MCM.sde() functionR> MCM.sde(model, statistic, R = 1000, time, exact = NULL, names = NULL,level = 0.95,
+ parallel = c("no", "multicore", "snow"),ncpus = getOption("ncpus", 1L), cl = NULL, ...)The main arguments of MCM.sde() function in Sim.DiffProc package consist:
model: an object from classes snssde1d(), snssde2d() and snssde3d().statistic: a function which when applied to the model (SDEs) returns a vector containing the statistic(s) of interest. Any further arguments can be passed to statistic(s) through the ... argument.R: number of Monte Carlo replicates (R batches), this will be a single positive integer.time: fixed time at which the estimate of the statistic(s).exact: a named list giving the exact statistic(s), if it exists the bias calculation will be performed.names: named the statistic(s) of interest. Default names=c("mu1","mu2",...).level: confidence level of the required interval(s).parallel: the type of parallel operation to be used. "multicore" does not work on Microsoft Windows operating systems, but on Unix is allowed and uses parallel operations. Default parallel="no".ncpus: an integer value specifying the number of cores to be used in the parallelized procedure. Default is 1 core of the machine.cl: an optional parallel cluster for use if parallel = "snow". Default cl = makePSOCKcluster(rep("localhost", ncpus)).R> plot(x,index = 1,type=c("all","hist","qqplot","boxplot","CI"), ...)This takes a MCM.sde() object and produces plots for the R replicates of the interesting quantity.
x: an object from the class MCM.sde().index: the index of the variable of interest within the output of class MCM.sde().type: type of plots. Default type="all".R> theta = 0.75; x0 = 1
R> fx <- expression( 0.5*theta^2*x )
R> gx <- expression( theta*x )
R> mod1 <- snssde1d(drift=fx,diffusion=gx,x0=x0,M=500,type="ito")
R> mod2 <- snssde1d(drift=fx,diffusion=gx,x0=x0,M=500,type="str")
R> ## True values of means and variance for mod1 and mod2
R> E.mod1 <- function(t) x0*exp(0.5*theta^2*t)
R> E.mod2 <- function(t) x0
R> V.mod1 <- function(t) x0^2*exp(theta^2*t)*(exp(theta^2*t)-1)
R> V.mod2 <- function(t) x0^2 *(exp(theta^2*t)-1)
R> ## function of the statistic(s) of interest.
R> sde.fun1d <- function(data, i){
+ d <- data[i, ]
+ return(c(mean(d),var(d)))
+ }
R> # Parallel MOnte Carlo for mod1
R> mcm.mod1 = MCM.sde(model=mod1,statistic=sde.fun1d,R=20, exact=list(m=E.mod1(1),S=V.mod1(1)),parallel="snow",ncpus=2)
R> mcm.mod1Itô Sde 1D:
| dX(t) = 0.5 * theta^2 * X(t) * dt + theta * X(t) * dW(t)
| t in [0,1] with mesh equal to 0.001
PMCM Based on 20 Batches with 500-Realisations at time 1:
Exact Estimate Bias Std.Error RMSE CI( 2.5 % , 97.5 % )
m 1.3248 1.3227 -0.00205 0.01438 0.06270 ( 1.29455 , 1.35091 )
S 1.3252 1.3354 0.01024 0.07996 0.34869 ( 1.17868 , 1.49212 )R> # Parallel MOnte Carlo for mod2
R> mcm.mod2 = MCM.sde(model=mod2,statistic=sde.fun1d,R=20, exact=list(m=E.mod2(1),S=V.mod2(1)),parallel="snow",ncpus=2)
R> mcm.mod2Stratonovich Sde 1D:
| dX(t) = 0.5 * theta^2 * X(t) * dt + theta * X(t) o dW(t)
| t in [0,1] with mesh equal to 0.001
PMCM Based on 20 Batches with 500-Realisations at time 1:
Exact Estimate Bias Std.Error RMSE CI( 2.5 % , 97.5 % )
m 1.00000 1.7869 0.78688 0.01597 0.78996 ( 1.75558 , 1.81818 )
S 0.75505 2.6798 1.92479 0.15066 2.03374 ( 2.38455 , 2.97513 )R> mu=1;sigma=0.5;theta=2
R> x0=0;y0=0;init=c(x0,y0)
R> f <- expression(1/mu*(theta-x), x)
R> g <- expression(sqrt(sigma),0)
R> OUI <- snssde2d(drift=f,diffusion=g,M=500,Dt=0.015,x0=c(x=0,y=0))
R> ## true values of first and second moment at time 10
R> Ex <- function(t) theta+(x0-theta)*exp(-t/mu)
R> Vx <- function(t) 0.5*sigma*mu *(1-exp(-2*(t/mu)))
R> Ey <- function(t) y0+theta*t+(x0-theta)*mu*(1-exp(-t/mu))
R> Vy <- function(t) sigma*mu^3*((t/mu)-2*(1-exp(-t/mu))+0.5*(1-exp(-2*(t/mu))))
R> covxy <- function(t) 0.5*sigma*mu^2 *(1-2*exp(-t/mu)+exp(-2*(t/mu)))
R> tvalue = list(m1=Ex(10),m2=Ey(10),S1=Vx(10),S2=Vy(10),C12=covxy(10))
R> ## function of the statistic(s) of interest.
R> sde.fun2d <- function(data, i){
+ d <- data[i,]
+ return(c(mean(d$x),mean(d$y),var(d$x),var(d$y),cov(d$x,d$y)))
+ }
R> ## Parallel Monte-Carlo of 'OUI' at time 10
R> mcm.mod2d = MCM.sde(OUI,statistic=sde.fun2d,time=10,R=20,exact=tvalue,parallel="snow",ncpus=2)
R> mcm.mod2dItô Sde 2D:
| dX(t) = 1/mu * (theta - X(t)) * dt + sqrt(sigma) * dW1(t)
| dY(t) = X(t) * dt + 0 * dW2(t)
| t in [0,15] with mesh equal to 0.015
PMCM Based on 20 Batches with 500-Realisations at time 10:
Exact Estimate Bias Std.Error RMSE
m1 1.99991 1.99647 -0.00344 0.00483 0.02135
m2 18.00009 18.00058 0.00049 0.02262 0.09858
S1 0.25000 0.24924 -0.00076 0.00258 0.01129
S2 4.25005 4.31008 0.06003 0.06261 0.27943
C12 0.24998 0.24965 -0.00033 0.01129 0.04921
CI( 2.5 % , 97.5 % )
m1 ( 1.987 , 2.00594 )
m2 ( 17.95625 , 18.04491 )
S1 ( 0.24418 , 0.2543 )
S2 ( 4.18737 , 4.43279 )
C12 ( 0.22752 , 0.27178 )R> mu=0.5;sigma=0.25
R> fx <- expression(mu*y,0,0)
R> gx <- expression(sigma*z,1,1)
R> Sigma <-matrix(c(1,0.3,-0.5,0.3,1,0.2,-0.5,0.2,1),nrow=3,ncol=3)
R> modtra <- snssde3d(drift=fx,diffusion=gx,M=500,type="str",corr=Sigma)
R> ## function of the statistic(s) of interest.
R> sde.fun3d <- function(data, i){
+ d <- data[i,]
+ return(c(mean(d$x),median(d$x),Mode(d$x)))
+ }
R> ## Monte-Carlo at time = 10
R> mcm.mod3d = MCM.sde(modtra,statistic=sde.fun3d,R=10,parallel="snow",ncpus=2)
R> mcm.mod3dStratonovich Sde 3D:
| dX(t) = mu * Y(t) * dt + sigma * Z(t) o dB1(t)
| dY(t) = 0 * dt + 1 o dB2(t)
| dZ(t) = 0 * dt + 1 o dB3(t)
| t in [0,1] with mesh equal to 0.001
| Correlation structure:
1.0 0.3 -0.5
0.3 1.0 0.2
-0.5 0.2 1.0
PMCM Based on 10 Batches with 500-Realisations at time 1:
Estimate Std.Error CI( 2.5 % , 97.5 % )
mu1 -0.07146 0.00319 ( -0.07771 , -0.06521 )
mu2 -0.05744 0.00512 ( -0.06748 , -0.0474 )
mu3 -0.02513 0.02182 ( -0.0679 , 0.01764 )MEM.sde() functionR> MEM.sde(drift, diffusion, corr = NULL, type = c("ito", "str"), solve = FALSE, parms = NULL, init = NULL, time = NULL, ...)The main arguments of MEM.sde() function in Sim.DiffProc package consist:
drift: an R vector of expressions which contains the drift specification (1D, 2D and 3D).diffusion: an R vector of expressions which contains the diffusion specification (1D, 2D and 3D).corr: the correlation coefficient ‘|corr|<=1’ of \(W_{1}(t)\) and \(W_{2}(t)\) (2D) must be an expression length equal 1. And for 3D \((W_{1}(t),W_{2}(t),W_{3}(t))\) an expressions length equal 3.type: type of SDEs to be used "ito" for Ito form and "str" for Stratonovich form. The default type="ito".solve: if solve=FALSE only the symbolic computational of system will be made. And if solve=TRUE a numerical approximation of the obtained system will be performed.parms: parameters passed to drift and diffusion.init: initial (state) values of system.time: time sequence (vector) for which output is sought; the first value of time must be the initial time....: arguments to be passed to ode() function available in deSolve package, if solve=TRUE.R> fx <- expression( 0.5*theta^2*x )
R> gx <- expression( theta*x )
R> start = c(m=1,S=0)
R> t = seq(0,1,by=0.001)
R> mem.mod1 = MEM.sde(drift=fx,diffusion=gx,type="ito",solve = TRUE,parms = c(theta=0.75), init = start, time = t)
R> mem.mod1Itô Sde 1D:
| dX(t) = 0.5 * 0.75^2 * X(t) * dt + 0.75 * X(t) * dW(t)
| t in [0,1].
Moment equations:
| dm(t) = 0.28125 * m(t)
| dS(t) = 0.5625 * m(t)^2 + 1.125 * S(t)
Approximation of moment at time 1
| m(1) = 1.3248
| S(1) = 1.3252R> mem.mod2 = MEM.sde(drift=fx,diffusion=gx,type="str",solve = TRUE,parms = c(theta=0.75), init = start, time = t)
R> mem.mod2Stratonovich Sde 1D:
| dX(t) = 0.5 * 0.75^2 * X(t) * dt + 0.75 * X(t) o dW(t)
| t in [0,1].
Moment equations:
| dm(t) = 0.5625 * m(t)
| dS(t) = 0.5625 * m(t)^2 + 1.6875 * S(t)
Approximation of moment at time 1
| m(1) = 1.755
| S(1) = 2.3257R> plot(mem.mod1$sol.ode, mem.mod2$sol.ode,ylab = c("m(t)"),select="m", xlab = "Time",main="",col = 2:3,lty=1)
R> legend("topleft",c(expression(m[mod1](t),m[mod2](t))),inset = .05, col=2:3,lty=1)
R> plot(mem.mod1$sol.ode, mem.mod2$sol.ode,ylab = c("S(t)"),select="S", xlab = "Time",main="",col = 2:3,lty=1)
R> legend("topleft",c(expression(S[mod1](t),S[mod2](t))),inset = .05, col=2:3,lty=1)R> fx <- expression(1/mu*(theta-x), x)
R> gx <- expression(sqrt(sigma),0)
R> start = c(m1=0,m2=0,S1=0,S2=0,C12=0)
R> t = seq(0,10,by=0.001)
R> mem.mod2d = MEM.sde(drift=fx,diffusion=gx,type="ito",solve = TRUE,parms = c(mu=1,sigma=0.5,theta=2), init = start, time = t)
R> mem.mod2dItô Sde 2D:
| dX(t) = 1/1 * (2 - X(t)) * dt + sqrt(0.5) * dW1(t)
| dY(t) = X(t) * dt + 0 * dW2(t)
| t in [0,10].
Moment equations:
| dm1(t) = 2 - m1(t)
| dm2(t) = m1(t)
| dS1(t) = 0.5 - 2 * S1(t)
| dS2(t) = 2 * C12(t)
| dC12(t) = S1(t) - C12(t)
Approximation of moment at time 10
| m1(10) = 1.9999 | S1(10) = 0.25 | C12(10) = 0.24998
| m2(10) = 18.0001 | S2(10) = 4.25 R> matplot.0D(mem.mod2d$sol.ode,main="")R> fx <- expression(mu*y,0,0)
R> gx <- expression(sigma*z,1,1)
R> RHO <- expression(0.75,0.5,-0.25)
R> start = c(m1=5,m2=0,m3=0,S1=0,S2=0,S3=0,C12=0,C13=0,C23=0)
R> t = seq(0,1,by=0.001)
R> mem.mod3d = MEM.sde(drift=fx,diffusion=gx,corr=RHO,type="ito",solve = TRUE,parms = c(mu=0.5,sigma=0.25), init = start, time = t)
R> mem.mod3dItô Sde 3D:
| dX(t) = 0.5 * Y(t) * dt + 0.25 * Z(t) * dB1(t)
| dY(t) = 0 * dt + 1 * dB2(t)
| dZ(t) = 0 * dt + 1 * dB3(t)
| t in [0,1].
| Correlation structure: E(dB1dB2) = 0.75 * dt
: E(dB1dB3) = 0.5 * dt
: E(dB2dB3) = -0.25 * dt
Moment equations:
| dm1(t) = 0.5 * m2(t)
| dm2(t) = 0
| dm3(t) = 0
| dS1(t) = 0.0625 * S3(t) + 0.0625 * m3(t)^2 + C12(t)
| dS2(t) = 1
| dS3(t) = 1
| dC12(t) = 0.1875 * m3(t) + 0.5 * S2(t)
| dC13(t) = 0.125 * m3(t) + 0.5 * C23(t)
| dC23(t) = -0.25
Approximation of moment at time 1
| m1(1) = 5 | S1(1) = 0.11458 | C12(1) = 0.2500
| m2(1) = 0 | S2(1) = 1.00000 | C13(1) = -0.0625
| m3(1) = 0 | S3(1) = 1.00000 | C23(1) = -0.2500R> matplot.0D(mem.mod3d$sol.ode,main="",select=c("m1","m2","m3"))
R> matplot.0D(mem.mod3d$sol.ode,main="",select=c("S1","S2","S3"))
R> matplot.0D(mem.mod3d$sol.ode,main="",select=c("C12","C13","C23"))snssdekd() & dsdekd() & rsdekd()- Monte-Carlo Simulation and Analysis of Stochastic Differential Equations.bridgesdekd() & dsdekd() & rsdekd() - Constructs and Analysis of Bridges Stochastic Differential Equations.fptsdekd() & dfptsdekd() - Monte-Carlo Simulation and Kernel Density Estimation of First passage time.MCM.sde() & MEM.sde() - Parallel Monte-Carlo and Moment Equations for SDEs.TEX.sde() - Converting Sim.DiffProc Objects to LaTeX.fitsde() - Parametric Estimation of 1-D Stochastic Differential Equation.Guidoum AC, Boukhetala K (2020). Performing Parallel Monte Carlo and Moment Equations Methods for Ito and Stratonovich Stochastic Differential Systems: R Package Sim.DiffProc. Accept Submission to Journal of Statistical Software.
Guidoum AC, Boukhetala K (2020). Sim.DiffProc: Simulation of Diffusion Processes. R package version 4.6, URL https://cran.r-project.org/package=Sim.DiffProc.
Department of Probabilities & Statistics, Faculty of Mathematics, University of Science and Technology Houari Boumediene, BP 32 El-Alia, U.S.T.H.B, Algeria, E-mail (acguidoum@usthb.dz)↩
Faculty of Mathematics, University of Science and Technology Houari Boumediene, BP 32 El-Alia, U.S.T.H.B, Algeria, E-mail (kboukhetala@usthb.dz)↩