BiGQD.mcmc: MCMC Inference on Bivariate Generalized Quadratic Diffusions...
In eta21/DiffusionRgqd: Inference and Analysis for Generalized Quadratic Diffusions

Description Usage Arguments Value Syntactical jargon Warning Note Author(s) References See Also Examples

BiGQD.mcmc() uses parametrised coefficients (provided by the user as R-functions) to construct a C++ program in real time that allows the user to perform Bayesian inference on the resulting diffusion model. Given a set of starting parameters and other input parameters, a MCMC chain is returned for further analysis. BiGQD.density generates approximate transitional densities for a class of bivariate diffusion processes with SDE:

where

and

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2
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BiGQD.mcmc(X, time, mesh=10, theta, sds, updates, burns=min(round(updates/2),25000),
           RK.order=4, exclude=NULL, plot.chain=TRUE, Tag=NA, Dtype='Saddlepoint',
           recycle=FALSE, rtf=runif(2), wrt=FALSE, print.output=TRUE, palette = "mono")

`X`	A matrix containing rows of data points to be modelled. Although observations are allowed to be non-equidistant, observations in both dimensions are assumed to occur at the same time epochs (i.e. `time` gives the time signature for both dimensions).
`time`	A vector containing the time epochs at which observations were made.
`mesh`	The number of mesh points in the time discretization.
`theta`	The parameter vector of the process. theta are taken as the starting values of the MCMC chain and gives the dimension of the parameter vector used to calculate the DIC. Care should be taken to ensure that each element in theta is in fact used within the coefficient-functions, otherwise redundant parameters will be counted in the calculation of the DIC.
`sds`	Proposal distribution standard deviations. That is, for the i-th parameter the proposal distribution is ~ Normal(...,`sds`[i]^2).
`updates`	The number of MCMC updates/iterations to perform (including burn-in).
`burns`	The number of MCMC updates/iterations to burn.
`exclude`	Vector indicating which transitions to exclude from the analysis. Default = `NULL`.
`plot.chain`	If = TRUE (default), a trace plot of the MCMC chain will be made along with a trace of the acceptance rate.
`RK.order`	The order of the Runge-Kutta solver used to approximate the trajectories of cumulants. Must be 4 (default) or 10.
`Tag`	`Tag` can be used to name (tag) an MCMC run e.g. `Tag='Run_1'`
`Dtype`	The density approximant to use. Valid types are `"Saddlepoint"` (default), `"Edgeworth"` or `"Normal"`.
`recycle`	Whether or not to recycle the roots calculated for the saddlepoint approximation over succesive updates.
`rtf`	Starting vector for internal use.
`wrt`	If `TRUE` a .cpp file will be written to the current directory. For bug report diagnostics.
`print.output`	If `TRUE` information about the model and algorithm is printed to the console.
`palette`	Colour palette for drawing trace plots. Default `palette = 'mono'`, otherwise a qualitative palette will be used.

`par.matrix`	A matrix containing the MCMC chain on `theta`.
`acceptence.rate`	A vector containing the acceptance rate of the MCMC at every iteration.
`model.info`	A list of variables pertaining to inference calculations.
`model.info$elapsed.time`	The runtime, in h/m/s format,of the MCMC procedure (excluding compile time).
`model.info$time.homogeneous`	‘No’ if the model has time-homogeneous coefficients and ‘Yes’ otherwise.
`model.info$p`	The dimension of `theta`.
`model.info$DIC`	Calculated Deviance Information Criterion.
`model.info$pd`	Effective number of parameters (see `model.info$DIC`).

Synt. [1]: The coefficients of the 2D GQD may be parameterized using the reserved variable theta. For example:

a00 <- function(t){theta[1]*(theta[2]+sin(2*pi*(t-theta[3])))}.

Synt. [2]: Due to syntactical differences between R and C++ special functions have to be used when terms that depend on t. When the function cannot be separated in to terms that contain a single t, the prod(a,b) function must be used. For example:

a00 <- function(t){0.1*(10+0.2*sin(2*pi*t)+0.3*prod(sqrt(t),1+cos(3*pi*t)))}.

Here sqrt(t)*cos(3*pi*t) constitutes the product of two terms that cannot be written i.t.o. a single t. To circumvent this isue, one may use the prod(a,b) function.

Synt. [3]: Similarly, the ^ - operator is not overloaded in C++. Instead the pow(x,p) function may be used to calculate x^p. For example sin(2*pi*t)^3 in:

a00 <- function(t){0.1*(10+0.2*pow(sin(2*pi*t),3))}.

Warning [1]: The parameter mesh is used to discretize the transition horizons between successive observations. It is thus important to ensure that mesh is not too small when large time differences are present in time. Check output for max(dt) and divide by mesh.

Note [1]: When plot.chain is TRUE, a trace plot is created of the resulting MCMC along with the acceptance rate at each update. This may save time when scrutinizing initial MCMC runs.

Etienne A.D. Pienaar etiannead@gmail.com

Updates available on GitHub at https://github.com/eta21.

Daniels, H.E. 1954 Saddlepoint approximations in statistics. Ann. Math. Stat., 25:631–650.

Eddelbuettel, D. and Romain, F. 2011 Rcpp: Seamless R and C++ integration. Journal of Statistical Software, 40(8):1–18,. URL http://www.jstatsoft.org/v40/i08/.

Eddelbuettel, D. 2013 Seamless R and C++ Integration with Rcpp. New York: Springer. ISBN 978-1-4614-6867-7.

Eddelbuettel, D. and Sanderson, C. 2014 Rcpparmadillo: Accelerating R with high-performance C++ linear algebra. Computational Statistics and Data Analysis, 71:1054–1063. URL http://dx.doi.org/10.1016/j.csda.2013.02.005.

Feagin, T. 2007 A tenth-order Runge-Kutta method with error estimate. In Proceedings of the IAENG Conf. on Scientifc Computing.

Varughese, M.M. 2013 Parameter estimation for multivariate diffusion systems. Comput. Stat. Data An., 57:417–428.

GQD.remove, BiGQD.mle, GQD.mcmc, GQD.mle, GQD.passage and GQD.TIpassage.

#===============================================================================
# This example simulates a bivariate time homogeneous diffusion and shows how
# to conduct inference using BiGQD.mcmc(). We fit two competing models and then
# use the output to select a winner.
#-------------------------------------------------------------------------------

  data(SDEsim2)
  data(SDEsim2)
  attach(SDEsim2)
  # Have a look at the time series:
  plot(Xt~time,type='l',col='blue',ylim=c(2,10),main='Simulated Data',xlab='Time (t)',ylab='State',
       axes=FALSE)
  lines(Yt~time,col='red')
  expr1=expression(dX[t]==2(Y[t]-X[t])*dt+0.3*sqrt(X[t]*Y[t])*dW[t])
  expr2=expression(dY[t]==(5-Y[t])*dt+0.5*sqrt(Y[t])*dB[t])
  text(50,9,expr1)
  text(50,8.5,expr2)
  axis(1,seq(0,100,5))
  axis(1,seq(0,100,5/10),tcl=-0.2,labels=NA)
  axis(2,seq(0,20,2))
  axis(2,seq(0,20,2/10),tcl=-0.2,labels=NA)

 #------------------------------------------------------------------------------
 # Define the coefficients of a proposed model
 #------------------------------------------------------------------------------
  GQD.remove()
  a00 <- function(t){theta[1]*theta[2]}
  a10 <- function(t){-theta[1]}
  c00 <- function(t){theta[3]*theta[3]}

  b00 <- function(t){theta[4]}
  b01 <- function(t){-theta[5]}
  f00 <- function(t){theta[6]*theta[6]}

  theta.start <- c(3,3,3,3,3,3)
  prop.sds    <- c(0.15,0.16,0.04,0.99,0.19,0.04)
  updates     <- 50000
  X           <- cbind(Xt,Yt)
  
  # Define prior distributions:
  priors=function(theta){dunif(theta[1],0,100)*dunif(theta[4],0,100)}
  
  # Run the MCMC procedure
  m1=BiGQD.mcmc(X,time,10,theta.start,prop.sds,updates)

 #------------------------------------------------------------------------------
 # Remove old coefficients and define the coefficients of a new model
 #------------------------------------------------------------------------------
  GQD.remove()
  a10 <- function(t){-theta[1]}
  a01 <- function(t){theta[1]*theta[2]}
  c11 <- function(t){theta[3]*theta[3]}

  b00 <- function(t){theta[4]*theta[5]}
  b01 <- function(t){-theta[4]}
  f01 <- function(t){theta[6]*theta[6]}

  theta.start <- c(3,3,3,3,3,3)
  prop.sds    <- c(0.16,0.02,0.01,0.18,0.12,0.01)

  # Define prior distributions:
  priors=function(theta){dunif(theta[1],0,100)*dunif(theta[4],0,100)}

  # Run the MCMC procedure
  m2=BiGQD.mcmc(X,time,10,theta.start,prop.sds,updates)

 # Compare estimates:
  GQD.estimates(m1)
  GQD.estimates(m2)

 #------------------------------------------------------------------------------
 # Compare the two models
 #------------------------------------------------------------------------------

  GQD.dic(list(m1,m2))


#===============================================================================