如何使用 python 中的最小化来将数据与方程拟合以获得 model 参数

Question

I'm new to Python and programming. I made the below code to get optimum model parameters (R0, t_inc, t_rec, ex, teta) by minimizing the error between the data and the model (several differential equations). I am stuck at how to define the error function as seen in the code below

import numpy as np
import pandas as pd
from scipy.integrate import odeint
import matplotlib.pyplot as plt
import datetime
from lmfit import Parameters, fit_report, minimize

totaldays = 93  # as of today                
# This code is only to adjust the R0, t_infective, t_incubation to match data until to date
n_to_start = 0 # start data to fit
n_to_fit = totaldays # end data to fit -1, 
NumberofTest = 136.73*n_to_fit**2 - 17609*n_to_fit + 605936

# getting data till to date
DataMalaysia = pd.read_csv('DataMalaysia.csv')
Dates = DataMalaysia.iloc[:,0].values
TotalCase = DataMalaysia.iloc[:,3].values
TotalRecovered = DataMalaysia.iloc[:, 5].values
TotalDeath = DataMalaysia.iloc[:,4].values
Days = DataMalaysia.iloc[:, 1].values
ActiveCase = DataMalaysia.iloc[:, 7].values
Daystofit = Days[:n_to_fit] 
Dates = Dates[:n_to_fit]
start_date = datetime.date(2020,1,22)
ActiveCasetofit = ActiveCase[:n_to_fit]
TotalRecoveredtofit = TotalRecovered[:n_to_fit]
TotalDeathtofit = TotalDeath[:n_to_fit]


# parameter values including death and immigration
N = NumberofTest           # number of population
i_initial = 4       # 4 people is infected at the beginning 25th Jan 2020
# parameters around the Susceptible population (possible to get infected)
immigrating_s = 0   # fraction of population immigrating into the infected location
death_s = 0         # fraction of population died due to other diseases
# parameters around the Exposed or Infected people (but not yet Infecting)
immigrating_e = 0   # fraction of the infected people immigrating into the infected location
death_e = 0         # fraction of the infected/exposed people die due to other diseases
#parameters around the Infectious population
immigrating_i = 0   # fraction of the infectious people immigrating into the infected location
# death_i_MCO = 0.0157      # fraction of the infectious people die due to the virus
# mitigation effort

# variables to fit
R0 = 2.96   # reproduction number. This number is relatively high
t_inc = 11.93  # incubation period (5-6 is most reported one)
t_rec = 1.24   # infectious period, gamma = 1/t_infectious is the recovery rate, typical 3-4 days
ex = 0.016 #death fraction
teta = 0.1 # recovered fraction without getting ill

# using population
e0 = 0
i0 = i_initial
r0 = 0
d0 = 0
rprime0 = 0
s0 = N - e0 - i0 - r0 - d0 - rprime0

# SEIR model including MCO
def SEIR(x, t, R0, t_inc, t_rec, ex, teta):
    # introduction of the variables to calculate
    s, e, i, r, rprime, d = x
    alpha = 1/t_inc
    gamma = 1/t_rec
    R0t = R0/N
    beta = R0t*gamma
    # the differential equations
    dsdt = -(1-u)*beta * s * i + (immigrating_s - death_s)*s
    dedt = (1-u)*beta * s * i - alpha*e - teta*e + (immigrating_e - death_e)*e
    didt = alpha * e - gamma * i + (immigrating_i - ex)*i
    drdt = gamma*i
    drprimedt = teta*e
    dddt = ex*i

    return [dsdt, dedt, didt, drdt, drprimedt, dddt]

# integrating the SEIR model
def integrate_i(t, R0, t_inc, t_rec, ex, teta):
    x0 = s0, e0, i0, r0, rprime0, d0
    solution = odeint(SEIR, x0, t, args = (R0, t_inc, t_rec, ex, teta)).T
    solutiona = solution.T
    return solutiona[:, 2]

def integrate_r(t, R0, t_inc, t_rec, ex, teta):
    x0 = s0, e0, i0, r0, rprime0, d0
    solution = odeint(SEIR, x0, t, args = (R0, t_inc, t_rec, ex, teta)).T
    solutiona = solution.T
    return solutiona[:, 3]

def integrate_d(t, R0, t_inc, t_rec, ex, teta):
    x0 = s0, e0, i0, r0, rprime0, d0
    solution = odeint(SEIR, x0, t, args = (R0, t_inc, t_rec, ex, teta)).T
    solutiona = solution.T
    return solutiona[:, 5]

def integrate_total(t_total, R0, t_inc, t_rec, ex, teta):
    #slicing the time frame to each integration
    ti = t_total[:n_to_fit]
    td = t_total[len(ti)+len(ti):]
    tr = t_total[len(ti):len(t_total)-len(td)]
    result_i = integrate_i(ti, R0, t_inc, t_rec, ex, teta)
    result_r = integrate_r(tr, R0, t_inc, t_rec, ex, teta)
    result_d = integrate_d(td, R0, t_inc, t_rec, ex, teta)
    return np.concatenate([result_i, result_r, result_d])


def error(t_total, R0, t_inc, t_rec, ex, teta):
    R0 = 2.96   # reproduction number. This number is relatively high
    t_inc = 11.93  # incubation period (5-6 is most reported one)
    t_rec = 1.24   # infectious period, gamma = 1/t_infectious is the recovery rate, typical 3-4 days
    ex = 0.016 #death fraction
    teta = 0.1
    total_error = (np.sum((integrate_total(t_total, R0, t_inc, t_rec, ex, teta)-y_total)**2))

    return total_error

# fitting predictions with data points
start = n_to_start
end = n_to_fit
t = np.linspace(start, end, end)
ta = np.array(t)
# t_total = np.append(ta, ta, ta)
t_total = np.concatenate([ta, ta, ta])
y_total = np.concatenate([ActiveCasetofit, TotalRecoveredtofit, TotalDeathtofit])
p0=[R0, t_inc, t_rec, ex, teta]

params, extras = minimize(error, p0 ,method='BFGS',
                          options={'disp':True})

# Getting the optimized variables to plot what happens after to date
# getting the optimum values of R0, t_inc, t_rec, ex, teta
R0 = params[0]
t_incubation = params[1]
t_recovery = params[2]
death_ratio = params[3]
recovery_ratio = params[4]

#generation of the fitting curve
Predicted_ActiveCase = integrate_i(t, *params)
Predicted_RecoveredCase = integrate_r(t, *params)
Predicted_Death = integrate_d(t, *params)
print("Optimum R0 is {0:.2f}".format(params[0]))
print("Optimum Incubation Period is {0:.2f} days".format(params[1]))
print("Optimum Recovery Period is {0:.2f} days".format(params[2]))
print("Optimum Death Ratio is {0:.4f} ".format(params[3]))
print("Optimum Recovery Ratio Without Getting Ill is {0:.4f} ".format(params[4]))

# plotting data and results
fig1 = plt.figure()
t_fit = np.array([start_date+datetime.timedelta(days=i) for i in range(n_to_fit)])
plt.scatter(t_fit, ActiveCasetofit, c='red', label ='Data To Date')
plt.plot(t_fit, Predicted_ActiveCase, "r", label ='Fitted Active Case To Date')
plt.scatter(t_fit, TotalRecoveredtofit, c='blue', label ='Data To Date')
plt.plot(t_fit, Predicted_RecoveredCase, "b", label ='Fitted Total Recovered Case To Date')
plt.scatter(t_fit, TotalDeathtofit, c='green', label ='Data To Date')
plt.plot(t_fit, Predicted_Death, "g", label ='Fitted Death To Date')
plt.title('Fitting Data to Date')
plt.xlabel('Time/days')
plt.ylabel('Population')
plt.legend(loc='best')

It gives me the following error, which I think this is because I don't know how to input the arguments into the code. This is the whole error:

runfile('C:/.../Python/SEIR_v8.py', wdir='C:/.../Python') Traceback (most recent call last):

File "C:...\Python\SEIR_v8.py", line 171, in params, extras = minimize(error, p0, method='BFGS')

File "C:\Users...\AppData\Local\Continuum\anaconda3\envs\PythonNew\lib\site-packages\lmfit\minimizer.py", line 2393, in minimize return fitter.minimize(method=method)

File "C:\Users...\AppData\Local\Continuum\anaconda3\envs\PythonNew\lib\site-packages\lmfit\minimizer.py", line 2176, in minimize return function(**kwargs)

File "C:\Users...\AppData\Local\Continuum\anaconda3\envs\PythonNew\lib\site-packages\lmfit\minimizer.py", line 931, in scalar_minimize ret = scipy_minimize(self.penalty, variables, **fmin_kws)

File "C:\Users...\AppData\Local\Continuum\anaconda3\envs\PythonNew\lib\site-packages\scipy\optimize_minimize.py", line 604, in minimize return _minimize_bfgs(fun, x0, args, jac, callback, **options)

File "C:\Users...\AppData\Local\Continuum\anaconda3\envs\PythonNew\lib\site-packages\scipy\optimize\optimize.py", line 1003, in _minimize_bfgs old_fval = f(x0)

File "C:\Users...\AppData\Local\Continuum\anaconda3\envs\PythonNew\lib\site-packages\scipy\optimize\optimize.py", line 327, in function_wrapper return function(*(wrapper_args + args))

File "C:\Users...\AppData\Local\Continuum\anaconda3\envs\PythonNew\lib\site-packages\lmfit\minimizer.py", line 598, in penalty r = self.__residual(fvars, apply_bounds_transformation)

File "C:\Users...\AppData\Local\Continuum\anaconda3\envs\PythonNew\lib\site-packages\lmfit\minimizer.py", line 530, in __residual out = self.userfcn(params, *self.userargs, **self.userkws)

TypeError: error() missing 5 required positional arguments: 'R0', 't_inc', 't_rec', 'ex', and 'teta'

Please help.

Regards,

Zulfan

Answer 1

You are calling minimize() incorrectly. I'm too tired now to figure out the details. I suggest you carefully read the scipi documentation. You need to pass in a vector x0 as an initial guess as well as args which are the fixed parameters of the function you are minimizing. As it stands, you are only passing in x0 , but your error function expects additional parameters.

Answer 2

I modified the error function as below and it works.

def error(params, t_total, y_total):
    R0 = params['R0'].value
    t_inc = params['t_inc'].value
    t_rec = params['t_rec'].value
    ex = params['ex'].value
    teta = params['teta'].value

    y_model = integrate_total(t_total, R0, t_inc, t_rec, ex, teta)
    return y_model - y_total


params = lmfit.Parameters()
params.add('R0', 3.65, min=0, max=5.0)
params.add('t_inc', 15, min=0, max=20.0)
params.add('t_rec', 1, min=0, max=2.0)
params.add('ex', 0.02, min=0, max=1.0)
params.add('teta', 0.1, min=0, max=1.0)

# fitting predictions with data points
start = n_to_start
end = n_to_fit
t = np.linspace(start, end, end)
ta = np.array(t)
t_total = np.concatenate([ta, ta, ta])
y_total = np.concatenate([ActiveCasetofit, TotalRecoveredtofit, TotalDeathtofit])

o1 = lmfit.minimize(error, params, args=(t_total, y_total), method='powell')
print("# Fit using leastsq:")
lmfit.report_fit(o1)

Then, I can vary the solver and initial points to get better fits.