Python和lmfit：如何使用共享参数拟合多个数据集？

Question

我想使用lmfit模块来使函数适合可变数量的数据集，并带有一些共享参数和一些单独参数。

这是生成高斯数据并分别适合每个数据集的示例：

import numpy as np
import matplotlib.pyplot as plt
from lmfit import minimize, Parameters, report_fit

def func_gauss(params, x, data=[]):
    A = params['A'].value
    mu = params['mu'].value
    sigma = params['sigma'].value
    model = A*np.exp(-(x-mu)**2/(2.*sigma**2))

    if data == []:
        return model
    return data-model

x  = np.linspace( -1, 2, 100 )
data = []
for i in np.arange(5):
    params = Parameters()
    params.add( 'A'    , value=np.random.rand() )
    params.add( 'mu'   , value=np.random.rand()+0.1 )
    params.add( 'sigma', value=0.2+np.random.rand()*0.1 )
    data.append(func_gauss(params,x))

plt.figure()
for y in data:
    fit_params = Parameters()
    fit_params.add( 'A'    , value=0.5, min=0, max=1)
    fit_params.add( 'mu'   , value=0.4, min=0, max=1)
    fit_params.add( 'sigma', value=0.4, min=0, max=1)
    minimize(func_gauss, fit_params, args=(x, y))
    report_fit(fit_params)

    y_fit = func_gauss(fit_params,x)
    plt.plot(x,y,'o',x,y_fit,'-')
plt.show()


# ideally I would like to write:
#
# fit_params = Parameters()
# fit_params.add( 'A'    , value=0.5, min=0, max=1)
# fit_params.add( 'mu'   , value=0.4, min=0, max=1)
# fit_params.add( 'sigma', value=0.4, min=0, max=1, shared=True)
# minimize(func_gauss, fit_params, args=(x, data))
#
# or:
#
# fit_params = Parameters()
# fit_params.add( 'A'    , value=0.5, min=0, max=1)
# fit_params.add( 'mu'   , value=0.4, min=0, max=1)
#
# fit_params_shared = Parameters()
# fit_params_shared.add( 'sigma', value=0.4, min=0, max=1)
# call_function(func_gauss, fit_params, fit_params_shared, args=(x, data))

Answer 1

我认为您就在那儿。您需要将数据集放到一个数组或结构中，以供您使用一个单独的全局目标函数，该函数提供给minimum（）并使用所有数据集的单个参数集来适合所有数据集。您可以根据需要在数据集之间共享此集。稍微扩展一下示例，下面的代码确实可以对5种不同的高斯函数进行一次拟合。对于跨数据集绑定参数的示例，我对5个数据集的sigma使用了几乎相同的值。我创建了5个不同的sigma参数（“
sig_1”，“ sig_2”，…，“ sig_5”），然后使用数学约束将其设置为相同的值。因此，问题中有11个变量，而不是15个。

import numpy as np
import matplotlib.pyplot as plt
from lmfit import minimize, Parameters, report_fit

def gauss(x, amp, cen, sigma):
    "basic gaussian"
    return amp*np.exp(-(x-cen)**2/(2.*sigma**2))

def gauss_dataset(params, i, x):
    """calc gaussian from params for data set i
    using simple, hardwired naming convention"""
    amp = params['amp_%i' % (i+1)].value
    cen = params['cen_%i' % (i+1)].value
    sig = params['sig_%i' % (i+1)].value
    return gauss(x, amp, cen, sig)

def objective(params, x, data):
    """ calculate total residual for fits to several data sets held
    in a 2-D array, and modeled by Gaussian functions"""
    ndata, nx = data.shape
    resid = 0.0*data[:]
    # make residual per data set
    for i in range(ndata):
        resid[i, :] = data[i, :] - gauss_dataset(params, i, x)
    # now flatten this to a 1D array, as minimize() needs
    return resid.flatten()

# create 5 datasets
x  = np.linspace( -1, 2, 151)
data = []
for i in np.arange(5):
    params = Parameters()
    amp   =  0.60 + 9.50*np.random.rand()
    cen   = -0.20 + 1.20*np.random.rand()
    sig   =  0.25 + 0.03*np.random.rand()
    dat   = gauss(x, amp, cen, sig) + np.random.normal(size=len(x), scale=0.1)
    data.append(dat)

# data has shape (5, 151)
data = np.array(data)
assert(data.shape) == (5, 151)

# create 5 sets of parameters, one per data set
fit_params = Parameters()
for iy, y in enumerate(data):
    fit_params.add( 'amp_%i' % (iy+1), value=0.5, min=0.0,  max=200)
    fit_params.add( 'cen_%i' % (iy+1), value=0.4, min=-2.0,  max=2.0)
    fit_params.add( 'sig_%i' % (iy+1), value=0.3, min=0.01, max=3.0)

# but now constrain all values of sigma to have the same value
# by assigning sig_2, sig_3, .. sig_5 to be equal to sig_1
for iy in (2, 3, 4, 5):
    fit_params['sig_%i' % iy].expr='sig_1'

# run the global fit to all the data sets
result = minimize(objective, fit_params, args=(x, data))
report_fit(result.fit_params)

# plot the data sets and fits
plt.figure()
for i in range(5):
    y_fit = gauss_dataset(result.fit_params, i, x)
    plt.plot(x, data[i, :], 'o', x, y_fit, '-')

plt.show()

对于它的价值，我会考虑将多个数据集保存在字典或DataSet类列表中，而不是多维数组。无论如何，我希望这能帮助您进入真正需要做的事情。