Heaviside and related functions

Functions which compute the Heaviside and related functions. These include the Heaviside function, the sign function, the delta function, the boxcar function, and the ramp function.

Usage

Heaviside(x, a = 0)
Sign(x, a = 0)
Delta(x, a = 0)
Boxcar(x, a = 0.5)
Ramp(x, a = 0)

Arguments

x: a numeric vector.
a: a numeric value, the location of the break.

Details

Heaviside computes the Heaviside unit step function. Heaviside is 1 for x > a, 1/2 for x = a, and 0 for x < a.

Sign computes the sign function. Sign is 1 for x > a, 0 for x = a, and -1 for x < a.

Delta computes the delta function. Delta is defined as: Delta(x) = d/dx H(x-a).

Boxcar computes the boxcar function. Boxcar is defined as: Boxcar(x) = H(x+a) - H(x-a).

Ramp computes ramp function. The ramp function is defined as: Ramp(x) = (x-a) * H(x-a).

Value

numeric vector

Note

The Heaviside function is used in the implementation of the skew Normal, Student-t, and Generalized Error distributions, distributions functions which play an important role in modelling GARCH processes.

References

Weisstein W. (2004); http://mathworld.wolfram.com/HeavisideStepFunction.html, Mathworld.

Examples

## Heaviside -
   x = sort(round(c(-1, -0.5, 0, 0.5, 1, 5*rnorm(5)), 2))
   h = Heaviside(x)
    
## Sign -
   s = Sign(x)
   
## Delta -
   d = Delta(x)
   
## Boxcar -
   Pi = Boxcar(x)

## Ramp - 
   r = Ramp(x)
   cbind(x = x, Step = h, Signum = s, Delta = d, Pi = Pi, R = r)        
#>           x Step Signum Delta   Pi    R
#>  [1,] -1.00  0.0     -1     0  0.0 0.00
#>  [2,] -0.82  0.0     -1     0  0.0 0.00
#>  [3,] -0.50  0.0     -1     0 -0.5 0.00
#>  [4,]  0.00  0.5      0   Inf -1.0 0.00
#>  [5,]  0.50  1.0      1     0 -0.5 0.50
#>  [6,]  1.00  1.0      1     0  0.0 1.00
#>  [7,]  3.48  1.0      1     0  0.0 3.48
#>  [8,]  4.01  1.0      1     0  0.0 4.01
#>  [9,]  4.30  1.0      1     0  0.0 4.30
#> [10,]  5.40  1.0      1     0  0.0 5.40