所以这里面临的问题是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是这样的。
dXdt = (u * S * X )(K + S)
dSdt = (((-1Y) * u * S * X )(K + S)
这些方程使用scipy odeint函数进行求解。整合后的结果被存储到两个向量中,一个是增长,另一个是退化。下一步是将这个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。代码运行后,会产生以下误差。
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"
为方便起见,曲线拟合部分被注释掉了 所以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()
如果有任何关于如何处理这个问题的建议,并继续进行下去,我将非常感激。先谢谢你了。
的结果是 f()
需要与您输入的实验数据形状相同。curve_fit
作为第三个参数。在最后一行的 f()
你只需取两个ODE的解的t=0s值并返回,但你应该返回完整的解。当使用以下方法同时拟合几组数据时 curve_fit
,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]))
并像调用curve_fit那样调用
k, kcov = curve_fit(f, t_exp, np.hstack([X_exp, S_exp]), p0=(1, 2, 2))
你也要调整剧本的情节部分。
compute = f(t_mod, u, K, Y)
compute = compute.reshape((2,-1))