BiGQD.mle: Calculate Maximum Likelihood Estimates for a 2D GQD Model.

In DiffusionRgqd: Inference and Analysis for Generalized Quadratic Diffusions

Description

BiGQD.mle() uses parametrised coefficients (provided by the user as R-functions) to construct a C++ program in real time that allows the user to perform maximum likelihood inference on the resulting diffusion model. BiGQD.density generates approximate transitional densities for a class of bivariate diffusion processes with SDE:

where

and

Usage

BiGQD.mle(X, time, mesh=10, theta, control=NULL, method='Nelder-Mead', RK.order=4,
          exclude=NULL, Tag=NA, Dtype='Saddlepoint', rtf= runif(2,-1,1), wrt=FALSE,
          print.output=TRUE)

Arguments

X	A matrix containing rows of data points to be modelled. Though observations are allowed to be non-equidistant, observations in both dimensions are assumed to occur at the same time epochs.
time	A vector containing the time epochs at which observations were made.
mesh	The number of mesh points in the time discretisation.
theta	The parameter vector starting values.
control	List of control variables to be used by optim.
method	Method to be used by optim.
exclude	Vector indicating which transitions to exclude from the analysis. Default = NULL.
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".
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.

Value

opt	The output from optim.
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.

Syntactical jargon

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

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.

Warning [2]: Note that minus the likelihood is minimized, as such the optim output (hessian) needs to be adjusted accordingly if used for calculating confidence intervals. Furthermore, GQD.mle may be temperamental under certain conditions

Author(s)

Etienne A.D. Pienaar: [e-mail]

References

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.

See Also

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

Examples

#===============================================================================
# This example simulates a bivariate time homogeneous diffusion and shows how
# to conduct inference using BiGQD.mle(). 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)
  X           <- cbind(Xt,Yt)

  # Calculate MLEs
  m1=BiGQD.mle(X,time,10,theta.start)

 #------------------------------------------------------------------------------
 # 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)

  # Calculate MLEs
  m2=BiGQD.mle(X,time,10,theta.start)

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

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

  GQD.aic(list(m1,m2))


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

DiffusionRgqd documentation built on May 2, 2019, 3:26 a.m.