Keep these three objects separate
1 · Annual winning event
For each box and year, test every contained 72-hour period and choose the period with the greatest cell-area-weighted 5×5 mean. Freeze all 25 cell totals from that same event.
2 · CAM field
At each fixed relative cell position, average that cell across the 45 selected annual events. The result is one 5×5 field with 25 CAM values.
3 · Spatial L-moments
Sort those 25 CAM values and compute L-mean, L-CV, L-skewness and L-kurtosis. The observations at this stage are cells in space, not years in time.
From a rainfall field to L₁, L₂, L₃ and L₄
This integrates the useful pieces of the earlier teaching pages into one chain rather than making you mentally teleport among three files.
A small synthetic CAM
After annual event selection and cell-by-cell averaging, suppose a five-cell CAM is:
Sort it:
The rank-weighted averages
b₀ weights every rank equally. As r rises, higher ranks receive progressively greater weight. These are building blocks, not new rainfall observations.
Dimensional L-moments
The four SLAM descriptors
See the real domain before interpreting any statistic
Interactive metric map: all 2,376 real boxes
A real geographic signal
Across the 2,376 boxes, longitude and L-mean correlate at r ≈ +0.860. Eastward boxes are substantially wetter in the 45-year CAM magnitude. The other L-moment ratios do not simply reproduce that gradient.
| Approx. longitude third | n | L-mean | L-CV | L-skew | L-kurt |
|---|---|---|---|---|---|
| West third | 800 | 62.58 | 0.0177 | -0.031 | 0.083 |
| Central third | 810 | 71.58 | 0.0187 | -0.024 | 0.089 |
| East third | 766 | 80.70 | 0.0163 | -0.073 | 0.089 |
Why this matters for regionalization
A regionalization based only on magnitude would mostly rediscover the east–west precipitation gradient. SLAM asks for agreement in magnitude, relative spread, asymmetry and tail structure. The extra dimensions are what prevent “same average rainfall” from being mistaken for “same spatial storm texture.”
Box Laboratory: watch 45 annual winners collapse into one CAM
Selected annual 72-hour winner
45-year CAM for the same 25 relative cells
Observed CAM L-moments
One bootstrap CAM, for intuition only
L-kurtosis is not “how pointy the map looks”
For n = 25, L₄ has a rank signature
Because L₄ is a linear combination of b₀…b₃, each sorted CAM value gets one net L₄ coefficient. Center the values by subtracting L₁ and the pattern becomes interpretable:
The coefficient sum is zero, so shifting every cell by +50 mm does not change L₄. The dry extreme has a negative coefficient and is below the mean, so their product contributes positively; the wet extreme has a positive coefficient and is above the mean, so it also contributes positively. Shoulder ranks tend to contribute in the opposite direction. Scaling all values scales L₄ and L₂ together, leaving τ₄ unchanged.
The ratio that SLAM uses
L₄ is in millimeters. L₂ is the spatial L-scale in millimeters. Dividing makes τ₄ dimensionless: tail/shoulder structure relative to the field's overall spread.
High τ₄ does not mean “the box has huge spread.” A box can have ordinary L-CV but high L-kurtosis if most cells cluster tightly while a few outer-ranked cells sit distinctly away from that core.
Kurtosis example
Sorted CAM values and L₄ contributions
Actual low-L-kurtosis box
B00827_R0239_C1119 · τ₄ = −0.0988. Its 25 CAM values progress fairly smoothly from about 60.0 to 65.9 mm. The shoulder ranks carry much of the spread, so the outermost ranks do not dominate after normalization.
Actual high-L-kurtosis box
B02180_R0205_C1173 · τ₄ = +0.4203. Most CAM cells cluster roughly around 72.8–75.1 mm, while the far upper ranks jump to about 79.5 and 81.9 mm and the low end reaches about 69.6–70.1 mm. That separation of outer ranks from the shoulders drives positive L₄.
Paste any CAM values and expose every calculation
This combines the earlier estimator calculator with the real-box laboratory. Load a representative 25-cell CAM or type your own values.
Probability-weighted moments
Convert to L₁–L₄
Rank weights and contributions
| i | x(i) | w₁ | xw₁ | w₂ | xw₂ | w₃ | xw₃ |
|---|
What “regionalization” will mean in this PRISM adaptation
Target uncertainty
Select the exact target/reference box. Resample its 45 complete annual 25-cell event fields with replacement. Rebuild one CAM per replicate and compute four L-moment statistics. Repeat 10,000 times with a fixed seed.
Translated comparisons
Every accepted translated 5×5 box already has its observed L-mean, L-CV, L-skewness and L-kurtosis. Compare each observed statistic with the corresponding target bootstrap distribution to obtain two-sided empirical p-values.
Spatial domain
Control false discoveries for each statistic, combine the four acceptance fields, then keep the connected accepted region associated with the target. That connected set is the inferential transposition region.
Pre-bootstrap similarity sandbox
What this sandbox teaches correctly
Regionalization is an intersection problem. A box can look similar in mean rainfall and still fail because its within-box contrast, asymmetry or tail structure differs. Adding dimensions usually reduces the candidate set.
What this sandbox does not claim
Standard-deviation tolerances are not SLAM p-values, do not represent target sampling uncertainty, do not apply FDR, and do not enforce the final connected-component rule. This exists to build intuition before the inferential machinery is turned on.