将常微分方程与 python 最小sq 拟合时，当初始条件作为参数传递时，会出现转换错误

Question

我有一组数据，我正在尝试使用 scipy 的 minimumsq 函数将其拟合到 ODE 模型中。我的 ODE 有参数 beta 和 gamma ，因此它看起来像这样：

# dS/dt = -betaSI
# dI/dt = betaSI - gammaI
# dR/dt = gammaI
# with y0 = y(t=0) = (S(0),I(0),R(0))

这个想法是找到beta和gamma，以便我的 ODE 系统的最佳数值积分接近数据。如果我知道初始条件中的所有点，我就可以使用 minimumsq y0 很好地做到这一点。

现在，我正在尝试做同样的事情，但现在将y0的一个条目作为额外的参数传递。这就是我和蟒蛇停止交流的地方......我做了一个函数，现在我传递给 minimumsq 的参数的第一个条目是我的变量 R 的初始条件。我收到以下消息：

*Traceback (most recent call last):
  File "/Users/Laura/Dropbox/SHIV/shivmodels/test.py", line 73, in <module>
    p1,success = optimize.leastsq(errfunc, initguess, args=(simpleSIR,[y0[0]],[Tx],[mydata]))
  File "/Library/Frameworks/Python.framework/Versions/7.2/lib/python2.7/site-packages/scipy/optimize/minpack.py", line 283, in leastsq
    gtol, maxfev, epsfcn, factor, diag)
TypeError: array cannot be safely cast to required type*

这是我的代码。对于此示例，它需要的内容要复杂一些，因为实际上我想拟合另一个具有 7 个参数的颂歌，并希望一次拟合多个数据集。但我想在这里发布一些更简单的东西......任何帮助将不胜感激！谢谢！

import numpy as np
from matplotlib import pyplot as plt
from scipy import optimize
from scipy.integrate import odeint
#define the time span for the ODE integration:
Tx = np.arange(0,50,1)
num_points = len(Tx)
#define a simple ODE to fit:
def simpleSIR(y,t,params):
    dydt0 = -params[0]*y[0]*y[1]
    dydt1 = params[0]*y[0]*y[1] - params[1]*y[1]
    dydt2 = params[1]*y[1]
    dydt = [dydt0,dydt1,dydt2]
    return dydt

#generate noisy data:
y0 = [1000.,1.,0.]
beta = 12*0.06/1000.0
gamma = 0.25
myparam = [beta,gamma]
sir = odeint(simpleSIR, y0, Tx, (myparam,))
mydata0 = sir[:,0] + 0.05*(-1)**(np.random.randint(num_points,size=num_points))*sir[:,0]
mydata1 = sir[:,1] + 0.05*(-1)**(np.random.randint(num_points,size=num_points))*sir[:,1]
mydata2 = sir[:,2] + 0.05*(-1)**(np.random.randint(num_points,size=num_points))*sir[:,2]
mydata = np.array([mydata0,mydata1,mydata2]).transpose()

#define a function that will run the ode and fit it, the reason I am doing this
#is because I will use several ODE's to see which one fits the data the best.
def fitfunc(myfun,y0,Tx,params):
    myfit = odeint(myfun, y0, Tx, args=(params,))
    return myfit
#define a function that will measure the error between the fit and the real data:
def errfunc(params,myfun,y0,Tx,y):
    """
    INPUTS:
    params are the parameters for the ODE
    myfun is the function to be integrated by odeint
    y0 vector of initial conditions, so that y(t0) = y0
    Tx is the vector over which integration occurs, since I have several data sets and each 
    one has its own vector of time points, Tx is a list of arrays.
    y is the data, it is a list of arrays since I want to fit to multiple data sets at once
    """
    res = []
    for i in range(len(y)):
        V0 = params[0][i]
        myparams = params[1:]
        initCond = np.zeros([3,])
        initCond[:2] = y0[i]
        initCond[2] = V0
        myfit = fitfunc(myfun,initCond,Tx[i],myparams)
        res.append(myfit[:,0] - y[i][:,0])
        res.append(myfit[:,1] - y[i][:,1])
        res.append(myfit[1:,2] - y[i][1:,2])
    #end for
    all_residuals = np.hstack(res).ravel()
    return all_residuals
#end errfunc

#example of the problem:
V0 = [0]
params = [V0,beta,gamma]  
y0 = [1000,1]
#this is just to test that my errfunc does work well.
errfunc(params,simpleSIR,[y0],[Tx],[mydata])
initguess = [V0,0.5,0.5]
p1,success = optimize.leastsq(errfunc, initguess, args=(simpleSIR,[y0[0]],[Tx],[mydata])) 

Answer 1

问题出在变量 initguess 上。 函数 optimize.minimumsq 具有以下调用签名：

http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.leastsq.html

第二个参数 x0 必须是一个数组。 您的列表

initguess = [v0,0.5,0.5]

不会转换为数组，因为 v0 是列表而不是 int 或浮点数。 因此，当您尝试在 minimumsq 函数中将 initguess 从列表转换为数组时，会出现错误。

我会调整变量参数从

def errfunc(params,myfun,y0,Tx,y):

所以它是一个一维数组。 使前几个条目成为 v0 的值，然后将 beta 和 gamma 附加到该条目。