DiffusionRgqd (version 0.1.3)

GQD.mcmc: MCMC Inference on Generalized Quadratic Diffusion Models (GQDs).

Description

GQD.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 jump diffusion model. Given a set of starting parameters, a MCMC chain is returned for further analysis.

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.mcmc(X, time, mesh=10, theta, sds, updates, burns=min(round(updates/2),25000), Dtype='Saddle', Trunc=c(4,4), RK.order=4, P=200, alpha=0, lower=min(na.omit(X))/2, upper=max(na.omit(X))*2, exclude=NULL, plot.chain=TRUE, Tag=NA, wrt=FALSE, print.output=TRUE, palette = 'mono')

Arguments

X

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 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 chain updates (including burned updates) to perform.

burns

The number of updates to burn. That is, the first burns values are omitted from the inference, although the entire chain is returned.

exclude

Vector indicating which transitions to exclude from the analysis. Default = NULL.

plot.chain

If TRUE (default), a trace plot is made of the resulting MCMC chain (see details).

RK.order

The order of the Runge-Kutta solver used to approximate the trajectories of cumulants. 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).

P

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.

palette

Colour palette for drawing trace plots. Default palette = 'mono', otherwise a qualitative palette will be used.

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))}.

Details

GQD.mcmc() operates by searching the workspace for functions with names that match the coefficients of the predefined stochastic differential equation. Only the required coefficients need to be specified e.g. G0(t),G1(t) and Q0(t) for an Ornstein-Uhlenbeck model. Unspecified coefficients are ignored. When a new model is to be defined, the current model may be removed from the workspace by using the GQD.remove function, after which the new coefficients may be supplied.

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

Examples

#===============================================================================
# This example simulates a time inhomogeneous diffusion and shows how to conduct
# inference using GQD.mcmc
#-------------------------------------------------------------------------------
 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 parameterized coefficients of the process, and set up starting
# parameters.
# True model: dX_t = 2X_t(5+3sin(0.25 pi t)-X_t)dt+0.5X_tdW_t
#------------------------------------------------------------------------------

# Remove any existing coeffients. If none are pressent NAs will be returned, but
# this is a safeguard against overlapping.
GQD.remove()

# Define time dependant coefficients. Note that all functions have a single argument.
# This argument has to be `t' in order for the dependancy to be recognized.
# theta does not have to be defined as an argument.

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
proposal.sds <- c(0.4,0.3,0.2,0.1)*1/2        # Std devs for proposal distributions
mesh.points  <- 10                            # Number of mesh points
updates      <- 50000                         # Perform 50000 updates

#------------------------------------------------------------------------------
# Run the MCMC procedure for the model defined above
#------------------------------------------------------------------------------

m1 <- GQD.mcmc(Xt,time,mesh=mesh.points,theta=theta.start,sds=proposal.sds,
               updates=updates)

# Calculate estimates:
GQD.estimates(m1,thin=200)
#===============================================================================