将曲线拟合（python）与高斯函数结合使用，可获得均值和方差

我在使用

scipy

中的

curve\u fit

时遇到问题。或者至少它没有按照我期望的方式工作

我有一些来自直方图的数据集，x值是直方图的大小，作为numpy数组。如果需要，我将在以后添加数据。直方图几乎是高斯形状，我想用两个自由参数来拟合高斯函数，但它不起作用

使用三个参数，代码工作正常：

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

hist=np.load('Rinnen.npy') 
faktor = np.sum(hist)

norm_hist=hist/faktor # values from the histogram are normed

ref_werte = np.arange(0,1,0.001)

def gauss(x, *p):
  a, b, c = p
  y = a*np.exp(-0.5*((x - b)/c)**2.)

  return y

p_initial = [0.1, 0.0, 0.1] 

popt, pcov = curve_fit(gauss, ref_werte, norm_hist, p0=p_initial)

print(popt) #zeigen der koeffizienten

plt.figure()
plt.plot(ref_werte, norm_hist, linewidth=2.0, color='b')
plt.plot(ref_werte, gauss(ref_werte, *popt), 'b-', linewidth=2.0, color='r')
plt.xlabel('Reflektanzen') 
plt.ylabel('normierte Häufigkeit')
plt.show()

但我的目标是使用高斯分布，它是正态分布的PDF（参见）。但是，当我更改代码并使用一个新的函数定义（如下面的定义）时，它会把一切都搞砸，根本不起作用

def gauss(x, *p):
  b, c = p
  kons = np.sqrt(2.*np.pi)
  y = (1./(c*kons))*np.exp(-0.5*((x - b)/c)**2.)

  return y

即使我使用非常接近直方图的

p_initial

值，比如

p_initial=[0.08,0.02]

也不起作用，我真的不明白为什么

如果有人能帮助我，我会非常高兴的

编辑：代码示例

hist

的一个示例数组是：

array([    0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     4,
          30,   224,  2257,  3603,  2029,  2412,  1391,  2269,  3789,
        8279,  9091,  6617,  7087,  5071,  2316,  2675,  2273,  3913,
        2299,  3573,  1761,  2445,  2426,  3261,  5881,  8408, 11659,
       15174, 21250, 19644, 32068, 25315, 19329, 23333, 17168, 15748,
       15744, 15045, 14274, 11566, 13887, 10144,  8532, 10696,  8531,
        9687,  9493,  9424, 10294,  8869,  9509,  8445,  7723,  8515,
        7137,  7464,  8006,  6440,  6457,  4999,  5364,  4519,  4361,
        3976,  3366,  3352,  2833,  2475,  2332,  1881,  1905,  1639,
        1568,  1318,  1141,  1130,  1010,   907,   906,   823,   789,
         745,   726,   674,   692,   630,   610,   568,   575,   589,
         535,   538,   522,   511,   513,   534,   467,   446,   445,
         337,   441,   454,   451,   438,   417,   388,   456,   405,
         408,   399,   356,   404,   371,   412,   404,   401,   389,
         354,   342,   358,   317,   306,   303,   295,   303,   294,
         288,   251,   256,   226,   178,   241,   213,   196,   215,
         210,   184,   165,   208,   200,   181,   171,   136,   156,
         147,   137,   102,   119,   116,    89,   117,   104,    85,
          77,    74,    52,    69,    34,    47,    44,    50,    32,
          27,    34,    38,    24,    21,    28,    24,    22,    25,
          19,    17,    15,    17,    18,    14,    11,    12,     5,
           9,     9,     9,     6,     5,     5,     8,     7,     4,
           4,     2,     1,     4,     0,     2,     2,     2,     3,
           2,     3,     1,     1,     2,     1,     2,     2,     0,
           1,     1,     2,     1,     1,     2,     0,     0,     0,
           0,     0,     1,     1,     0,     0,     0,     0,     0,
           0,     0,     1,     0,     0,     0,     0,     0,     0,
           1,     0,     0,     0,     1,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
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           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
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           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,     0])

---

有了这个数组，我的代码与有三个参数的版本一起工作，结果得到了

[0.02697915 0.08060284 0.01334016]

。如果我将代码更改为具有相同数组和所有内容的双参数版本，则。。。那它就不合适了，根本不起作用。我使用了另一个版本的

p_initial=[0.08，0.02]

，它使用了平均值和标准导数。结果一点也不真实：

[-1.62281493 0.53329897]

正态分布是一个连续的分布，因此你不能规范化直方图的和，而是积分

这是一个适合我的代码版本。我唯一改变的是用柱状图除以箱子的宽度。它返回

[0.08083458 0.01470529]

：

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


hist = np.array([    0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     4,
          30,   224,  2257,  3603,  2029,  2412,  1391,  2269,  3789,
        8279,  9091,  6617,  7087,  5071,  2316,  2675,  2273,  3913,
        2299,  3573,  1761,  2445,  2426,  3261,  5881,  8408, 11659,
       15174, 21250, 19644, 32068, 25315, 19329, 23333, 17168, 15748,
       15744, 15045, 14274, 11566, 13887, 10144,  8532, 10696,  8531,
        9687,  9493,  9424, 10294,  8869,  9509,  8445,  7723,  8515,
        7137,  7464,  8006,  6440,  6457,  4999,  5364,  4519,  4361,
        3976,  3366,  3352,  2833,  2475,  2332,  1881,  1905,  1639,
        1568,  1318,  1141,  1130,  1010,   907,   906,   823,   789,
         745,   726,   674,   692,   630,   610,   568,   575,   589,
         535,   538,   522,   511,   513,   534,   467,   446,   445,
         337,   441,   454,   451,   438,   417,   388,   456,   405,
         408,   399,   356,   404,   371,   412,   404,   401,   389,
         354,   342,   358,   317,   306,   303,   295,   303,   294,
         288,   251,   256,   226,   178,   241,   213,   196,   215,
         210,   184,   165,   208,   200,   181,   171,   136,   156,
         147,   137,   102,   119,   116,    89,   117,   104,    85,
          77,    74,    52,    69,    34,    47,    44,    50,    32,
          27,    34,    38,    24,    21,    28,    24,    22,    25,
          19,    17,    15,    17,    18,    14,    11,    12,     5,
           9,     9,     9,     6,     5,     5,     8,     7,     4,
           4,     2,     1,     4,     0,     2,     2,     2,     3,
           2,     3,     1,     1,     2,     1,     2,     2,     0,
           1,     1,     2,     1,     1,     2,     0,     0,     0,
           0,     0,     1,     1,     0,     0,     0,     0,     0,
           0,     0,     1,     0,     0,     0,     0,     0,     0,
           1,     0,     0,     0,     1,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
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           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
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           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,
           0,     0,     0,     0,     0,     0,     0,     0,     0,     0])

faktor = np.sum(hist)

ref_werte = np.arange(0,1,0.001)

bin_width = 0.001

norm_hist=hist/(bin_width*faktor) # values from the histogram are normed

def gauss(x, *p):
  b, c = p
  kons = np.sqrt(2.*np.pi)
  y = (1./(c*kons))*np.exp(-0.5*((x - b)/c)**2.)

  return y

p_initial = [0, 1] 

popt, pcov = curve_fit(gauss, ref_werte, norm_hist, p0=p_initial)

print(popt) #zeigen der koeffizienten

plt.figure()
plt.plot(ref_werte, norm_hist, linewidth=2.0, color='b')
plt.plot(ref_werte, gauss(ref_werte, *popt), 'b-', linewidth=2.0, color='r')
plt.xlabel('Reflektanzen') 
plt.ylabel('normierte Häufigkeit')
plt.show()

---

如果平均值接近0.08，那么使用

np.arange（0,1,0.001）

作为bin中心是没有意义的。也许你可以用你的数据样本创建一个。我已经添加了它。：）我将使用代码的结果相互比较不同的直方图。。。“因此，规范它是很重要的。”玛拉，我想你误解了。我不是说你不应该正常化。我想说的是我怀疑你没有正常化。你似乎以这样一种方式规范化了，总和变成了1。这是不对的。你希望积分为1，这意味着你必须考虑箱子的宽度。啊，是的。我误解了你。但是两个版本中的错误是相同的，我认为这并不能解释为什么一个版本可以工作，而另一个版本不能。@Mara它确实解释了差异：在三参数拟合中，比例是拟合参数（a）之一。因此，在这种情况下，即使缩放错误，拟合程序也会通过调整a进行补偿。在双参数版本中，这是不可能的。无论如何，对于我来说，将直方图除以

0.001

有效，请参阅更新的帖子。霍克，谢谢你。：）我想我需要读一点关于规范化的内容。