On Wed, Feb 4, 2015 at 12:18 AM, Warren Weckesser < warren.weckesser@gmail.com> wrote:
On Tue, Feb 3, 2015 at 11:14 PM, Sturla Molden <sturla.molden@gmail.com> wrote:
Warren Weckesser <warren.weckesser@gmail.com> wrote:
0 if x < 0 heaviside(x) = 0.5 if x == 0 1 if x > 0
This is not correct. The discrete form of the Heaviside step function has the value 1 for x == 0.
heaviside = lambda x : 1 - (x < 0).astype(int)
By "discrete form", do you mean discrete time (i.e. a function defined on the integers)? Then I agree, the discrete time unit step function is defined as
u(k) = 0 k < 0 1 k >= 0
for integer k.
The domain of the proposed Heaviside function is not discrete; it is defined for arbitrary floating point (real) arguments. In this case, the choice heaviside(0) = 0.5 is a common convention. See for example,
* http://mathworld.wolfram.com/HeavisideStepFunction.html * http://www.mathworks.com/help/symbolic/heaviside.html * http://en.wikipedia.org/wiki/Heaviside_step_function, in particular http://en.wikipedia.org/wiki/Heaviside_step_function#Zero_argument
Other common conventions are the right-continuous version that you prefer (heavisde(0) = 1), or the left-continuous version (heaviside(0) = 0).
We can accommodate the alternatives with an additional argument that sets the value at 0:
heaviside(x, zero_value=0.5)
What's the usecase for a heaviside function? I don't think I have needed one since I was using mathematica or maple. (x < 0).astype(...) (x <= 0).astype(...) np.sign(x, dtype) look useful enough for most cases, or not? (What I wish numpy had is conditional place that doesn't calculate all the values. (I think there is a helper function in scipy.stats for that)) Josef
Warren
Sturla
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