Choose your reading route
eduKateSG · Why Mathematics? · Spatial Block Bootstrap: Neighbourhoods, Geometry and Dependence Range
Start with the question, audit the worked example, or jump to the assumptions and practice. Every chapter stays open for uninterrupted reading.
Understand the question
Define the quantity and follow the mechanism.Route 2 · Chapters 8–14
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 15–21
Check assumptions
See where design and calibration matter.Route 4 · Chapters 22–28
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 29–34
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why is mathematics important when nearby locations share weather, soil, traffic or social conditions? The spatial block bootstrap resamples two-dimensional neighbourhoods rather than isolated points. Block shape and side length determine which local covariance patterns survive, making geometry part of the inferential method.
Did You Know? A square block with side length 3 grid cells contains 9 locations. Doubling the side to 6 quadruples the area to 36, so spatial block tuning changes information much faster than a one-dimensional block length.
Full chapter index
Chapters 1–7 · Understand the question
Chapters 8–14 · Reproduce the example
Chapters 15–21 · Check assumptions
Chapters 22–28 · Explore applications
Chapters 29–34 · Study and verify
Row resampling treats locations as exchangeable and independent. Spatial data often have covariance that decays with distance, so nearby observations must move together.
Nordman, Lahiri and Fridley studied optimal block size for spatial block-bootstrap variance estimation and emphasised dependence on block mechanism and covariance structure. Read the research paper copy.
Define square or rectangular blocks of neighbouring grid cells, often allowing overlap. Sample blocks with replacement and assemble enough selected values for a bootstrap field or statistic.
A square with side l=3 cells contains l²=9 grid locations. With l=6 it contains 36, four times as many locations.
| Side cells | Locations |
|---|---|
| 3 | 9 |
| 6 | 36 |
If grid spacing is 2 km and l=3, the block spans 6 km along its side under a three-cell-width convention. The physical interpretation must match the coordinate convention.
| Grid spacing | Side cells | Illustrative span |
|---|---|---|
| 2 km | 3 | 6 km |
Square, rectangular and circular neighbourhoods preserve different directional relations. Anisotropic processes may need longer blocks along the direction of stronger dependence.
Overlapping blocks supply more candidate neighbourhoods and can improve variance estimation, but selected blocks share observations in the candidate library.
Partition blocks are simpler and cheaper but depend strongly on arbitrary boundaries. Shifting the origin can materially change results.
Blocks near domain boundaries may be truncated, wrapped on a torus, reflected or excluded. Each rule changes inclusion probabilities and scientific plausibility.
Small blocks break long-range dependence and underestimate uncertainty. Large blocks leave few effectively distinct pieces and increase Monte Carlo variability.
Report block dimensions in metres or kilometres as well as cell counts. A three-cell block means different science at 10 m and 10 km resolution.
For scattered coordinates, neighbourhoods can be defined by distance, tessellations or moving windows. Naively mapping to a grid may distort density and barriers.
Wind, rivers and roads can make dependence direction-specific. A single isotropic side length can preserve the wrong covariance structure.
Resample spatial blocks, calculate each field mean and use replicate variation for uncertainty. The assembly rule must prevent dense overlap from silently overweighting locations.
Spatial regression pairs outcomes and predictors at each location. Blocks should move complete location rows while preserving the design and any fixed coordinate features.
Random holdouts scattered across space test interpolation near training points. Holding out spatial blocks tests prediction in unseen areas and answers a harder transport question.
Time-series blocks have one order and natural direction. Spatial blocks have shape, orientation, boundary and irregular-location choices.
Administrative regions can be clusters, but spatial correlation may cross their borders. Geometric blocks target distance dependence rather than membership alone.
A parametric spatial model estimates a covariance function and simulates fields. Block bootstrap is less globally parametric but still assumes local stationarity and an appropriate block geometry.
Plot variograms or correlograms, vary block side and shape, test edge rules, simulate known spatial fields, measure coverage and compare with covariance-model alternatives.
Blocks may fail under strong nonstationarity, barriers, sparse irregular sampling, long-range dependence, tiny domains or a block size chosen after seeing favourable results.
Store coordinates, block origin, shape, selected locations and replicate weights. Verify physical dimensions and map every selected block before calculating statistics.
Construct candidate neighbourhoods, sample block IDs and unit-test l=3 gives 9 cells and l=6 gives 36. Save coordinate reference, boundary rule and seed.
Spatial bootstrap methods appear in climate, ecology, public health, agriculture and urban planning. Careers require geometry, geostatistics, coding and substantive map knowledge.
Simulate a Gaussian spatial field, compare row and block bootstrap intervals over block sizes, and add anisotropy to test rectangular blocks.
A postcode is not automatically a valid spatial block, larger blocks are not always safer, and a pretty resampled map does not prove correct coverage.
CHAPTER 27 OF 34 · Explore applications
27. Connection to recent mathematics articles
The cluster-bootstrap guide explains membership dependence. The moving-block-bootstrap guide explains one-dimensional blocks.
Students should calculate block area and translate cells into kilometres. Parents can ask what distance dependence the block is intended to preserve and how edge locations were treated.
- Calculate l² for two sizes.
- Convert cells to physical distance.
- Diagnose an edge block.
- Explain anisotropy.
- Compare regions and geometric blocks.
### What is resampled? Spatial neighbourhoods.
### Does shape matter? Yes.
### Why report physical units? Cell size defines real distance.
### Are administrative clusters enough? Not always.
Report coordinate system, grid or neighbourhood rule, block side and physical size, shape, overlap, edge handling, B, seed, target statistic, spatial diagnostics and sensitivity results.
In two dimensions, square-block area grows with l². This makes block-size sensitivity especially sharp: a modest side-length change can greatly reduce the number of distinct neighbourhoods.
Confirm coordinate units, justified geometry, preserved local dependence, transparent edges, anisotropy checks, sensitivity across sizes and no assumption that borders stop spatial correlation.
Continue through the Mathematics Learning Hub. Reproduce one cluster, stratum, autoregressive or spatial-block calculation by hand before trusting a large automated run.
Extended practice laboratory
Practice case 1: Map coordinates
Setup: Start from the article’s spatial-block-bootstrap mechanism for case 1. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, verify the grouping, recursion or geometry identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 2: Choose grid
Setup: Start from the article’s spatial-block-bootstrap mechanism for case 2. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, verify the grouping, recursion or geometry identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 3: Choose block shape
Setup: Start from the article’s spatial-block-bootstrap mechanism for case 3. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, verify the grouping, recursion or geometry identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 4: Choose side length
Setup: Start from the article’s spatial-block-bootstrap mechanism for case 4. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, verify the grouping, recursion or geometry identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 5: Calculate area
Setup: Start from the article’s spatial-block-bootstrap mechanism for case 5. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, verify the grouping, recursion or geometry identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 6: Convert to kilometres
Setup: Start from the article’s spatial-block-bootstrap mechanism for case 6. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, verify the grouping, recursion or geometry identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 7: List overlapping blocks
Setup: Start from the article’s spatial-block-bootstrap mechanism for case 7. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, verify the grouping, recursion or geometry identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 8: Draw block IDs
Setup: Start from the article’s spatial-block-bootstrap mechanism for case 8. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, verify the grouping, recursion or geometry identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 9: Assemble field
Setup: Start from the article’s spatial-block-bootstrap mechanism for case 9. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, verify the grouping, recursion or geometry identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 10: Handle edges
Setup: Start from the article’s spatial-block-bootstrap mechanism for case 10. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, verify the grouping, recursion or geometry identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 11: Use torus wrap
Setup: Start from the article’s spatial-block-bootstrap mechanism for case 11. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, verify the grouping, recursion or geometry identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 12: Use truncation
Setup: Start from the article’s spatial-block-bootstrap mechanism for case 12. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, verify the grouping, recursion or geometry identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 13: Add anisotropy
Setup: Start from the article’s spatial-block-bootstrap mechanism for case 13. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, verify the grouping, recursion or geometry identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 14: Use rectangle
Setup: Start from the article’s spatial-block-bootstrap mechanism for case 14. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, verify the grouping, recursion or geometry identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 15: Use irregular sites
Setup: Start from the article’s spatial-block-bootstrap mechanism for case 15. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, verify the grouping, recursion or geometry identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 16: Calculate field mean
Setup: Start from the article’s spatial-block-bootstrap mechanism for case 16. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, verify the grouping, recursion or geometry identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 17: Fit spatial regression
Setup: Start from the article’s spatial-block-bootstrap mechanism for case 17. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, verify the grouping, recursion or geometry identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 18: Use block holdout
Setup: Start from the article’s spatial-block-bootstrap mechanism for case 18. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, verify the grouping, recursion or geometry identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 19: Compare row bootstrap
Setup: Start from the article’s spatial-block-bootstrap mechanism for case 19. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, verify the grouping, recursion or geometry identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 20: Compare clusters
Setup: Start from the article’s spatial-block-bootstrap mechanism for case 20. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, verify the grouping, recursion or geometry identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 21: Increase side length
Setup: Start from the article’s spatial-block-bootstrap mechanism for case 21. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, verify the grouping, recursion or geometry identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 22: Change seed
Setup: Start from the article’s spatial-block-bootstrap mechanism for case 22. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, verify the grouping, recursion or geometry identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 23: Audit variogram
Setup: Start from the article’s spatial-block-bootstrap mechanism for case 23. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, verify the grouping, recursion or geometry identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 24: Name limits
Setup: Start from the article’s spatial-block-bootstrap mechanism for case 24. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, verify the grouping, recursion or geometry identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 25: Write conclusion
Setup: Start from the article’s spatial-block-bootstrap mechanism for case 25. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, verify the grouping, recursion or geometry identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.