Validation

Table 16.2 and Table A18 — the four-point sequence

Source.

Gilbert, R.O. (1987). Statistical Methods for Environmental Pollution Monitoring. Van Nostrand Reinhold, New York — chapter 16, pages 208 to 222. Section 16.4.1 and Table 16.2, page 211; Table A18.

Method.

The book scores the sequence 10, 15, 14, 20 pair by pair in a printed table and then reads its probability off Table A18. Both are computed here from the same four numbers.

• Four points is the smallest sequence the test is defined on and the probabilities are exact combinatorics rather than an approximation, so this case validates the sign count and the exact tail and nothing else. The normal approximation is reached only by the examples below it.

Asserted by src/tools/mann-kendall/__tests__/stats.test.ts

Example 16.2 — the tie-corrected variance and the continuity correction

Source.

Gilbert, R.O. (1987). Statistical Methods for Environmental Pollution Monitoring. Van Nostrand Reinhold, New York — chapter 16, pages 208 to 222. Example 16.2, pages 212 to 213.

Method.

Twenty-two monthly uranium-234 concentrations with one tied group of six and two of two. The book prints the variance term by term, its square root, and the corrected Z; all three are computed from the tie structure it states.

• The twenty-two concentrations themselves are not reproduced, because the book does not print them — the example works from a scatter plot, and reading values off a scanned figure would be pinning an act of eyesight rather than a published number. What is reproduced is the arithmetic the book prints: n, the tie groups, the variance, its root and Z.
• Z is printed to two significant figures. The unrounded value here is 3.0545, so one decimal place is the honest comparison and the assertion says so.

Asserted by src/tools/mann-kendall/__tests__/stats.test.ts

Example 16.3 — ties in the data and in the time index at once

Source.

Gilbert, R.O. (1987). Statistical Methods for Environmental Pollution Monitoring. Van Nostrand Reinhold, New York — chapter 16, pages 208 to 222. Example 16.3, pages 214 to 215, and Equation 16.5.

Method.

Eight observations across five time periods, three of them sharing period 1 and two sharing period 3, with three tied values. The book states the score, the number of comparable pairs, both tie structures, the variance and Z.

Asserted by src/tools/mann-kendall/__tests__/stats.test.ts

Example 16.5 — Sen’s slope and its confidence limits

Source.

Gilbert, R.O. (1987). Statistical Methods for Environmental Pollution Monitoring. Van Nostrand Reinhold, New York — chapter 16, pages 208 to 222. Example 16.5, pages 218 to 219; Table A1.

Method.

The same eight observations. The book lists all 24 pairwise slopes, takes their median, and finds the 90 per cent limits by interpolating between ranked slopes rather than by rounding the ranks.

Asserted by src/tools/mann-kendall/__tests__/stats.test.ts

An independent worked S, from a document a regulator accepted

Source.

HydroGeoLogic, Inc. (2005). Mann-Kendall Analysis for the Fort Ord Site — Appendix D of the OU-1 Annual Groundwater Monitoring Report, prepared for the U.S. Army Corps of Engineers. Section 2.2.

Method.

The appendix works a fictitious monitoring well — 1.5, 2.0, 1.75 and 3.0 µg/L of TCE in four consecutive years — through the same arithmetic, and separately states how non-detects are grouped.

• This is a second independent statement of the same formulae rather than a second independent result — the sequence is the same shape as Gilbert’s, deliberately. Its value is that it is a document written for a regulator, which is the audience the tool is for.

Asserted by src/tools/mann-kendall/__tests__/stats.test.ts

The exact null distribution at n = 5

Source.

Kendall, M.G. (1975). Rank Correlation Methods, 4th edition, Charles Griffin, London — the tabulated tail probabilities of S, reproduced as Table A18 in Gilbert (1987) and in every rank-correlation table since.

Method.

The exact distribution is built by dynamic programming and its upper tail read at the three attainable values of S that the published table prints for five observations.

• The exact branch is used only on untied series of at most 22 observations. A tied dataset’s null distribution is not the permutation distribution of distinct values, and the 22 is chosen to match where ProUCL switches; above it, and whenever anything is tied, the normal approximation is used and the tool says which it used.

Asserted by src/tools/mann-kendall/__tests__/stats.test.ts

Pearson against a NIST certified value

Source.

National Institute of Standards and Technology, Statistical Reference Datasets, linear regression, dataset "Norris" (Norris, J., NIST, calibration of ozone monitors), 36 observations. Certified R-Squared 0.999993745883712.

Method.

The 36 published pairs are transcribed into the test file and the Pearson coefficient computed on them; for a simple linear regression R² is exactly r², so the certified value is a certified correlation.

Asserted by src/tools/correlation-matrix/__tests__/reference.test.ts

Anscombe’s quartet

Source.

Anscombe, F.J. (1973). Graphs in Statistical Analysis. The American Statistician 27(1), 17–21 — four datasets sharing, to the precision printed, the same correlation of 0.816.

Method.

All four sets are transcribed and correlated independently; the published figure is three digits, so the assertion is that each set lands within a thousandth of it and that the four agree with each other far more closely than that.

• Three digits is all Anscombe printed. The four sets compute to between 0.81624 and 0.81652, so set IV does not even round to 0.816 at three places, and an assertion at four decimals would be asserting digits the source does not have. The bound in the third column is therefore what is claimed, and the value in the second is what was published — they are not the same statement.
• The rank correlations of the same four sets are asserted in the test file too, and are deliberately absent from this table: they are this implementation’s own output, not anything Anscombe printed, and a validation page is the wrong place for a number with no source behind it.

Asserted by src/tools/correlation-matrix/__tests__/reference.test.ts

Spearman with a tie, against a documented library output

Source.

SciPy reference documentation for `scipy.stats.spearmanr`, the worked example on that page: `spearmanr([1, 2, 3, 4, 5], [5, 6, 7, 8, 7])` returning a statistic of 0.8207826816681233.

Method.

The same five pairs are correlated here. The second series ties on 7, so this exercises the correction that the widely-taught rank shortcut omits, and the printed output is sixteen digits long — enough to pin the convention rather than merely be consistent with it.

• This reproduces another implementation, not a paper. That is worth exactly what it sounds like: it establishes that this tool uses the same tie convention as SciPy and R, which is what somebody checking a result against either of them needs, and it establishes nothing about whether that convention is the right one for their data.

Asserted by src/tools/correlation-matrix/__tests__/reference.test.ts

A tied four-point case, worked by hand

Source.

No published source. The arithmetic is written out in full in the comment above the assertion in the test file named below, and is small enough to check by eye.

Method.

Four pairs, with a tie in the second series, chosen so that the taught shortcut and the definition give visibly different answers.

• Nothing here is published and the entry says so in its citation rather than in a footnote. It is included because a hand derivation a reader can follow is a different kind of evidence from a library output, and because it is the case that would catch a plausible-looking wrong implementation that the SciPy example alone would not.

Asserted by src/tools/correlation-matrix/__tests__/reference.test.ts

R’s autocorrelation of 1 to 5

Source.

R Core Team, `stats` package documentation for `acf` and `ccf`, and the C routine `acf0` in src/library/stats/src/filter.c, which defines the estimator: each series is centred on its own mean, the products are divided by the pair count PLUS the lag, and the result is normalised and clipped into [−1, 1].

Method.

Cross-correlating the sequence 1, 2, 3, 4, 5 with itself makes the cross-correlation an autocorrelation, so the answer can be read against R’s printed output for `acf(1:5)`.

Asserted by src/tools/cross-correlation/__tests__/compute.test.ts

An asymmetric impulse pair, in closed form

Source.

No published source. The estimator is R’s, cited above; these three values are derived from it by hand in the comment above the assertion, on series short enough that a reader can check them with a pencil.

Method.

Two four-point impulses, the first spiking one step before the second, which makes the function deliberately asymmetric — an implementation that had transposed the index would swap the two outer values and report that the ground warms before the air does.

• A hand derivation is not a reproduction of a published result and this table does not pretend otherwise. It is a stronger test than a pasted coefficient would have been, because a pasted coefficient cannot tell you which part of the arithmetic is wrong, and it is a weaker CLAIM, because nobody else has checked it.

Asserted by src/tools/cross-correlation/__tests__/compute.test.ts

The direction convention

Source.

Shumway, R.H. & Stoffer, D.S. Time Series Analysis and Its Applications — the Southern Oscillation Index and Recruitment example, whose `ccf(soi, rec)` peaks near lag −6 and is read in the text as the index leading recruitment. · R Core Team, `stats` documentation for `acf`: "The lag k value returned by ccf(x, y) estimates the correlation between x[t+k] and y[t]."

Method.

A deterministic non-symmetric signal is shifted by a known number of steps and fed in both ways round, and the recovered lag is required to carry the sign the convention demands.

• The correlation at the recovered lag is bounded below at 0.97 rather than at 0.99, and that is the estimator working rather than noise: R’s divisor shrinks a lag-3 estimate by three parts in the series length even when the alignment is perfect.

Asserted by src/tools/cross-correlation/__tests__/compute.test.ts

Example 18 — reference evapotranspiration at Uccle from a single day of data

Source.

Allen, R.G., Pereira, L.S., Raes, D. & Smith, M. (1998). Crop evapotranspiration — Guidelines for computing crop water requirements. FAO Irrigation and Drainage Paper 56, Food and Agriculture Organization of the United Nations, Rome. Chapter 4, Example 18 (Uccle, Brussels, 6 July).

Method.

The example states a maximum of 21.5 °C, a minimum of 12.3 °C, relative humidity of 84 and 63 per cent, 2.78 m s⁻¹ measured by a 10 m anemometer, 9.25 hours of bright sunshine, latitude 50°48′ N and an elevation of 100 m. Those inputs are fed in and every quantity the example prints on its way to the answer is compared with the printed value.

• The printed wind speed and the printed u₂ cannot both be exact. Equation 47 at 10 m is a factor of 0.747951, and 2.78 × 0.747951 is 2.0793, not the 2.078 the example prints; working backwards, 2.078 implies a measured 2.7783 m s⁻¹. It is the input the example rounded, not the output, and the test asserts both readings of it.
• This is the only worked example in the paper that exercises what the daily tool actually does — one day, soil heat flux taken as zero, humidity from the two relative-humidity extremes, and a 10 m anemometer. It is therefore the strongest single row on this page and also the narrowest: one day, one site, one set of inputs.

Asserted by src/tools/et0/__tests__/fao56.test.ts

Example 17 — reference evapotranspiration at Bangkok from monthly means

Source.

Allen, R.G., Pereira, L.S., Raes, D. & Smith, M. (1998). Crop evapotranspiration — Guidelines for computing crop water requirements. FAO Irrigation and Drainage Paper 56, Food and Agriculture Organization of the United Nations, Rome. Chapter 4, Example 17 (Bangkok, April, from monthly mean data).

Method.

A hot humid tropical site, which is the opposite corner of the parameter space from Uccle: maximum 34.8 °C, minimum 25.6 °C, ea 2.85 kPa, wind 2.0 m s⁻¹, 8.5 hours of sunshine, latitude 13°44′ N, elevation 2 m. The same chain is run and compared with the printed intermediates.

• The soil heat flux is supplied to the equation rather than computed. Monthly data means G is not zero — the example derives 0.14 MJ m⁻² day⁻¹ from the March and April means by Equation 44 — and the monthly soil-heat-flux equations are out of scope for a daily tool. What this example validates is the assembled radiation chain and Equation 6 itself, on a site nothing like the other one.
• The psychrometric constant is checked against the pressure the paper PRINTED rather than the one the chain computes. At 2 m elevation Equation 7 gives 101.276 kPa and the example prints 101.3; Equation 8 on the printed value gives the printed 0.0674 and on the unrounded value misses a four-decimal assertion by a whisker. That is a rounding cascade in the paper rather than a defect in Equation 8, and the unrounded chain is separately asserted to be within a tenth of a per cent.

Asserted by src/tools/et0/__tests__/fao56.test.ts

The chapter-three examples each term comes from

Source.

Allen, R.G., Pereira, L.S., Raes, D. & Smith, M. (1998). Crop evapotranspiration — Guidelines for computing crop water requirements. FAO Irrigation and Drainage Paper 56, Food and Agriculture Organization of the United Nations, Rome. Chapter 3, Examples 2, 3, 5, 6, 8, 9, 10, 11, 12, 14, 15 and 16.

Method.

Each intermediate equation is exercised on its own, on the inputs the example states, and compared with the value that example prints — so that a fault in one term does not have to be found by bisecting a finished ET₀.

• Example 12 chains its solar radiation from Example 10 rather than taking the 14.5 the box prints. Feeding the printed value into Equation 38 gives 11.165, which the paper would have printed as 11.2, and it prints 11.1 — so it carried the unrounded 14.450 across. Pinning the test to the printed input would fail against a correct implementation of Equation 38.
• Two behaviours the paper implies but works no example for are asserted separately and appear in no row above: the daylight-hour equation returning a full day rather than NaN above the Arctic circle in midsummer, and the clamp that stops a pyranometer reading above the modelled clear-sky value inflating the longwave loss.

Asserted by src/tools/et0/__tests__/fao56.test.ts

The simple-average convention, as the EPA indicator states it

Source.

United States Environmental Protection Agency (2024). Climate Change Indicators: Growing Degree Days — technical documentation, growing-degree-days_documentation.pdf.

Method.

The documentation states that the daily maximum and minimum are averaged, a base temperature is subtracted, and a negative value is taken as zero; and it gives one worked pair, a day averaging 70 °F above a 50 °F base being 20 growing degree days.

• The third row is the one worth reading twice. A 30 to 60 °F day is zero under the EPA convention, because the mean of 45 is below the base and the finished number is floored; it is 5 under the modified method, which raises the minimum to the base first. This tool implements the EPA convention and says so on the page. Neither is wrong — they are different definitions, and a season total computed under one cannot be compared with a season total computed under the other.

Asserted by src/tools/gdd/__tests__/methods.test.ts

The single-sine method against a printed worked day

Source.

Baskerville, G.L. & Emin, P. (1969). Rapid estimation of heat accumulation from maximum and minimum temperatures. Ecology 50(3), 514–517. · Zalom, F.G., Goodell, P.B., Wilson, L.T., Barnett, W.W. & Bentley, W.J. (1983). Degree Days: The Calculation and Use of Heat Units in Pest Management. University of California Division of Agriculture and Natural Resources Leaflet 21373 — the six-case table the closed forms are transcribed from. · University of California Agriculture and Natural Resources, `degday` R package README (github.com/UCANR-IGIS/degday) — the printed single-sine result reproduced below.

Method.

The package publishes the single-sine degree-days for the Esparto.A station on 2 January 2020: a minimum of 36 °F, a maximum of 67 °F, and Navel Orangeworm thresholds of 55 and 93.9 °F. The same four numbers are fed to the closed forms here.

• The closed forms are a hand transcription of a printed table of six trigonometric cases, which is exactly the kind of thing that is wrong in one case and right in the other five. That is why the second table exists: agreement with one published day would not have caught a transcription error in a case that day does not reach.

Asserted by src/tools/gdd/__tests__/methods.test.ts

The Utah weighting table

Source.

Richardson, E.A., Seeley, S.D. & Walker, D.R. (1974). A model for estimating the completion of rest for Redhaven and Elberta peach trees. HortScience 9(4), 331–332. · Luedeling, E. chillR: Statistical Methods for Phenology Analysis in Temperate Fruit Trees — R/temp_models.R, the `step_model` and `Utah_Model` functions, read verbatim.

Method.

The seven bands and their weights are asserted as a list, and then every band boundary is approached from both sides — because the bands are half-open in chillR’s own selection, and a prose transcription of the table always loses that.

• The table is checked against chillR’s transcription of it rather than against the 1974 paper directly. chillR is the de-facto reference in this literature and its source is quoted in the test file, but it is a second-hand source and this row is weaker than a row checked against a paper.

Asserted by src/tools/chill-hours/__tests__/models.test.ts

The Dynamic model parameter set as published

Source.

Luedeling, E. & Brown, P.H. (2011). A global analysis of the comparability of winter chill models for fruit and nut trees. International Journal of Biometeorology 55, 411–421 — the printed parameter set. · Fishman, S., Erez, A. & Couvillon, G.A. (1987). The temperature dependence of dormancy breaking in plants: computer simulation of processes studied under controlled temperatures. Journal of Theoretical Biology 126, 309–321.

Method.

Luedeling & Brown print the six parameters as a sentence; each is asserted as an exact equality, because a parameter that has drifted produces a curve that still looks like a chill curve.

Asserted by src/tools/chill-hours/__tests__/models.test.ts

Agreement with chillR’s reference implementations

Source.

Luedeling, E. chillR: Statistical Methods for Phenology Analysis in Temperate Fruit Trees — R/temp_models.R. The `Chilling_Hours`, `step_model` and `Dynamic_Model` functions are transcribed into the test file in their original indexing and run alongside this implementation.

Method.

Three reference functions are re-implemented from chillR’s own R, in its indexing rather than paraphrased into something that agrees by construction, and run against this implementation on deterministic pseudo-random temperature series.

• This is agreement with a reference implementation, not reproduction of a printed result. It is strong evidence that the model here is the model that literature runs, and it is no evidence at all that either of them describes a real orchard.
• The chilling-hours window is 0 to 7.2 °C, and 7.2 is the ROUNDED metric form of 45 °F, which is 7.2222 °C. The rounded value is what every published implementation uses, so it is what this uses — and the hours between the two are why a grower checking against a Fahrenheit calculator finds a small discrepancy. Both numbers are asserted.

Asserted by src/tools/chill-hours/__tests__/models.test.ts

Equation 11 at the temperatures FAO-56 prints

Source.

Allen, R.G., Pereira, L.S., Raes, D. & Smith, M. (1998). Crop evapotranspiration — Guidelines for computing crop water requirements. FAO Irrigation and Drainage Paper 56, Rome. Chapter 3, Equation 11 and the worked Examples 3, 4 and 5.

Method.

Each temperature the chapter prints a saturation vapour pressure for is fed to the equation and the result rounded to the number of decimals the paper printed — not to more, because a constant recalled to more digits than its source printed is not a reference value.

• The third row is the example’s own warning made executable rather than an extra check. Taking e° at the mean temperature gives 2.3 kPa where the mean of the two extremes gives 2.39, an underestimate of about four per cent that goes straight into the deficit. The tool takes the mean of the two extremes when it has both.

Asserted by src/tools/vpd/__tests__/compute.test.ts

The size of the gap to Buck (1981)

Source.

Buck, A.L. (1981). New equations for computing vapor pressure and enhancement factor. Journal of Applied Meteorology 20(12), 1527–1532. · Allen, R.G., Pereira, L.S., Raes, D. & Smith, M. (1998). Crop evapotranspiration — Guidelines for computing crop water requirements. FAO Irrigation and Drainage Paper 56, Rome. Chapter 3, Equation 11 and the worked Examples 3, 4 and 5.

Method.

Both equations are evaluated across the range any user of this tool will see and the worst relative difference between them is measured, so that the choice of one over the other can be argued about with a number rather than an adjective.

• Outside 0 to 40 °C nothing is claimed. Below freezing the two equations diverge because Buck publishes a separate form over ice and Equation 11 does not, and a greenhouse tool had no reason to go there.

Asserted by src/tools/vpd/__tests__/compute.test.ts

Inverting Equation 11 at FAO-56’s printed pressures

Source.

Allen, R.G., Pereira, L.S., Raes, D. & Smith, M. (1998). Crop evapotranspiration — Guidelines for computing crop water requirements. FAO Irrigation and Drainage Paper 56, Rome. Chapter 3, Equation 11 and the worked Examples 3, 4 and 5.

Method.

Every saturation vapour pressure the chapter prints is fed to the inverse, and the temperature it came from is required back, rounded to two decimals.

• This is a reproduction of the paper’s printed table read backwards, not of a printed dew point: FAO-56 works no example whose answer is a dew point. The claim it supports is that the inverse is the inverse, which is what a reader comparing this page against the site’s own saturation vapour pressures needs it to be.

Asserted by src/tools/dew-point/__tests__/compute.test.ts

The two documented alternatives, measured

Source.

Allen, R.G., Pereira, L.S., Raes, D. & Smith, M. (1998). Crop evapotranspiration — Guidelines for computing crop water requirements. FAO Irrigation and Drainage Paper 56, Rome. Chapter 3, Equation 11 and the worked Examples 3, 4 and 5. Annex 3, Equation 3-11 — the same inverse printed with its two derived constants rounded to two decimals. · Buck, A.L. (1981). New equations for computing vapor pressure and enhancement factor. Journal of Applied Meteorology 20(12), 1527–1532.

Method.

The exact inverse is evaluated against the paper’s printed approximation and against Buck across the terrestrial range, and the worst departure of each is measured.

Asserted by src/tools/dew-point/__tests__/compute.test.ts