Version 4.7

Author: ac-guidoum

inst/doc/bridgesde.R

@@ -8,141 +8,59 @@ options(prompt="R> ",scipen=16,digits=5,warning=FALSE, message=FALSE,
 ## -------------------------------------------------------------------
 f <- expression((1-x)/(1-t))
 g <- expression(x)
-mod <- bridgesde1d(drift=f,diffusion=g,x0=3,y=1,M=1000,method="milstein")
+mod <- bridgesde1d(drift=f,diffusion=g,x0=3,y=1,M=1000)
 mod
-summary(mod) ## default: summary at time = (T-t0)/2
 
 ## ----01,fig.env='figure*', fig.cap=' ',eval=FALSE, include=TRUE-----
 #  plot(mod,ylab=expression(X[t]))
 #  lines(time(mod),apply(mod$X,1,mean),col=2,lwd=2)
 #  legend("topleft","mean path",inset = .01,col=2,lwd=2,cex=0.8,bty="n")
 
-## ----33, echo=FALSE, fig.cap=' Bridge sde 1D ', fig.env='figure*',fig.width=7,fig.height=7----
-knitr::include_graphics("Figures/fig03.png")
-
-## ----02, fig.cap='  ', fig.env='figure*'----------------------------
-m  <- apply(mod$X,1,mean) 
-S  <- apply(mod$X,1,var)
-out <- data.frame(m,S)
-matplot(time(mod), out, type = "l", xlab = "time", ylab = "", col=2:3,lwd=2,lty=2:3,las=1)
-legend("topright",c(expression(m(t),S(t))),col=2:3,lty=2:3,lwd=2,bty="n")
-
 ## -------------------------------------------------------------------
-s = 0.55
-mean(mod, at = s)
-moment(mod, at = s , center = TRUE , order = 2) ## variance
-Median(mod, at = s)
-Mode(mod, at = s)
-quantile(mod , at = s)
-kurtosis(mod , at = s)
-skewness(mod , at = s)
-cv(mod , at = s )
-min(mod , at = s)
-max(mod , at = s)
-moment(mod, at = s , center= TRUE , order = 4)
-moment(mod, at = s , center= FALSE , order = 4)
+x <- rsde1d(object = mod, at = 0.55) 
+head(x, n = 3)
 
-## -------------------------------------------------------------------
-summary(mod, at = 0.55)
+## ----04,fig.env='figure*', fig.cap='  ',eval=FALSE, include=TRUE----
+#  dens <- dsde1d(mod, at = 0.55)
+#  plot(dens,hist=TRUE) ## histgramme
+#  plot(dens,add=TRUE)  ## kernel density
 
-## -------------------------------------------------------------------
-x <- rsde1d(object = mod, at = s) 
-head(x, n = 10)
-summary(x)
-
-## ----03 ,fig.env='figure*', fig.cap='  '----------------------------
-plot(time(mod),mod$X[,1],type="l",ylab="X(t)",xlab="time",axes=F,lty=3)
-points(s,x[1],pch=19,col=2,cex=0.5)
-lines(c(s,s),c(0,x[1]),lty=2,col=2)
-lines(c(0,s),c(x[1],x[1]),lty=2,col=2)
-axis(1, s, bquote(at==.(s)),col=2,col.ticks=2)
-axis(2, x[1], bquote(X[t==.(s)]),col=2,col.ticks=2)
-legend('topright',col=2,pch=19,legend=bquote(X[t==.(s)]==.(x[1])),bty = 'n')
-box()
-
-## ----04,fig.env='figure*', fig.cap='  '-----------------------------
-dens <- dsde1d(mod, at = s)
-dens
-plot(dens,hist=TRUE) ## histgramme
-plot(dens,add=TRUE)  ## kernel density
-
-## ----05,fig.env='figure*', fig.cap=' Transitional densitie at time $t-s = 0.25,0.75$ '----
-plot(dsde1d(mod,at=0.75))
-plot(dsde1d(mod,at=0.25),add=TRUE)
-legend('topright',col=c('#0000FF4B','#FF00004B'),pch=15,legend=c("t-s=0.25","t-s=0.75"),bty = 'n')
+## ----33, echo=FALSE, fig.cap='Bridge sde 1D. Histgramme and kernel density estimation for $X_{t}|X_{0}=3,X_{T}=1$', fig.env='figure*',fig.width=7,fig.height=7----
+knitr::include_graphics(c("Figures/fig03.png","Figures/fig1008.png"))
 
 ## -------------------------------------------------------------------
 fx <- expression(-(1+y)*x , -(1+x)*y)
 gx <- expression(0.2*(1-y),0.1*(1-x))
-mod2 <- bridgesde2d(drift=fx,diffusion=gx,x0=c(1,-0.5),y=c(1,0.5),Dt=0.01,M=1000,type="str",method="rk1")
+Sigma <-matrix(c(1,0.3,0.3,1),nrow=2,ncol=2)
+mod2 <- bridgesde2d(drift=fx,diffusion=gx,x0=c(1,-0.5),y=c(1,0.5),Dt=0.01,M=1000,type="str",corr=Sigma)
 mod2
-summary(mod2) ## default: summary at time = (T-t0)/2
 
 ## ----06,fig.env='figure*', fig.cap=' ',eval=FALSE, include=TRUE-----
 #  plot(mod2,col=c('#FF00004B','#0000FF82'))
 
-## ----333, echo=FALSE, fig.cap='  Bridge sde 2D ', fig.env='figure*',fig.width=7,fig.height=7----
+## ----333, echo=FALSE, fig.cap='  Bridge sde 2D ', fig.env='figure*',fig.width=5,fig.height=5----
 knitr::include_graphics("Figures/fig04.png")
 
-## ----07, fig.cap='  ', fig.env='figure*'----------------------------
-m1  <- apply(mod2$X,1,mean) 
-m2  <- apply(mod2$Y,1,mean) 
-S1  <- apply(mod2$X,1,var)
-S2  <- apply(mod2$Y,1,var)
-C12 <- sapply(1:dim(mod2$X)[1],function(i) cov(mod2$X[i,],mod2$Y[i,]))
-out2 <- data.frame(m1,m2,S1,S2,C12)
-matplot(time(mod2), out2, type = "l", xlab = "time", ylab = "", col=2:6,lwd=2,lty=2:6,las=1)
-legend("top",c(expression(m[1](t),m[2](t),S[1](t),S[2](t),C[12](t))),col=2:6,lty=2:6,lwd=2,bty="n")
-
 ## -------------------------------------------------------------------
-s = 6.75
-mean(mod2, at = s)
-moment(mod2, at = s , center = TRUE , order = 2) ## variance
-Median(mod2, at = s)
-Mode(mod2, at = s)
-quantile(mod2 , at = s)
-kurtosis(mod2 , at = s)
-skewness(mod2 , at = s)
-cv(mod2 , at = s )
-min(mod2 , at = s)
-max(mod2 , at = s)
-moment(mod2 , at = s , center= TRUE , order = 4)
-moment(mod2 , at = s , center= FALSE , order = 4)
+x2 <- rsde2d(object = mod2, at = 5) 
+head(x2, n = 3)
 
-## -------------------------------------------------------------------
-summary(mod2, at = 6.75)
+## ----09,fig.env='figure*', fig.cap='  ',eval=FALSE, include=TRUE----
+#  ## Marginal
+#  denM <- dsde2d(mod2,pdf="M",at = 5)
+#  plot(denM, main="Marginal Density")
+#  ## Joint
+#  denJ <- dsde2d(mod2, pdf="J", n=100,at = 5)
+#  plot(denJ,display="contour",main="Bivariate Transition Density at time t=5")
 
-## -------------------------------------------------------------------
-x2 <- rsde2d(object = mod2, at = s) 
-head(x2, n = 10)
-summary(x2)
-
-## ----08,fig.env='figure*', fig.cap=' '------------------------------
-plot(ts.union(mod2$X[,1],mod2$Y[,1]),col=1:2,lty=3,plot.type="single",type="l",ylab= "",xlab="time",axes=F)
-points(s,x2$x[1],pch=19,col=3,cex=0.8)
-points(s,x2$y[1],pch=19,col=4,cex=0.8)
-lines(c(s,s),c(-10,x2$x[1]),lty=2,col=6)
-lines(c(0,s),c(x2$x[1],x2$x[1]),lty=2,col=3)
-lines(c(0,s),c(x2$y[1],x2$y[1]),lty=2,col=4)
-axis(1, s, bquote(at==.(s)),col=6,col.ticks=6)
-axis(2, x2$x[1], bquote(X[t==.(s)]),col=3,col.ticks=3)
-axis(2, x2$y[1], bquote(Y[t==.(s)]),col=4,col.ticks=4)
-legend('topright',legend=bquote(c(X[t==.(s)]==.(x2$x[1]),Y[t==.(s)]==.(x2$y[1]))),bty = 'n')
-box()
-
-## ----09,fig.env='figure*', fig.cap='  '-----------------------------
-denM <- dsde2d(mod2,pdf="M",at =s)
-denM
-plot(denM, main="Marginal Density")
-
-## ----10,fig.env='figure*', fig.cap='  '-----------------------------
-denJ <- dsde2d(mod2, pdf="J", n=100,at =s)
-denJ
-plot(denJ,display="contour",main="Bivariate Transition Density at time t=6.755")
-plot(denJ,display="image",main="Bivariate Transition Density at time t=6.755")
-
-## ----11,fig.env='figure*', fig.cap='  '-----------------------------
-plot(denJ,main="Bivariate Transition Density at time t=6.75")
+## ----103, echo=FALSE, fig.cap='The marginal and joint density of $X_{t}|X_{0}=1,X_{T}=1$ and $Y_{t}|Y_{0}=-0.5,Y_{T}=0.5$ at time $t=5$', fig.env='figure*',fig.width=7,fig.height=7----
+knitr::include_graphics(c("Figures/fig1009.png","Figures/fig1010.png"))
+
+## ----11,fig.env='figure*', fig.cap='  ',eval=FALSE, include=TRUE----
+#  plot(denJ,main="Bivariate Transition Density at time t=5")
+
+## ----33303, echo=FALSE, fig.cap='$3$D plot of the transition density of $X_{t}|X_{0}=1,X_{T}=1$ and $Y_{t}|Y_{0}=-0.5,Y_{T}=0.5$ at time $t=5$  ', fig.env='figure*',fig.width=5,fig.height=5----
+knitr::include_graphics("Figures/fig1011.png")
 
 ## ----eval=FALSE, include=TRUE---------------------------------------
 #  for (i in seq(1,9,by=0.005)){
@@ -154,74 +72,23 @@ fx <- expression(-4*(1+x)*y, 4*(1-y)*x, 4*(1-z)*y)
 gx <- rep(expression(0.2),3)
 mod3 <- bridgesde3d(x0=c(0,-1,0.5),y=c(0,-2,0.5),drift=fx,diffusion=gx,M=1000)
 mod3
-summary(mod3) ## default: summary at time = (T-t0)/2
 
 ## ----12,fig.env='figure*', fig.cap=' ',eval=FALSE, include=TRUE-----
 #  plot(mod3) ## in time
 #  plot3D(mod3,display = "persp",main="3D Bridge SDE's") ## in space
 
-## ----3333, echo=FALSE, fig.cap=' ', fig.env='figure*',fig.width=7,fig.height=7----
-knitr::include_graphics("Figures/fig05.png")
-
-## ----33333, echo=FALSE, fig.cap='  Bridge sde 3D ', fig.env='figure*',fig.width=7,fig.height=7----
-knitr::include_graphics("Figures/fig06.png")
-
-## ----13, fig.cap='  ', fig.env='figure*'----------------------------
-m1  <- apply(mod3$X,1,mean) 
-m2  <- apply(mod3$Y,1,mean) 
-m3  <- apply(mod3$Z,1,mean) 
-S1  <- apply(mod3$X,1,var)
-S2  <- apply(mod3$Y,1,var)
-S3  <- apply(mod3$Z,1,var)
-out3 <- data.frame(m1,m2,m3,S1,S2,S3)
-matplot(time(mod3), out3, type = "l", xlab = "time", ylab = "", col=2:7,lwd=2,lty=2:7,las=1)
-legend("bottom",c(expression(m[1](t),m[2](t),m[3](t),S[1](t),S[2](t),S[3](t))),col=2:7,lty=2:7,lwd=2,bty="n")
-
-## -------------------------------------------------------------------
-s = 0.75
-mean(mod3, at = s)
-moment(mod3, at = s , center = TRUE , order = 2) ## variance
-Median(mod3, at = s)
-Mode(mod3, at = s)
-quantile(mod3 , at = s)
-kurtosis(mod3 , at = s)
-skewness(mod3 , at = s)
-cv(mod3 , at = s )
-min(mod3 , at = s)
-max(mod3 , at = s)
-moment(mod3 , at = s , center= TRUE , order = 4)
-moment(mod3 , at = s , center= FALSE , order = 4)
-
-## -------------------------------------------------------------------
-summary(mod3, at = 0.75)
+## ----3333, echo=FALSE, fig.cap=' Bridge sde 3D ', fig.env='figure*',fig.width=7,fig.height=7----
+knitr::include_graphics(c("Figures/fig05.png","Figures/fig06.png"))
 
 ## -------------------------------------------------------------------
-x3 <- rsde3d(object = mod3, at = s) 
-head(x3, n = 10)
-summary(x3)
-
-## ----14,fig.env='figure*', fig.cap=' '------------------------------
-plot(ts.union(mod3$X[,1],mod3$Y[,1],mod3$Z[,1]),col=1:3,lty=3,plot.type="single",type="l",ylab= "",xlab="time",axes=F)
-points(s,x3$x[1],pch=19,col=4,cex=0.8)
-points(s,x3$y[1],pch=19,col=5,cex=0.8)
-points(s,x3$z[1],pch=19,col=6,cex=0.8)
-lines(c(s,s),c(-10,x3$x[1]),lty=2,col=7)
-lines(c(0,s),c(x3$x[1],x3$x[1]),lty=2,col=4)
-lines(c(0,s),c(x3$y[1],x3$y[1]),lty=2,col=5)
-lines(c(0,s),c(x3$z[1],x3$z[1]),lty=2,col=6)
-axis(1, s, bquote(at==.(s)),col=7,col.ticks=7)
-axis(2, x3$x[1], bquote(X[t==.(s)]),col=4,col.ticks=4)
-axis(2, x3$y[1], bquote(Y[t==.(s)]),col=5,col.ticks=5)
-axis(2, x3$z[1], bquote(Z[t==.(s)]),col=6,col.ticks=6)
-legend("bottomleft",legend=bquote(c(X[t==.(s)]==.(x3$x[1]),Y[t==.(s)]==.(x3$y[1]),Z[t==.(s)]==.(x3$z[1]))),bty = 'n',cex=0.75)
-box()
-
-## ----15,fig.env='figure*', fig.cap='  '-----------------------------
-denM <- dsde3d(mod3,pdf="M",at =s)
-denM
-plot(denM, main="Marginal Density")
-
-## ----111,fig.env='figure*', fig.cap='  ',eval=FALSE, include=TRUE----
+x3 <- rsde3d(object = mod3, at = 0.75) 
+head(x3, n = 3)
+
+## ----15,fig.env='figure*', fig.cap='  ',eval=FALSE, include=TRUE----
+#  ## Marginal
+#  denM <- dsde3d(mod3,pdf="M",at =0.75)
+#  plot(denM, main="Marginal Density")
+#  ## Joint
 #  denJ <- dsde3d(mod3,pdf="J",at=0.75)
 #  plot(denJ,display="rgl")
 

inst/doc/fitsde.R

@@ -114,7 +114,7 @@ Parms  <- data.frame(Theta1,Theta2,Theta3,Theta4,row.names = c("True mod",
                       "Comp mod1","Comp mod2","Comp mod3"))
 Parms
 
-## ----01,fig.env='figure*', fig.cap='  '-----------------------------
+## ----01,fig.env='figure*', fig.cap=' The U.S. Interest Rates monthly form $06/1964$ to $12/1989$ ',fig.width=6,fig.height=4----
 data(Irates)
 rates <- Irates[, "r1"]
 X <- window(rates, start = 1964.471, end = 1989.333)
@@ -136,7 +136,7 @@ names(Info) <- c("logLik","AIC","BIC")
 Coef
 Info
 
-## ----02,fig.env='figure*', fig.cap='  '-----------------------------
+## ----02,fig.env='figure*', fig.cap='The path mean of the solution of the CKLS model with the estimated parameters and real data ',fig.width=6,fig.height=4----
 f <- expression( (2.076-0.263*x) )
 g <- expression( 0.130*x^1.451 )
 mod <- snssde1d(drift=f,diffusion=g,x0=X[1],M=500, N=length(X),t0=1964.471, T=1989.333)