# Examples of Nonstandard Copulas – “Wild Animals”

#### Marius Hofert and Martin Mächler

#### 2015-10-26

## 1) Example from Marius Hofert and Frederik Vrins

### Lambda and its inverse

    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 \(M_i, i=1,2\)):

    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 \(S_i, i=1,2\))

    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 \(p_i(t_k), i = 1,2\)):

    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:

    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 \(C\)

    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) \(u_2 = S2(S1^{-1}(u_1))\) :

    s.comp <- function(u1, H, S1Inv, S2){
        if(u1 == 0) 0 else S2 (S1Inv(u1,H), H)
    }

Generate one bivariate random vector from `C`

    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 \(C\)

    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 :

    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

    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

    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

\(X :=\) Rotate \(A\) by \(-45^{o} = -\pi/4\) :

    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")`:

    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:

    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 …):

    par(pty="s")
    plot(U, pch=".", xlab = quote(italic(U[1]) %~% ~ "U[0,1]"),
         asp = 1,    ylab = quote(italic(U[2]) %~% ~ "U[0,1]"))