GQD.mle: MLEs for Generalized Quadratic Diffusion Models (GQDs).

Description

GQD.mle() constructs a C++ program in real time that allows the user to perform maximum likelihood inference on scalar GQDs. Given a set of starting parameters, the maximum likelihood estimates are calculated via minimization of minus the likelihood function via the built-in R-function optim.

GQD.mcmc() performs inference using the Metropolis-Hastings algorithm for jump diffusions of the form:

ScalarEqn1.png

where

ScalarEqn2.png

and

ScalarEqn3.png

Usage

GQD.mle(X, time, mesh=10, theta, control=NULL, method='Nelder-Mead', Dtype='Saddle',  Trunc=c(4,4), RK.order=4, P=200, alpha=0, lower=min(X)/2, upper=max(X)*2, exclude=NULL, Tag=NA, wrt=FALSE, print.output=TRUE)

Arguments

Time series (vector) of discretely observed points of the process of interest. These may be non-equidistant observations (see time).

time

A vector of time-stamps associated with each observation in X.

mesh

The number mesh points between any two given data points.

theta

The set of starting parameters for the optimization routine.

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 scheme used. Value must be 4 or (default) 10.

Dtype

Character string indicating the type of density approximation (see details) to use. Types: 'Saddlepoint', 'Normal', 'Gamma', 'InvGamma' and 'Beta' are supported (default = 'Saddlepoint').

Trunc

Vector of length 2 containing the cumulant truncation order and the density truncation order respectively. May take on values 4, 6 and 8 with the constraint that Trunc[1] >= Trunc[2]. Default is c(4,4).

Normalization parameter indicating the number of points to use when normalizing members of the Pearson system (see details)

alpha

Normalization parameter controlig the mesh concentration when normalizing members of the Pearson system (see details). Increasing alpha decreases concentration around the mean and vice versa (default alpha = 0).

lower,upper

Lower and upper bounds for the normalization range.

Tag

Tag can be used to name (tag) an MCMC run e.g. Tag='Run_1'

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

Syntactical jargon

Synt. [1]: The coefficients of the GQD may be parameterized using the reserved variable theta. For example: G0 <- 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: G0 <- 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: G0 <- 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

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.

Examples

Run this code


#===============================================================================
# Simulate a time inhomogeneous diffusion.
#-------------------------------------------------------------------------------

  data(SDEsim1)
  attach(SDEsim1)
  par(mfrow=c(1,1))
  expr1=expression(dX[t]==2*(5+3*sin(0.5*pi*t)-X[t])*dt+0.5*sqrt(X[t])*dW[t])
  plot(Xt~time,type='l',col='blue',xlab='Time (t)',ylab=expression(X[t]),main=expr1)

 #------------------------------------------------------------------------------
 # Define coefficients of the process.
 #------------------------------------------------------------------------------

  GQD.remove()
  G0 <- function(t){theta[1]*(theta[2]+theta[3]*sin(0.25*pi*t))}
  G1 <- function(t){-theta[1]}
  Q0 <- function(t){theta[4]*theta[4]}

  theta.start  <- c(1,1,1,1)                      # Starting values for the chain
  mesh.points  <- 10                              # Number of mesh points

  m1 <- GQD.mle(Xt,time,mesh=mesh.points,theta=theta.start)

  GQD.remove()

  G0 <- function(t){theta[1]*(theta[2]+theta[3]*sin(0.25*pi*t))}
  G1 <- function(t){-theta[1]}
  Q1 <- function(t){theta[4]*theta[4]}

  theta.start  <- c(1,1,1,1)                      # Starting values for the chain
  mesh.points  <- 10                              # Number of mesh points

  m2 <- GQD.mle(Xt,time,mesh=mesh.points,theta=theta.start)

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

  # Compare models:
  GQD.aic(list(m1,m2))

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

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