library(Sim.DiffProc) library(knitr) knitr::opts_chunk$set(comment="",prompt=TRUE, fig.show='hold',warning=FALSE, message=FALSE) options(prompt="R> ",scipen=16,digits=5,warning=FALSE, message=FALSE,width = 80)
TEX.sde()
functionTEX.sde(object,...)
produces the related LATEX code (table and mathematic expression) for Sim.DiffProc environment, which can be copied and pasted in a scientific article.
object
: an objects from class MCM.sde()
and MEM.sde()
. Or an R
vector of expressions of SDEs, i.e., drift and diffusion coefficients....
: arguments to be passed to kable()
function available in knitr package (Xie, 2015), if object
from class MCM.sde()
.MCM.sde
The Monte Carlo results of MCM.sde
class can be presented in terms of LaTeX tables.
\begin{equation}\label{eq01} \begin{cases} dX_t = -\frac{1}{\mu} X_t dt + \sqrt{\sigma} dW_t\ dY_t = X_{t} dt \end{cases} \end{equation}
mu=1;sigma=0.5;theta=2 x0=0;y0=0;init=c(x0,y0) f <- expression(1/mu*(theta-x), x) g <- expression(sqrt(sigma),0) mod2d <- snssde2d(drift=f,diffusion=g,M=500,Dt=0.015,x0=c(x=0,y=0)) ## true values of first and second moment at time 10 Ex <- function(t) theta+(x0-theta)*exp(-t/mu) Vx <- function(t) 0.5*sigma*mu *(1-exp(-2*(t/mu))) Ey <- function(t) y0+theta*t+(x0-theta)*mu*(1-exp(-t/mu)) Vy <- function(t) sigma*mu^3*((t/mu)-2*(1-exp(-t/mu))+0.5*(1-exp(-2*(t/mu)))) covxy <- function(t) 0.5*sigma*mu^2 *(1-2*exp(-t/mu)+exp(-2*(t/mu))) tvalue = list(m1=Ex(15),m2=Ey(15),S1=Vx(15),S2=Vy(15),C12=covxy(15)) ## function of the statistic(s) of interest. sde.fun2d <- function(data, i){ d <- data[i,] return(c(mean(d$x),mean(d$y),var(d$x),var(d$y),cov(d$x,d$y))) } ## Parallel Monte-Carlo of 'OUI' at time 10 mcm.mod2d = MCM.sde(mod2d,statistic=sde.fun2d,time=15,R=10,exact=tvalue,parallel="snow",ncpus=2) mcm.mod2d$MC
In R we create simple LaTeX table for this object using the following code:
TEX.sde(object = mcm.mod2d, booktabs = TRUE, align = "r", caption ="LaTeX table for Monte Carlo results generated by `TEX.sde()` method.")
For inclusion in LaTeX documents, and optionally if we use booktabs = TRUE
in the previous function, the LaTeX add-on package booktabs
must be loaded into the .tex
document.
kable(mcm.mod2d$MC, format = "html",booktabs = TRUE,align = "r", caption ="LaTeX table for Monte Carlo results generated by `TEX.sde()` method.")
MEM.sde
we want to automatically generate the LaTeX code appropriate to moment equations obtained from the previous model using TEX.sde()
method.
mem.oui <- MEM.sde(drift = f, diffusion = g) mem.oui
In R we create LaTeX mathematical expressions for this object using the following code:
TEX.sde(object = mem.oui)
that can be typed with LaTeX to produce a system:
\begin{equation} \begin{cases} \begin{split} \frac{d}{dt} m_{1}(t) ~&= \frac{\left( \theta - m_{1}(t) \right)}{\mu} \ \frac{d}{dt} m_{2}(t) ~&= m_{1}(t) \ \frac{d}{dt} S_{1}(t) ~&= \sigma - 2 \, \left( \frac{S_{1}(t)}{\mu} \right) \ \frac{d}{dt} S_{2}(t) ~&= 2 \, C_{12}(t) \ \frac{d}{dt} C_{12}(t) &= S_{1}(t) - \frac{C_{12}(t)}{\mu} \end{split} \end{cases} \end{equation}
Note that it is obvious the LaTeX package amsmath
must be loaded into the .tex
document.
In this section, we will convert the R expressions of a SDEs, i.e., drift and diffusion coefficients into their LaTeX mathematical equivalents with the same procedures previous. An example sophisticated that will make this clear.
f <- expression((alpha*x *(1 - x / beta)- delta * x^2 * y / (kappa + x^2)), (gamma * x^2 * y / (kappa + x^2) - mu * y^2)) g <- expression(sqrt(sigma1)*x*(1-y), abs(sigma2)*y*(1-x)) TEX.sde(object=c(drift = f, diffusion = g))
under LaTeX will create this system:
\begin{equation} \begin{cases} \begin{split} dX_{t} &= \left( \alpha \, X_{t} \, \left( 1 - \frac{X_{t}}{\beta} \right) - \frac{\delta \, X_{t}^2 \, Y_{t}}{\left( \kappa + X_{t}^2 \right)} \right) \:dt + \sqrt{\sigma_{1}} \, X_{t} \, \left( 1 - Y_{t} \right) \:dW_{1,t} \ dY_{t} &= \left( \frac{\gamma \, X_{t}^2 \, Y_{t}}{\left( \kappa + X_{t}^2 \right)} - \mu \, Y_{t}^2 \right) \:dt + \left| \sigma_{2}\right| \, Y_{t} \, \left( 1 - X_{t} \right) \:dW_{2,t} \end{split} \end{cases} \end{equation}
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.Xie Y (2015). Dynamic Documents with R and knitr. 2nd edition. Chapman and Hall/CRC, Boca Raton, Florida. ISBN 978-1498716963, URL https://yihui.org/knitr/
Wickham H (2015). Advanced R. Chapman & Hall/CRC The R Series. CRC Press. ISBN 9781498759809.
Guidoum AC, Boukhetala K (2020). "Performing Parallel Monte Carlo and Moment Equations Methods for Itô and Stratonovich Stochastic Differential Systems: R Package Sim.DiffProc". Journal of Statistical Software, 96(2), 1--82. https://doi.org/10.18637/jss.v096.i02
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