Ramp function

Applications

The ramp function has numerous applications in engineering, such as in the theory of digital signal processing.

In finance, the payoff of a call option is a ramp (shifted by strike price). Horizontally flipping a ramp yields a put option, while vertically flipping (taking the negative) corresponds to selling or being "short" an option. In finance, the shape is widely called a "hockey stick", due to the shape being similar to an ice hockey stick.

In statistics, hinge functions of multivariate adaptive regression splines (MARS) are ramps, and are used to build regression models.

Analytic properties

Non-negativity

In the whole domain the function is non-negative, so its absolute value is itself, i.e. $${\displaystyle \mathrm{\forall }x\in \mathbb{R}:R(x)\ge 0}$$ and $${\displaystyle \left|R(x)\right|=R(x)}$$

Proof

by the mean of definition 2, it is non-negative in the first quarter, and zero in the second; so everywhere it is non-negative.

Derivative

Its derivative is the Heaviside step function: $${\displaystyle R^{\prime }(x)=H(x)\text{for }x\ne 0.}$$

Second derivative

The ramp function satisfies the differential equation: $${\displaystyle \frac{d^2}{d{x}^2}R(x-x_0)=\delta (x-x_0),}$$ where δ(x) is the Dirac delta. This means that R(x) is a Green's function for the second derivative operator. Thus, any function, f(x), with an integrable second derivative, f″(x), will satisfy the equation: $${\displaystyle f(x)=f(a)+(x-a)f^{\prime }(a)+{\int }_a^{b}R(x-s)f^{″}(s)ds\text{for }a<x<b.}$$

Fourier transform

$${\displaystyle \mathcal{F}\{R(x)\}(f)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}R(x)e^{-2\pi ifx}dx=\frac{i{\delta }^{\prime }(f)}{4\pi }-\frac{1}{4{\pi }^2{f}^2},}$$ where δ(x) is the Dirac delta (in this formula, its derivative appears).

Laplace transform

The single-sided Laplace transform of R(x) is given as follows, $${\displaystyle \mathcal{L}\{R(x)\}(s)={\int }_0^{\mathrm{\infty }}{e}^{-sx}R(x)dx=\frac{1}{s^2}.}$$

Algebraic properties

Iteration invariance

Every iterated function of the ramp mapping is itself, as $${\displaystyle R(R(x))=R(x).}$$

Proof

$${\displaystyle R(R(x)):=\frac{R(x)+|R(x)|}{2}=\frac{R(x)+R(x)}{2}=R(x).}$$ This applies the non-negative property.