Compositional data occur in many disciplines: geology, nutrition, economics, and ecology, to name a few. Data are compositional when each sample is sum-constrained. For example, mineral compositions describe a mineral in terms of the weight percentage coming from various elements; or taxonomic compositions break down a community by the fraction of community memebers that come from a particular species. In ecology in particular, the covariance between features is often of interest to determine which species possibly interact with each other. However, the sum constraint of compositional data makes naive measures inappropriate.
BAnOCC is a package for analyzing compositional covariance while
accounting for the compositional structure. Briefly, the model assumes
that the unobserved counts are log-normally distributed and then
infers the correlation matrix of the log-basis (see the [The Model]
section for a more detailed explanation). The inference is made using
No U-Turn Sampling for Hamiltonian Monte Carlo [@HoffmanAndGelman2014]
as implemented in the rstan
R package [@StanSoftware2015].
There are three options for installing BAnOCC:
This is not yet available
if (!requireNamespace("BiocManager", quietly=TRUE)) install.packages("BiocManager") BiocManager::install("BAnOCC")
This is not yet available
Clone the repository using git clone
, which downloads the package as
its own directory called banocc
.
git clone https://<your-user-name>@bitbucket.org/biobakery/banocc.git
Then, install BAnOCC's dependencies. If these are already installed on your machine, this step can be skipped.
Rscript -e "install.packages(c('rstan', 'mvtnorm', 'coda', 'stringr'))"
Lastly, install BAnOCC using R CMD INSTALL
. Note that this will
not automatically install the dependencies, so they must be installed
first.
R CMD INSTALL banocc
We first need to load the package:
library(banocc)
The BAnOCC package contains four things:
banocc_model
, which is the BAnOCC model in the rstan
formatrun_banocc
, a wrapper function for rstan::sampling
that samples
from the model and returns a list with various useful elementsget_banocc_output
, which takes as input a stanfit
object outputted by
run_banocc
, and outputs various statistics from the chains.Several test datasets which are included both as counts and as the corresponding compositions:
Dataset Description | Counts | Composition
----------------------------- | ------------------ | ------------
No correlations in the counts | counts_null
| compositions_null
No correlations in the counts | counts_hard_null
| compositions_hard_null
Positive corr. in the counts | counts_pos_spike
| compositions_pos_spike
Negative corr. in the counts | counts_neg_spike
| compositions_neg_spike
For a full and complete description of the possible parameters for
run_banocc
and get_banocc_output
, their default values, and the
output, see
?run_banocc ?get_banocc_output
There are only two required inputs to run_banocc
:
C
. This is assumed to be $N \times P$, with $N$
samples and $P$ features. The row sums are therefore required to be
less than one for all samples.compiled_banocc_model
. The compiled model is
required so that run_banocc
doesn't need to waste time compiling the
model every time it is called. To compile, use
rstan::stan_model(model_code=banocc::banocc_model)
.The simplest way to run the model is to load a test dataset, compile the model, sample from it (this gives a warning because the default number of iterations is low), and get the output:
rerun <- 0
Sys.setenv("PKG_CXXFLAGS"="-std=c++11") data(compositions_null) compiled_banocc_model <- rstan::stan_model(model_code = banocc::banocc_model) b_fit <- banocc::run_banocc(C = compositions_null, compiled_banocc_model=compiled_banocc_model) b_output <- banocc::get_banocc_output(banoccfit=b_fit)
The hyperparameter values can be specified as input to
run_banocc
. Their names correspond to the parameters in the plate
diagram figure (see section [The Model]). For example,
Sys.setenv("PKG_CXXFLAGS"="-std=c++11") p <- ncol(compositions_null) b_fit_hp <- banocc::run_banocc(C = compositions_null, compiled_banocc_model = compiled_banocc_model, n = rep(0, p), L = 10 * diag(p), a = 0.5, b = 0.01)
There are several options to control the behavior of the HMC sampler
within run_banocc
. This is simply a call to rstan::sampling
, and
so many of the parameters are the same.
The number of chains, iterations, and warmup iterations as well as the
rate of thinning for run_banocc
can be specified using the same
parameters as for rstan::sampling
and rstan::stan
. For example,
the following code gives a total of three iterations from each of two
chains. These parameters are used only for brevity and are NOT
recommended in practice.
b_fit_sampling <- banocc::run_banocc(C = compositions_null, compiled_banocc_model = compiled_banocc_model, chains = 2, iter = 11, warmup = 5, thin = 2)
The number of cores used for sampling on a multi-processor machine can also be specified, which allows chains to run in parallel and therefore decreases computation time. Since its purpose is running chains in parallel, computation time will decrease as cores are added up to when the number of cores and the number of chains are equal.
# This code is not run b_fit_cores <- banocc::run_banocc(C = compositions_null, compiled_banocc_model = compiled_banocc_model, chains = 2, cores = 2)
By default, the initial values for $m$ and $\lambda$ are sampled from the priors and the initial values for $O$ are set to the identity matrix of dimension $P$. Setting the initial values for $O$ to the identity helps ensure a parsimonious model fit. The initial values can also be set to a particular value by using a list whose length is the number of chains and whose elements are lists of initial values for each parameter:
init <- list(list(m = rep(0, p), O = diag(p), lambda = 0.02), list(m = runif(p), O = 10 * diag(p), lambda = runif(1, 0.1, 2))) b_fit_init <- banocc::run_banocc(C = compositions_null, compiled_banocc_model = compiled_banocc_model, chains = 2, init = init)
More specific control of the sampler's behavior comes from the
control
argument to rstan::sampling
. Details about this argument
can be found in the help for the rstan::stan
function:
?stan
There are several parameters that control the type of output which is
returned by get_banocc_output
.
The width of the returned credible intervals is controlled by
conf_alpha
. A $100\% * (1-\alpha_\text{conf})$ credible interval is
returned:
# Get 90% credible intervals b_out_90 <- banocc::get_banocc_output(banoccfit=b_fit, conf_alpha = 0.1) # Get 99% credible intervals b_out_99 <- banocc::get_banocc_output(banoccfit=b_fit, conf_alpha = 0.01)
Convergence is evaluated automatically, and in this case the credible
intervals, estimates, and any additional output in section [Additional
Output] is missing. This behavior can be turned off using the
eval_convergence
option. But be careful!
# Default is to evaluate convergence b_out_ec <- banocc::get_banocc_output(banoccfit=b_fit) # This can be turned off using `eval_convergence` b_out_nec <- banocc::get_banocc_output(banoccfit=b_fit, eval_convergence = FALSE)
# Iterations are too few, so estimates are missing b_out_ec$Estimates.median # Convergence was not evaluated, so estimates are not missing b_out_nec$Estimates.median
Two types of output can be requested for each correlation that are not included by default:
# Get the smallest credible interval width that includes zero b_out_min_width <- banocc::get_banocc_output(banoccfit=b_fit, get_min_width = TRUE) # Get the scaled neighborhood criterion b_out_snc <- banocc::get_banocc_output(banoccfit=b_fit, calc_snc = TRUE)
Detailed statements about the function's execution can also be printed
using the verbose
argument. The relative indentation of the verbose
output indicates the nesting level of the function. The starting
indentation can be set with num_level
.
There are many ways of assessing convergence, but the two most easily implemented using BAnOCC are:
Traceplots of parameters, which show visually what values of a parameter have been sampled across all iterations. At convergence, the sampler should be moving rapidly across the space, and the chains should overlap well. In other words, it should look like grass.
The Rhat statistic [@GelmanAndRubin1992], which measures agreement between all the chains. It should be close to one at convergence.
Traceplots can be directly accessed using the traceplot
function in
the rstan
package, which creates a ggplot2
object that can be
further maniuplated to 'prettify' the plot. The traceplots so
generated are for the samples drawn after the warmup period. For
example, we could plot the traceplots for the inverse covariances of
feature 1 with all other features. There is overlap between some of
the chains, but not all and so we conclude that we need more samples
from the posterior to be confident of convergence.
# The inverse covariances of feature 1 with all other features rstan::traceplot(b_fit$Fit, pars=paste0("O[1,", 2:9, "]"))
We could also see the warmup period samples by using
inc_warmup=TRUE
. This shows that some of the chains have moved from
very different starting points to a similar distribution, which is a
good sign of convergence.
# The inverse covariances of feature 1 with all other features, including warmup rstan::traceplot(b_fit$Fit, pars=paste0("O[1,", 2:9, "]"), inc_warmup=TRUE)
The Rhat values can also be directly accessed using the summary
function in the rstan
package. It measures the degree of agreement
between all the chains. At convergence, the Rhat statistics should be
approximately one for all parameters. For example, the Rhat values for
the correlation between feature 1 and all other features (the same as
those plotted above), agree with the traceplots that convergence has
not yet been reached.
# This returns a named vector with the Rhat values for all parameters rhat_all <- rstan::summary(b_fit$Fit)$summary[, "Rhat"] # To see the Rhat values for the inverse covariances of feature 1 rhat_all[paste0("O[1,", 2:9, "]")]
The hyperparameters for the model (see section [The Model]) need to be chosen appropriately.
The prior on the precision matrix $O$ is a GLASSO prior from [@Wang2012] with parameter $\lambda$ [see also section [The Model]]. As $\lambda$ decreases, the degree of shrinkage correspondingly increases.
# The above plot is from Dropbox/hutlab/Emma/paper1/writeup/figures/supplemental/lambbda_behavior.png
We recommend using an uninformative prior for the log-basis mean: centered at zero and with large variance.
We recommend using a prior with large probability mass close to zero; because $\lambda$ has a gamma prior, this means that the shape parameter $a$ should be less than one. The rate parameter $b$ determines the variability; in cases with either small (order of 10) or very large ($p > n$) numbers of features $b$ should be large so that the variance of the gamma distribution, $a / b^2$, is small. Otherwise, a small value of $b$ will make the prior more uninformative.
A pictoral representation of the model is shown below. Briefly, the basis (or unobserved, unrestricted counts) for each sample is assumed to be a lognormal distribution with parameters $m$ and $S$. The prior on $m$ is a normal distribution parametrized by mean $n$ and variance-covariance matrix $L$. Since we are using a graphical LASSO prior, we parametrize the model with precision matrix $O$. The prior on $O$ is a graphical LASSO prior [@Wang2012] with shrinkage parameter $\lambda$. To circumvent the necessity of choosing $\lambda$, a gamma hyperprior is placed on $\lambda$, with parameters $a$ and $b$.
If we print the model, we can actually see the code. It is written in
the format required by the rstan
package, since banocc
uses this
package to sample from the model. See [@StanManual2015] for more
detailed information on this format.
# This code is not run cat(banocc::banocc_model)
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