Monod生长/降解方程对实验数据的曲线拟合

Question

因此，这里面临的问题是Monod方程与实验数据的曲线拟合。细菌生长和降解有机碳的模型如下所示：

dX/dt = (u *S*X )/(K + S)

dS/dt = ((-1/Y) *u*S*X )/(K + S)

这些方程是使用scipy odeint函数求解的。集成后的结果被存储到两个向量中，一个用于增长，另一个用于降级。下一步是将该模型与实验观察到的数据进行曲线拟合，并估计模型参数: u、K和Y。一旦运行代码，就会产生以下错误：

File "C:\ProgramData\Anaconda3\lib\site-packages\scipy\optimize\minpack.py", line 392, in leastsq
    raise TypeError('Improper input: N=%s must not exceed M=%s' % (n, m))

TypeError: Improper input: N=3 must not exceed M=2"

为了方便起见，曲线拟合部分被注释掉，这样就可以生成预期结果的曲线图。下面是代码示例：

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint
from scipy.optimize import curve_fit

"""Experimental data!"""
t_exp = np.array([0, 8, 24, 32, 48, 96, 168])
S_exp = np.array([5.5, 4.7, 3.7, 2.5, 1.5, 0.7, 0.5])
X_exp = np.array([10000, 17000, 30000, 40000, 60000, 76000, 80000])

"Model of the microbial growth and the TOC degradation"
# SETTING UP THE MODEL
def f(t, u, K, Y):
     'Function that returns mutually dependent variables X and S'
     def growth(x, t):
         X = x[0]
         S = x[1]
         "Now differential equations are defined!"
         dXdt = (u * S * X )/(K + S)
         dSdt = ((-1/Y) * u * S * X )/(K + S)
         return [dXdt, dSdt]
     # INTEGRATING THE DIFFERENTIAL EQUATIONS
     "initial Conditions"
     init = [10000, 5]
     results = odeint(growth, init, t)
     "Taking out desired column vectors from results array"
     return results[:,0], results[:,1]

# CURVE FITTING AND PARAMETER ESTIMATION
"""k, kcov = curve_fit(f, t_exp, [X_exp, S_exp], p0=(1, 2, 2))
u = k[0]
K = k[1]
Y = k[2]"""

# RESULTS OF THE MODEL WITH THE ESTIMATED MODEL PARAMETERS
t_mod = np.linspace(0, 168, 100)
compute = f(t_mod, 0.8, 75, 13700)# these fit quite well, but estimated manually
X_mod = compute[0]
S_mod = compute[1]

# PLOT OF THE MODEL AND THE OBSERVED DATA
fig = plt.figure()
ax1 = fig.add_subplot(111)
ax1.plot(t_exp, X_exp, "yo")
ax1.plot(t_mod, X_mod, "g--", linewidth=3)
ax1.set_ylabel("X")

ax2 = ax1.twinx()
ax2.plot(t_exp, S_exp, "mo", )
ax2.plot(t_mod, S_mod, "r--", linewidth=3)
ax2.set_ylabel("S", color="r")
for tl in ax2.get_yticklabels():
    tl.set_color("r")
plt.show()

任何关于如何处理这个问题并进一步进行的建议都将受到高度赞赏。提前谢谢。

Answer 1

f()的结果需要与作为第三个参数输入到curve_fit中的实验数据具有相同的形状。在f()的最后一行，您只需获取两个ODE的解决方案的t=0s值并返回该值，但您应该返回完整的解决方案。当使用curve_fit一次拟合多组数据时，只需将它们连接起来(水平堆叠)，即

def f(t, u, K, Y):
   .....
   return np.hstack((results[:,0], results[:,1]))

并像这样调用curve_fit

k, kcov = curve_fit(f, t_exp, np.hstack([X_exp, S_exp]), p0=(1, 2, 2))

您还必须调整脚本的绘图部分：

compute = f(t_mod, u, K, Y)
compute = compute.reshape((2,-1))