confidence interval with leastsq fit in scipy python

Question

How to calculate confidence interval for the least square fit (scipy.optimize.leastsq) in python?

Answer 1

I would use bootstrapping method.
See here: http://phe.rockefeller.edu/LogletLab/whitepaper/node17.html

Simple example for noisy gaussian:

x = arange(-10, 10, 0.01)

# model function
def f(p):
    mu, s = p
    return exp(-(x-mu)**2/(2*s**2))

# create error function for dataset    
def fff(d):
    def ff(p):
        return d-f(p)
    return ff

# create noisy dataset from model
def noisy_data(p):
    return f(p)+normal(0,0.1,len(x))

# fit dataset to model with least squares    
def fit(d):
    ff = fff(d)
    p = leastsq(ff,[0,1])[0]
    return p

# bootstrap estimation        
def bootstrap(d):
    p0 = fit(d)
    residuals = f(p0)-d
    s_residuals = std(residuals)

    ps = []
    for i in range(1000):
        new_d = d+normal(0,s_residuals,len(d))
        ps.append(fit(new_d))

    ps = array(ps)
    mean_params = mean(ps,0)
    std_params = std(ps,0)

    return mean_params, std_params

data = noisy_data([0.5, 2.1])
mean_params, std_params = bootstrap(data)

print "95% confidence interval:"
print "mu: ", mean_params[0], " +/- ", std_params[0]*1.95996
print "sigma: ", mean_params[1], " +/- ", std_params[1]*1.95996