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Figure S1 Contours of the half-taxi distance on the 2-simplex at intervals of 0.05 around the point (0.7, 0.1, 0.2).

Figure S2 (a) The Dirichlet (2, 3, 4) distribution. (b) The Dirichlet (0.75, 0.9, 2) distribution.

Figure S3 The Epanechnikov kernel v(d) with bandwidth h, as a function of distance d.

Figure S4 Numerical solutions (red lines) of the deterministic model specified by eqn S9. Blue circles are initial values. Redrawn from Mumby et  al. (2007). Parameter values: a = 0.1, g = 0.3, g = 0.8, μ = 0.44, r = 1.

Figure S5 One simulated realization (red line) of the stochastic model specified by eqn S11 for each member of a grid of initial values (blue circles). Parameter values: ψA = ψB = 0.4. Other parameters as in Figure S4.

Figure S6 Relationships between coefficient of variation (CV) and mean coral cover for the data in Table S1, sorted by number of points per unit. Blue lines, CV from multinomial model (eqn S14). Blue circles, line point intercept; pink triangles, photoquadrats; green crosses, video transects.

Figure S7 Local estimates of the Dirichlet α parameters for each of the three regions (a–c, Caribbean; d–f, Kenya; g–i GBR), using proportions of neighbours (Caribbean 0.664, Kenya 0.627, GBR 0.288) selected by cross-validation as described in section S1.6. Colour scales are truncated at 25 to show detail. In each row, α1 is the Dirichlet parameter associated with coral, α2 is associated with algae and α3 is associated with others.

Figure S8 Simulated data under the models for (a) the Caribbean, (b) Kenya, and (c) the Great Barrier Reef. Blue circles are real observations for which there is an observation on the same reef in the next year. For each of these, the red dot connected by a black line is a single sample from the estimated transition kernel at the observation.

Figure S9 Predicted temporal change in the Caribbean. (a) Current distribution of reef states (red dots: most recent observation on each reef). (b–d) Predicted differences between the estimated stationary distribution and the distribution for the points in part (a) after 1, 2 and 3 years respectively. Pink values are where the probability density after a given time is greater than in the stationary distribution, and cyan values are where the probability density is less than in the stationary distribution. The colour scales differ between this and the following two figures, in order to show detail for individual regions.

Figure S10 Predicted temporal change in Kenya. See Figure S9 for details.

Figure S11 Predicted temporal change in the Great Barrier Reef. See Figure S9 for details.

Figure S12 Boxplots of composite residuals Ci (a) and univariate standardized residuals rij (b: coral, j = 1, c: algae, j = 2, d: others, j = 3) for each observation i, grouped by time for the Caribbean. Year is the date of the observation we are predicting. The vertical scales are different for each panel.

Figure S13 Boxplots of composite residuals Ci (a) and univariate standardized residuals rij (b: coral, j = 1, c: algae, j = 2, d: others, j = 3) for each observation i, grouped by time for Kenya. Year is the date of the observation we are predicting. 1998 is omitted because even though observations were made at roughly 12-month intervals, they were made in December up until 1997, then in January from 1999 onwards. The vertical scales are different for each panel.

Figure S14 Boxplots of composite residuals Ci (a) and univariate standardized residuals rij (b: coral, j = 1, c: algae, j = 2, d: others, j = 3) for each observation i, grouped by time for the GBR. Year is the date of the observation we are predicting. The vertical scales are different for each panel.

Figure S15 Standardized univariate residuals for Caribbean data against composition in the observation x. Points are values of standardized residuals for coral (a, d, g), algae (b, e, h) and others (c, f, i) against the cover of coral (a, b, c), algae (d, e, f), and others (g, h, i) in the observation x. The red  lines are LOWESS smoothers with span 0.4.

Figure S16 Standardized univariate residuals for Kenyan data against composition in the observation x. Points are values of standardized residuals for coral (a, d, g), algae (b, e, h) and others (c, f, i) against the cover of coral (a, b, c), algae (d, e, f), and others (g, h, i) in the observation x. The red lines are LOWESS smoothers with span 0.2.

Figure S17 Standardized univariate residuals for Great Barrier Reef data against composition in the observation x. Points are values of standardized residuals for coral (a, d, g), algae (b, e, h) and others (c, f, i) against the cover of coral (a, b, c), algae (d, e, f), and others (g, h, i) in the observation x. The red lines are LOWESS smoothers with span 0.1.

Figure S18 Standardized univariate residuals for Caribbean data against composition the year before the observation x. Points are values of standardized residuals for coral (a, d, g), algae (b, e, h) and others (c, f, i) against the cover of coral (a, b, c), algae (d, e, f), and others (g, h, i) the year before the observation x. The red lines are LOWESS smoothers with span 0.4.

Figure S19 Standardized univariate residuals for Kenyan data against composition the year before the observation x. Points are values of standardized residuals for coral (a, d, g), algae (b, e, h) and others (c, f, i) against the cover of coral (a, b, c), algae (d, e, f), and others (g, h, i) the year before the observation x. The red lines are LOWESS smoothers with span 0.2.

Figure S20 Standardized univariate residuals for Great Barrier Reef data against composition the year before the observation x. Points are values of standardized residuals for coral (a, d, g), algae (b, e, h) and others (c, f, i) against the cover of coral (a, b, c), algae (d, e, f), and others (g, h, i) the year before the observation x. The red lines are LOWESS smoothers with span 0.1.

Figure S21 Standardized univariate residuals for Caribbean data by reef. For each component (a: coral, b: algae, c: others), the residuals for each reef (row) are shown as black crosses, with a red circle at the median. Reefs are identified by numerical codes and sorted from top to bottom in descending order of median composite residual, where the composite residual is the sum of the squared standardized univariate residuals.

Figure S22 Standardized univariate residuals for Kenya data by reef. See Figure S21 for explanation. Reefs in this region are identified by name.

Figure S23 Standardized univariate residuals for Great Barrier Reef data by reef. See Figure S21 for explanation.

Figure S24 Estimated equilibrium distributions for (a) Caribbean (b) Kenya, and (c) GBR, omitting reefs with potentially unusual patterns of residuals. Lighter colours are higher probability densities, with a colour scale truncated at 25, as in the main text

Figure S25 Jacknife means (a, c, e) and standard errors (b, d, e) of the equilibrium distributions for the Caribbean (a, b), Kenya (c, d) and GBR (e, f). Lighter colours are higher values. In a, c, and e, the colour scale is truncated at 25, as in the main text. In b, d, and f, the maximum of the colour scale is at 2 (just above the largest jacknife standard error at any grid point).

Figure S26 Mean (a) and standard error (b) of the stationary distributions of 100 replicates simulated under the model used to produce Figures S4 and S5, without measurement error. Lighter colours are higher probability densities. The maximum of the colour scale is set to 25, as for the real data, although the highest mean density is 72 and the highest standard error of density is 75.

Figure S27 Mean (a) and standard error (b) of the stationary distributions of 100 replicates simulated under the model used to produce Figures S4 and S5, with multinomial measurement error, m = 20 points. Lighter colours are higher probability densities. The maximum of the colour scale is set to 25, as for the real data.

Figure S28 Mean (a) and standard error (b) of the stationary distributions of 100 replicates simulated under the model used to produce Figures S4 and S5, with multinomial measurement error, = 160 points. Lighter colours are higher probability densities. The maximum of the colour scale is set to 25, as for the real data.

Figure S29 Mean (a) and standard error (b) of the stationary distributions of 100 replicates simulated under the model used to produce Figures S4 and S5, with multinomial measurement error, m = 1000 points. Lighter colours are higher probability densities. The maximum of the colour scale is set to 25, as for the real data.

Figure S30 Mean (a) and standard error (b) of the stationary distributions of 100 replicates simulated under the model used to produce Figures S4 and S5, with multinomial measurement error, m = 5000 points. Lighter colours are higher probability densities. The maximum of the colour scale is set to 25, as for the real data.

Figure S31 1000 samples (black dots) of 160 points each from multinomial distributions with true states (red circles) x = [1/3, 1/3, 1/3] (centre), [0.01, 0.49, 0.50] (upper left), and [0.01, 0.01, 0.98] (lower left).

Figure S32 Estimated stationary distributions for (a) Caribbean and (b) Great Barrier Reef, with Reef Check data omitted. Lighter colours are higher probability densities. The maximum of the colour scale is set to 25.

Table S1 Literature data on mean and coefficient of variation (CV) of coral cover, from video transects, photoquadrats, and Line Point Intercept (LPI). Points per unit is the number of points at which the state of the substrate is recorded in each sampling unit (quadrat or transect).

Table S2 Sample means and standard deviations of standardized univariate residuals. Each entry is mean, standard deviation of the standardized difference between the observed and predicted proportion of each category (section S1.10). Thus, although both observed and predicted proportions are constrained to be between 0 and 1, the residuals may be either negative (if there is less of a category than predicted) or positive (if there is more).

Table S3 Effects of multinomial measurement error on the shapes of estimated stationary distributions. Sampling effort m is the number of points in the multinomial sample.

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