dream: Differential Evolution Adaptive Metropolis (DREAM) algorithm

Description Usage Arguments Details Value Note Author(s) References

View source: R/dream.R

Description

Differential Evolution Adaptive Metropolis (DREAM) algorithm

Usage

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dream(fun, ..., lik = NULL, par.info = list(initial = NULL, min = NULL, max
  = NULL, mu = NULL, cov = NULL, val_ini = NULL, bound = NULL, names = NULL,
  prior = "uniform"), nc, t, d, burnin = 0, adapt = 0.1,
  updateInterval = 10, delta = 3, c_val = 0.1, c_star = 1e-12,
  nCR = 3, p_g = 0.2, beta0 = 1, thin = 1, outlier_check = TRUE,
  obs = NULL, abc_rho = NULL, abc_e = NULL, glue_shape = NULL,
  lik_fun = NULL, checkConvergence = FALSE, verbose = TRUE)

Arguments

fun

character. Name of a function(x, ...) which is evaluated at x being a d-dimensional parameter vector. Returns a likelihood, log-likelihood, simulation values or summary statistics depending on argument lik.

...

Additional arguments for fun.

lik

integer. Flag specifying a likelihood function to be used, see details.

par.info

A list of parameter information.

nc

numeric. Number of chains evolved in parallel.

t

numeric. Number of samples from the Markov chain.

d

numeric. Number of parameters.

burnin

numeric. Length of the burn-in period as portion of t (burnin period = burnin * t). These samples from the Markov chain will not be included in the output. Default: 0.

adapt

numeric. Length of the adaptation period as portion of t (adaptation period = adapt * t). Will be used for the update of crossover probabilities and the replacement of outlier chains. Default: 0.1.

updateInterval

integer. Interval for crossover probability updates during the adaptation period.

delta

integer. Maximum number of chain pairs used to generate the jump (default: 3).

c_val

numeric. Lambda value is sampled from U[-c_val,c_val] (default: 0.1).

c_star

numeric. Zeta value sampled from N[0,c_star]. Should be small compared to target (i.e. in this case the normal) distribution. Default: 1e-12.

nCR

integer. Length of vector with crossover probabilities for parameter subspace sampling (default: 3).

p_g

numeric. Probability for gamma, the jump rate, being equal to 1. Default: 0.2.

beta0

numeric. Reduce jump distance, e.g. if the average acceptance rate is low (less than 15 %). 0 < beta0 <= 1. Default: 1 (i.e. jump distance is not adjusted).

thin

integer. Thinning to be applied to output in case of large t. See below.

outlier_check

logical. Shall outlier chains be identified and removed? Default: TRUE.

obs

numeric vector of observations to be compared with output of fun. Only needed for some realisations of lik (see details).

abc_rho

character. Name of an ABC distance function(sim, obs) calculating the distance between simulated ('sim', output of fun) and observed (obs) diagnostic values. Needed for ABC methods of lik.

abc_e

numeric vector of length of obs specifying the ABC tolerance value (lik = 22) or representing the standard deviations for each summary statistics (e.g. streamflow signatures) returned by fun (lik = 21).

glue_shape

numeric scalar value used for GLUE-based informal likelihood functions (see details).

lik_fun

character specifying the name of a user-defined likelihood function(sim, obs) with sim being the output of fun and obs a vector of corresponding observations (argument obs above).

checkConvergence

logical. Shall convergence of the MCMC chain be checked? Currently implemented: Calculating the Gelman-Rubin diagnostic. Takes a lot of time! Default: FALSE.

verbose

logical. Print progress bar to console? Default: TRUE.

initial

character. Method for prior sampling. One of: uniform - sampling from a uniform distribution; normal - a (multivariate) normal distribution; latin - latin hypercube sampling; user - value(s) given by the user.

min

numeric. A d-dimensional vector of minimum values for each parameter to sample from if initial is 'uniform' or 'latin'. Also defines bounding region for the proposal, see bound.

max

numeric. A d-dimensional vector of maximum values for each parameter to sample from if initial is 'uniform' or 'latin'. Also defines bounding region for the proposal, see bound.

mu

numeric. d-dimensional vector of parameter means if initial is 'normal'.

cov

numeric. d-by-d positive-definite symmetric matrix of parameter covariances if initial is 'normal'.

val_ini

numeric nc-by-d-dimensional matrix of prior values if initial is 'user'.

bound

character. What to do if the proposal parameter is outside the defined min-max limits. One of: bound - proposal is set to min/max value if it is smaller/larger than the defined limit. reflect - parameter value is reflected at the boundary towards the feasible space by the amount of boundary violation. fold - upper bound of each parameter dimension is connect the its respective lower bound (NOTE: this ensures detailed balance of the MCMC simulation but may lead to inflation of acceptance rates of prposals). NULL (default): nothing is done and proposals outside the feasible parameter range will be evaluated as well.

names

character vector of length d with names for the parameters. These can be used within fun (in this case, parameter input x of fun is a named vector) and will appear in the output list element 'chain'.

prior

character specifying the prior pdf. One of: 'uniform' (prior is assumed to be uniformly distributed between min and max, i.e. non-informative); 'normal' (prior assumed to be normally distributed with mu and cov); name of a user-defined function(x) with x being a d-dimensional parameter vector and returning the log-density of the d-variate prior distribution at x.

Details

To understand the notation (e.g. what is lambda, nCR etc.), have a look at Sect. 3.3 of the reference paper (see below).

Likelihood options (argument lik):

1: fun returns the likelihood of parameter realisation x given some observations.

2: fun returns the log-likelihood.

21: ABC diagnostic model evaluation using a continuous fitness kernel, a variation of so-called noisy-ABC.

22: ABC diagnostic model evaluation using a Boxcar likelihood. fun has to return m diagnostic values to be compared with observations. Requires arguments obs, abc_rho, and abc_e. Modification in computation of Metropolis probability is used (Eq. 13 of Sadegh and Vrugt, 2014). Prior pdf set to zero!

31: GLUE with informal likelihood function based on the NSE: glue_shape * log(NSE). fun needs to return a time series of model simulations and obs should contain a time series of corresponding observations. glue_shape affects the sampling: small values result in high parameter uncertainty, large values produce narrow posterior pdfs of parameters. Should be in the range of 1-100 (experiment!).

99: A user-defined function, see argument lik_fun.

Value

list with named elements:

chain: a (1-burnin)*t/thin-by-d+3-by-nc array of parameter realisations and the log-pdfs of the prior ('lp'), likelihood ('ll'), and posterior ('lpost') distribution for each iteration and Markov chain;

fx: a (1-burnin)*t/thin-by-nc-by-[length of fun's output] array of raw output of fun, corresponds with parameter realisations in chain;

runtime: time of function execution;

outlier: a list with adapt*t vectors of outlier indices in nc (value of 0 means no outliers);

AR: a (1-burnin)*t/thin-by-nCR matrix giving the acceptance rate for each sample number and crossover value (first element is NA due to computational reasons);

CR: a (1-burnin)*t/thin-by-nCR matrix giving the selection probability for each sample number and crossover value (first element is NA due to computational reasons).

IF checkConvergence == TRUE:

R_stat: a (1-burnin)*t/thin-50+1-by-d matrix giving the Gelman-Rubin convergence diagnostic (note that at least 50 observations are used to compute R_stat).

Note

If you want to use a non-implemented likelihood function with nuisance variables to be calibrated along with the actual model parameters, it is suggested to implement the likelihood calculation directly into fun.

Author(s)

Tobias Pilz tpilz@uni-potsdam.de

References

Code based on 'Algorithm 5' and 'Algorithm 6' of:

Vrugt, J. A.: "Markov chain Monte Carlo simulation using the DREAM software package: Theory, concepts, and MATLAB implementation." Environmental Modelling & Software, 2016, 75, 273 – 316, http://dx.doi.org/10.1016/j.envsoft.2015.08.013.

For ABC method see:

Sadegh, M. and J. A. Vrugt: "Approximate Bayesian computation using Markov Chain Monte Carlo simulation: DREAM_ABC". Water Resources Research, 2014, 50, 6767 – 6787, http://dx.doi.org/10.1002/2014WR015386.


tpilz/HydroBayes documentation built on May 6, 2019, 3:44 p.m.