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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).

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.

See also

GarchDistribution, GarchDistributionFits

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