衰减余弦函数曲线拟合的可能方法[回归]
问题描述
我有一组数据，我想用衰减余弦函数来拟合这些数据，即"；A*.cos(K*x).exp(-Bx)"；。为此，我使用了以下代码，但适配性非常差。有没有人能建议我找一件最合适的？X和Y数据如下：
x=[0, 1.3, 1.7, 1.72, 1.84, 1.98, 2.02, 2.16, 2.2, 2.2, 2.3, 2.38, 2.5, 2.55, 2.75, 2.8, 2.82, 2.84, 2.9, 2.92, 3.1, 3.13, 3.18, 3.19, 3.22, 3.3, 3.38, 3.44, 3.49, 3.62, 3.64, 3.72, 3.72, 3.75, 3.8, 3.82, 3.86, 3.92, 4.0, 4.07, 4.1, 4.1, 4.13, 4.14, 4.14, 4.17, 4.21, 4.24, 4.24, 4.24, 4.28, 4.3, 4.38, 4.49, 4.62, 4.62, 4.67, 4.72, 4.73, 4.74, 4.76, 4.76, 4.81, 4.81, 4.88, 4.89, 4.9, 4.9, 4.94, 4.96, 5.03, 5.05, 5.06, 5.1, 5.1, 5.15, 5.16, 5.16, 5.19, 5.22, 5.22, 5.3, 5.37, 5.41, 5.46, 5.56, 5.63, 5.65, 5.65, 5.73, 5.76, 5.81, 5.86, 5.91, 5.98, 6.03, 6.05, 6.05, 6.06, 6.11, 6.14, 6.22, 6.25, 6.27, 6.27, 6.3, 6.3, 6.31, 6.36, 6.42, 6.42, 6.47, 6.48, 6.5, 6.51, 6.58, 6.59, 6.62, 6.65, 6.66, 6.67, 6.69, 6.72, 6.77, 6.8, 6.84, 6.87, 6.91, 6.94, 6.94, 6.94, 7.05, 7.14, 7.17, 7.22, 7.23, 7.24, 7.32, 7.32, 7.35, 7.38, 7.4, 7.41, 7.42, 7.44, 7.45, 7.49, 7.5, 7.52, 7.54, 7.6, 7.72, 7.75, 7.81, 7.9, 7.92, 7.95, 7.97, 7.98, 7.99, 8.02, 8.03, 8.03, 8.05, 8.06, 8.07, 8.1, 8.12, 8.14, 8.19, 8.2, 8.21, 8.24, 8.25, 8.28, 8.28, 8.29, 8.32, 8.38, 8.38, 8.43, 8.49, 8.52, 8.54, 8.54, 8.57, 8.7, 8.75, 8.75, 8.78, 8.79, 8.88, 8.88, 8.93, 8.95, 9.0, 9.01, 9.02, 9.03, 9.06, 9.07, 9.11, 9.14, 9.16, 9.17, 9.18, 9.19, 9.2, 9.3, 9.33, 9.44, 9.46, 9.59, 9.62, 9.62, 9.64, 9.66, 9.71, 9.73, 9.73, 9.75, 9.76, 9.76, 9.79, 9.88, 9.9, 9.93, 9.93, 9.95, 9.99, 10.01, 10.03, 10.04, 10.05, 10.07, 10.11, 10.13, 10.18, 10.22, 10.22, 10.31, 10.37, 10.38, 10.41, 10.42, 10.44, 10.5, 10.52, 10.55, 10.56, 10.56, 10.58, 10.6, 10.66, 10.68, 10.68, 10.69, 10.7, 10.73, 10.75, 10.81, 10.93, 10.96, 10.98, 10.98, 11.02, 11.04, 11.1, 11.14, 11.15, 11.15, 11.17, 11.19, 11.21, 11.23, 11.24, 11.28, 11.3, 11.31, 11.32, 11.33, 11.4, 11.42, 11.48, 11.5, 11.51, 11.6, 11.62, 11.62, 11.63, 11.65, 11.72, 11.74, 11.74, 11.94, 11.95, 11.98, 12.02, 12.02, 12.03, 12.04, 12.09, 12.11, 12.17, 12.2, 12.23, 12.26, 12.3, 12.31, 12.33, 12.33, 12.37, 12.38, 12.61, 12.63, 12.69, 12.7, 12.74, 12.79, 12.8, 12.84, 12.87, 12.9, 12.91, 12.92, 12.94, 13.0, 13.19, 13.2, 13.26, 13.29, 13.3, 13.31, 13.31, 13.34, 13.35, 13.36, 13.44, 13.48, 13.52, 13.59, 13.78, 13.83, 13.88, 13.98, 14.02, 14.05, 14.07, 14.1, 14.14, 14.19, 14.25, 14.33, 14.36, 14.38, 14.41, 14.46, 14.47, 14.53, 14.54, 14.57, 14.69, 14.72, 14.77, 14.78, 14.78, 14.8, 14.82, 14.82, 14.91, 14.92, 14.96, 14.96, 15.05, 15.09, 15.17, 15.2, 15.21, 15.25, 15.26, 15.31, 15.32, 15.36, 15.36, 15.4, 15.4, 15.4, 15.41, 15.47, 15.52, 15.6, 15.61, 15.61, 15.63, 15.65, 15.71, 15.77, 15.8, 15.84, 15.86, 15.88, 15.94, 15.94, 15.97, 15.98, 16.02, 16.03, 16.27, 16.43, 16.56, 16.64, 16.64, 16.64, 16.68, 16.88, 16.91, 16.92, 16.93, 16.97, 16.99, 17.0, 17.01, 17.02, 17.05, 17.13, 17.21, 17.32, 17.45, 17.59, 17.79, 17.8, 17.81, 17.87, 17.9, 17.92, 17.93, 17.93, 17.97, 17.98, 18.02, 18.05, 18.08, 18.11, 18.2, 18.24, 18.4, 18.48, 18.5, 18.51, 18.59, 18.65, 18.76, 18.76, 18.86, 18.86, 18.86, 18.87, 18.9, 18.92, 18.93, 19.05, 19.06, 19.17, 19.26, 19.27, 19.41, 19.47, 19.48, 19.54, 19.6, 19.66, 19.67, 19.68, 19.8, 19.9, 20.01, 20.04, 20.1, 20.49, 20.49, 20.5, 20.56, 20.65, 20.65, 20.7, 20.71, 20.78, 20.91, 21.11, 21.19, 21.2, 21.28, 21.29, 21.58, 21.62, 21.7, 21.7, 21.76, 21.76, 21.84, 21.85, 21.87, 21.9, 21.94, 22.0, 22.02, 22.09, 22.16, 22.3, 22.3, 22.41, 22.51, 22.53, 22.71, 22.77, 22.94, 23.17, 23.25, 23.33, 23.72, 23.87, 24.12, 24.14, 24.19, 24.34, 24.4, 24.6, 24.62, 24.62, 24.8, 25.01, 25.13, 25.4, 25.42, 25.81, 25.85, 25.89, 26.03, 26.17, 26.22, 26.41, 26.98, 27.01, 27.02, 27.06, 27.17, 27.49, 27.73, 28.14, 28.23, 28.37, 28.56, 28.83, 28.84, 30.32, 30.57, 31.95, 33.23, 33.46, 33.81, 33.85, 34.44]
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from scipy.optimize import curve_fit
import numpy as np
from scipy.optimize import least_squares
x = np.array(x)
y = np.array(y)
def fit_cosine(x, y):
guess_amp = 1 # A
guess_wave_no = .05# K
guess_dampcoeff = 0 # B
guess = np.array([guess_amp, guess_wave_no, guess_dampcoeff])
def cos_func(x, A, K, B): return A * np.cos(K*x)*np.exp(-1*B*x)
popt, pcov = curve_fit(cos_func, x, y, p0=guess, maxfev=5000)
A, K, B = popt
fitfunc = lambda x: A * np.cos(K*x)* np.exp(-1*B*x)
return {"amp": A, "omega": K, "decay": B, "fitfunc": fitfunc, "maxcov": np.max(pcov), "rawres": (guess, popt, pcov)}
import matplotlib.pyplot as plt
res = fit_cosine(x, y)
print("Amplitude=%(amp)s, WaveNumber =%(omega)s, DampingCoefficient = %(decay)s, Max. Cov.=%(maxcov)s" % res)
plt.figure()
plt.scatter(x, y, label="Coherence")
#plt.plot(x, res["fitfunc"](x), "r-", label="y fit curve", linewidth=2)
plt.legend(loc="best")
plt.show()
from scipy.optimize import curve_fit, least_squares
import numpy as np
import itertools
import matplotlib.pyplot as plt
x = np.array(x)
y = np.array(y)
A=1
K = .2
B = .03# B
guess = np.array([A, K, B])
def cos_func(x, A, K, B):
y = (1-A+A*np.cos(K * x))*np.exp(-1 * B * x)
return y
# Method 02: Non-linear least-squares with bounds on the variables
def model(x,A,K,B):
return (1-A+A*np.cos(K * x))*np.exp(-1 * B * x)
def error(params):
A, K, B = params
model_func = model(x, A, K, B)
z = y - model_func
return z
f_scale = 0.1
ls_bound = least_squares(error, x0, loss='soft_l1', f_scale=f_scale)
y_fitting = model(x, *ls_bound.x)
#########################
# non-linear least squares to fit the damped cosine function
fit_val, fit_cov = curve_fit(cos_func, x, y, p0=guess, method='trf')
fit_cosine = cos_func(x, *fit_val)
plt.figure()
colors = itertools.cycle(["r", "b", "c", "m", "c", "r"])
plt.scatter(x, y, color=next(colors), s=6, label='Data')
plt.plot(x, fit_cosine, "k--", linewidth=1, label=(f'Non-linear_LSM'))
plt.plot(x, y_fitting, "b--", linewidth=1, label=(f'Non-linear_LSM_bounds,f_scale = {f_scale}'))
plt.ylim(-1, 1.1)
plt.xlim(0, 35)
plt.rcParams.update({'font.size': 17})
plt.xlabel("Distance [Meter]")
plt.ylabel("Real part of coherence")
plt.title('Curve fitting of a damped cosine function')
plt.grid(True)
plt.legend(loc='best')
剧情点评：
- 从图中可以看出，带边界的非线性最小二乘法在区域(‘x=3’到‘x=14’)中的拟合效果更好，而另一种方法在绘图的其余部分表现更好。
- 这两条曲线都是由相同的初始条件生成的
- 如果我降低‘f_Scale’，曲线图将显示脉动性质(趋于不稳定！)。
- 我相信有很多技术可以很好地拟合数据，但我并不了解所有这些技术。任何人都可以站出来向我展示/建议我更好地拟合数据。
我们将感谢您在这些上下文中提出的任何建议。
解决方案
由于高度分散，很难进行拟合。
在方程中加入参数C后，拟合度明显提高。
此外，为了回答一些评论：
导致上述结果的方法包括两个步骤。
第一步：非迭代的非常规方法不需要参数的初始值。本文阐述了其一般原理：https://fr.scribd.com/doc/14674814/Regressions-et-equations-integrales
由于本文中没有明确处理函数y(X)=(A*cos(K*x)+C)*exp(-B*x)的情况，因此该函数的应用如下：
不熟悉这种方法的人在编码时出错并说演算失败的情况并不少见。在错误被发现和纠正之前，很多时间都浪费在讨论上。为了避免浪费时间，下面提供了一张测试表。使用非常小的数据，用户很容易在将每个中间数值与正确的值进行比较时检查他的代码。
然后，该方法可以用于OP给出的大数据。结果是：
参数的值与我开始回答时第一个图中给出的值不完全相同，这并不奇怪。这是因为拟合的标准不同。
在他的问题中，OP没有指定拟合标准(LMSE、LMAE或LMSRE等)。对于每个拟合标准，对应不同的结果。当散布很大时，结果可能会与其他结果截然不同。由于在目前的情况下，散布非常大，所以人们不可避免地要选择特定的拟合标准。如果不是，结果就不是唯一的。这就是为什么在目前的情况下第二步是必要的。但这不是普遍的必需品。
第二步(最终)：
我们必须选择一个适合的标准。例如，最小均方误差。
必须使用非线性回归方法(其中实现了所选标准)。它们是大量的软件。微积分是迭代的，用户必须给出一些猜测的初始值才能开始迭代。在大散布的情况下，收敛并不总是很好。如果初始值与未知的正确值不接近，则最终失败的结果可能远远不是很好。由于上述第一步，这是(部分)避免的。可以使用上述K、B、A、C的值作为相当好的初始值。这就是为了计算我答案中第一个数字上写的值所做的事情。这解释了第一个数字与最后一个数字不同的原因。
注意：
老实说，必须承认上面的方法并不是万无一失的，特别是在散布很大的情况下。我很惊讶能得到一个不太差的结果。对于三次数值积分，我预计会有更大的困难。当然，大量的积分是有利的。也许我们有这个数据是幸运的。使用其他数据集可能会产生更差的结果。相关文章