Ramp function


The ramp function is a unary real function, whose graph is shaped like a ramp. It can be expressed by numerous [|definitions], for example "0 for negative inputs, output equals input for non-negative inputs". The term "ramp" can also be used for other functions obtained by scaling and shifting, and the function in this article is the unit ramp function.
This function has numerous [|applications] in mathematics and engineering, and goes by various names, depending on the context.

Definitions

The ramp function may be defined analytically in several ways. Possible definitions are:
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. Horizontally flipping a ramp yields a put option, while vertically flipping corresponds to selling or being "short" an option. In finance, the shape is widely called a "hockey stick", due the shape being similar to an ice hockey stick.
In statistics, hinge functions of multivariate adaptive regression splines are ramps, and are used to build regression models.
In machine learning, it is commonly known as the rectifier used in rectified linear units.

Analytic properties

Non-negativity

In the whole domain the function is non-negative, so its absolute value is itself, i.e.
and
Its derivative is the Heaviside function:

Second derivative

The ramp function satisfies the differential equation:
where is the Dirac delta. This means that is a Green's function for the second derivative operator. Thus, any function,, with an integrable second derivative,, will satisfy the equation:

[Fourier transform]

where is the Dirac delta.

[Laplace transform]

The single-sided Laplace transform of is given as follows,

Algebraic properties

Iteration invariance

Every iterated function of the ramp mapping is itself, as
This applies the [|non-negative property].