Intensity analysis: how fast, and which classes

A transition matrix says how much changed. It does not say whether that is a lot. Forest losing 35 km² is alarming on a small island and negligible on a continent; built land gaining 30 km² in six years is fast if the next eight years add the same. In this tutorial you will run the intensity analysis of Aldwaik and Pontius (2012) on two landscapes: Plum Island, in Massachusetts, mapped at three dates (two periods), and Sibuyan Island, from part 1. You will learn to read its three levels — the interval, the category and the transition — and to say, with numbers, which period changed fast, which classes were active, and which classes a transition targeted or avoided.

The idea#

Intensity analysis compares every observed rate of change with the rate it would have if change were spread uniformly. It grew out of an earlier idea, comparing each observed transition with the transition expected from the sizes of the classes (Pontius et al., 2004). "Uniform" is a deliberately simple null model: if change were blind to time, every period would change at the same annual rate; if it were blind to classes, every class would gain and lose in proportion to its size; if a gaining class took land at random, it would take from every other class in proportion to that class's size. Where the observed rate departs from the uniform one, something non-random is at work, and that is where an ecologist or a geographer should look for causes.

Safaa Aldwaik and Robert Gilmore Pontius Jr. (2012) organised the comparison in three levels, each answering one question and each nested in the previous one:

1. Interval level. Was the change in each period fast or slow, compared with the whole time span?
2. Category level. In each period, which classes gained or lost actively (faster than the period's rate) and which were dormant?
3. Transition level. For a class that gained, did it target some classes (take from them faster than their size explains) and avoid others? For a class that lost, which classes took its losses?

The same idea runs through all three levels: a rate is an area of change divided by the time and by the area that could have changed, so that small classes and long periods are compared fairly with large classes and short periods. All rates are in % per year.

With more than two periods, the analysis also tells whether a pattern is stationary: the same in every period (a class active in all of them, a transition targeted in all of them). Terraflux's change.intensity computes all of this as OpenLand's intensityAnalysis does, the R package of Exavier and Zeilhofer (2020).

When to use it#

• To compare periods of unequal length. Censuses and satellite archives rarely give evenly spaced dates. The interval level puts 6 years and 8 years on the same scale.
• To compare classes of very different sizes. A large forest loses many hectares even when it is barely touched; a small class of wetlands can lose few hectares and still be disappearing fast. The category level divides by each class's own size.
• To find the processes behind a transition matrix. Urban growth that targets agriculture rather than forest, deforestation that feeds pasture rather than crops: the transition level says which, and whether the pattern held in every period. Teixeira et al. (2016), for example, used it on CORINE land cover maps of a Portuguese watershed to show which transitions deviated from uniform in each period.

It is not a test of statistical significance: a rate above uniform is a fact about the maps, not proof of a process. Map errors alone can create deviations from uniform (Aldwaik & Pontius, 2013), so small deviations deserve caution. For how much changed, see part 1; for whether a simulated map got the change right, part 3.

What you need#

A series of two or more maps of classes on the same grid, with the date of each. Two files from Terraflux's example data are used here:

• pie_landuse.tif: Plum Island Ecosystems, Massachusetts, a long-term ecological research site north of Boston. Three bands, named 1985, 1991 and 1999; 434 × 497 cells of about 100 m (EPSG:26986, the Massachusetts state plane in metres); classes 1 forest, 2 built, 3 other (mostly open land and wetlands); 255 no data. 113,563 cells with data: 1,134 km².
• sibuyan_landuse.tif: Sibuyan Island in 1997 and 2011, as in part 1 (1 forest, 2 coconut, 3 grassland, 4 rice, 5 other; cells of 250 m; modified for teaching).

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The time between the dates matters: it is read from the layers' names when they are numbers (as here), or given with the times setting (for example times = [1985, 1991, 1999]). Rates are "per unit of time": per year when the times are years.

Build the model#

The model is short. A raster.read node (library: IO) with path = "pie_landuse.tif" and no band gives, on its raster output, the three dates as layers named by their years. Connect that output to the maps input of change.intensity (library: Change detection). Add change.transitions on the same stack to have the areas behind the rates. Click dates to see the maps in the Viewer, a layer at a time, and keep the view with Plots › Save as a new plot (here titled Plum Island, 1985-1999).

change.intensity needs two settings: gain, the class whose gains the transition level examines, and loss, the class whose losses it examines. Here the question is urban growth: gain = 2 (built) and loss = 1 (forest). Its other settings: times and cell_area (in m², by default from the grid's cell sides) as for change.transitions, and area (true by default) to give the areas in km² rather than in cells. The rates do not depend on area.

format = "terraflux/1"

[[nodes]]
id = "dates"
op = "raster.read"
params = { path = "pie_landuse.tif" }
note = "Three layers: 1985, 1991, 1999 (1 forest, 2 built, 3 other)."

[[nodes]]
id = "transitions"
op = "change.transitions"
inputs = { maps = "dates.raster" }
note = "Every transition of each period, in km²."

[[nodes]]
id = "intensity"
op = "change.intensity"
inputs = { maps = "dates.raster" }
params = { gain = 2, loss = 1 }
note = "Interval, category and transition levels; built's gains, forest's losses."

[[plots]]
id = "plot"
title = "Plum Island, 1985-1999"
layers = [{ from = "dates.raster" }]

Select the intensity node and press Ctrl+Enter. It has nine outputs, all tables, which the Viewer shows when you pick them in its header:

The column names are those of OpenLand, so that results can be compared with R. Classes without any gain (or loss) in a period have no row in that period's category table: read a missing row as an intensity of 0.

[outputs.interval]
from = "intensity.interval"

[outputs.category_gain]
from = "intensity.category_gain"

[outputs.category_loss]
from = "intensity.category_loss"

[outputs.transition_gain]
from = "intensity.transition_gain"

terraflux run plum_island_intensity.toml --out output

Read the results#

The charts below follow Aldwaik and Pontius's convention: a bar per observed intensity, a dashed line for the uniform intensity it is compared with.

Interval level: was a period fast or slow?#

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From 1985 to 1991, 3.59% of the area changed class in 6 years: 0.598% a year. From 1991 to 1999, 4.19% changed in 8 years: 0.523% a year. More land changed in the second period, but it was longer; per year, the first period was fast and the second slow, against a uniform intensity of 0.556% a year (the 7.78% that changed in the two periods, over 14 years). The difference is small: both periods are within 8% of the uniform rate.

Category level: which classes were active?#

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In each period the dashed line is that period's rate \(S_t\). Built land gained actively in both periods: 1.35% of its (final) area a year from 1985 to 1991, more than twice the period's rate, and 0.93% from 1991 to 1999. Forest and other land gained dormantly. On the loss side, forest and other land lost actively in both periods, and built land almost never lost (0.02 and 0.04% a year): once land is built, it stays built.

Other land is a lesson in why the category level divides by each class's size. Forest lost more km² than other land in both periods (23.4 against 17.0 km² from 1985 to 1991), yet other land lost at a higher intensity (1.03% against 0.80% a year), because it was the smaller class.

The stationarity tables summarise the two periods. From category_gain_stationarity and category_loss_stationarity:

Every class behaved the same way in both periods: the category level is stationary. In the tables, a row gives the class, the number of periods it was active (above = 1) or dormant (above = 0), the number of periods N, and test = 1 when that held in every period.

Transition level: what did built land take?#

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The left column answers "which classes did built land take?". From 1985 to 1991, built land took other land at 0.81% of it a year and forest at 0.65%, against a uniform intensity of 0.71%: it targeted other land and avoided forest. From 1991 to 1999 the pattern reversed: forest at 0.58% a year, above the uniform 0.55%, and other land at 0.51%, below it. The transition level is not stationary: transition_gain_stationarity lists forest and other land each as targeted in 1 of 2 periods and avoided in the other. A plausible reading is that development first took the open land that was easiest to build on, then moved into the forest when that ran short; testing it would need data on land prices, zoning or soils.

The right column answers "which classes took the forest?". Built land took forest's losses at 0.80% and 0.63% of built land's area a year, above the uniform intensities (0.59% and 0.48%): forest's losses targeted built land in both periods and avoided other land. Forest was lost to development, not to farming or wetlands, and that was stationary.

The two columns look at the same transition, forest → built, from its two sides, and need not agree. Built land avoided forest in the first period (seen from its gains), while forest's losses targeted built land (seen from forest's losses). Both are true: built land took more from other land than from forest relative to their sizes, and of the forest that was lost, more went to built land than built land's size alone would predict.

Sibuyan: one interval, many classes#

With only two dates, Sibuyan has one interval, and the interval level is trivial: the period's rate equals the uniform rate (10.28% of the island changed in 14 years: 0.734% a year). The category and transition levels are not trivial.

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Rice is the most dynamic class: it gained at 3.27% and lost at 2.17% of its area a year, both far above the period's 0.73%. It is a small class (14 km² in 1997) moving around the coastal plain, which part 1 found as a large swap. Forest lost actively (0.85% a year) and never gained. Grassland both gained (2.26%) and lost (0.80%) actively.

At the transition level, the slider shows the gains of each class of farmland in turn (the gain setting): which classes did each one take?

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Gaining class (gain)

The picture is of a frontier: grassland and rice advance into the forest, coconut follows by taking the grassland and rice behind them. Coconut avoided forest even though it took 7.9 km² of it, because forest is so large that 7.9 km² is a small share of it. Rice's targeting of grassland is a deviation of 0.001% a year from uniform: a real fact of these maps, but one that a single misclassified cell could reverse (see the pitfalls).

And forest's losses (loss = 1): rice took them at 2.88% and grassland at 2.10% of their own area a year, against a uniform 1.31%; coconut (0.64%) and other land (0.03%) were avoided.

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Settings that change the answer#

• gain and loss. They do not change the interval and category levels; they choose which class the transition level examines. Run the node once per class of interest (as the slider above), or add one change.intensity node per class.
• times. The rates are per unit of time. Give the dates when the layers' names are not numbers: without them, the layers are numbered 0, 1, 2 and the rates become "per period", which hides periods of unequal length.
• Which dates. The intervals are those between consecutive layers. Adding a date splits a period and can reverse a conclusion: Plum Island's 14 years, analysed as one interval, would have hidden that built land's target switched from other land to forest.
• The legend. Merging classes changes every level. A class "other" that mixes open land and wetlands averages two different behaviours; at the transition level, built land may target one and avoid the other.
• cell_area and area. They change the areas reported (GG_km2, T_i2n_km2...), never the intensities.

Common pitfalls#

• Deviations within the margin of map error. A class barely above uniform (rice taking grassland at 0.136% against 0.135%) is not evidence of anything. Aldwaik and Pontius (2013) show how to compute the map error that would explain a deviation; at the least, flag deviations that a few cells could reverse.
• Reading active as large. An active class changed fast relative to its size. A small class can be very active and still contribute little area; report the areas beside the intensities.
• Gain denominators at the end, loss denominators at the start. \(G_{tj}\) divides by the class's area at the end of the period, \(L_{ti}\) by its area at the start. A class that grows quickly has a large end area, which lowers its gain intensity.
• Missing rows. A class with no gain in a period has no row in category_gain for it; the charts must show it as 0, not leave it out (forest in Sibuyan).
• Unequal maps. The analysis assumes the maps are comparable: same legend, same method, same grid. Changes in the method between dates show up as change, often as swap at class boundaries.
• Dates read from the wrong place. Check the interval table's year_from and year_to: if they read 0 and 1, the layers' names were not numbers and times is needed.

Going further#

• Part 1: the transition matrix, gains, losses, quantity, exchange and shift.
• Part 3: comparing a map with a reference — quantity and allocation disagreement, several resolutions, kappa and the estimation of areas.
• Part 4: the change of a forest seen by GuidosToolbox.
• The template Land change analysis (menu Templates → Land change) runs the transitions, components and intensity analysis together on three dates of a landscape drawn by chance.
• Operators: change.intensity, change.transitions, change.count.

References#

• Aldwaik, S. Z., & Pontius, R. G., Jr. (2012). Intensity analysis to unify measurements of size and stationarity of land changes by interval, category, and transition. Landscape and Urban Planning, 106(1), 103–114. https://doi.org/10.1016/j.landurbplan.2012.02.010
• Aldwaik, S. Z., & Pontius, R. G., Jr. (2013). Map errors that could account for deviations from a uniform intensity of land change. International Journal of Geographical Information Science, 27(9), 1717–1739. https://doi.org/10.1080/13658816.2013.787618
• Exavier, R., & Zeilhofer, P. (2020). OpenLand: software for quantitative analysis and visualization of land use and cover change. The R Journal, 12(2), 359. https://doi.org/10.32614/RJ-2021-021
• Pontius, R. G., Jr., Shusas, E., & McEachern, M. (2004). Detecting important categorical land changes while accounting for persistence. Agriculture, Ecosystems & Environment, 101(2–3), 251–268. https://doi.org/10.1016/j.agee.2003.09.008
• Teixeira, Z., Marques, J. C., & Pontius, R. G., Jr. (2016). Evidence for deviations from uniform changes in a Portuguese watershed illustrated by CORINE maps: an intensity analysis approach. Ecological Indicators, 66, 382–390. https://doi.org/10.1016/j.ecolind.2016.01.018