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次联合线性回归。更具体地说，我希望所有拟合中的一个参数都相等，此处示意性地描述了 y 轴交点。

在 Google 搜索了一段时间后，我既找不到任何执行此操作的 Python (Scipy) 例程，也找不到任何一般文献来说明如何完成此操作。

理想情况下，我不仅希望在简单线性回归的情况下执行这些联合拟合，而且还希望针对更一般的拟合函数（例如，联合指数的幂律拟合）执行这些联合拟合。

Answer 1

该lmfit模块允许您执行此操作，如常见问题解答中所述：

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