BMB {BayesMassBal} | R Documentation |
Bayesian Mass Balance
Description
Allows the user to specify the covariance structure for a Bayesian mass balance, simulates draws from reconciled masses and relevant covariance matrix, and approximates the log-marginal likelihood.
Usage
BMB(
X,
y,
cov.structure = c("indep", "component", "location"),
priors = "default",
BTE = c(500, 20000, 1),
lml = FALSE,
ybal = TRUE,
diagnostics = TRUE,
verb = 1
)
Arguments
X |
A matrix that maps constrained masses to observed masses. Can be built from the function |
y |
A list of matrices of observed mass flow rates. Each matrix is a separate sample component. The rows of each matrix index the sampling location, and the columns index the sample set number. Can be specified using the |
cov.structure |
Character string. |
priors |
List or character string. When the default value |
BTE |
Numeric vector giving |
lml |
Logical indicating if the log-marginal likelihood should be approximated. Default is |
ybal |
Logical indicating if the mass balanced samples for each |
diagnostics |
Logical or list indicating if diagnostic functions |
verb |
Numeric indicating verbosity of progress printed to R-console. The default of 1 prints messages and a progress bar to the console during all iterative methods. |
Details
See vignette("Two_Node_Process", package = "BayesMassBal")
for further details on how to use function outputs.
When the priors
argument is left unspecified, a set of default conjugate priors are used, which are chosen to allow BMB()
to work well in a general setting. In the current version of the BayesMassBal
package, only the conjugate priors stated below can be used, but hyperparameter values can be specified by the user using the priors
argument.
The prior distribution on beta
is a normal distribution truncated at 0. The mean of this distribution before truncation is the ordinary least squares (OLS) estimate of \beta
. OLS estimates less than 0, are changed to 0. The prior variance, before truncation, of each element of \beta
is set to:
10^{\mathrm{number of integer digits of an element of } \beta + 6}
Currently, there is only support for diagonal prior covariance matrices for \beta
When cov.structure = "indep"
the error of all observations in a sample set are independent. An inverse gamma prior distribution, with \alpha_0 = 0.000001
and \beta_0 = 0.000001
, is placed on the variance of the mass flow rate for each sample component at each sample location.
When cov.structure = "component"
or "location"
, the prior distribution on \Sigma_i
is inverse Wishart (\nu_0, \nu_0 \times S_0)
. The degrees of freedom parameter, \nu_0
, is equal to the dimension of \Sigma_i
. The scale matrix parameter is equal to a matrix, S_0
, with the sample variance of the relevant observation on the diagonal, multiplied by \nu_0
.
The user is able to specify the prior hyperparameters of the mean and variance of beta
, \alpha_0
and \beta_0
for each \sigma^2
, and the degrees of freedom and scale matrix for each \Sigma_i
using the priors
argument. It is advisable for the user to specify their own prior hyperparameters for p(\sigma^2)
if the variance of any element is well under 1, or p(\beta)
if the there is a wide range in the magnitude of observations.
When priors = "Jeffreys"
Jeffreys priors are used for the prior distribution of the variance and covariance parameters. Priors used are p(\sigma^2) \propto \frac{1}{\sigma^2}
and p(\Sigma) \propto |\Sigma|^{-(p+1)/2}
, as listed in (Yang and Berger 1996). The Jeffreys prior for a \beta
with infinite support is p(\beta) \propto 1
. To preserve the prior information that \beta > 0
, p(\beta)\propto I\lbrack \beta > 0 \rbrack
is chosen. It is not possible to calculate log-marginal likelihood using the methods in (Chib 1995) with Jeffreys priors. Therefore, if priors = "Jeffreys"
and lml = TRUE
, the lml
argument will be ignored and a warning will be printed.
lml
is reported in base e
. See here for some guidance on how to interpret Bayes Factors, but note log base 10 is used on Wikipedia.
Value
Returns a list of outputs
beta |
List of matrices of samples from the distribution of reconciled data. Each matrix in the list is a separate sample component. Each column of a matrix in |
Sig |
List of matrices containing draws from the distribution of each covariance matrix. If
|
priors |
List of prior hyperparameters used in generating conditional posterior distributions and approximating log-marginal likelihood. The structure of the input argument |
cov.structure |
Character string containing the covariance structure used. |
y.cov |
List of character matrices indicating details for the structure of each covariance matrix. Only returned when |
lml |
Numeric of the log-marginal likelihood approximation. Returns |
diagnostics |
List containing results from diagnostic functions |
ybal |
List of samples from the distribution of reconciled mass flow rates, in the same format as the function argument |
X |
The function argument |
type |
Character string used by |
References
Chib S (1995). “Marginal likelihood from the Gibbs output.” Journal of the american statistical association, 90(432), 1313–1321. Casella G, George EI (1992). “Explaining the Gibbs sampler.” The American Statistician, 46(3), 167–174. Plummer M, Best N, Cowles K, Vines K (2006). “CODA: Convergence Diagnosis and Output Analysis for MCMC.” R News, 6(1), 7–11. https://journal.r-project.org/archive/. Yang R, Berger JO (1996). A catalog of noninformative priors. Institute of Statistics and Decision Sciences, Duke University.
Examples
y <- importObservations(file = system.file("extdata", "twonode_example.csv",
package = "BayesMassBal"),
header = TRUE, csv.params = list(sep = ";"))
C <- matrix(c(1,-1,0,-1,0,0,1,-1,0,-1), byrow = TRUE, ncol = 5, nrow = 2)
X <- constrainProcess(C = C)
BMB_example <- BMB(X = X, y = y, cov.structure = "indep",
BTE = c(10,300,1), lml = FALSE, verb=0)
summary(BMB_example)