sigmoidal regression with scipy, numpy, python, etc

Question

I have two variables (x and y) that have a somewhat sigmoidal relationship with each other, and I need to find some sort of prediction equation that will enable me to predict th

Answer 1

As pointed out by @unutbu above scipy now provides scipy.optimize.curve_fit which possess a less complicated call. If someone wants a quick version of how the same process would look like in those terms I present a minimal example below:

from scipy.optimize import curve_fit

def sigmoid(x, k, x0):

    return 1.0 / (1 + np.exp(-k * (x - x0)))

# Parameters of the true function
n_samples = 1000
true_x0 = 15
true_k = 1.5
sigma = 0.2

# Build the true function and add some noise
x = np.linspace(0, 30, num=n_samples)
y = sigmoid(x, k=true_k, x0=true_x0) 
y_with_noise = y + sigma * np.random.randn(n_samples)

# Sample the data from the real function (this will be your data)
some_points = np.random.choice(1000, size=30)  # take 30 data points
xdata = x[some_points]
ydata = y_with_noise[some_points]

# Fit the curve
popt, pcov = curve_fit(sigmoid, xdata, ydata)
estimated_k, estimated_x0 = popt

# Plot the fitted curve
y_fitted = sigmoid(x, k=estimated_k, x0=estimated_x0)

# Plot everything for illustration
fig = plt.figure()
ax = fig.add_subplot(111)
ax.plot(x, y_fitted, '--', label='fitted')
ax.plot(x, y, '-', label='true')
ax.plot(xdata, ydata, 'o', label='samples')

ax.legend()

The result of this is shown in the next figure: