small example makes the mechanism visible. Choose a meaningful increased rate.

A concrete situation

λ1=8 per day.

What the mathematics does not license

A convenient target may lack relevance. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Tie it to action. 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. Likelihood ratio

This is where modelling choices begin to affect evidence. Compare Poisson probabilities under both rates.

A concrete situation

The count contributes evidence for 8 versus 4.

What the mathematics does not license

Both models can be wrong. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Check 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.

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6. Score formula

The central mathematical idea is concrete. Compute Y log(λ1/λ0)−(λ1−λ0).

A concrete situation

For Y=9, score≈2.238.

What the mathematics does not license

Logs require positive design rates. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Validate inputs. 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. Recursive sum

Here the geometry becomes operational. Add the score to yesterday’s statistic.

A concrete situation

S0=0 gives S1≈2.238.

What the mathematics does not license

State initialization affects early behaviour. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Publish S0. 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. Reset rule

This step deserves a careful pause. Take max with zero.

A concrete situation

Negative totals return to zero.

What the mathematics does not license

Reset loses evidence for decreases. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Use a separate lower chart. 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. Decision limit

The useful question is not only how to calculate, but what the calculation assumes. Signal when S exceeds h.

A concrete situation

h is chosen before monitoring.

What the mathematics does not license

Lower h means more false alarms. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Calibrate run lengths. 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. Run length

A small example makes the mechanism visible. Count observations until the signal.

A concrete situation

A simulation records τ=min{t:St>h}.

What the mathematics does not license

One run is noisy. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Use many 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.

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11. False-alarm rate

This is where modelling choices begin to affect evidence. Simulate under λ0.

A concrete situation

Average run length summarizes quiet operation.

What the mathematics does not license

The average hides tail behaviour. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Report quantiles. 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. Detection delay

The central mathematical idea is concrete. Simulate a change to λ1.

A concrete situation

Delay begins at the change point.

What the mathematics does not license

Immediate large spikes and small shifts differ. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Test several shapes. 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. Fast initial response

Here the geometry becomes operational. Consider a head start if justified.

A concrete situation

S0 can be above zero.

What the mathematics does not license

It raises early false-alarm risk. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Compare transparently. 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. Overdispersion

This step deserves a careful pause. Compare variance with the mean.

A concrete situation

Variance exceeds four.

What the mathematics does not license

Plain Poisson limits become liberal. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Use robust 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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15. Dependence

The useful question is not only how to calculate, but what the calculation assumes. Check serial correlation and weekday effects.

A concrete situation

Counts cluster across days.

What the mathematics does not license

Independence calibration fails. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Simulate realistic dependence. 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. Alarm review

A small example makes the mechanism visible. Pair signals with an investigation protocol.

A concrete situation

Analysts inspect reporting and cases.

What the mathematics does not license

CUSUM cannot identify cause. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Log the disposition. 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

A Poisson CUSUM is a sequential detector for count streams. Instead of judging each day alone, it accumulates log-likelihood evidence that the rate has shifted from a reference value λ0 to a larger value λ1. Evidence that favours the baseline pushes the statistic down, and the reset at zero prevents old negative evidence from creating permanent debt.

For count Yt, one likelihood-score form is Yt log(λ1/λ0)−(λ1−λ0), with St=max(0,St−1+score). If λ0=4, λ1=8 and Yt=9, the score is 9 log 2−4≈2.238. Starting from zero, the statistic becomes about 2.238. It signals only after crossing a separately calibrated decision limit h.

The design values λ0, λ1 and h express the monitoring question. λ0 represents the in-control rate, λ1 the shift worth detecting and h the evidence needed before action. Average run length, detection probability and delay depend on all three, as well as estimation error, overdispersion, seasonality and dependence.

A Poisson assumption equates mean and variance and treats the defined exposure interval as comparable. Real surveillance counts may be overdispersed, autocorrelated or driven by varying denominators. Before interpreting a CUSUM, analysts should model exposure, test residual behaviour, simulate the complete operating environment and attach investigation rules to alarms.

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Frequently asked questions

What does a Poisson CUSUM monitor?

It monitors sequential count evidence for a declared shift in a Poisson rate.

What are λ0 and λ1?

λ0 is the baseline rate and λ1 is the increase the chart is designed to detect efficiently.

Why use a log-likelihood score?

It adds evidence for λ1 over λ0 on a common sequential scale.

Why reset at zero?

The reset discards accumulated evidence favouring the baseline so a later increase can be detected.

What is the decision limit h?

It is the accumulated-evidence threshold that triggers a signal and controls the run-length trade-off.

What if counts are overdispersed?

A plain Poisson chart may signal too often; the variance and exposure model must be checked or replaced.

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Primary source and fact-check note

James M. Lucas, “Counted Data CUSUM’s,” Technometrics 27(2), 129–144 (1985), DOI 10.1080/00401706.1985.10488030. 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
• Knorr–Held space–time interaction models
• 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

Rate audit

Justify λ0 and λ1.

Exposure audit

Check comparable opportunities at risk.

Score audit

Reproduce every log-likelihood increment.

Limit audit

Calibrate h by run-length simulation.

Variance audit

Test Poisson, dispersion and dependence.

Delay audit

Evaluate several shift sizes and shapes.

Workflow audit

Connect signals to investigation.

Final claim

Report accumulated evidence without causal overreach.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Counting unit

Begin with this mechanism: Fix the event and interval. Recreate the concrete case: Daily incidents are counted once. Write counts, exposure units, baseline rates, target rates, likelihood scores, recursive sums, reset indicators, decision limits, run lengths, false-alarm rates, detection delays and investigation outcomes 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: Duplicate records inflate evidence. Complete the transfer step: Deduplicate. Change exactly one observed count, reporting date, historical window, baseline mean, standard deviation, guard-band length, dispersion factor, trend term, seasonal factor, robust weight, predictive quantile, baseline rate, target rate, likelihood score, reset value, decision limit, EWMA weight, starting value, variance model, calendar effect, exposure denominator, random seed, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, baseline design, model, computation, threshold, uncertainty, interpretation and independent investigation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 2: Exposure

Begin with this mechanism: Record opportunities at risk. Recreate the concrete case: Each day has comparable coverage. Write counts, exposure units, baseline rates, target rates, likelihood scores, recursive sums, reset indicators, decision limits, run lengths, false-alarm rates, detection delays and investigation outcomes 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: Varying denominators alter the rate. Complete the transfer step: Use an offset or rate model. Change exactly one observed count, reporting date, historical window, baseline mean, standard deviation, guard-band length, dispersion factor, trend term, seasonal factor, robust weight, predictive quantile, baseline rate, target rate, likelihood score, reset value, decision limit, EWMA weight, starting value, variance model, calendar effect, exposure denominator, random seed, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, baseline design, model, computation, threshold, uncertainty, interpretation and independent investigation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 3: Baseline λ0

Begin with this mechanism: Estimate the in-control mean count. Recreate the concrete case: λ0=4 per day. Write counts, exposure units, baseline rates, target rates, likelihood scores, recursive sums, reset indicators, decision limits, run lengths, false-alarm rates, detection delays and investigation outcomes 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: Contaminated baselines bias design. Complete the transfer step: Use stable history. Change exactly one observed count, reporting date, historical window, baseline mean, standard deviation, guard-band length, dispersion factor, trend term, seasonal factor, robust weight, predictive quantile, baseline rate, target rate, likelihood score, reset value, decision limit, EWMA weight, starting value, variance model, calendar effect, exposure denominator, random seed, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, baseline design, model, computation, threshold, uncertainty, interpretation and independent investigation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 4: Target λ1

Begin with this mechanism: Choose a meaningful increased rate. Recreate the concrete case: λ1=8 per day. Write counts, exposure units, baseline rates, target rates, likelihood scores, recursive sums, reset indicators, decision limits, run lengths, false-alarm rates, detection delays and investigation outcomes 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 convenient target may lack relevance. Complete the transfer step: Tie it to action. Change exactly one observed count, reporting date, historical window, baseline mean, standard deviation, guard-band length, dispersion factor, trend term, seasonal factor, robust weight, predictive quantile, baseline rate, target rate, likelihood score, reset value, decision limit, EWMA weight, starting value, variance model, calendar effect, exposure denominator, random seed, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, baseline design, model, computation, threshold, uncertainty, interpretation and independent investigation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 5: Likelihood ratio

Begin with this mechanism: Compare Poisson probabilities under both rates. Recreate the concrete case: The count contributes evidence for 8 versus 4. Write counts, exposure units, baseline rates, target rates, likelihood scores, recursive sums, reset indicators, decision limits, run lengths, false-alarm rates, detection delays and investigation outcomes 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: Both models can be wrong. Complete the transfer step: Check residuals. Change exactly one observed count, reporting date, historical window, baseline mean, standard deviation, guard-band length, dispersion factor, trend term, seasonal factor, robust weight, predictive quantile, baseline rate, target rate, likelihood score, reset value, decision limit, EWMA weight, starting value, variance model, calendar effect, exposure denominator, random seed, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, baseline design, model, computation, threshold, uncertainty, interpretation and independent investigation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 6: Score formula

Begin with this mechanism: Compute Y log(λ1/λ0)−(λ1−λ0). Recreate the concrete case: For Y=9, score≈2.238. Write counts, exposure units, baseline rates, target rates, likelihood scores, recursive sums, reset indicators, decision limits, run lengths, false-alarm rates, detection delays and investigation outcomes 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: Logs require positive design rates. Complete the transfer step: Validate inputs. Change exactly one observed count, reporting date, historical window, baseline mean, standard deviation, guard-band length, dispersion factor, trend term, seasonal factor, robust weight, predictive quantile, baseline rate, target rate, likelihood score, reset value, decision limit, EWMA weight, starting value, variance model, calendar effect, exposure denominator, random seed, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, baseline design, model, computation, threshold, uncertainty, interpretation and independent investigation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 7: Recursive sum

Begin with this mechanism: Add the score to yesterday’s statistic. Recreate the concrete case: S0=0 gives S1≈2.238. Write counts, exposure units, baseline rates, target rates, likelihood scores, recursive sums, reset indicators, decision limits, run lengths, false-alarm rates, detection delays and investigation outcomes 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: State initialization affects early behaviour. Complete the transfer step: Publish S0. Change exactly one observed count, reporting date, historical window, baseline mean, standard deviation, guard-band length, dispersion factor, trend term, seasonal factor, robust weight, predictive quantile, baseline rate, target rate, likelihood score, reset value, decision limit, EWMA weight, starting value, variance model, calendar effect, exposure denominator, random seed, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, baseline design, model, computation, threshold, uncertainty, interpretation and independent investigation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 8: Reset rule

Begin with this mechanism: Take max with zero. Recreate the concrete case: Negative totals return to zero. Write counts, exposure units, baseline rates, target rates, likelihood scores, recursive sums, reset indicators, decision limits, run lengths, false-alarm rates, detection delays and investigation outcomes 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: Reset loses evidence for decreases. Complete the transfer step: Use a separate lower chart. Change exactly one observed count, reporting date, historical window, baseline mean, standard deviation, guard-band length, dispersion factor, trend term, seasonal factor, robust weight, predictive quantile, baseline rate, target rate, likelihood score, reset value, decision limit, EWMA weight, starting value, variance model, calendar effect, exposure denominator, random seed, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, baseline design, model, computation, threshold, uncertainty, interpretation and independent investigation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 9: Decision limit

Begin with this mechanism: Signal when S exceeds h. Recreate the concrete case: h is chosen before monitoring. Write counts, exposure units, baseline rates, target rates, likelihood scores, recursive sums, reset indicators, decision limits, run lengths, false-alarm rates, detection delays and investigation outcomes 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: Lower h means more false alarms. Complete the transfer step: Calibrate run lengths. Change exactly one observed count, reporting date, historical window, baseline mean, standard deviation, guard-band length, dispersion factor, trend term, seasonal factor, robust weight, predictive quantile, baseline rate, target rate, likelihood score, reset value, decision limit, EWMA weight, starting value, variance model, calendar effect, exposure denominator, random seed, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, baseline design, model, computation, threshold, uncertainty, interpretation and independent investigation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 10: Run length

Begin with this mechanism: Count observations until the signal. Recreate the concrete case: A simulation records τ=min{t:St>h}. Write counts, exposure units, baseline rates, target rates, likelihood scores, recursive sums, reset indicators, decision limits, run lengths, false-alarm rates, detection delays and investigation outcomes 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: One run is noisy. Complete the transfer step: Use many replicates. Change exactly one observed count, reporting date, historical window, baseline mean, standard deviation, guard-band length, dispersion factor, trend term, seasonal factor, robust weight, predictive quantile, baseline rate, target rate, likelihood score, reset value, decision limit, EWMA weight, starting value, variance model, calendar effect, exposure denominator, random seed, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, baseline design, model, computation, threshold, uncertainty, interpretation and independent investigation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 11: False-alarm rate

Begin with this mechanism: Simulate under λ0. Recreate the concrete case: Average run length summarizes quiet operation. Write counts, exposure units, baseline rates, target rates, likelihood scores, recursive sums, reset indicators, decision limits, run lengths, false-alarm rates, detection delays and investigation outcomes 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 average hides tail behaviour. Complete the transfer step: Report quantiles. Change exactly one observed count, reporting date, historical window, baseline mean, standard deviation, guard-band length, dispersion factor, trend term, seasonal factor, robust weight, predictive quantile, baseline rate, target rate, likelihood score, reset value, decision limit, EWMA weight, starting value, variance model, calendar effect, exposure denominator, random seed, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, baseline design, model, computation, threshold, uncertainty, interpretation and independent investigation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 12: Detection delay

Begin with this mechanism: Simulate a change to λ1. Recreate the concrete case: Delay begins at the change point. Write counts, exposure units, baseline rates, target rates, likelihood scores, recursive sums, reset indicators, decision limits, run lengths, false-alarm rates, detection delays and investigation outcomes 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: Immediate large spikes and small shifts differ. Complete the transfer step: Test several shapes. Change exactly one observed count, reporting date, historical window, baseline mean, standard deviation, guard-band length, dispersion factor, trend term, seasonal factor, robust weight, predictive quantile, baseline rate, target rate, likelihood score, reset value, decision limit, EWMA weight, starting value, variance model, calendar effect, exposure denominator, random seed, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, baseline design, model, computation, threshold, uncertainty, interpretation and independent investigation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 13: Fast initial response

Begin with this mechanism: Consider a head start if justified. Recreate the concrete case: S0 can be above zero. Write counts, exposure units, baseline rates, target rates, likelihood scores, recursive sums, reset indicators, decision limits, run lengths, false-alarm rates, detection delays and investigation outcomes 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: It raises early false-alarm risk. Complete the transfer step: Compare transparently. Change exactly one observed count, reporting date, historical window, baseline mean, standard deviation, guard-band length, dispersion factor, trend term, seasonal factor, robust weight, predictive quantile, baseline rate, target rate, likelihood score, reset value, decision limit, EWMA weight, starting value, variance model, calendar effect, exposure denominator, random seed, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, baseline design, model, computation, threshold, uncertainty, interpretation and independent investigation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 14: Overdispersion

Begin with this mechanism: Compare variance with the mean. Recreate the concrete case: Variance exceeds four. Write counts, exposure units, baseline rates, target rates, likelihood scores, recursive sums, reset indicators, decision limits, run lengths, false-alarm rates, detection delays and investigation outcomes 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: Plain Poisson limits become liberal. Complete the transfer step: Use robust alternatives. Change exactly one observed count, reporting date, historical window, baseline mean, standard deviation, guard-band length, dispersion factor, trend term, seasonal factor, robust weight, predictive quantile, baseline rate, target rate, likelihood score, reset value, decision limit, EWMA weight, starting value, variance model, calendar effect, exposure denominator, random seed, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, baseline design, model, computation, threshold, uncertainty, interpretation and independent investigation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 15: Dependence

Begin with this mechanism: Check serial correlation and weekday effects. Recreate the concrete case: Counts cluster across days. Write counts, exposure units, baseline rates, target rates, likelihood scores, recursive sums, reset indicators, decision limits, run lengths, false-alarm rates, detection delays and investigation outcomes 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: Independence calibration fails. Complete the transfer step: Simulate realistic dependence. Change exactly one observed count, reporting date, historical window, baseline mean, standard deviation, guard-band length, dispersion factor, trend term, seasonal factor, robust weight, predictive quantile, baseline rate, target rate, likelihood score, reset value, decision limit, EWMA weight, starting value, variance model, calendar effect, exposure denominator, random seed, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, baseline design, model, computation, threshold, uncertainty, interpretation and independent investigation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.

Workshop 16: Alarm review

Begin with this mechanism: Pair signals with an investigation protocol. Recreate the concrete case: Analysts inspect reporting and cases. Write counts, exposure units, baseline rates, target rates, likelihood scores, recursive sums, reset indicators, decision limits, run lengths, false-alarm rates, detection delays and investigation outcomes 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: CUSUM cannot identify cause. Complete the transfer step: Log the disposition. Change exactly one observed count, reporting date, historical window, baseline mean, standard deviation, guard-band length, dispersion factor, trend term, seasonal factor, robust weight, predictive quantile, baseline rate, target rate, likelihood score, reset value, decision limit, EWMA weight, starting value, variance model, calendar effect, exposure denominator, random seed, monitoring horizon or validation period. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, baseline design, model, computation, threshold, uncertainty, interpretation and independent investigation on separate notebook lines. Reserve complete periods outside tuning when testing transfer.