JGQD.mcmc: MCMC Inference on Jump Generalized Quadratic Diffusion Models (JGQDs).

Description

JGQD.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. The structure of the model is predefined and coefficients may be provided for models nested within the generalized quadratic diffusion framework.

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

ScalarEqn1.png

where

ScalarEqn2.png

ScalarEqn3.png

and

ScalarEqn4.png

describes a Poisson process with jumps of the form:

ScalarEqn6.png

arriving with intensity

ScalarEqn5.png

subject to a jump distribition of the form:

ScalarEqn7.png

Usage

JGQD.mcmc(X, time, mesh = 10, theta, sds, updates = 1000, burns = min(round(updates/2), 25000), Jtype = "Add", Jdist = "Normal", Dtype = "Saddlepoint", RK.order = 4, exclude = NULL, plot.chain = TRUE, wrt = FALSE, Tag = NA, factorize = TRUE, print.output = TRUE, decode = TRUE, palette = 'mono')

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 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' is supported in the current version of the software.

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.

Jdist

Valid entries are 'Normal', 'Expnential', 'Gamma' and 'Laplace'.

Jtype

Valid types are 'Add' or 'Mult'.

factorize

Should factorization be used (default = TRUE).

print.output

If TRUE, model information is printed to the console.

decode

Should the algorithm estimate jump detection probabilities? Default value is TRUE.

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 JGQD 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

JGQD.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 JGQD.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.

Examples

Run this code



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

 # For detailed notes and examples on how to use the JGQD.mcmc() function, see
 # the following vignette:

 RShowDoc('Part_4_Likelihood_Inference',type='html','DiffusionRjgqd')

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

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