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.
\[\begin{equation}\label{eq01} \begin{cases} dX_t = -\frac{1}{\mu} X_t dt + \sqrt{\sigma} dW_t\\ dY_t = X_{t} dt \end{cases} \end{equation}\]
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> mod2d <- 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(15),m2=Ey(15),S1=Vx(15),S2=Vy(15),C12=covxy(15))
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(mod2d,statistic=sde.fun2d,time=15,R=10,exact=tvalue,parallel="snow",ncpus=2)
R> mcm.mod2d$MC
Exact Estimate Bias Std.Error RMSE CI( 2.5 % , 97.5 % )
m1 2.00 1.99996 -0.00004 0.00578 0.01735 ( 1.98863 , 2.01129 )
m2 28.00 27.98526 -0.01474 0.04641 0.14000 ( 27.8943 , 28.07622 )
S1 0.25 0.24766 -0.00234 0.00383 0.01173 ( 0.24015 , 0.25517 )
S2 6.75 6.69702 -0.05298 0.14770 0.44625 ( 6.40753 , 6.98651 )
C12 0.25 0.27466 0.02466 0.02525 0.07966 ( 0.22517 , 0.32415 )
In R we create simple LaTeX table for this object using the following code:
R> TEX.sde(object = mcm.mod2d, booktabs = TRUE, align = "r", caption ="LaTeX
+ table for Monte Carlo results generated by `TEX.sde()` method.")
%%% LaTeX table generated in R 4.3.3 by TEX.sde() method
%%% Copy and paste the following output in your LaTeX file
\begin{table}
\caption{\label{tab:unnamed-chunk-2}LaTeX
table for Monte Carlo results generated by `TEX.sde()` method.}
\centering
\begin{tabular}[t]{lrrrrrr}
\toprule
& Exact & Estimate & Bias & Std.Error & RMSE & CI( 2.5 \% , 97.5 \% )\\
\midrule
$m_{1}(t)$ & 2.00 & 1.99996 & -0.00004 & 0.00578 & 0.01735 & ( 1.98863 , 2.01129 )\\
$m_{2}(t)$ & 28.00 & 27.98526 & -0.01474 & 0.04641 & 0.14000 & ( 27.8943 , 28.07622 )\\
$S_{1}(t)$ & 0.25 & 0.24766 & -0.00234 & 0.00383 & 0.01173 & ( 0.24015 , 0.25517 )\\
$S_{2}(t)$ & 6.75 & 6.69702 & -0.05298 & 0.14770 & 0.44625 & ( 6.40753 , 6.98651 )\\
$C_{12}(t)$ & 0.25 & 0.27466 & 0.02466 & 0.02525 & 0.07966 & ( 0.22517 , 0.32415 )\\
\bottomrule
\end{tabular}
\end{table}
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.
Exact | Estimate | Bias | Std.Error | RMSE | CI( 2.5 % , 97.5 % ) | |
---|---|---|---|---|---|---|
m1 | 2.00 | 1.99996 | -0.00004 | 0.00578 | 0.01735 | ( 1.98863 , 2.01129 ) |
m2 | 28.00 | 27.98526 | -0.01474 | 0.04641 | 0.14000 | ( 27.8943 , 28.07622 ) |
S1 | 0.25 | 0.24766 | -0.00234 | 0.00383 | 0.01173 | ( 0.24015 , 0.25517 ) |
S2 | 6.75 | 6.69702 | -0.05298 | 0.14770 | 0.44625 | ( 6.40753 , 6.98651 ) |
C12 | 0.25 | 0.27466 | 0.02466 | 0.02525 | 0.07966 | ( 0.22517 , 0.32415 ) |
MEM.sde
we want to automatically generate the LaTeX code appropriate to
moment equations obtained from the previous model using
TEX.sde()
method.
Itô Sde 2D:
| dX(t) = 1/mu * (theta - X(t)) * dt + sqrt(sigma) * dW1(t)
| dY(t) = X(t) * dt + 0 * dW2(t)
| t in [t0,T].
Moment equations:
| dm1(t) = (theta - m1(t))/mu
| dm2(t) = m1(t)
| dS1(t) = sigma - 2 * (S1(t)/mu)
| dS2(t) = 2 * C12(t)
| dC12(t) = S1(t) - C12(t)/mu
In R we create LaTeX mathematical expressions for this object using the following code:
%%% LaTeX equation generated in R 4.3.3 by TEX.sde() method
%%% Copy and paste the following output in your LaTeX file
\begin{equation}\label{eq:}
\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}
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.
R> f <- expression((alpha*x *(1 - x / beta)- delta * x^2 * y / (kappa + x^2)),
+ (gamma * x^2 * y / (kappa + x^2) - mu * y^2))
R> g <- expression(sqrt(sigma1)*x*(1-y), abs(sigma2)*y*(1-x))
R> TEX.sde(object=c(drift = f, diffusion = g))
%%% LaTeX equation generated in R 4.3.3 by TEX.sde() method
%%% Copy and paste the following output in your LaTeX file
\begin{equation}\label{eq:}
\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}
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
Department of Mathematics and Computer Science, Faculty of Sciences and Technology, University of Tamanghasset, Algeria, E-mail ([email protected])↩︎
Faculty of Mathematics, University of Science and Technology Houari Boumediene, BP 32 El-Alia, U.S.T.H.B, Algeria, E-mail ([email protected])↩︎