[Numpy-discussion] Combining Sigmoid Curves

[Angus McMorland]

2008/5/2 Rich Shepard <rshepard at appl-ecosys.com>:
>    When I last visited I was given excellent advice about Gaussian and other
>  bell-shaped curves. Upon further reflection I realized that the Gaussian
>  curves will not do; the curve does need to have y=0.0 at each end.
>
>    I tried to apply a Beta distribution, but I cannot correlate the alpha and
>  beta parameters with the curve characteristics that I have.
>
>    What will work (I call it a pi curve) is a matched pair of sigmoid curves,
>  the ascending curve on the left and the descending curve on the right. Using
>  the Boltzmann function for these I can calculate and plot each individually,
>  but I'm having difficulty in appending the x and y values across the entire
>  range. This is where I would appreciate your assistance.
>
>    The curve parameters passed to the function are the left end, right end,
>  and midpoint. The inflection point (where y = 0.5) is half-way between the
>  ends and the midpoint.
>
>    What I have in the function is this:
>
>         def piCurve(ll, lr, mid):
>           flexL = (mid - ll)/2.0
>           flexR = (lr - mid)/2.0
>           tau = (mid - ll)/10.0
>
>           x = []
>           y = []
>
>           xL = nx.arange(ll,mid,0.1)
>           for i in xL:
>             x.append(xL[i])
>           yL = 1.0 / (1.0 + nx.exp(-(xL-flexL)/tau))
>           for j in yL:
>             y.append(yL[j])
>
>           xR = nx.arange(mid,lr,0.1)
>           for i in xR:
>             x.append(xR[i])
>           yR = 1 - (1.0 / (1.0 + nx.exp(-(xR-flexR)/tau)))
>           for j in yR:
>             y.append(yR[j])
>
>           appData.plotX = x
>           appData.plotY = y
>
>  Python complains about adding to the list:
>
>      yL = 1.0 / (1.0 + nx.exp(-(x-flexL)/tau))
>  TypeError: unsupported operand type(s) for -: 'list' and 'float'
>
>    What is the appropriate way to generate two sigmoid curves so that the x
>  values range from the left end to the right end and the y values rise from
>  0.0 to 1.0 at the midpoint, then lower to 0.0 again?

How about multiplying two Boltzmann terms together, ala:

f(x) = 1/(1+exp(-(x-flex1)/tau1)) * 1/(1+exp((x-flex2)/tau2))

You'll find if your two flexion points get too close together, the
peak will drop below the maximum for each individual curve, but the
transition will be continuous.

Angus.
-- 
AJC McMorland, PhD candidate
Physiology, University of Auckland

(Nearly) post-doctoral research fellow
Neurobiology, University of Pittsburgh