Ramp function

The ramp function () may be defined analytically in several ways. Possible definitions are: • A piecewise function: R(x) := \begin{cases} x, & x \ge 0; \\ 0, & x • Using the Iverson bracket notation: R(x) := x \cdot [x \geq 0] or R(x) := x \cdot [x > 0] • The max function: R(x) := \max(x,0) • The mean of an independent variable and its absolute value (a straight line with unity gradient and its modulus): R(x) := \frac{x+|x|}{2} this can be derived by noting the following definition of , \max(a,b) = \frac{a + b + |a - b|}{2} for which and • The Heaviside step function multiplied by a straight line with unity gradient: R\left( x \right) := x H(x) • The convolution of the Heaviside step function with itself: R\left( x \right) := H(x) * H(x) • The integral of the Heaviside step function: R(x) := \int_{-\infty}^{x} H(\xi)\,d\xi • Macaulay brackets: R(x) := \langle x\rangle • The positive part of the identity function: R := \operatorname{id}^+ • As a limit function: R\left( x \right) := \lim_{a\to \infty} \begin{cases} \frac{1}{a} ,\quad x=0 \\ \dfrac{x}{1-e^{-ax}},\quad x\neq 0\end{cases} It could approximated as close as desired by choosing an increasing positive value a>0 . == 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. with a knot at x=3.1 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. \forall x \in \Reals: R(x) \geq 0 and \left| R (x) \right| = R(x) Derivative Its derivative is the Heaviside step function: R'(x) = H(x)\quad \mbox{for } x \ne 0. Second derivative The ramp function satisfies the differential equation: \frac{d^2}{dx^2} R(x - x_0) = \delta(x - x_0), 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: f(x) = f(a) + (x-a) f'(a) + \int_{a}^b R(x - s) f''(s) \,ds \quad \mbox{for }a === Fourier transform === \mathcal{F}\big\{ R(x) \big\}(f) = \int_{-\infty}^{\infty} R(x) e^{-2\pi ifx} \, dx = \frac{i\delta '(f)}{4\pi}-\frac{1}{4 \pi^2 f^2}, where is the Dirac delta (in this formula, its derivative appears). === Laplace transform === The single-sided Laplace transform of is given as follows, \mathcal{L}\big\{R(x)\big\} (s) = \int_{0}^{\infty} e^{-sx}R(x)dx = \frac{1}{s^2}. == Algebraic properties ==

Iteration invariance Every iterated function of the ramp mapping is itself, as R \big( R(x) \big) = R(x) . {{math proof | proof = R \big( R(x) \big) := \frac{R(x)+|R(x)|}{2} = \frac{R(x)+R(x)}{2} = R(x) . This applies the non-negative property.}} ==See also==