Supplementary methods
Road map to statistical computation
A simple example of the Metropolis-Hastings algorithm
Figure S1
Supplementary references
Supplementary methods
The Markov chains iteratively sample parameters and update the latent states of the model. In
our configuration, we proceed in the following order: (1) update the latent states (i.e. choose
the winner for each position, given current parameter estimates for the distribution of canopy
extension and the logistic regression relating horizontal distance between contenders and the
focal position to the probability of winning the position); (2) sample the parameters for canopy
extension; (3) sample the parameters for the logistic regression. All of these steps complete a
single iteration of the Gibbs sampler.
The latent states are sampled from,
𝑀𝑢𝑙𝑡𝑖𝑛𝑜𝑚(𝑃𝑖 ),
where 𝑃𝑖 is the vector of probabilities that each contender is the winner within the 5 × 5 pixel
neighborhood. There is a different vector for each neighborhood. These probabilities can be
calculated directly using estimates of the parameters for canopy extension and the logistic
regression from the current iteration of the Gibbs sampler.
For all models that assumed a gamma distribution for canopy extension, we placed gamma
priors on the rate and shape parameters of 0.01. We estimated the rate and shape parameters
using a Metropolis-Hastings step, where proposals were generated from,
𝐺𝑎𝑚(𝑎 = 750, 𝑏 = 750⁄𝑏 ∗ ),
where 𝑏 ∗ is the current estimate of the rate or shape parameter that is being sampled from the
previous step of the Markov chain (Clark 2007). This distribution produces values that are
centered on 𝑏 ∗ with a variance dictated by the magnitude of 𝑎.
For models that assumed a log-normal distribution for canopy extension we placed priors on
the mean and variance of log-transformed canopy extension. Because this is a normal
distribution, the mean and variance can be estimated directly using conjugate normal and
inverse-gamma priors on the mean and variance respectively (Clark 2007). We sampled
conditional posteriors from,
𝑁(𝜇|𝜃1 , 𝜏12 ),
where,
1
𝑛 −1
𝜏12 = (𝜏2 + 𝜎2 ) ,
0
and,
𝜃
𝑛𝑦̅
𝜃1 = 𝜏12 (𝜏20 + 𝜎2 ),
0
and,
𝑛
1
𝐼𝐺 (𝜎 2 | 2 + 𝛼 + 1, 𝛽 + 2 ∑𝑛𝑖=1(𝑦𝑖 − 𝜇)2 ).
The prior on 𝜃0 was 0, and the prior on 𝜏02 was 100. The prior on 𝛼 and 𝛽 was 0.01.
The logistic regression requires priors on intercept and slope parameters. We used normal
priors on the logit scale. For the intercept, the prior was 𝑁(𝑙𝑜𝑔𝑖𝑡(0.95), √0.5). For the slope
parameters, the prior was 𝑁(0, √0.5). We sampled these parameters using a Metropolis step,
where proposals were generated from,
𝑁(𝜇 ∗ , 𝜎 2 ),
where 𝜇 ∗ is the estimate of the intercept or slope parameter at the current iteration of the
Gibbs sampler, and 𝜎 2 depended on the parameter being sampled.
We ran each statistical model in 16 parallel Markov chains. Models H1G, H3G, H1LN, H2LN and
H3LN appeared to converge to conditional posterior distributions rapidly and the chains were
well mixed. We ran theses simulations for 15,000 iterations and discarded the first 5,000. We
then stored every 100th iteration. Gamma parameters for H2G mixed slowly, so we ran this
model in 16 parallel chains for 100,000 iterations. We discarded the first 25,000 and then
stored every 750th iteration.
Road map to statistical computation
Below we provide an explanation of statistical computation using the simple example of a single
neighborhood. We include code that can be used in the R programming environment for
statistical computation (R Development Core Team 2012). R Code is indicated by courier
font.
We begin by considering the fate of a hypothetical position that transitioned from 2 m to 20 m
tall (Fig. 2 main text, Fig. S1). In this example, the heights of pixels in a 5 × 5 pixel neighborhood
are indicated at two points in time. Each pixel is a square whose centers are separated by 1.25
m. Panel A is the first time point, and panel B is the second time point. Panel C shows an
arbitrary index that we use to refer to individual positions. For example, position 13 is the focal
position. Panel D shows the straight-line distance (m) between the center of each position and
the focal position. Only the focal position, indicated in green, has changed in height between
the first and second measurements in this example. Heights of the neighbors immediately to
the left and right correspond to the graphical example in Fig. 2 of the main text.
The data indicate that the position increased in height, but the data alone provide incomplete
information to infer whether this height change was due to the vertical growth of the
vegetation that occupied the position, corresponding to 18 m of vertical growth, or to lateral
capture of the position by one of the neighbors. Data provide some insight and constraints on
possible outcomes (for example, we might conclude that over a 2-year interval, vertical growth
of this magnitude is unlikely), but deciding which of these outcomes was most plausible, and
which of the contenders won the position, requires a framework for statistical inference.
The model described below proceeds in steps. These are (1) calculate the probability that each
candidate was the winner and randomly select the winner using a multinomial draw and the
corresponding probabilities. (2) Given the position and height of the winner, calculate the
distance between the height of the winner at the time of the first measurement and the height
of the focal position at the second measurement. We refer to this distance as canopy extension
(Fig. S1, panel E). When the winner is the focal position, canopy extension is simply vertical
growth. When the winner is a neighbor that was exactly the same height at the time of the first
measurement as the focal position at the time of the second measurement, canopy extension is
only horizontal lateral growth. When the winner is a neighbor that was not the same height at
the time of the first measurement as the focal position at the time of the second measurement,
canopy extension is both vertical and lateral growth, the magnitude of which can be calculated
by imaging a right triangle that connects the height of the neighbor at the time of the first
measurement to the height of the focal position at the time of the second measurement. (3)
Given the magnitude of canopy extension, update the parameter estimates for the distribution
of canopy extension. In the example below we use the gamma distribution (i.e. this is model
H1G from the main text). In a Bayesian framework, this step can be achieved using the
Metropolis-Hastings algorithm (Metropolis et al. 1953; Hastings 1970). (4) Given covariates of
the winner, update the parameters for the logistic regression. In the example below we will use
the horizontal distance between the focal position and winner (Fig. S1, panel D). In the main
text we considered the horizontal distance between the focal position and winner (H1G and
H1LN), the height of the winner at the time of the first measurement (H2G and H2LN), and both
the horizontal distance between the focal position and winner and the height of the winner at
the time of the first measurement (H3G and H3LN). This completes a single iteration of the
Gibbs sampler.
Implementation of the Gibbs sampler requires some summaries of the data. The probability
that each position was the winner is the joint probability of canopy extension and winning the
position according to the logistic regression. In R, the data are:
time1 <- matrix(c(7,6,10,8,19,16,14,8,8,10,10,20,2,5,5,2,22,8,9,10,8,10,6,22,4), 5, 5, byrow=T)
time2 <- time1
time2[3,3] <- 20 # Only the focal position changes in height
The horizontal distance between each position and the focal position (assuming 1.25 m pixels)
is (Fig. S1, panel D):
dx <- matrix(sort(rep(seq(1,(5*1.25),1.25),5)), nrow=5, ncol=5,byrow=F)
dy <- matrix(sort(rep(seq(1,(5*1.25),1.25),5)), nrow=5, ncol=5,byrow=T)
horizontal_distance <- sqrt(((dx-3.5)^2)+((dy-3.5)^2))
The matrix called horizontal_distance can be interpreted as the bases of a set of right triangles
that connect each position to the focal position. We can calculate candidate canopy extension
using the Pythagorean Theorem, which requires the bases and heights of these triangles. The
heights can be calculated like this:
delta_height <- time2[3,3] - time1
Candidate canopy extension is then (Fig. S1, panel E):
candidate_canopy_extension <- sqrt(delta_height^2 + horizontal_distance^2)
This matrix contains the distance from the focal position at the time of the second
measurement to each candidate at the time of the first measurement. One can think of these
numbers as the growth that would have occurred, had any given contender been the winner.
Examining the delta_height matrix indicates that some positions were taller at the time of the
first measurement than the focal position at the time of the second measurement. We assume
that these contenders did not win the position, because doing so would have required an
apparent reduction in height. Although we do not do so here, this assumption could be relaxed,
for example, to allow winners to emerge from the subcanopy. Therefore, we set each of these
positions equal to -9 in the candidate_canopy_extension matrix, so that they will have a gamma
probability density of 0. These locations are blacked out in portions of Fig. S1.
candidate_canopy_extension[which(delta_height<0)] <- -9
The probability of canopy extension is (Fig. S1, panel F):
p_canopy_extension <- dgamma(x=candidate_canopy_extension, shape=0.997, rate=0.518)
p_canopy_extension <- p_canopy_extension/sum(p_canopy_extension)
In this example and in Fig. S1, we calculate probabilities using the posterior medians for the
shape and rate of the gamma distribution (H1G in the main text), which are 0.997 and 0.518,
respectively. In practice, these would be the estimates associated with an iteration of the Gibbs
sampler.
The probability of winning the position based on the logistic regression (Fig. S1, panel G)
requires an inverse logit function, again using posterior medians for the slope and intercept of
the logistic regression:
inv.logit <- function(x){p <- exp(x)/(1+(exp(x))); return(p)}
p_winning_logistic <-
inv.logit(0.860-4.277*horizontal_distance)
p_winning_logistic <- p_winning_logistic/sum(p_winning_logistic)
The probability that each position was the winner (step 1) is in Fig. S1, panel H:
p_winner <- p_canopy_extension*p_winning_logistic
p_winner <- p_winner/sum(p_winner)
Next we use these probabilities to choose a winner.
the_winner <- rmultinom(n=1, size=1, t(p_winner))
The result of this multinomial draw is a matrix with 25 rows and 1 column that contains 24
entries of 0 and one entry of 1 at the position of the randomly selected winner. The relationship
we used between these positions and the 5 × 5 pixel neighborhood is in Fig. S1, panel C. Using
this location, extract the candidate canopy extension value.
location_of_the_winner <- which(the_winner==1)
canopy_extension <- t(candidate_canopy_extension)[location_of_the_winner]
Step (3) is to update parameter estimates for the distribution of canopy extension. We do this
for the rate and shape using the Metropolis-Hastings algorithm. Assuming we are at the gth
iteration of the Gibbs sampler, start by setting the new value for the parameter (at the gth
iteration) equal to the previous value (the g-1th iteration). This value will be overwritten later if
the proposed new value is accepted by the Metropolis-Hastings algorithm. If not, we will simply
keep the previous value. Proceed sequentially (see the section below, ‘A simple example of the
Metropolis-Hastings algorithm’):
1. Propose a new value for the shape. We do this using a gamma proposal density with
a small variance, described above (Clark 2007).
2. Calculate the joint probability of the data, given the current estimate of the shape
and prior.
3. Calculate the joint probability of the data, given the proposed estimate of the shape
and prior.
4. Accept or reject the proposal using Metropolis-Hastings.
5. Repeat this procedure for the rate, being sure to use the current value for the shape
(i.e. the updated value, if it was updated).
Step (4) is to update the parameters for the logistic regression. This is similar to the procedure
for updating the parameters for the distribution of canopy extension. However, because we use
a normal proposal density, these parameters can be updated using the Metropolis algorithm,
rather than Metropolis-Hastings.
1. Propose a new value for the intercept. We do this using a normal proposal density
with a small variance.
2. Calculate the joint probability of the data, given the current estimate of the
intercept and prior.
3. Calculate the joint probability of the data, given the proposed estimate of the
intercept and prior.
4. Accept or reject the proposal using the Metropolis algorithm.
5. Repeat this procedure for the slope(s), being sure to use the current value for other
parameters (i.e. the updated value, if parameters were updated).
This completes a single iteration of the Gibbs sampler. Next go back to step (1) and repeat the
entire process with the current parameter estimates. Repeat this process a large number of
times until the parameter estimates converge to the conditional posterior distribution.
An example of the Metropolis-Hastings algorithm
Here we provide a simple example of one implementation of the Metropolis-Hastings algorithm
(Metropolis et al. 1953; Hastings 1970). We do this so that readers can understand the
structure of a working version of the algorithm without the complexity of the statistical model.
Begin by generating a simulated data set under a gamma distribution with a shape of 1 and a
rate of 0.5.
x <- rgamma(1000,shape=1, rate=0.5)
hist(x)
Assume that you know the shape is 1, but do not know the rate, so that we can focus on just
one parameter. We will estimate the rate using the Metropolis-Hastings algorithm.
myRate <- matrix(NA, nrow=5000, ncol=1)
# A matrix to hold parameter estimates
myRate[1,1] <- 1
# A starting guess of 1 (we start at the wrong value to test the algorithm)
for (i in 2:length(myRate)){
# A loop
myRate[i,1] <- myRate[(i-1),1]
# set current = previous
proposal <- rgamma(1, shape=500, rate=500/myRate[(i-1),1])
# A proposed value (can be tuned)
pnow <-sum(dgamma(x,shape=1,rate=myRate[(i-1),1],log=T))+dgamma(myRate[(i-1),1],0.1,0.1,log=T)
pnew <-sum(dgamma(x,shape=1,rate=proposal,log=T))+dgamma(proposal,0.1,0.1,log=T)
#
#
#
#
the two lines above are the log likelihood of the data and the prior under the current (now)
and proposed (new) values of the rate
note that the prior here is a gamma with rate and shape of 0.1
this is an uninformative prior distribution
q1 <- dgamma(myRate[(i-1),1], shape=500, rate=500/proposal, log=T)
q2 <- dgamma(proposal, shape=500, rate=500/myRate[(i-1),1], log=T)
a <- exp(pnew-pnow)/exp(q2-q1)
# Acceptance probability
zz <- runif(1,0,1)
if (zz < a) {myRate[i,1] <- proposal}
}
par(mfrow=c(3,1))
plot(myRate, type="l")
abline(h=0.5, col="Red")
Figure S1. Inferring the fate of a single 5 × 5 pixel neighborhood at two points in time (panels A,
and B). Between the first and second measurements, the focal position (indicated by green)
experienced a height transition from 2 m to 20 m tall. The heights of the neighbors to the left
and right of this position correspond to the graphical example in Fig. 2 of the main text. Panel C
shows index locations for each position. These numbers are arbitrary, but are needed to refer
to individual positions. For example, position number 13 is the focal position. Panel D shows the
horizontal distance between the center of each position and the focal position. This is simply
the hypotenuse of a right triangle whose base and height are the distances between pixel
centers on the X and Y axes, respectively. In this example, we assume that pixels are square
with a side length of 1.25 m. Panel E shows the three dimensional distance between each
position and the focal position. This is the hypotenuse of a right triangle whose base is an
element in panel D, and whose height is the height difference between the given contender and
focal position. For example, position 14 was 10 m tall at the time of the first measurement, and
to win the focal position by the time of the second measurement it needed to grow vertically
and laterally to a height of 20 m tall. Thus, position 14 would have needed to extend its
vegetation by 10 m vertically and 1.25 m laterally, which corresponds to √101.5625, or
approximately 10.08 m. Panel F shows the probability associated with each candidate canopy
extension from Panel E. These numbers are simply a normalized gamma density, given the
posterior median shape and rate parameters. In practice, parameter estimates are updated
sequentially using a Gibbs sampler. Panel G shows the probability that any given contender will
win as a function of the horizontal distance between the contender and focal position. These
numbers were back-transformed from the logit scale, using posterior medians for the slope and
intercept of a logistic regression, and then normalized so that they sum to 1. Panel H is the
normalized product of panels G and F. Note that the focal position has a high probability of
winning based on distance alone (panel G), but a small probability of winning based on the
observed height change (panel F, i.e. a change of 18 m is statistically very unlikely if growth
follows a gamma distribution with shape and rate parameters equal to the posterior medians).
Combining these two sets of probabilities indicates that this position was most likely to have
been captured laterally by position 12. If position 12 was the winner, this position was
associated with lateral growth of 1.25 m.
References
Clark, J.S. (2007). Models for Ecological Data. Princeton University Press, Princeton, NJ.
Hastings, W.K. (1970). Monte-Carlo Sampling Methods Using Markov Chains and Their
Applications. Biometrika, 57, 97-&.
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H. & Teller, E. (1953). Equation of
State Calculations by Fast Computing Machines. J Chem Phys, 21, 1087-1092.
R Development Core Team (2012). R: a language and environment for statistical computing. R
foundation for statistical computing Vienna, Austria.