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Last data update: 2014.03.03

R: MLEs for Generalized Quadratic Diffusion Models (GQDs).

GQD.mle

R Documentation

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:

where

and

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

`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 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)`.
`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.

Value

`opt`	The output from `optim`.
`model.info`	A list of variables pertaining to inference calculations.
`model.info$elapsed.time`	The run-time, in h/m/s format,of the MCMC procedure (excluding compile time).
`model.info$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 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

Author(s)

Etienne A.D. Pienaar: etiennead@gmail.com

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


#===============================================================================
# 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))

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

Results


R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.

R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.

Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> library(DiffusionRgqd)
> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/DiffusionRgqd/GQD.mle.Rd_%03d_medium.png", width=480, height=480)
> ### Name: GQD.mle
> ### Title: MLEs for Generalized Quadratic Diffusion Models (GQDs).
> ### Aliases: GQD.mle
> ### Keywords: syntax C++ maximum likelihood
> 
> ### ** Examples
> 
> ## No test: 
> #===============================================================================
> # 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()
[1] "Removed :  NA "
>   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)
Compiling C++ code. Please wait.                                                                                                           
 ================================================================
                  Generalized Ornstein-Uhlenbeck                 
 ================================================================
 _____________________ Drift Coefficients _______________________
 G0 : theta[1]*(theta[2]+theta[3]*sin(0.25*pi*t))                
 G1 : -theta[1]                                                  
 G2                                                              
 ___________________ Diffusion Coefficients _____________________
 Q0 : theta[4]*theta[4]                                          
 Q1                                                              
 Q2                                                              
                                                                 
 _______________________ Model/Chain Info _______________________
 Time Homogeneous    : No                                        
 Data Resolution     : Homogeneous: dt=0.25                      
 # Removed Transits. : None                                      
 Density approx.     : 2nd Ord. Truncation + Std Normal Dist.    
 Elapsed time        : 00:00:00                                  
 ...   ...   ...   ...   ...   ...   ...   ...   ...   ...   ... 
 dim(theta)          : 4                                         
 ----------------------------------------------------------------
> 
>   GQD.remove()
[1] "Removed :  G0 G1 Q0"
> 
>   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)
Compiling C++ code. Please wait.                                                                                                           
 ================================================================
                  Generalized Quadratic Diffusion (GQD)          
 ================================================================
 _____________________ Drift Coefficients _______________________
 G0 : theta[1]*(theta[2]+theta[3]*sin(0.25*pi*t))                
 G1 : -theta[1]                                                  
 G2                                                              
 ___________________ Diffusion Coefficients _____________________
 Q0                                                              
 Q1 : theta[4]*theta[4]                                          
 Q2                                                              
                                                                       
 _______________________ Model/Chain Info _______________________      
 Time Homogeneous    : No                                              
 Data Resolution     : Homogeneous: dt=0.25                            
 # Removed Transits. : None                                            
 Density approx.     : 4 Ord. Truncation +4th Ord. Saddlepoint Appr.   
 Elapsed time        : 00:00:00                                        
 ...   ...   ...   ...   ...   ...   ...   ...   ...   ...   ...       
 dim(theta)          : 4                                               
 ----------------------------------------------------------------      
> 
>   # Check estimates:
>   GQD.estimates(m1)
         Estimate Lower_95 Upper_95
theta[1]    2.087    1.805    2.369
theta[2]    5.016    4.907    5.124
theta[3]    2.898    2.737    3.059
theta[4]    1.138    1.052    1.223
>   GQD.estimates(m2)
         Estimate Lower_95 Upper_95
theta[1]    2.036    1.773    2.299
theta[2]    5.014    4.906    5.121
theta[3]    2.911    2.756    3.066
theta[4]    0.498    0.460    0.535
> 
>   # Compare models:
>   GQD.aic(list(m1,m2))
        Convergence p min.likelihood           AIC           BIC   N
Model 1           0 4       246.5406      501.0812      517.0570 401
Model 2           0 4       225.0633  [=] 458.1267  [=] 474.1025 401
> 
> #===============================================================================
> ## End(No test)
> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>