Leastsq 不适合给定的数据

如何解决Leastsq 不适合给定的数据

我正在尝试使用最小二乘拟合通过一些收集的数据拟合低音模型。虽然我的代码没有报错，但是leastsq给出的参数值和我的数据根本不匹配。有没有人知道如何解决这个问题？ 我很确定问题出在 leastsq 函数中，因为 varfinal[0] 的值太低了。我已经在其他数据集上尝试过代码，它在那里工作得很好...... 我正在使用的代码如下：

from scipy.optimize import leastsq
from scipy.optimize import curve_fit
import matplotlib.pyplot as plt
import numpy as np
#time intervals
t = np.arange(1960,2021,1)
t0 = np.arange(0,len(t),1)
#t= [1973,1974,1975,1976,1977,1978,1979,1980,1981,1982,1983,1984,1985,1986,1987,1988,1989,1990,1991,1992,1993,1994,1995,1996,1997,1998,1999,2000,2001,2002,2003,2004,2005,2006,2007,2008,2009,2010,2011,2012,2013,2014,2015,2016,2017,2018,2019,2020]

#xlabels
x = np.arange(1960,2080,20)
x_len = len(np.arange(1960,2100,1))
#places of xlabels
x0 = np.arange(0,x_len,20)

# papers vector
#keywords AND
patents=np.array([0,2,1,8,7,24,33,30,36,78,86,96,67,82,107,90,123,160,116,177,143,246,271,552,736,1137,1857,2681,3329,4044,5064,6318,6928,8843,9531,13560,14665,16801,19945,21421,25584,34608,43150,49742,53006,56061,61777,66665,64216,60540])
# cumulatice papers
cum_patents=np.cumsum(patents)+3
# initial variables(M,P & Q)
vars = [600000,0.03,0.38]

# residual (error) function
def residual(vars,t,papers):
    M = vars[0]
    P = vars[1]
    Q = vars[2]
    Bass = M * (((P+Q)**2/P)*np.exp(-(P+Q)*t))/(1+(Q/P)*np.exp(-(P+Q)*t))**2 
    return (Bass - (papers))
                 
# non linear least square fitting
varfinal,success = leastsq(residual,vars,args=(t0,patents))

# estimated coefficients
m = varfinal[0]
p = varfinal[1]
q = varfinal[2]


print(varfinal)
#time interpolation
tp=(np.linspace(1.0,110.0,num=100))
cofactor= np.exp(-(p+q) * tp)

# Cumulative sales (cdf)
patents_cdf= m*(1-cofactor)/(1+(q/p)*cofactor)
plt.figure(dpi=100)
plt.plot(tp,patents_cdf,"r")
plt.grid(True)
plt.scatter(t0,cum_patents)
plt.title('Total patent applicants')
plt.legend(['Fit','True'])
plt.ylabel("Cumulative patent applicants")
plt.xlabel("Years")
plt.xticks(x0,x)
plt.savefig("lib_number_of_patents_bass_fit.png")
plt.show()

变量的输出为：

[24.24333383  0.07548748 -0.32937682]

True data vs leastsq fit