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 (slope 1, starting at 0).

In mathematics, the ramp function is also known as the positive part.

In machine learning, it is commonly known as a ReLU activation function or a rectifier in analogy to half-wave rectification in electrical engineering. In statistics (when used as a likelihood function) it is known as a tobit model.

This function has numerous applications in mathematics and engineering, and goes by various names, depending on the context. There are differentiable variants of the ramp function.

Definitions

The ramp function () may be defined analytically in several ways. Possible definitions are:

• A piecewise function

• Using the Iverson bracket

• The max function

• The mean of an independent variable and its absolute value this can be derived by noting the following definition of for which and

• The Heaviside step function

• The convolution

• The integral of the Heaviside step function:

• Macaulay brackets

• The positive part of the identity 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

Derivative

Its derivative is the Heaviside step function

Second derivative

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,

Antiderivative

Ramp function has the following antiderivative:

is the integration constant.

Fourier transformFourier transform

where is the Dirac delta (in this formula, its derivative appears).

Laplace transformLaplace transform

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

Algebraic properties

Iteration invariance

Every iterated function

See also

• Tobit model

• Rectifier (neural networks)

References