[SciPy-User] [Matplotlib-users] Sigmoid Curve Fitting

[Gökhan Sever]

Hi,
Does Matplotlib/Numpy/Scipy contain the ability to fit a sigmoid curve
to a set of data points?
Regards,
Chris

I am very curious to know how this is done. I have seen in many Cloud
Condensation Nuclei (CCN) counter calibration experiments [e.g. Rose et. al.
(2008) - p1156, fig2
http://www.atmos-chem-phys.net/8/1153/2008/acp-8-1153-2008.html] people use
sigmoidal fits. Never seen a code is shared so far or how it is obtained.

from scipy.optimize import leastsq
could be a good point to start. To me this fit is very likely a least square
minimization problem. You have your tanh(x) or 1/(1+e^-x) as a sigmoidal
function, and you should find the points that will make the error minimal
for a given discrete points/measurements.

from scipy.optimize import curve_fit might be another way to approach.

[Gökhan Sever]

Hi,
Does Matplotlib/Numpy/Scipy contain the ability to fit a sigmoid curve
to a set of data points?

You could also experiment at zunzun.com. Choose 2D sigmoidal from the 2D
function scroll list.

[Chris Spencer]

Hi,

Does Scipy contain the ability to fit a sigmoid curve to a set of data points?

I found some Numpy code
(http://pingswept.org/2009/01/24/least-squares-polynomial-fitting-in-python/)
for fitting curves using the least squares method, but it only seems
to fit parabolas to my sigmoid data.

Regards,
Chris

[Warren Weckesser]

Post by Chris Spencer
Hi,
Does Scipy contain the ability to fit a sigmoid curve to a set of data points?
I found some Numpy code
(http://pingswept.org/2009/01/24/least-squares-polynomial-fitting-in-python/)
for fitting curves using the least squares method, but it only seems
to fit parabolas to my sigmoid data.

You can use curve_fit (scipy.optimize.curve_fit).

Which family of sigmoid functions do you want to use?
See http://en.wikipedia.org/wiki/Sigmoid_function for a few possibilities.

If, for example, you want to fit the following family to your data:

f(x) = 1/(1 + exp(-k*(x-x0)))

(which has two parameters, k and x0), you can do something like this:

-----
import numpy as np
import pylab
from scipy.optimize import curve_fit

def sigmoid(x, x0, k):
y = 1 / (1 + np.exp(-k*(x-x0)))
return y

xdata = np.array([0.0, 1.0, 3.0, 4.3, 7.0, 8.0, 8.5, 10.0, 12.0])
ydata = np.array([0.01, 0.02, 0.04, 0.11, 0.43, 0.7, 0.89, 0.95, 0.99])

popt, pcov = curve_fit(sigmoid, xdata, ydata)
print popt

x = np.linspace(-1, 15, 50)
y = sigmoid(x, *popt)


pylab.plot(xdata, ydata, 'o', label='data')
pylab.plot(x,y, label='fit')
pylab.ylim(0, 1.05)
pylab.legend(loc='best')
pylab.show()
-----

This script generates the attached plot.


Warren

[Chris Spencer]

Thank you. Since that feature only appears to be in the bleeding-edge
version, it took me a while to track-down and compile all the
libraries on Ubuntu 10.04, but it works perfectly.

Regards,
Chris

On Mon, Sep 20, 2010 at 11:54 PM, Warren Weckesser

Post by Warren Weckesser

Post by Chris Spencer
Hi,
Does Scipy contain the ability to fit a sigmoid curve to a set of data points?
I found some Numpy code
(http://pingswept.org/2009/01/24/least-squares-polynomial-fitting-in-python/)
for fitting curves using the least squares method, but it only seems
to fit parabolas to my sigmoid data.

You can use curve_fit (scipy.optimize.curve_fit).
Which family of sigmoid functions do you want to use?
See http://en.wikipedia.org/wiki/Sigmoid_function for a few possibilities.
f(x) = 1/(1 + exp(-k*(x-x0)))
-----
import numpy as np
import pylab
from scipy.optimize import curve_fit
   y = 1 / (1 + np.exp(-k*(x-x0)))
   return y
xdata = np.array([0.0,   1.0,  3.0, 4.3, 7.0,   8.0,   8.5, 10.0, 12.0])
ydata = np.array([0.01, 0.02, 0.04, 0.11, 0.43,  0.7, 0.89, 0.95, 0.99])
popt, pcov = curve_fit(sigmoid, xdata, ydata)
print popt
x = np.linspace(-1, 15, 50)
y = sigmoid(x, *popt)
pylab.plot(xdata, ydata, 'o', label='data')
pylab.plot(x,y, label='fit')
pylab.ylim(0, 1.05)
pylab.legend(loc='best')
pylab.show()
-----
This script generates the attached plot.
Warren
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[Chris Spencer]

Is it possible to get it to determine the asymptotes as well? It seems
to assume the curve will be bounded between y=0 and 1, whereas my data
can have arbitrary limits. I tried changing sigmoid() to:

def sigmoid(x, x0, k, a):
y = a * 1 / (1 + np.exp(-k*(x-x0)))
return y

but that only results in a curve of f(x)=0.5.

Regards,
Chris

On Mon, Sep 20, 2010 at 11:54 PM, Warren Weckesser

Post by Warren Weckesser

Post by Chris Spencer
Hi,
Does Scipy contain the ability to fit a sigmoid curve to a set of data points?
I found some Numpy code
(http://pingswept.org/2009/01/24/least-squares-polynomial-fitting-in-python/)
for fitting curves using the least squares method, but it only seems
to fit parabolas to my sigmoid data.

You can use curve_fit (scipy.optimize.curve_fit).
Which family of sigmoid functions do you want to use?
See http://en.wikipedia.org/wiki/Sigmoid_function for a few possibilities.
f(x) = 1/(1 + exp(-k*(x-x0)))
-----
import numpy as np
import pylab
from scipy.optimize import curve_fit
   y = 1 / (1 + np.exp(-k*(x-x0)))
   return y
xdata = np.array([0.0,   1.0,  3.0, 4.3, 7.0,   8.0,   8.5, 10.0, 12.0])
ydata = np.array([0.01, 0.02, 0.04, 0.11, 0.43,  0.7, 0.89, 0.95, 0.99])
popt, pcov = curve_fit(sigmoid, xdata, ydata)
print popt
x = np.linspace(-1, 15, 50)
y = sigmoid(x, *popt)
pylab.plot(xdata, ydata, 'o', label='data')
pylab.plot(x,y, label='fit')
pylab.ylim(0, 1.05)
pylab.legend(loc='best')
pylab.show()
-----
This script generates the attached plot.
Warren
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SciPy-User mailing list
http://mail.scipy.org/mailman/listinfo/scipy-user

[Warren Weckesser]

Post by Chris Spencer
Is it possible to get it to determine the asymptotes as well? It seems
to assume the curve will be bounded between y=0 and 1, whereas my data
y = a * 1 / (1 + np.exp(-k*(x-x0)))
return y
but that only results in a curve of f(x)=0.5.

The following is a variation that includes more parameters in the family
of sigmoid functions. But bear in mind, I chose this family of
functions just as a demonstration of curve_fit. I don't know if it
makes sense to use this family for your data. The appropriate family to
use depends on the nature of the data.

-----
import numpy as np
import pylab
from scipy.optimize import curve_fit

def sigmoid(x, x0, k, a, c):
y = a / (1 + np.exp(-k*(x-x0))) + c
return y

xdata = np.array([0.0, 1.0, 3.0, 4.3, 7.0, 8.0, 8.5, 10.0,
12.0, 14.0])
ydata = np.array([0.11, 0.12, 0.14, 0.21, 0.83, 1.45, 1.78, 1.9,
1.98, 2.02])

popt, pcov = curve_fit(sigmoid, xdata, ydata)
print "Fit:"
print "x0 =", popt[0]
print "k =", popt[1]
print "a =", popt[2]
print "c =", popt[3]
print "Asymptotes are", popt[3], "and", popt[3] + popt[2]

x = np.linspace(-1, 15, 50)
y = sigmoid(x, *popt)


pylab.plot(xdata, ydata, 'o', label='data')
pylab.plot(x,y, label='fit')
pylab.ylim(0, 2.05)
pylab.legend(loc='upper left')
pylab.grid(True)
pylab.show()
-----


Warren

Post by Chris Spencer
Regards,
Chris
On Mon, Sep 20, 2010 at 11:54 PM, Warren Weckesser

Post by Warren Weckesser

Post by Chris Spencer
Hi,
Does Scipy contain the ability to fit a sigmoid curve to a set of data points?
I found some Numpy code
(http://pingswept.org/2009/01/24/least-squares-polynomial-fitting-in-python/)
for fitting curves using the least squares method, but it only seems
to fit parabolas to my sigmoid data.

You can use curve_fit (scipy.optimize.curve_fit).
Which family of sigmoid functions do you want to use?
See http://en.wikipedia.org/wiki/Sigmoid_function for a few possibilities.
f(x) = 1/(1 + exp(-k*(x-x0)))
-----
import numpy as np
import pylab
from scipy.optimize import curve_fit
y = 1 / (1 + np.exp(-k*(x-x0)))
return y
xdata = np.array([0.0, 1.0, 3.0, 4.3, 7.0, 8.0, 8.5, 10.0, 12.0])
ydata = np.array([0.01, 0.02, 0.04, 0.11, 0.43, 0.7, 0.89, 0.95, 0.99])
popt, pcov = curve_fit(sigmoid, xdata, ydata)
print popt
x = np.linspace(-1, 15, 50)
y = sigmoid(x, *popt)
pylab.plot(xdata, ydata, 'o', label='data')
pylab.plot(x,y, label='fit')
pylab.ylim(0, 1.05)
pylab.legend(loc='best')
pylab.show()
-----
This script generates the attached plot.
Warren
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[Chris Spencer]

On Tue, Sep 21, 2010 at 12:16 PM, Warren Weckesser

Post by Warren Weckesser
The following is a variation that includes more parameters in the family
of sigmoid functions.  But bear in mind, I chose this family of
functions just as a demonstration of curve_fit.  I don't know if it
makes sense to use this family for your data.  The appropriate family to
use depends on the nature of the data.

I see what you mean. That modification only fits a low-to-high
sigmoid, but that's close enough for me to adapt by reversing my data
set. Thank you for the excellent example.

Regards,
Chris

[Chris Spencer]

I found this modification allows for the inversion of the estimated
sigmoid curve:

def sigmoid(x, x0, k, a, c, d):
y = 1 / (1 + np.exp(-k*(x-x0)))
y = (1 - y)*(1 - d) + y*d
y = a * y + c
return y

Regards,
Chris

Post by Chris Spencer
On Tue, Sep 21, 2010 at 12:16 PM, Warren Weckesser

Post by Warren Weckesser
The following is a variation that includes more parameters in the family
of sigmoid functions.  But bear in mind, I chose this family of
functions just as a demonstration of curve_fit.  I don't know if it
makes sense to use this family for your data.  The appropriate family to
use depends on the nature of the data.

I see what you mean. That modification only fits a low-to-high
sigmoid, but that's close enough for me to adapt by reversing my data
set. Thank you for the excellent example.
Regards,
Chris

[Warren Weckesser]

Post by Chris Spencer
I found this modification allows for the inversion of the estimated
y = 1 / (1 + np.exp(-k*(x-x0)))
y = (1 - y)*(1 - d) + y*d
y = a * y + c
return y

A negative value of k "reverses" the sigmoid shape, so you shouldn't
have to define a new function. If you prefer to have k be positive, you
could use

def sigmoid(x, x0, k, a, c):
y = a / (1 + np.exp(k*(x-x0))) + c
return y

(I changed "-k" to "k".)


Warren

Post by Chris Spencer
Regards,
Chris

Post by Chris Spencer
On Tue, Sep 21, 2010 at 12:16 PM, Warren Weckesser

Post by Warren Weckesser
The following is a variation that includes more parameters in the family
of sigmoid functions. But bear in mind, I chose this family of
functions just as a demonstration of curve_fit. I don't know if it
makes sense to use this family for your data. The appropriate family to
use depends on the nature of the data.

I see what you mean. That modification only fits a low-to-high
sigmoid, but that's close enough for me to adapt by reversing my data
set. Thank you for the excellent example.
Regards,
Chris

_______________________________________________
SciPy-User mailing list
http://mail.scipy.org/mailman/listinfo/scipy-user

[Chris Spencer]

Right, I noticed that negating k should theoretically have the same
effect. However, when I reversed your sample data (i.e. ydata =
ydata[::-1]), I was surprised to find that curve_fit gives me
f(x)=0.92 instead of a proper sigmoid curve. Adding my parameter seems
to work around this problem.

Regards,
Chris

On Tue, Sep 21, 2010 at 2:21 PM, Warren Weckesser

Post by Warren Weckesser

Post by Chris Spencer
I found this modification allows for the inversion of the estimated
     y = 1 / (1 + np.exp(-k*(x-x0)))
     y = (1 - y)*(1 - d) + y*d
     y = a * y + c
     return y

A negative value of k "reverses" the sigmoid shape, so you shouldn't
have to define a new function.  If you prefer to have k be positive, you
could use
    y = a / (1 + np.exp(k*(x-x0))) + c
    return y
(I changed "-k" to "k".)
Warren

Post by Chris Spencer
Regards,
Chris

Post by Chris Spencer
On Tue, Sep 21, 2010 at 12:16 PM, Warren Weckesser

Post by Warren Weckesser
The following is a variation that includes more parameters in the family
of sigmoid functions.  But bear in mind, I chose this family of
functions just as a demonstration of curve_fit.  I don't know if it
makes sense to use this family for your data.  The appropriate family to
use depends on the nature of the data.

I see what you mean. That modification only fits a low-to-high
sigmoid, but that's close enough for me to adapt by reversing my data
set. Thank you for the excellent example.
Regards,
Chris

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[Warren Weckesser]

Post by Chris Spencer
Right, I noticed that negating k should theoretically have the same
effect. However, when I reversed your sample data (i.e. ydata =
ydata[::-1]), I was surprised to find that curve_fit gives me
f(x)=0.92 instead of a proper sigmoid curve.

Ah, in that case, it can help to give curve_fit a better initial guess
for the parameters. For example, if I use

popt, pcov = curve_fit(sigmoid, xdata, ydata, p0=(1.0, -1.0, 1.0, 0.0))

with the original function, it works as expected.


Warren

Post by Chris Spencer
Adding my parameter seems
to work around this problem.
Regards,
Chris
On Tue, Sep 21, 2010 at 2:21 PM, Warren Weckesser

Post by Warren Weckesser

Post by Chris Spencer
I found this modification allows for the inversion of the estimated
y = 1 / (1 + np.exp(-k*(x-x0)))
y = (1 - y)*(1 - d) + y*d
y = a * y + c
return y

A negative value of k "reverses" the sigmoid shape, so you shouldn't
have to define a new function. If you prefer to have k be positive, you
could use
y = a / (1 + np.exp(k*(x-x0))) + c
return y
(I changed "-k" to "k".)
Warren

Post by Chris Spencer
Regards,
Chris

Post by Chris Spencer
On Tue, Sep 21, 2010 at 12:16 PM, Warren Weckesser

Post by Warren Weckesser
The following is a variation that includes more parameters in the family
of sigmoid functions. But bear in mind, I chose this family of
functions just as a demonstration of curve_fit. I don't know if it
makes sense to use this family for your data. The appropriate family to
use depends on the nature of the data.

I see what you mean. That modification only fits a low-to-high
sigmoid, but that's close enough for me to adapt by reversing my data
set. Thank you for the excellent example.
Regards,
Chris

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[Sturla Molden]

Post by Chris Spencer
Hi,
Does Scipy contain the ability to fit a sigmoid curve to a set of data points?
I found some Numpy code
(http://pingswept.org/2009/01/24/least-squares-polynomial-fitting-in-python/)
for fitting curves using the least squares method, but it only seems
to fit parabolas to my sigmoid data.

scipy.optimize.leastsq can fit any non-linear regression curve using
Levenberg-Marquardt. You will have to supply a function that computes the
residuals, and optionally a function that returns their Jacobian unless
you want it estimated.

Sturla

[Gökhan Sever]

Post by Sturla Molden
scipy.optimize.leastsq can fit any non-linear regression curve using
Levenberg-Marquardt. You will have to supply a function that computes the
residuals, and optionally a function that returns their Jacobian unless
you want it estimated.

[Bruce Southey]

Post by Sturla Molden

Post by Chris Spencer
Hi,
Does Scipy contain the ability to fit a sigmoid curve to a set of data points?
I found some Numpy code
(http://pingswept.org/2009/01/24/least-squares-polynomial-fitting-in-python/)
for fitting curves using the least squares method, but it only seems
to fit parabolas to my sigmoid data.

scipy.optimize.leastsq can fit any non-linear regression curve using
Levenberg-Marquardt. You will have to supply a function that computes the
residuals, and optionally a function that returns their Jacobian unless
you want it estimated.
Sturla

Ignoring that you data is the opposite, there are a lot of 'growth
curves' such as Richard's (growth) curve or generalized logistic
function.
http://en.wikipedia.org/wiki/Generalised_logistic_function

Also, you can fit a linear model if your model can be linearized. The
downside is that the errors are multiplicative rather than additive.
For nonlinear models, it usually helps to standardize your data.

Bruce