Here is a fully configurable sigmoid function. I post the full code, try the plotting option. I add furthermore two other functions derivated from the sigmoid, just in case:

```
def sigmosym(x, Lambda = 0.5, mirror = False, plotting = False):
# Symetric sigmoid function f from [0 1] to [0 1], with the
# property that f(0) = 0, f(1) = 1, f'(0) = f'(1) = 0, and f is
# symetric about the point (1/2, 1/2).
# If mirror is set to True, then sigmosym returns 1 - f.
#
# INPUT:
# * x: a scalar
# * Lambda (default 1): a real number > 0; if lambda <=1, then
# the sigmoid is convex at the first half of the curve, and
# concave at the second half, that is, it has only one
# inflexion point; if Lambda >1, it will be concave between 0 and
# 1/2 and convex between 1/2 and 1, with 4 inflexion points,
# extending to the horizontal around the center.
# * mirror (default False): if set to True, then the function returns
# 1-f, where f is the function returned if mirror = False
# * plotting (default False): if set to True, the function is plotted,
# and no value is returned.
if plotting:
import matplotlib.pyplot as plt
x = list(np.arange(0, 1, 0.01))
y = [sigmosym(t, Lambda, mirror = mirror, plotting = False) for t in x]
fig1 = plt.figure()
ax = fig1.add_subplot(1,1,1)
ax.grid(True)
ax.plot(x, y)
return None
f = 2 * np.abs(x) - 1;
f = (np.sign(f) * (np.abs(f)**Lambda) + 1) / 2
f = np.sin(np.pi * f/2)**2
if mirror:
f = 1 - f
return f
def sigmocurve(x, stiffness = 0.5, inflexion = 1,
mirror = False, plotting = False):
# A positive sigmoidal function f extending from 0 to +infty, with the
# property that f(0) = f'(0) = 0 = f'(+inf), f(+inf) = 1,
# f is convex between 0 and a point x_infl very near to "inflexion", f is
# concave between x_infl and +inf.
# If mirror is set to True, then sigmocurve returns 1 - f.
#
# INPUT:
# * x: a scalar
# * stiffness (default 0.5): a real number between 0 and 1 (>0, <= 1):
# the more "stiffness" is close to 0, the more abrubtly the function
# climbs from x_infl to 1.
# * inflexion: the approximative inflexion point of the curve, that
# is, the point where the curve climbs to upper_bound; "inflexion"
# must be > 0.
# * mirror (default False): if set to True, then the function returns
# 1-f, where f is the function returned if mirror = False
# * plotting (default False): if set to True, the function is plotted,
# and no value is returned.
#
if plotting:
import matplotlib.pyplot as plt
intvl = 60
x = list(np.arange(0, intvl, 0.01))
y = [sigmocurve(t, stiffness = stiffness, inflexion = inflexion,
mirror = mirror, plotting = False) for t in x]
fig2 = plt.figure()
ax = fig2.add_subplot(1,1,1)
ax.plot(x, y)
ax.grid(True)
return None
a_infl = 1/inflexion
t = a_infl * x / (1 + a_infl * x)
f = sigmosym(t, stiffness, mirror = False)
if mirror:
f = 1 - f
return f
def sigmobell(x, mu = 0, sigma = 1, Lambda = 1, plotting = False):
# A bell shaped sygmoid based function f with center mu, and extent sigma
# (it is 0 outside of [mu-sigma, mu+ sigma]). f is symmetrical about mu,
# and reach its maximum f = 1 at x = mu. The two halves of the bell are
# built using the sygmosym function.
#
# INPUT:
# * x : a scalar
# * mu : the center of the bell (default 0)
# * sigma: a number > 0 the radius of the bell (default 1)
# * Lambda: a number between 0 and 1, that controls the shape of
# the bell (default 1): the more Lambda is close to 0, the more
# the bell resembles to a rectangle, and the more it is close
# to 1, the more it resembles a well shaped bell.
# * plotting (default False): if set to True, the function is plotted,
# and no value is returned.
#
if plotting:
import matplotlib.pyplot as plt
x = list(np.arange(mu - sigma - 1, mu + sigma + 1, 0.01))
y = [sigmobell(t, mu = mu, sigma = sigma, Lambda = Lambda,
plotting = False) for t in x]
fig3 = plt.figure()
ax = fig3.add_subplot(1,1,1)
ax.grid(True)
ax.plot(x, y)
return None
Delta = (x - mu) / sigma
if np.abs(Delta) > 1:
Delta = 1
f = 1 - sigmosym(Delta, Lambda = Lambda, mirror = False)
return f
if __name__ == '__main__': # PLOT THE FUNCTIONS FOR SOME PARAMETERS
sigmosym(None, Lambda = 0.5, mirror = False, plotting = True)
sigmocurve(None, stiffness = 0.2, inflexion = 12,
mirror = False, plotting = True)
sigmobell(None, mu = 0, sigma = 2, Lambda = 1, plotting = True)
plt.show()
```