File:BrownscheBewegung.png

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This chart was created with R.

R-Quelltext:

set.seed(303898)
H=14                         #Anzahl der Halbierungsschritte
if (exists("B")) {           #nach dem ersten Durchlauf als Variable vorhanden
  mBM=range(B)               #max und min der BM, für ylim des plots;
} else {
  mBM=c(-1, 0.8)
}

B=array(data=0,dim=2)        #Startpunkt der Brownschen Bewegung
B[2]=rnorm(n=1, mean=0, sd=1)
png(filename = "BrownscheBewegung.png", width=1600, height=1200, pointsize=12)
par(mfrow = c(2,3), oma = c(0,0,3,0), bg="whitesmoke")
plot(x=seq(from=0,to=1,length=2), y=B, type="o", col="red", xlab="Zeit", ylab="BM", ylim=mBM, lwd=3)
title(main="i=1, 1 Punkt",cex.main=1.1)
for (i in 2:H) {   #iteriere über jede Halbierung
  old=B
  for (t in 1:2^(i-2)) {  #für jeden alten Zeitpunkt gibt es zwei neue (außer dem letzten)
    B[2*t-1] = old[t]
    B[2*t]   = rnorm(n=1, mean=0.5*(old[t]+old[t+1]), sd=1/(sqrt(2)^i) )
  }
  B[2^(i-1)+1]=old[2^(i-2)+1]     # letzter Zeitpunkt
  if (i<6) {#zeichne nur die ersten 5
    plot(x=seq(from=0,to=1,length=2^(i-1)+1),y=B,type="o",col="red",xlab="Zeit",ylab="BM",ylim=mBM,lwd=3)
    points(x=seq(from=0,to=1,length=2^(i-2)+1),y=old,type="l",col="blue")
    title(main=paste("i=",i,", ",2^(i-1)+1-1," Punkte",sep=""),cex.main=1.1)
  }
  Sys.sleep(0.4)     # um es quasi "live" in R zu sehen: "png()" und "dev.off()" auskommentieren!
}
plot(x=seq(from=0,to=1,length=2^(H-1)+1),y=B,type="l",col="blue",xlab="Zeit",ylab="BM",ylim=mBM,lwd=1)
title(main=paste("i=",H,", ",2^(H-1)+1-1," Punkte",sep=""),cex.main=1.1)
title(main="Brownsche Brücke - Konstruktion einer Brownsche Bewegung", cex.main=2, outer=TRUE)
dev.off()