Chapter 5

Landscape metrics

Landscape metrics are numbers that summarise the pattern of a categorical map: how much of each class there is, how many patches, how large, how complex, how clumped, how diverse. They let us compare landscapes, follow them over time and relate pattern to ecological responses. They are also easy to misuse: hundreds exist, many measure the same thing, and all depend on the map. This chapter explains what the main metrics measure, computes a few by hand, and gives rules for choosing and interpreting them.

In this chapter

• composition and configuration; patches, classes and landscapes;
• the families of metrics, with the formulas of the most used ones, checked by hand on a tiny map;
• redundancy: why a handful of metrics carries most of the information;
• common pitfalls, and a parsimonious set to start from;
• sampling designs: whole maps, plots, windows and zones.

A short history

Quantitative indices of landscape pattern appeared in the 1980s (O'Neill et al., 1988). The program FRAGSTATS (McGarigal & Marks, 1995; McGarigal et al., 2012) gathered them into a standard library with standard definitions, and became the reference for thousands of studies. The R package landscapemetrics reimplemented them in open source (Hesselbarth et al., 2019), and added information-theory metrics of complexity (Nowosad & Stepinski, 2019). Landschaft reproduces the definitions of landscapemetrics and FRAGSTATS (accuracy), so its values can be compared with the literature.

Composition and configuration

Every metric describes one of two aspects of pattern (McGarigal & Marks, 1995; Gustafson, 1998):

• Composition: what is in the landscape and in which amounts, with no reference to where. The share of each class, the number of classes, their diversity. Composition does not change if you shuffle the cells.
• Configuration: how it is arranged in space. The size and shape of patches, their isolation, their aggregation or interspersion. Configuration changes when you shuffle the cells.

Chapter 3 gave a reason to keep them apart: amount of habitat (composition) and fragmentation per se (configuration) have different ecological effects, and configuration metrics are often correlated with composition.

Patches, classes and landscapes

A patch is a group of connected cells of the same class. "Connected" needs a rule: the 4-neighbour rule joins cells sharing a side; the 8-neighbour rule also joins cells touching at a corner. The rule changes the number and size of patches (chapter 3), so report it. FRAGSTATS and landscapemetrics default to 8, as does Landschaft (directions=8).

Metrics exist at three levels:

Level	One value per	Examples
patch	patch	area, perimeter, shape index, core area, nearest-neighbour distance
class	class	class area, percentage of landscape, number of patches, edge density, aggregation
landscape	map	total edge, diversity, contagion

Class and landscape metrics are either computed directly from the map (ed, ai, shdi) or summarise the patch metrics with a statistic: mean (_mn), area-weighted mean (_am), median (_md), standard deviation (_sd), coefficient of variation (_cv) and range (_ra); the area-weighted mean, median and range need extras=True. The area-weighted mean deserves attention: it describes the patch in which a random cell of the class falls, which is often more meaningful ecologically than the mean over patches, dominated by the many small ones.

Metrics by hand

A tiny map of 6 × 6 cells of 10 m (0.01 ha each) with three classes is enough to see how the main metrics work:

import numpy as np
import pandas as pd
import landschaft as ls

toy = np.array([
    [1, 1, 1, 2, 2, 2],
    [1, 1, 1, 2, 2, 2],
    [1, 1, 3, 3, 2, 2],
    [1, 1, 3, 3, 2, 2],
    [3, 1, 1, 1, 1, 1],
    [3, 3, 1, 1, 1, 1],
])
small = ls.Landscape(toy, res=10)
small.patch_metrics(["area", "perim"])

Four patches: one of class 1, one of class 2 and two of class 3 (the block in the middle and the three cells in the corner, which do not touch). Areas are in hectares, perimeters in metres.

Area and edge

The percentage of the landscape (PLAND) is composition in its simplest form; edge density (ED) is the length of the boundaries between classes per unit area:

PLAND_i = 100 × A_i / A      ED = E / A A_i: area of class i; A: total area (ha); E: total length of edge (m) between cells of different classes, not counting the map border.

Counting the edges between horizontally and vertically adjacent cells of different classes gives the total edge:

horizontal = (toy[:, 1:] != toy[:, :-1]).sum()
vertical = (toy[1:, :] != toy[:-1, :]).sum()
total_edge = 10 * (horizontal + vertical)                 # metres
total_area = toy.size * 0.01                              # hectares

pland_1 = 100 * (toy == 1).sum() / toy.size
{"TE": int(total_edge), "ED": total_edge / total_area, "PLAND class 1": pland_1}

small.landscape_metrics(["te", "ed"])

The values match: 16 cell sides lie between different classes, 160 m of edge in 0.36 ha, 444 m/ha. Edge density is among the most used metrics because edges are where edge effects happen.

Diversity

The Shannon diversity index (SHDI), from information theory (Shannon, 1948), combines the number of classes (richness) and how evenly the area is shared among them:

SHDI = − Σ_i P_i ln P_i      SHEI = SHDI / ln m P_i: proportion of the landscape in class i; m: number of classes. SHDI is 0 with a single class and grows with more, and more evenly shared, classes. SHEI, the evenness, is 1 when all classes have the same area.

p = np.array([(toy == c).mean() for c in (1, 2, 3)])
shdi = -(p * np.log(p)).sum()
{"SHDI by hand": shdi, "SHEI by hand": shdi / np.log(3)}

small.landscape_metrics(["shdi", "shei", "pr"])

Diversity metrics are pure composition: shuffle the cells of the toy map and SHDI stays 1.01.

Aggregation

The aggregation index (AI) of a class compares the number of adjacencies between its own cells with the maximum possible for that number of cells (He et al., 2000):

AI_i = 100 × g_ii / max g_ii g_ii: number of shared sides between cells of class i; max g_ii: the number it would have if its cells formed the most compact shape possible.

small.class_metrics(["np", "ai"])

Class 2 scores 100: its ten cells are as compact as ten cells can be. Class 3, split in two, scores lowest. AI is independent of the amount of the class, which makes it useful to compare aggregation between classes and landscapes with different amounts (chapter 3).

The families of metrics

Beyond the examples above, metrics fall into six families. Landschaft lists them with ls.list_metrics(), and every tool selects them by the same level, metric and type filters.

catalogue = ls.list_metrics()
catalogue.groupby(["type", "level"]).size().unstack(fill_value=0)

Family	Measures	Key metrics
area and edge	size of patches and classes, amount of edge	area, ca, pland, lpi, te, ed
shape	complexity of patch outlines	shape (perimeter over the most compact perimeter, 1 = square), frac (fractal dimension), para, circle, contig
core area	interior habitat after removing an edge depth	core, cai (core as % of the patch), tca, ndca
aggregation	subdivision, isolation, clumping, interspersion	np, pd, enn (nearest-neighbour distance), ai, clumpy, cohesion, division, mesh, split, contag, iji
diversity	richness and evenness of classes	pr, shdi, shei, sidi, siei
complexity	information-theory measures of the whole pattern	ent, condent, joinent, mutinf, relmutinf

A few deserve a sentence each:

• Largest patch index (lpi): the share of the landscape in the largest patch. Sensitive to percolation (chapter 3).
• Euclidean nearest-neighbour distance (enn): edge-to-edge distance from each patch to the nearest patch of the same class. The simplest isolation metric; it ignores everything but the nearest neighbour.
• Patch cohesion (cohesion): the physical connectedness of a class, shown to relate to the dispersal success of simulated organisms (Schumaker, 1996).
• Contagion (contag) and interspersion and juxtaposition (iji): whether classes form large clumps or are finely mixed (O'Neill et al., 1988; Li & Reynolds, 1993; McGarigal & Marks, 1995).
• Complexity metrics separate the diversity of classes (marginal entropy, ent) from the complexity of their arrangement (conditional entropy, condent) and their aggregation (mutual information, mutinf), in one consistent framework (Nowosad & Stepinski, 2019).
• Core-area metrics depend on an edge depth (edge_depth, in cells): choose it from the depth of edge effects, as for MSPA (chapter 4).

Redundancy: fewer metrics than you think

With more than 130 metrics available, it is tempting to compute them all. But many are mathematically related, and in real landscapes most are strongly correlated. Factor analyses of dozens of metrics over many landscapes found that a handful of independent dimensions (about six or seven) explain most of the variation: roughly amount, subdivision, edge, shape, aggregation, isolation and diversity (Riitters et al., 1995; Cushman et al., 2008).

We can see it on the example map, cut into a grid of 100 zones of 5 km:

landcover = ls.load_raster(ls.example_data("clc"))
zones = ls.grid_zones(landcover, 5000)
names = ["np", "pd", "te", "ed", "lsi", "lpi", "ai", "contag",
         "division", "mesh", "shdi", "sidi", "pr"]
per_zone = ls.zonal_metrics(landcover, zones, level="landscape", metric=names)

corr = per_zone[names].corr(method="spearman")
upper = corr.where(np.triu(np.ones(corr.shape, dtype=bool), k=1)).stack()
print(f"{(upper.abs() > 0.7).sum()} of {len(upper)} pairs have |r| > 0.7")
upper[upper.abs() > 0.95].round(3)

Some pairs are identical by definition over zones of equal area: np and pd (patch density is the number of patches per area), te, ed and lsi, division and mesh. Others are nearly so because they measure the same property: aggregation (ai) is the mirror image of edge density, the largest patch (lpi) of division. Reporting all of them adds columns, not information. ls.calculate_correlation and ls.show_correlation compute and draw such matrices for any metric table.

Pitfalls

Landscape metrics are easily misused (Li & Wu, 2004; Kupfer, 2012). The most frequent problems:

1. Metrics without a question. A metric is useful when it measures a property linked to the process studied. Start from the process (dispersal, edge effects, disturbance spread), then pick the metric; not the reverse.
2. Scale dependence. Values change with grain, extent and legend (chapter 2). Compare only maps with the same grain, extent and classes.
3. Confounding with amount. Many configuration metrics change with the amount of the class alone (Neel et al., 2004). Include the amount (pland) in analyses and interpret configuration conditional on it.
4. Map error. Misclassified cells create spurious patches and edges; patch counts and edge metrics are the most affected (Langford et al., 2006). Know the accuracy of the map, and smooth or filter very small patches if they are mostly noise.
5. The map border. Patches cut by the border look small and simple. Buffer the study area, or be consistent in how the border is treated (count_boundary, consider_boundary).
6. No sampling distribution. A metric is a descriptive value of one map; whether two values differ "significantly" requires replicate landscapes or a model of the pattern (Remmel & Csillag, 2003).
7. Binary patch thinking. Patch metrics assume sharp boundaries. For gradual landscapes, consider continuous (surface) measures (McGarigal et al., 2009) or the moving-window tools of chapter 3.

A parsimonious starting set

No list fits every study, but one metric per independent dimension is a defensible start, to be adapted to the question:

Dimension	Class level (one habitat)	Landscape level (the mosaic)
amount	pland	–
subdivision	np or pd, mesh	np, mesh
edge	ed	ed
aggregation	ai or cohesion	contag or ai
isolation	enn_mn (or enn_am)	–
shape	shape_am	shape_am
interior	cai_am (with a justified edge_depth)	–
diversity	–	pr, shdi

Area-weighted means (_am) are FRAGSTATS statistics that landscapemetrics does not have; ask for them with extras=True, e.g. Landscape(landcover).class_metrics(["shape_am", "cai_am"], extras=True).

Sampling designs

The same metrics can be computed over different spatial units, each suited to a kind of question:

Design	Unit	Question	Tool
whole map	the study area	How does this landscape compare with others, or with itself later?	Landscape, landscape_metrics
plots	buffers around sites	How does the landscape around each site relate to what was observed there?	sample_metrics, scale_sample, extract_metrics
moving window	a window around every cell	Where does the structure change across the region?	window_metrics
zones	polygons or a grid	How do administrative units, watersheds or grid cells compare?	zonal_metrics, grid_zones
many landscapes	a set of maps or zones	What types of landscape exist, and what gradients separate them?	SpatialSignatureAnalysis

Site-based designs are the core of landscape-scale ecological studies, and their design deserves as much care as the metrics. Landscapes around sites should not overlap (overlapping buffers share information, so they are not independent samples), should span a wide range of the variables of interest, and should decorrelate them as far as possible, for example selecting landscapes where habitat amount and fragmentation vary independently (Eigenbrod et al., 2011; Pasher et al., 2013). The buffer size should be chosen by the multi-scale approach of chapter 2.

Key points

• Composition (how much) and configuration (how arranged) are distinct aspects of pattern; measure both, and interpret configuration given composition.
• Patches depend on the neighbour rule; metrics exist at patch, class and landscape levels.
• Most metrics are simple counts and ratios; computing a few by hand is the best way to understand them.
• Metrics are highly redundant: choose a small set, one per dimension, guided by the process studied.
• Metrics are scale-dependent and sensitive to map error; compare like with like.

Check your understanding

1. Shuffling the cells of a map at random: which of pland, shdi, ed, ai and np change?
2. A study reports mean patch area (area_mn) of forest falling over time. Give two different landscape changes that could explain it.
3. Why is area_am often more informative than area_mn?

Answers

1. pland and shdi (composition) stay the same; ed increases, ai falls and np usually increases: configuration changes.
2. Loss of habitat from large patches (shrinkage, or large patches split in two), or the appearance of many new small patches (new plantations, regrowth) that pull the mean down while the large patches stay intact. The mean alone cannot tell them apart; look at np, lpi and pland.
3. The mean over patches is dominated by the many small patches; the area-weighted mean describes the patch where a random cell of habitat lies, i.e. the landscape as experienced by the habitat itself.

References

• Cushman, S. A., McGarigal, K., & Neel, M. C. (2008). Parsimony in landscape metrics: strength, universality, and consistency. Ecological Indicators, 8(5), 691–703.
• Eigenbrod, F., Hecnar, S. J., & Fahrig, L. (2011). Sub-optimal study design has major impacts on landscape-scale inference. Biological Conservation, 144(1), 298–305.
• Gustafson, E. J. (1998). Quantifying landscape spatial pattern: what is the state of the art? Ecosystems, 1(2), 143–156.
• He, H. S., DeZonia, B. E., & Mladenoff, D. J. (2000). An aggregation index (AI) to quantify spatial patterns of landscapes. Landscape Ecology, 15(7), 591–601.
• Hesselbarth, M. H. K., Sciaini, M., With, K. A., Wiegand, K., & Nowosad, J. (2019). landscapemetrics: an open-source R tool to calculate landscape metrics. Ecography, 42(10), 1648–1657.
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