## WHITE NOISE x<-rnorm(1000,0,1) ts.plot(x) --------------------------- ## LINEAR EQUATION WITH NOISE (STABLE CASE) a<- -1/2 b<- 0 sigma <- 1 x <- 1 N<-100 W<-rnorm(N,0,1) X<-1:N X[1]<-x for (n in 1:(N-1)) { X[n+1]<-a*X[n]+b+sigma*W[n] } ts.plot(X) ------------- ## LINEAR EQUATION WITH NOISE (UNSTABLE CASE) a<- 3/2 b<- 0 sigma <- 1 x <- 1 N<-10 W<-rnorm(N,0,1) X<-1:N X[1]<-x for (n in 1:(N-1)) { X[n+1]<-a*X[n]+b+sigma*W[n] } ts.plot(X) ----------------- ## RANDOM WALK (DIM=1) a<- 1 b<- 0 sigma <- 1 x <- 0 N<-1000 W<-rnorm(N,0,1) X<-1:N X[1]<-x for (n in 1:(N-1)) { X[n+1]<-a*X[n]+b+sigma*W[n] } ts.plot(X) ------------------------ ## RANDOM WALK (DIM=2) N<-10000 W1<-rnorm(N,0,1) W2<-rnorm(N,0,1) X1<-1:N X2<-1:N X1[1]<-0 X2[1]<-0 X1<-cumsum(W1) X2<-cumsum(W2) plot(X1,X2, type="l") ----------------------- ## MOTO BROWNIANO, UNA TRAIETTORIA L<-5000 W<-rnorm(L,0,sqrt(1/1000)) X<-1:L X[1]<-0 X<-cumsum(W) ts.plot(X) ------------------------ ## MOTO BROWNIANO, LA VARIABILE AL TEMPO T=5 N<- 1000 B<- 1:N L<-5000 X<-1:L X[1]<-0 s<-sqrt(1/1000) for (n in 1:N) { W<-rnorm(L,0,s) X<-cumsum(W) B[n]<-X[L] } hist(B,50) mean(B) sd(B) -------------------------- ## PROCESSO DI POISSON T <- rweibull(10,1,2) S <- cumsum(T) S plot(ecdf(S)) --------------------------- ## PROCESSO DI POISSON NEL PIANO L <- 10 N <- 100 X <- runif(N,-L,L) Y <- runif(N,-L,L) plot(X,Y) L <- 100 N <- 10000 X <- runif(N,-L,L) Y <- runif(N,-L,L) plot(X,Y) ----------------------------- ## ESEMPIO DI EDS: CASO LINEARE N<-2000 x0<-1 h<-0.01 s<-0.1 W<-rnorm(N,0,sqrt(h)) X.det<-1:N X<-1:N X.det[1]<-x0 X[1]<-x0 for (n in 1:(N-1)) { X.det[n+1] <- X.det[n] - h*X.det[n] X[n+1] <- X[n] - h*X[n] + s*W[n] } plot(c(0,N*h),c(-1,1)) lines(c(0,N*h),c(0,0), type="l") T<-(1:N)*h lines(T,X.det) lines(T,X,col="red") ----------------------------- ## ESEMPIO DI EDS: DUE BUCHE N<-2000 h<-0.01 s<-0.5 W<-rnorm(N,0,sqrt(h)) X<-1:N X.det.1<-1:N X.det.2<-1:N X.det.3<-1:N X.det.4<-1:N X.det.1[1]<-4 X.det.2[1]<-0.1 X.det.3[1]<- -0.1 X.det.4[1]<- -4 X[1]<-2 for (n in 1:(N-1)) { X.det.1[n+1] <- X.det.1[n] + h*X.det.1[n] - h*X.det.1[n]^3 X.det.2[n+1] <- X.det.2[n] + h*X.det.2[n] - h*X.det.2[n]^3 X.det.3[n+1] <- X.det.3[n] + h*X.det.3[n] - h*X.det.3[n]^3 X.det.4[n+1] <- X.det.4[n] + h*X.det.4[n] - h*X.det.4[n]^3 X[n+1] <- X[n] + h*X[n] - h*X[n]^3 + s*W[n] } plot(c(0,N*h),c(-2,2)) lines(c(0,N*h),c(1,1), type="l") lines(c(0,N*h),c(-1,-1), type="l") lines(c(0,N*h),c(0,0), type="l") T<-(1:N)*h lines(T,X.det.1) lines(T,X.det.2) lines(T,X.det.3) lines(T,X.det.4) lines(T,X,col="red") ------------------------------------- ------------------------------------- ## FILMATO DI EDS N<-2000 h<-0.01 s<-0.5 W<-rnorm(N,0,sqrt(h)) X<-1:N X[1]<-2 for (n in 1:(N-1)) { X[n+1] <- X[n] + h*X[n] - h*X[n]^3 + s*W[n] plot(c(0,N*h),c(-2,2)) lines(c(0,N*h),c(1,1), type="l") lines(c(0,N*h),c(-1,-1), type="l") lines(c(0,N*h),c(0,0), type="l") T<-(1:N)*h lines(T[1:n],X[1:n],col="red") }