Python lmfit 给出非常小的卡方；如何缩放残差

Question

我的 NIR 光谱 (x,y) 文件不提供错误信息。我正在做一个黑体加幂律拟合，代码如下；根据生成的参数值和相应的图，它似乎可以正常工作。然而，卡方值非常小，如下面的示例所示。文档说应该正确缩放残差。执行此操作的确切步骤是什么？感谢您的帮助。

def bb(x, T, const):
    from scipy.constants import h,k,c
    x = 1e-6 * x 
    return const*2*h*c**2 / (x**5 * (np.exp(h*c / (x*k*T)) - 1)) 

def powerlaw(x,A,p):
    return A*x**p


mod= Model(bb) + Model(powerlaw)
pars  = mod.make_params(T=2000,const=2*1e-21,A=2*np.average(y),p=-1.0)                          
result = mod.fit(y,pars,x=x)
print((result.fit_report()))

#Parameters
T=    (result.params['T'].value)
const=(result.params['const'].value)
A=    (result.params['A'].value)
p=    (result.params['p'].value)
---------------------------------
 T:      1403.30461 +/- 4.19860373 (0.30%) 
line 67  [[Model]]
    (Model(bb) + Model(powerlaw))
[[Fit Statistics]]
    # fitting method   = leastsq
    # function evals   = 69
    # data points      = 5362
    # variables        = 4
    chi-square         = 5.6981e-29
    reduced chi-square = 1.0635e-32
    Akaike info crit   = -394752.758
    Bayesian info crit = -394726.409
[[Variables]]
    T:      1403.30461 +/- 4.19860373 (0.30%) (init = 2000)
    const:  6.9272e-26 +/- 8.9056e-28 (1.29%) (init = 2e-21)
    A:      2.1975e-15 +/- 7.9268e-18 (0.36%) (init = 4.309166e-15)
    p:     -2.57314708 +/- 0.01807976 (0.70%) (init = -1)

Answer 1

嗯，lmfit 文档唯一说明不确定性规模的地方，它确实说明了这是如何完成的。

见：https://lmfit.github.io/lmfit-py/fitting.html#uncertainties-in-variable-parameters-and-their-correlations

引用：

In principle, the scale of the uncertainties in the Parameters is closely
tied to the goodness-of-fit statistics chi-square and reduced chi-square
(chisqr and redchi). The standard errors or  uncertainties are those that
increase chi-square by 1. Since a “good fit” should have redchi of around 1,
this requires that the data uncertainties (and to some extent the sampling
of the N data points) is correct. Unfortunately, it is often not the case
that one has high-quality estimates of the data uncertainties (getting the
data is hard enough!). Because of this common situation, the uncertainties
reported and held in stderr are not those that increase chi-square by 1, but
those that increase chi-square by reduced chi-square. This is equivalent to
rescaling the uncertainty in the data such that reduced chi-square would be
1. To be clear, this rescaling is done by default because if reduced 
chi-square is far from 1, this rescaling often makes the reported uncertainties 
sensible, and if reduced chi-square is near 1 it does little harm. If you 
have good scaling of the data uncertainty and believe the scale of the
residual array is correct, this automatic rescaling can be turned off using
scale_covar=False.