Chapter 14

Simulating land change

Chapter 10 looked back: what changed between two maps, by which process, and with what effect on habitat and connectivity. This chapter looks forward. A land change model takes what happened in an observed period, encodes it as rules, and asks where the landscape is going if those rules hold, or if a policy changes them. Its maps of the future are the inputs of the analyses of the previous chapters: the fragmentation, the connectivity or the habitat of a species in 2050 under one scenario or another. Done carelessly, it produces confident maps of nothing; done well, it is one of the few ways to compare the spatial consequences of decisions before they are taken (Verburg et al. 2004; Brown et al. 2013).

In this chapter

• the anatomy of a land change model: how much changes (demand), where (transition potential) and with what shape (allocation), under which rules;
• the main families of models and what each assumes;
• validation: the persistence null model, the figure of merit, the comparison of three maps at several resolutions, fuzzy similarity, and why kappa misleads;
• calibration by patterns, and the separation of calibration and validation periods;
• scenarios, replicates and the uncertainty of a single simulated map.

The anatomy of a land change model

Almost every model of land change answers three questions, and the clearest way to read one is to ask how it answers each (Verburg et al. 2004):

1. How much will change? The demand or quantity of change: hectares of cropland in 2030, a rate of deforestation, the housing a growing population needs. It usually comes from outside the map: trends, a transition matrix, economic models, a storyline.
2. Where will it happen? The transition potential (or suitability): for each cell, how likely each class, or each transition, is, given the drivers: distance to roads and to existing patches, slope, soil, protected areas.
3. With what shape? The allocation: how the demanded change is placed on the most likely cells: one cell at a time, in competition between classes, as new patches or as the growth of old ones.

Around them sit the rules: transitions that cannot happen (built land does not turn into primary forest in ten years), areas where nothing may change, the inertia of each class (an orchard resists conversion more than a pasture), succession after abandonment.

Family	Demand	Potential	Allocation	Typical use
Competition among classes (CLUE-S, Verburg et al. 2002)	areas per class and year	logistic regression per class	iterative: every cell takes the class of highest suitability plus an adjustable term, until the areas match	regional land use under scenarios
Land systems (CLUMondo, van Asselen & Verburg 2013)	goods and services	per land system	land systems that supply the demand, with intensification	agriculture, intensification against expansion
Transition-based cellular automata (DINAMICA, Soares-Filho et al. 2002)	Markov rates per transition	weights of evidence per transition	new patches and expansion of old ones, with size and shape	deforestation frontiers
Patch-generating CA (PLUS, Liang et al. 2021)	areas per class	random forests on the areas that expanded	competition with random seeds and a decreasing threshold	multi-class urban and rural change
Patch growing (FUTURES, Meentemeyer et al. 2013)	population per region	mixed-effects regression	patches of observed sizes grown with compactness	urban growth

Landschaft's ls.simulate builds all of them from the same pieces: a demand, a potential, an allocation and rules, put together in a Model. The pieces can be mixed: weights of evidence with the competition of CLUE-S, a random forest with Expander and Patcher.

How much: demand

The simplest demand is the trend of the observed period. A transition matrix (chapter 10) gives, for each class, the share of its cells that became each other class. As a Markov chain it projects the areas forward, assuming that the rates stay as they were. For yearly steps from a 14-year period, the yearly matrix is the 14th root of the period's matrix (a matrix whose 14th power is the observed one), not the period's rates divided by 14.

Sibuyan Island in the Philippines, mapped in 1997 and 2011 with five classes (forest, coconut, grassland, rice, other), is the classic example of CLUE-S:

import numpy as np
import landschaft as ls

S = ls.simulate
lu = ls.load_raster(S.example_data("sibuyan_landuse"))      # 1997 and 2011
drivers = ls.load_raster(S.example_data("sibuyan_drivers"))
drivers = drivers.sel(band=["dist_sea", "dist_roads", "elevation", "slope"])
t0, t1 = lu[0].where(lu[0] != 255), lu[1].where(lu[1] != 255)

chain = S.Markov.from_maps(t0, t1, steps=14)        # yearly rates
print(chain.multi.round(4))
print(chain.project({c: int((t1 == c).sum()) for c in chain.classes}, steps=10).round(0).iloc[[0, -1]])

The projection is only as good as the assumption that the next decade repeats the last. Most studies replace it with scenarios: demands from an economic model, a policy target (zero deforestation), or a storyline ("agricultural expansion", "protection"), each a table of areas per year (S.AreaDemand). The location and pattern parts of the model stay the same, which is what makes the scenarios comparable.

Where: transition potential

The potential of a cell for a class or transition is estimated from where change happened in the observed period. Three approaches dominate:

• Weights of evidence (Bonham-Carter 1994) cut each driver into ranges and give each range a weight: the log of how much more frequent the transition is there than elsewhere. The weights add up (assuming the drivers independent given the transition) into a probability. They are easy to read: the curve of weight against distance to roads is the effect of roads.
• Logistic regression of presence (or of change) on the drivers, the workhorse of CLUE-S and of many studies since.
• Machine learning (random forests, boosting): flexible responses and interactions, at the price of interpretability.

What the model learns matters as much as how. Fitted to where a class is, the model describes its current distribution, which mixes old and new land uses. Fitted to where the class appeared in the observed period (the land expansion analysis of PLUS), it describes where it expands, which is what a simulation needs:

from sklearn.ensemble import RandomForestClassifier

leas = S.Estimator(RandomForestClassifier(n_estimators=50, random_state=0),
                   target="expansion", classes=[2, 3], sample=0.3, stratify=True, seed=1)
leas.fit(t0, drivers, t1=t1)
print(leas.importance().round(3))     # which drivers explain where coconut (2) and grassland (3) expanded

Variables that the simulation itself changes, such as the distance to the nearest forest, the share of built land around a cell or the gravitational pull of the towns, should be recomputed at every step: these are the dynamic variables of Model(dynamic={...}) (S.distance_to, S.share_of, S.gravity_of). Roads can grow with the simulation too: a S.RoadBuilder (Dinamica's road constructor) adds roads from the network towards attractive places at every step, and its dynamic(network) is the distance to that growing network.

With what shape: allocation

The allocation decides the pattern of the simulated change, and two models with the same demand and potential can produce very different landscapes. The competitive allocation of CLUE-S gives each cell to the class of highest potential, which tends to scatter change over the many cells of near-equal potential. Patch-based allocations place change as new patches and as the growth of existing ones, with sizes taken from the observed change: the Expander and Patcher of DINAMICA, or the random patch seeds of PLUS. The share of the change that extends existing patches, and the size distribution of the new ones, can be measured on the observed maps:

patches = S.presets.observed_patches(t0, t1)
print(patches[["from", "to", "cells", "expansion", "new_patches", "new_mean"]].round(2))

A model whose patches are too small fragments the simulated landscape; one whose patches are too large consolidates it. Since the next steps of most studies are landscape metrics or connectivity, the shape of the simulated change is not a detail: it decides the answer.

Two kinds of change escape the class-by-class picture. Logging, fire and harvest take biomass, carbon or timber from the forest without turning it into another class: the Patcher of a continuous state of Dinamica EGO (S.ContinuousPatcher) opens patches that take the stock of their cells, drawn by probability, until a quota of stock is met. And change builds the infrastructure that drives more change: on Amazonian frontiers clearing follows the roads, and new roads follow the clearing (Soares-Filho et al., 2004, 2006). A road constructor (S.RoadBuilder) draws destinations among attractive cells at a range of distances from the roads and joins each to the network down the cheapest way over a friction surface, every step; the distance to the growing network is then a dynamic variable of the model.

Validation: is the simulation any good?

A simulation of the observed period, from the earlier map, can be compared with the later observed map. The comparison is less obvious than it seems.

Persistence is the null model. Most of a landscape does not change in a decade. A simulation that changes nothing at all agrees with the observed map in most cells (often more than 90 %), and a model that changes the right amount in the wrong places can agree less than doing nothing. Agreement must be measured on the change: the figure of merit (Pontius et al. 2008) is the share of correctly simulated change among all change, observed or simulated: hits / (hits + misses + wrong hits + false alarms). It ranges from 0 to 1; values of 0.1-0.3 are common for published models at the resolution of the data (Pontius et al. 2008).

Kappa misleads. It compares agreement with a random baseline that no one proposes as a model, and mixes errors of quantity and of location (Pontius & Millones 2011). Report instead the quantity disagreement (wrong amounts of each class) and the allocation disagreement (right amounts in the wrong places), and, if kappa is needed, kappa simulation, which measures agreement on the transitions (van Vliet et al. 2011).

Near misses are not misses. A patch of deforestation simulated two cells away from the real one is a better simulation than one on the other side of the map, but cell-by-cell measures score both as errors. Two answers: compare the maps at several resolutions (blocks of 2, 4, 8... cells; Pontius et al. 2011), or score each cell by how near the same change is in the other map, with a fuzzy similarity that decays with distance (Hagen 2003). Both show how far apart the simulated and observed change are:

model = S.presets.dinamica(t0, t1, drivers, steps=14, woe=500)
sim = model.run(t0, drivers, seed=1)
check = S.validate(t0, t1, sim.maps[-1], factors=[1, 2, 4, 8, 16], windows=[1, 3, 5, 7, 9, 11])
print(check.summary().round(3))
print(check.three_map.figure_of_merit()["overall"].round(3))

The figure of merit and the similarity grow with the resolution and the window: the model gets the general location of change right more often than the exact cells. That growth is information, not consolation: it tells at which resolution the simulated maps can be used.

Calibration: patterns, not parameters

The settings of a model (the size of its patches, their compactness, the inertia of each class, the weights of the drivers) are rarely measurable. Pattern-oriented modelling (Grimm et al. 2005) chooses them so that the simulation reproduces several observed patterns at once: the location of change (figure of merit), its neighbourhood (fuzzy similarity), and the structure of the resulting landscape (number of patches, edge density, aggregation). Matching one pattern is easy and uninformative; matching several independent ones constrains the model.

def build(isometry):
    return S.presets.dinamica(t0, t1, drivers, steps=14, woe=500, isometry=isometry)

cal = S.calibrate(build, t0, t1, drivers, params={"isometry": [0.5, 1.0, 1.5, 2.0]},
                  targets={"similarity_5": 1, "lsm": (["np", "ed"], 1)},
                  method="grid", replicates=2, seed=1)
print(cal.history.round(3))

Two rules keep calibration honest. Calibrate and validate on different periods: a model tuned to reproduce 1997-2011 and then "validated" on 1997-2011 has only been fitted. And average replicates: a stochastic model run once gives one draw; comparing two settings on single runs mostly compares their random numbers. Running each setting with the same seeds (common random numbers) makes the comparison fair.

Scenarios and uncertainty

A single simulated map of 2050 hides how much of it is chance. Running the model many times with different seeds gives, for each cell, the probability of each class: cells that change in every run are consequences of the drivers and the demand; cells that change in a few runs are where the allocation happened to fall. Scenarios are then compared on these probabilities, or on the distribution of a landscape metric across runs, rather than on one map each:

plus = S.presets.plus(t0, t1, drivers, steps=14, seed=1, step_size=20)     # LEAS + CARS
twice = S.Markov.from_maps(t0, t1, steps=7)        # the same change in half the time
ens = S.scenarios(plus, t1, drivers, scenarios={"trend": {}, "faster": {"demand": twice}},
                  replicates=10, steps=5, years=2011, seed=1)
coconut = ens.probability(2, year=2016)            # share of the runs where each cell is coconut
print(coconut.sel(scenario="trend").sum().item(), coconut.sel(scenario="faster").sum().item())
change = ens.change_probability().sel(year=2016)
print((change > 0.5).sum(dim=change.dims[1:]).values)     # cells that change in most runs

Good practice

1. Describe the three parts of the model separately (demand, potential, allocation) and justify each; most disagreements about a model are about one of them.
2. Compare with persistence and report the figure of merit, quantity and allocation disagreement, at several resolutions.
3. Use the shape of the observed change (patch sizes, expansion against new patches) and check the landscape metrics of the simulated maps.
4. Calibrate and validate on different periods, and on several patterns.
5. Run replicates, and report probabilities or distributions, not one map per scenario.
6. Treat demand as a scenario, not a forecast: the model says where and how, given how much.

Key points

• A land change model answers how much (demand), where (potential) and with what shape (allocation), under rules.
• Persistence is the null model: measure agreement on change (figure of merit), separate quantity from allocation disagreement, avoid kappa.
• Agreement depends on resolution; fuzzy similarity and multiple resolutions show how near the simulated change is to the observed.
• Calibrate on patterns, with replicates and common random numbers, and validate on another period.
• One simulated map is one draw: use replicates and probabilities.

Check your understanding

1. A simulation of 2001-2011 agrees with the observed 2011 map in 94 % of the cells. The observed change covered 5 % of the map. Is the model good?
2. Two models have the same figure of merit at the cell level; at blocks of 8 × 8 cells one reaches 0.6 and the other 0.25. What does that say?
3. Why fit the potential to where a class appeared rather than to where it is?

Answers

1. Not necessarily: a model that changes nothing agrees in 95 % of the cells. Look at the figure of merit, the quantity and allocation disagreement, and compare with persistence.
2. The first places its change near the observed change (it misses by a few cells), the second far from it: the first is useful at coarser resolutions, the second is not.
3. Because the distribution of a class mixes old and new land use; the simulation needs the conditions under which the class expands, which can differ (new cropland on flatter land than the old, near the newest roads).

References

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