FatTailsR (version 2.0.0)

kashp: Kashp Function

Description

kashp, which stands for kappa times arc-sine-hyperbola-power is the nonlinear transformation of x at the heart of power hyperbolas, power hyperbolic functions and symmetric Kiener distributions. dkashp_dx is its derivative with respect to x. ashp is provided for convenience.

Usage

kashp(x, k = 1)
dkashp_dx(x, k = 1)
ashp(x, k = 1)

Arguments

x

a numeric value, vector or matrix.

k

a numeric value or vector, preferably strictly positive.

Details

ashp function is defined for x in (-Inf, +Inf) by: $$ ashp(x, k) = asinh(x/k) $$ kashp function is defined for x in (-Inf, +Inf) by: $$ kashp(x, k) = k * asinh(x/k) $$ dkashp_dx function is defined for x in (-Inf, +Inf) by: $$ \frac{d}{dx}kashp(x, k) = \frac{1}{\sqrt{(x/k)^2 + 1}} = \frac{1}{\cosh(ashp(x, k))} $$

If k is a vector, then the use of the function outer is recommanded.

The undesired case k=0 returns 0 for kashp and dkashp_dx.

Examples

require(graphics)

### FUNCTIONS kashp, dkashp_dx, ashp
xx <- (-3:3)*3
x  <- (-9:9) ; names(x) <- x
k  <- c(9999, 8, 5, 3, 2, 1) ; names(k) <- k
mat1 <- outer(x, k, kashp)    ; mat1
mat2 <- outer(x, k,dkashp_dx) ; mat2
mat3 <- outer(x, k,  ashp)    ; mat3

### GRAPHICS
op <- par(mfcol = c(2,2), mar = c(3,3,2,1))
matplot(x, mat1, type="l", lwd=2, xaxt="n", yaxt="n", main="kashp") 
axis(1, at = xx) ; axis(2, at = xx, las = 1)
legend("topleft", title = expression(kappa), legend = colnames(mat1),
       lty = 1:6, col = 1:6, lwd = 2, inset = 0.02, cex = 0.7)

matplot(x, mat2, type="l", lwd=2, xaxt="n", main="dkashp_dx", las=1, ylim=c(0,1)) 
axis(1, at = xx)
legend("bottom", title = expression(kappa), legend = colnames(mat1),
       lty = 1:6, col = 1:6, lwd = 2, inset = 0.02, cex = 0.7)

matplot(x, mat3, type="l", lwd=2, xaxt="n", main="ashp", las=1) 
axis(1, at = xx)
legend("topleft", title = expression(kappa), legend = colnames(mat1),
       lty = 1:6, col = 1:6, lwd = 2, inset = 0.02, cex = 0.7)
par(op)