Aggregate indicator scores by the area of a radar diagram

Aggregates a vector of scored soil indicators into a single value using the area of the polygon they trace on a radar (star) diagram. This is a weight-free alternative to the weighted additive index, and so sidesteps the most contested step in the whole SQI pipeline: deciding how important each indicator is.

Usage

sqi_area(s, reference = NULL, na.rm = FALSE)

Arguments

s: Numeric vector of indicator scores, normally in [0,1] as produced by score_indicators. Must contain at least three values: fewer than three indicators do not describe a polygon.
reference: Optional numeric vector of reference scores, of the same length as s, describing a non-degraded reference soil. When supplied, the returned value is the ratio of the two areas. When NULL (the default) the absolute area is returned.
na.rm: Logical. If TRUE, missing scores are dropped before computing the area. Note that this changes $n$ and therefore the area, so it is FALSE by default: a missing indicator is a fact about the data, not something to silently absorb.

Value

A single numeric value: the absolute area when reference is NULL, otherwise the dimensionless ratio of the two areas. A ratio below 1 indicates the sample has lost function relative to the reference.

Details

The area is computed as $$A = 0.5 \sum_{i} s_i^2 \sin(2\pi / n)$$ where $s_i$ are the scored indicators and $n$ their number.

It is the square of each score, not the product of adjacent radii. The true area of the polygon would be $0.5 \sin(2\pi/n) \sum_i s_i s_{i+1}$, which depends on the order in which indicators are placed around the diagram – an arbitrary choice that would make the index arbitrary too. Kuzyakov's square form approximates the polygon as a sum of individual triangles and is order-independent, which is why it is used here. This is a deliberate deviation from the geometric area, and reviewers do ask about it.

The formula is designed to be used as a ratio. Kuzyakov standardises against a non-degraded reference soil (whose scores are all 1.0) and reports $A_{degraded} / A_{reference}$; the worked example in the paper gives 0.47, read as "half the soil function lost". Used absolutely, with reference = NULL, the value is standardised against nothing but your own sample and is not comparable to any other study:

Use	Standardised against	Comparable?
`reference = NULL`	your own sample	no
`reference` supplied	a non-degraded reference soil	claimed yes

The weight-independence that makes this route attractive is a consequence of taking a ratio, not a property of the formula itself.

Note also that the absolute area depends on $n$ through the $\sin(2\pi/n)$ term, so areas computed from different numbers of indicators are not comparable either. The ratio is insensitive to this, provided both vectors have the same length – which is why a mismatch warns.

References

Kuzyakov, Y. et al. (2020), eq. (2). Frontiers of Agricultural Science and Engineering 7(3):282-288. doi:10.15302/J-FASE-2020338

Examples

scores <- c(0.8, 0.6, 0.7, 0.9, 0.5)

# Absolute area -- not comparable across studies
sqi_area(scores)
#> [1] 1.212597

# The maximum attainable area for this number of indicators
sqi_area(rep(1, 5))
#> [1] 2.377641

# As a ratio against a non-degraded reference soil
sqi_area(scores, reference = rep(1, 5))
#> [1] 0.51

# Order-independence: the point of the square form
sqi_area(scores) == sqi_area(rev(scores))
#> [1] TRUE