Question

from lmfit import minimize, Parameters, Parameter, report_fit
from scipy.integrate import odeint

def f(xs, t, ps):
    """Receptor synthesis-internalization model."""
    try:
        a = ps['a'].value
        b = ps['b'].value
    except:
        a, b = ps
    x = xs

    return a - b*x

def g(t, x0, ps):
    """
    Solution to the ODE x'(t) = f(t,x,k) with initial condition x(0) = x0
    """
    x = odeint(f, x0, t, args=(ps,))
    return x

def residual(ps, ts, data):
    x0 = ps['x0'].value
    model = g(ts, x0, ps)
    return (model - data).ravel()

a = 2.0
b = 0.5
true_params = [a, b]
x0 = 10.0

t = np.linspace(0, 10, 10)
data = g(t, x0, true_params)
data += np.random.normal(size=data.shape)

# set parameters incluing bounds
params = Parameters()
params.add('x0', value=float(data[0]), min=0, max=100)
params.add('a', value= 1.0, min=0, max=10)
params.add('b', value= 1.0, min=0, max=10)

# fit model and find predicted values
result = minimize(residual, params, args=(t, data), method='leastsq')
final = data + result.residual.reshape(data.shape)

# plot data and fitted curves
plt.plot(t, data, 'o')
plt.plot(t, final, '--', linewidth=2, c='blue');

# display fitted statistics
report_fit(result)

[[Fit Statistics]]
    # function evals   = 29
    # data points      = 10
    # variables        = 3
    chi-square         = 10.080
    reduced chi-square = 1.440
[[Variables]]
    x0:   10.1714231 +/- 1.156777 (11.37%) (init= 10.54454)
    a:    2.56428320 +/- 1.700899 (66.33%) (init= 1)
    b:    0.52952597 +/- 0.296358 (55.97%) (init= 1)
[[Correlations]] (unreported correlations are <  0.100)
    C(a, b)                      =  0.989
    C(x0, b)                     =  0.453
    C(x0, a)                     =  0.416