---
title: Examples of Nonstandard Copulas -- "Wild Animals"
author: Marius Hofert and Martin M&auml;chler
date: '`r Sys.Date()`'
vignette: >
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteIndexEntry{Wild Animals: Examples of Nonstandard Copulas}
---
## 1) Example from Marius Hofert and Frederik Vrins

### Lambda and its inverse
```{r}
Lambda <- function(t) pmax(log(t), 0)
LambdaInv <- function(t) (t != 0)*exp(t) ## <=>  ifelse(t == 0, 0, exp(t))
```

### M and its inverse (for $M_i, i=1,2$):
<!--- MM FIXME: M2, M2Inv(): use `Vectorize()` - more elegant and faster! -->
```{r}
M1 <- function(t) 0.01*t
M1Inv <- function(t) t/0.01

M2 <- function(t) {
    f <- function(t.) {
      	ct <- ceiling(t.)
	if(ct %% 2) t. - (ct-1)/2 else ct/2
    }
    unlist(lapply(t, f))
}
M2Inv <- function(t) {
    f <- function(t.) {
        if(t. == 0) return(0)
        t. + ceiling(t) - 1
    }
    unlist(lapply(t, f))
}
```

### S and its inverse (for $S_i, i=1,2$)
```{r}
S1 <- function(t, H) exp(-(M1(t) + Lambda(t)*(1-exp(-H))))
S1Inv <- function(u, H, upper=1e6) unlist(lapply(u, function(u.)
    uniroot(function(x) S1(x,H=H)-u., interval=c(0, upper))$root))
S2 <- function(t, H) exp(-(M2(t) + Lambda(t)*(1-exp(-H))))
S2Inv <- function(u, H, upper=1e6) unlist(lapply(u, function(u.)
    uniroot(function(x) S2(x,H=H)-u., interval=c(0, upper))$root))
```

### Wrappers for $p_1$ and $p_2$ and their inverses:
p and its inverse (for $p_i(t_k-)$ and $p_i(t_k), i = 1,2$):
```{r, p1-p2}
p1 <- function(t, k, H) exp(-M1(t)-H*k)
p1Inv <- function(u, k, H, upper=1e6) unlist(lapply(u, function(u.)
    uniroot(function(x) p1(x,k=k,H=H)-u., interval=c(0, upper))$root))
p2 <- function(t, k, H) exp(-M2(t)-H*k)
p2Inv <- function(u, k, H, upper=1e6) unlist(lapply(u, function(u.)
    uniroot(function(x) p2(x,k=k,H=H)-u., interval=c(0, upper))$root))
```

and the wrappers, which work with `p1(), p2(), p1Inv(), p2Inv()` as arguments:
```{r, p-pinv}
p <- function(t, k, H, I, p1, p2){
    if((lI <- length(I)) == 0){
        stop("error in p")
    }else if(lI==1){
        if(I==1) p1(t, k=k, H=H) else p2(t, k=k, H=H)
    }else{ # lI == 2
        c(p1(t, k=k, H=H), p2(t, k=k, H=H))
    }
}
pInv <- function(u, k, H, I, p1Inv, p2Inv){
    if((lI <- length(I)) == 0){
        stop("error in pInv")
    }else if(lI==1){
        if(I==1) p1Inv(u, k=k, H=H) else p2Inv(u, k=k, H=H)
    }else{ # lI == 2
        c(p1Inv(u[1], k=k, H=H), p2Inv(u[2], k=k, H=H))
    }
}
```

### Define the copula $C$
```{r}
C <- function(u, H, Lambda, S1Inv, S2Inv) {
    if(all(u == 0)) 0 else
    u[1]*u[2] * exp(expm1(-H)^2 * Lambda(min(S1Inv(u[1],H), S2Inv(u[2],H))))
}
```
Compute the singular component (given $u_1=$ `u1`, find $u_2=$ `u2` on the singular component)
$u_2 = S2(S1^{-1}(u_1))$ :
```{r}
s.comp <- function(u1, H, S1Inv, S2){
    if(u1 == 0) 0 else S2 (S1Inv(u1,H), H)
}
```

Generate one bivariate random vector from `C`
<!-- MM: FIXME make faster: no need to grow  't_h', 'I_' all the time -->
```{r, rC1}
rC1 <- function(H, LambdaInv, S1, S2, p1, p2, p1Inv, p2Inv) {
    d <- 2 # dim = 2
    ## (1)
    U <- runif(d) # for determining the default times of all components
    ## (2) -- t_{h,0} := initial value for the occurrence of the homogeneous
    ##                   Poisson process with unit intensity
    t_h <- 0
    t <- 0 # t_0; initial value for the occurrence of the jump process
    k <- 1 # indices for the sets I
    I_ <- list(1:d) # I_k; indices for which tau has to be determined
    tau <- rep(Inf,d) # in the beginning, set all default times to Inf (= "no default")
    ## (3)
    repeat{
        ## (4)
        ## k-th occurrence of a homogeneous Poisson process with unit intensity:
        t_h[k] <- rexp(1) + if(k == 1) 0 else t_h[k-1]
        ## k-th occ. of non-homogeneous Poisson proc. with integrated rate function Lambda:
        t[k] <- LambdaInv(t_h[k])
        ## (5)
        pvec <- p(t[k], k=k, H=H, I=c(1,2), p1=p1, p2=p2) # c(p_1(t_k), p_2(t_k))
        I. <- (1:d)[U >= pvec] # determine all i in I
        I <- intersect(I., I_[[k]]) # determine I
        ## (6)--(10)
        if(length(I) > 0){
            default.at.t <- U[I] <= p(t[k], k=k-1, H=H, I=I, p1=p1, p2=p2)
            tau[I[default.at.t]] <- t[k]
            Ic <- I[!default.at.t] # I complement
            if(length(Ic) > 0) tau[Ic] <- pInv(U[Ic], k=k-1, H=H, I=Ic,
                                               p1Inv=p1Inv, p2Inv=p2Inv)
        }
        ## (11) --  define I_{k+1} := I_k \ I
        I_[[k+1]] <- setdiff(I_[[k]], I)
        ## (12)
        if(length(I_[[k+1]]) == 0) break else k <- k+1
    }
    ## (14)
    c(S1(tau[1], H=H), S2(tau[2], H=H))
}
```

### Draw n vectors of random variates from $C$
```{r}
rC <- function(n, H, LambdaInv, S1, S2, p1, p2, p1Inv, p2Inv){
    mat <- t(sapply(rep(H, n), rC1, LambdaInv=LambdaInv, S1=S1, S2=S2,
                    p1=p1, p2=p2, p1Inv=p1Inv, p2Inv=p2Inv))
    row.names(mat) <- NULL
    mat
}

## Generate copula data
n <- 2e4
H <- 10
set.seed(271)
U <- rC(n, H=H, LambdaInv=LambdaInv, S1=S1, S2=S2, p1=p1, p2=p2, p1Inv=p1Inv, p2Inv=p2Inv)

## Check margins of U
par(pty="s")
hist(U[,1], probability=TRUE, main="Histogram of the first component",
     xlab=expression(italic(U[1])))
hist(U[,2], probability=TRUE, main="Histogram of the second component",
     xlab=expression(italic(U[2])))

## Plot U (copula sample)
plot(U, cex=0.2, xlab=expression(italic(U[1])%~%~"U[0,1]"),
                 ylab=expression(italic(U[2])%~%~"U[0,1]"))
```

Wireframe plot to incorporate singular component :
```{r}
require(lattice)
wf.plot <- function(grid, val.grid, s.comp, val.s.comp, Lambda, S1Inv, S2Inv){
    wireframe(val.grid ~ grid[,1]*grid[,2], xlim=c(0,1), ylim=c(0,1), zlim=c(0,1),
              aspect=1, scales = list(arrows=FALSE, col=1), # remove arrows
              par.settings= list(axis.line = list(col="transparent"), # remove global box
                                 clip = list(panel="off")),
              pts = cbind(s.comp, val.s.comp), # <- add singular component
              panel.3d.wireframe = function(x, y, z, xlim, ylim, zlim, xlim.scaled,
                                            ylim.scaled, zlim.scaled, pts, ...) {
                  panel.3dwire(x=x, y=y, z=z, xlim=xlim, ylim=ylim, zlim=zlim,
                               xlim.scaled=xlim.scaled, ylim.scaled=ylim.scaled,
                               zlim.scaled=zlim.scaled, ...)
                  xx <- xlim.scaled[1]+diff(xlim.scaled)*(pts[,1]-xlim[1])/diff(xlim)
                  yy <- ylim.scaled[1]+diff(ylim.scaled)*(pts[,2]-ylim[1])/diff(ylim)
                  zz <- zlim.scaled[1]+diff(zlim.scaled)*(pts[,3]-zlim[1])/diff(zlim)
                  panel.3dscatter(x=xx, y=yy, z=zz, xlim=xlim, ylim=ylim, zlim=zlim,
                                  xlim.scaled=xlim.scaled, ylim.scaled=ylim.scaled,
                                  zlim.scaled=zlim.scaled, type="l", col=1, ...)
              },
              xlab = expression(italic(u[1])),
              ylab = expression(italic(u[2])),
              zlab = list(expression(italic(C(u[1],u[2]))), rot=90))
}

## Copula plot with singular component
u <- seq(0, 1, length.out=20) # grid points per dimension
grid <- expand.grid(u1=u, u2=u) # grid
val.grid <- apply(grid, 1, C, H=H, Lambda=Lambda, S1Inv=S1Inv, S2Inv=S2Inv) # copula values on grid
s.comp <- cbind(u, sapply(u, s.comp, H=H, S1Inv=S1Inv, S2=S2)) # pairs (u1, u2) on singular component
val.s.comp <- apply(s.comp, 1, C, H=H, Lambda=Lambda, S1Inv=S1Inv, S2Inv=S2Inv) # corresponding z-values
wf.plot(grid=grid, val.grid=val.grid, s.comp=s.comp, val.s.comp=val.s.comp,
        Lambda=Lambda, S1Inv=S1Inv, S2Inv=S2Inv)
```


## 2) A nice example from Wolfgang Trutschnig and Manuela Schreyer ############

For more details, see
Trutschnig, Fernandez Sanchez (2014)
*Copulas with continuous, strictly increasing singular conditional distribution functions*

Roughly, one defines an **Iterated Function System** whose attractor is the word
"Copula" and starts the chaos game.

### Define the Iterated Function System
```{r, ifs-def}
IFS <- local({ ## Using `local`, so `n` is part of IFS
    n <- 23
    list(function(x) c(3*x[1]/n,      x[2]/4),
         function(x) c(-(x[2]-1)/n,   x[1]/2+1/4),
         function(x) c(3*x[1]/n,      x[2]/4+3/4),
         function(x) c((3*x[1]+4)/n,  x[2]/4),
         function(x) c(-(x[2]-5)/n,   x[1]/2+1/4),
         function(x) c((3*x[1]+4)/n,  x[2]/4+3/4),
         function(x) c(-(x[2]-7)/n,   x[1]/2+1/4),
         function(x) c(-x[2]/n+9/n,   3*x[1]/4),
         function(x) c((3*x[1]+8)/n,  x[2]/4+3/4),
         function(x) c(x[1]/n+10/n,   x[2]/8+1/2+1/8),
         function(x) c(2*x[1]/n+9/n,  x[2]/4+1/4+1/8),
         function(x) c(-x[2]/n+13/n,  (3*x[1]+1)/4),
         function(x) c((3*x[1]+12)/n, x[2]/4),
         function(x) c((3*x[1]+12)/n, x[2]/4),
         function(x) c(-x[2]/n+15/n,  (3*x[1]+1)/4),
         function(x) c((3*x[1]+16)/n, x[2]/4),
         function(x) c(-(x[2]-21)/n,  3*x[1]/4),
         function(x) c((3*x[1]+20)/n, x[2]/4+3/4),
         function(x) c((x[1]+21)/n,   x[2]/4+1/4+1/8),
         function(x) c(-(x[2]-23)/n,  3*x[1]/4))
})
```

#### Run chaos game B times
```{r, run.chaos}
B <- 20 # replications
n.steps <- 20000 # number of steps
AA <- vector("list", length=B)
set.seed(271)
for(i in 1:B) {
    ind <- sample(length(IFS), size=n.steps, replace=TRUE) # (randomly) 'bootstrap' functions
    res <- matrix(0, nrow=n.steps+1, ncol=2) # result matrix (for each i)
    pt <- c(0, 0) # initial point
    for(r in seq_len(n.steps)) {
        res[r+1,] <- IFS[[ind[r]]](pt) # evaluate randomly chosen functions at pt
        pt <- res[r+1,] # redefine point
    }
    AA[[i]] <- res # keep this matrix
}

A <- do.call(rbind, AA) # rbind (n.steps+1, 2)-matrices
n <- nrow(A)
stopifnot(ncol(A) == 2, n == B*(n.steps+1)) # sanity check
```

<!-- ^{o}: \degrees in some LaTeX package ? -->
$X :=$  Rotate $A$ by $-45^{o} = -\pi/4$ :
```{r, rotate}
phi <- -pi/4
X <- cbind(cos(phi)*A[,1] - sin(phi)*A[,2]/3,
           sin(phi)*A[,1] + cos(phi)*A[,2]/3)
stopifnot(identical(dim(X), dim(A)))
```

Now transform the margings by their marginal ECDF's so we get **uniform** margins.
Note that, it is equivalent but faster to use `rank(*, ties.method="max")`:
```{r, transform}
U <- apply(X, 2, function(x) ecdf(x)(x))
## Prove equivalence:
stopifnot(all.equal(U,
                    apply(X, 2, rank, ties.method="max") / n,
                    tolerance = 1e-14))
```

Now, visually check the margins of `U`; they are *perfectly* uniform:
```{r, plot.margins}
par(pty="s")
sfsmisc::mult.fig(mfcol = c(1,2), main = "Margins are uniform")
hist(U[,1], probability=TRUE, main="Histogram of U[,1]", xlab=quote(italic(U[1])))
hist(U[,2], probability=TRUE, main="Histogram of U[,2]", xlab=quote(italic(U[2])))
```

whereas `U`, the copula sample, indeed is peculiar and contains the word
"COPULA" many times if you look closely (well, the "L" is defect ...):
```{r, plot.U}
par(pty="s")
plot(U, pch=".", xlab = quote(italic(U[1]) %~% ~ "U[0,1]"),
     asp = 1,    ylab = quote(italic(U[2]) %~% ~ "U[0,1]"))
```
