如何执行多条曲线的联合拟合（在 Python 中）？

Question

假设我通过简单的线性回归拟合一些数据点。现在我想对几组数据点执行几个联合线性回归。更具体地说，我希望一个参数在所有拟合中都相等，这里示意性地描述了 y 轴交点。

在 Google 上搜索了一段时间后，我既找不到任何 Python (Scipy) 例程可以做到这一点，也找不到任何一般文献，说明如何做到这一点。

理想情况下，我希望不仅在简单线性回归的情况下执行这些联合拟合，还希望对更一般的拟合函数（例如，幂律与联合指数拟合）进行拟合。

Answer 1

lmfit 模块允许您执行此操作，正如他们的FAQ 中所述：

from lmfit import minimize, Parameters, fit_report
import numpy as np

# residual function to minimize
def fit_function(params, x=None, dat1=None, dat2=None):

    model1 = params['offset'] + x * params['slope1']
    model2 = params['offset'] + x * params['slope2']

    resid1 = dat1 - model1
    resid2 = dat2 - model2
    return np.concatenate((resid1, resid2))

# setup fit parameters
params = Parameters()
params.add('slope1', value=1)
params.add('slope2', value=-1)
params.add('offset', value=0.5)

# generate sample data
x = np.arange(0, 10)
slope1, slope2, offset = 1.1, -0.9, 0.2
y1 = slope1 * x + offset
y2 = slope2 * x + offset

# fit
out = minimize(residual, params, kws={"x": x, "dat1": y1, "dat2": y2})
print(fit_report(out))
# [[Fit Statistics]]
#     # fitting method   = leastsq
#     # function evals   = 9
#     # data points      = 20
#     # variables        = 3
#     chi-square         = 1.4945e-31
#     reduced chi-square = 8.7913e-33
#     Akaike info crit   = -1473.48128
#     Bayesian info crit = -1470.49408
# [[Variables]]
#     slope1:  1.10000000 +/- 8.2888e-18 (0.00%) (init = 1)
#     slope2: -0.90000000 +/- 8.2888e-18 (0.00%) (init = -1)
#     offset:  0.20000000 +/- 3.8968e-17 (0.00%) (init = 0.5)
# [[Correlations]] (unreported correlations are < 0.100)
#     C(slope1, offset) = -0.742
#     C(slope2, offset) = -0.742
#     C(slope1, slope2) =  0.551

Answer 2

我认为这个图形代码示例可以满足您的需求，用一个共享参数拟合两个数据集。请注意，如果数据集的长度不等，则可以有效地将拟合加权到具有更多单个点的数据集。此示例将初始参数值明确设置为 1,0 - curve_fit() 默认值 - 并且不使用 scipy 的遗传算法来帮助找到初始参数估计值。

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

y1 = np.array([ 16.00,  18.42,  20.84,  23.26])
y2 = np.array([-20.00, -25.50, -31.00, -36.50, -42.00])
comboY = np.append(y1, y2)

x1 = np.array([5.0, 6.1, 7.2, 8.3])
x2 = np.array([15.0, 16.1, 17.2, 18.3, 19.4])
comboX = np.append(x1, x2)

if len(y1) != len(x1):
    raise(Exception('Unequal x1 and y1 data length'))
if len(y2) != len(x2):
    raise(Exception('Unequal x2 and y2 data length'))


def function1(data, a, b, c): # not all parameters are used here, c is shared
        return a * data + c

def function2(data, a, b, c): # not all parameters are used here, c is shared
        return b * data + c


def combinedFunction(comboData, a, b, c):
    # single data reference passed in, extract separate data
    extract1 = comboData[:len(x1)] # first data
    extract2 = comboData[len(x1):] # second data

    result1 = function1(extract1, a, b, c)
    result2 = function2(extract2, a, b, c)

    return np.append(result1, result2)


# some initial parameter values
initialParameters = np.array([1.0, 1.0, 1.0])

# curve fit the combined data to the combined function
fittedParameters, pcov = curve_fit(combinedFunction, comboX, comboY, initialParameters)

# values for display of fitted function
a, b, c = fittedParameters

y_fit_1 = function1(x1, a, b, c) # first data set, first equation
y_fit_2 = function2(x2, a, b, c) # second data set, second equation

plt.plot(comboX, comboY, 'D') # plot the raw data
plt.plot(x1, y_fit_1) # plot the equation using the fitted parameters
plt.plot(x2, y_fit_2) # plot the equation using the fitted parameters
plt.show()

print('a, b, c:', fittedParameters)