Strategies

Note

This text about available transformation strategies uses several terms and symbols one might be unfamiliar with. The Glossary should be able to explain those terms.

To examine how observed patterns change with grain and extent, seal systematically transforms the original study grid. Neighboring sampling units are combined into larger quadrats, increasing grain, while their spatial configuration determines the extent represented at each transformation level.

seal provides three transformation strategies. Each strategy defines how the original sampling units are combined and therefore how grain and extent change across successive levels.

Before starting an analysis, it must be specified in the taskfile what strategy should be used. Nested transects is the default.

Note

seal assumes a grid-based representation of the sampling domain with sampling units identified by their coordinates. If the study grid contains sampled quadrats with no species records, their locations must be provided explicitly using a quadrat lists specified by the quadrat-list key in the taskfile. This preserves the complete spatial structure of the sampled grid and distinguishes genuinely empty sampling units from those where sampling was not possible. Such units can be marked using quadrat-types key of the taskfile, and are omitted from the analysis.

Warning

Each strategy have specific constraints regarding the shape of study grid they can be applied to. adjust-grid subcommand can be used to modify the grid to fulfill these constrains. Depending on the original grid, this may require removing some sampling units and therefore reducing the coverage of the data.

The following diagrams show transformation of a hypothetical, 9×9 study grid of 81 original quadrats during an analysis using three levels. For each strategy, the grid is shown after adjustment to support the three levels of transformation. As each strategy imposes different constraints, the required adjustments differ.

During the transformation, original sampling units are grouped into larger quadrats under the Nested quadrats strategy or into transects under one of the transect-based strategies (Nested transects, Contiguous Transects). New coordinates are assigned to the resulting sampling units at each transformation level.

Three strategies are available. The following legend applies to all figures in the strategies description.

• Sampling unit included in the analysis of current level, possibly with new coordinates assigned during transformation.

• Sampling unit present in the original data but not included in the analysis of current level.

• Sampling unit present in the original data, but removed to fulfill the constraint of a strategy, either manually or using adjust-grid.

• Sampling unit coordinate in the original study grid.

• Coordinate assigned to a sampling unit in the transformed grid.

• Coordinate of a sampling unit removed during the adjustment of the study grid.

Nested quadrats

In this strategy, neighboring quadrats are progressively merged along both transects (x-axis) and :term::zones <zone> (y-axis).

The grid is divided into square sections with side length equal to \(2^{\max{L} - 1}\) original quadrats. In other words, the adjusted study grid is “tiled” by modules corresponding to the largest quadrats resulting from selected levels. For each level, a merged quadrat is placed on the bottom left corner of its section, with side length equal to \(2^{l - 1}\) quadrat(s). The remaining original quadrats within the module are omitted from the analysis of the current level. For the highest requested level \(l = \max{L}\), the merged quadrats fill the entire study grid. Each quadrat at level \(l\) is twice as long and twice as wide as a quadrat at level \(l-1\). Its area is therefore four times larger.

Level 1 quadrats represent the smallest observational units and are nested in the lower-left corners of higher-level quadrats, assuming the study grid is viewed as the first quadrant of a Cartesian coordinate system, with x increasing to the right and y increasing upwards. When the same cumulative area is represented using quadrats of lower level a larger number of such smaller quadrats is required. These quadrats are distributed further apart within the study grid and therefore span a greater extent. Conversely, higher-level quadrats have a larger grain and are more spatially aggregated. The two-dimensional structure of the grid allows extent to vary along both spatial axes.

Note

This strategy is the most similar to the one used by Palmer & White⁶ and forms an observational experiment that allows the influence of grain and extent to be systematically examined. It is best suited for study grids with equal numbers of transects and zones (\(t = z\)), although the physical dimensions of the sampled area do not have to be square.

Constraints:

• \(t \mod 2^{\max{L}-1} = 0\)

  (number of transects in the original study grid must be divisible by 2 to the power of maximum requested level, decreased by 1)

• \(z \mod 2^{\max{L}-1} = 0\)

  (number of zones in the original study grid must be divisible by 2 to the power of maximum requested level, decreased by 1)

Warning

The strategy can be used for rectangular grids. However, because the transformation is based on square modules, strongly unequal number of transects and zones may result in an unbalanced representation of the two spatial dimensions. For distances exceeding the shorter dimension of the grid, variation in extent is represented mainly along the longer axis. The more elongated the grid, the more pronounced this limitation. For elongated study grids, we recommend using transect-based strategies, aggregating sampling unit along one axis only.

Example:

my-taskfile.toml

levels = [ 1, 2, 3 ]
level-strategy = "nested-quadrats"

Used notation: L, l, t, z

Nested transects

This strategy first merges all quadrats with the same x-coordinate into transects extending along the y-axis. The entire study grid is then divided along the x-axis into \(\max{L}\) wide sections, each containing \(t'=\frac{t}{\max{L}}\) original transects.

At each level \(l\), the leftmost \(l\) original transects within each wide section are merged into one transformed transect. The remaining transects are omitted from the analysis of the current level. As \(l\) increases, each transformed transect becomes wider, thereby nesting smaller ones into larger sampling units. The number of transects remains constant for every level: \(t'=\frac{t}{\max{L}}\). In the highest requested level \(\max{L}\), every transect of the original grid is included, although they are merged and assigned new coordinates rather than analysed in their original form.

Note

This is the default strategy. We recommend using it for rectangular grids.

This strategy is similar in nature to nested quadrats strategy, but aggregation occurs solely along the x-axis. It therefore represents changes in extent primarily along one spatial dimension rather than across a two-dimensional area. This provides a practical option for data collected along transects, where each transect represents a one-dimensional sampling unit. The nested quadrats strategy can still be used for such grids, but its limitations for elongated grids should be considered when interpreting the results.

Constraints:

• \(t \mod \max{L} = 0\)

  (number of transects in the original study grid must be divisible by the maximum analysed level)

Example:

my-taskfile.toml

levels = [ 1, 2, 3 ]
level-strategy = "nested-transects"

Used notation: L, l, t, t’

Contiguous Transects

This strategy first merges all quadrats with the same x-coordinate into transects extending along the y-axis. Then, at each level \(l\), groups of \(l\) adjacent transects are merged into one transformed transect.

This process is done across the entire study grid, resulting in new grid with transects \(l\) quadrats wide (and containing \(t' = \frac{t}{l}\) transects in total). An encounter at \(q_{x,y}\) would become encounter at \(q_{\lfloor{x / l} \rfloor, y}\).

In other words, the study grid is “tiled” from left to right by transects with width being a multiple of the original width and the analysed level.

This strategy does not result in any original transects being omitted from the analysis. The transects are indexed left to right, and the merging always starts with the lowest x-coordinate (leftmost).

Note

This strategy does not examine the effect of grain and extent simultaneously.

This strategy always covers the entire grid and the total extent, therefore, remains constant, while grain changes through the aggregation of adjacent transects.

Because all original sampling units are retained at every level, the total set of species recorded across the entire grid also remains unchanged. However, their distribution among transformed transects and the resulting grain-dependent community patterns may differ among levels.

It serves as a supporting strategy, capturing the effect of grain.

Constraints:

• \(\forall l\in L: t\mod l = 0\)

  (number of transects in the original study grid must be divisible by every analysed level)

Example:

my-taskfile.toml

levels = [ 1, 2, 3 ]
level-strategy = "contiguous-transects"

Used notation: L, l, t, t’