[英]Fitting Log-normal distribution (Python plot)
我正在嘗試將對數正態分布擬合到直方圖數據。 我嘗試在 Stack Exchange 上關注其他問題的示例，但我沒有得到合適的結果，因為在這種情況下我的軸壞了。 我已經將斷軸放在 plot 上，我試圖防止數字在軸上重疊，我從重復軸上刪除了數字，我減小了第二個子圖的大小，但我無法適應日志-普通的。 如何擬合該數據集的對數正態分布？
代碼：
#amostra 17B (menor intervalo)
import pandas as pd
import matplotlib.pyplot as plt
import numpy as np
from scipy.stats import lognorm
import matplotlib.ticker as tkr
import scipy, pylab
import locale
import matplotlib.gridspec as gridspec
from scipy.stats import lognorm
locale.setlocale(locale.LC_NUMERIC, "de_DE")
plt.rcParams['axes.formatter.use_locale'] = True
frequencia_relativa=[0.000, 0.000, 0.038, 0.097, 0.091, 0.118, 0.070, 0.124, 0.097, 0.059, 0.059, 0.048, 0.054, 0.043,
0.032, 0.005, 0.027, 0.016, 0.005, 0.000, 0.005, 0.000, 0.005, 0.000, 0.000, 0.000, 0.000, 0.000,
0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000,
0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000,
0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000,
0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.005, 0.000, 0.000]
x=[0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, 0.90, 1.00, 1.10, 1.20, 1.30, 1.40,
1.50, 1.60, 1.70, 1.80, 1.90, 2.00, 2.10, 2.20, 2.30, 2.40, 2.50, 2.60, 2.70, 2.80,
2.90, 3.00, 3.10, 3.20, 3.30, 3.40, 3.50, 3.60, 3.70, 3.80, 3.90, 4.00, 4.10, 4.20,
4.30, 4.40, 4.50, 4.60, 4.70, 4.80, 4.90, 5.00, 5.10, 5.20, 5.30, 5.40, 5.50, 5.60,
5.70, 5.80, 5.90, 6.00, 6.10, 6.20, 6.30, 6.40, 6.50, 6.60, 6.70, 6.80, 6.90, 7.00,
7.10, 7.20, 7.30, 7.40, 7.50, 7.60, 7.70, 7.80, 7.90, 8.00]
plt.rcParams["figure.figsize"] = [20,8]
f, (ax,ax2) = plt.subplots(1,2, sharex=True, sharey=True, facecolor='w')
axes = f.add_subplot(111, frameon=False)
ax.spines['top'].set_color('none')
ax2.spines['top'].set_color('none')
gs = gridspec.GridSpec(1,2,width_ratios=[3,1])
ax = plt.subplot(gs[0])
ax2 = plt.subplot(gs[1])
ax.yaxis.tick_left()
ax.xaxis.tick_bottom()
ax2.xaxis.tick_bottom()
ax.tick_params(labeltop='off') # don't put tick labels at the top
ax2.yaxis.tick_right()
ax.bar(x, height=frequencia_relativa, alpha=0.5, width=0.1, align='edge', edgecolor='black', hatch="///")
ax2.bar(x, height=frequencia_relativa, alpha=0.5, width=0.1, align='edge', edgecolor='black', hatch="///")
ax.tick_params(axis = 'both', which = 'major', labelsize = 18)
ax.tick_params(axis = 'both', which = 'minor', labelsize = 18)
ax2.tick_params(axis = 'both', which = 'major', labelsize = 18)
ax2.tick_params(axis = 'both', which = 'minor', labelsize = 18)
ax2.xaxis.set_ticks(np.arange(7.0, 8.5, 0.5))
ax2.xaxis.set_major_formatter(tkr.FormatStrFormatter('%0.1f'))
plt.subplots_adjust(wspace=0.04)
ax.set_xlim(0,2.5)
ax.set_ylim(0,0.14)
ax2.set_xlim(7.0,8.0)
def func(x, pos): # formatter function takes tick label and tick position
s = str(x)
ind = s.index('.')
return s[:ind] + ',' + s[ind+1:] # change dot to comma
x_format = tkr.FuncFormatter(func)
ax.xaxis.set_major_formatter(x_format)
ax2.xaxis.set_major_formatter(x_format)
# hide the spines between ax and ax2
ax.spines['right'].set_visible(False)
ax2.spines['left'].set_visible(False)
# This looks pretty good, and was fairly painless, but you can get that
# cut-out diagonal lines look with just a bit more work. The important
# thing to know here is that in axes coordinates, which are always
# between 0-1, spine endpoints are at these locations (0,0), (0,1),
# (1,0), and (1,1). Thus, we just need to put the diagonals in the
# appropriate corners of each of our axes, and so long as we use the
# right transform and disable clipping.
d = .015 # how big to make the diagonal lines in axes coordinates
# arguments to pass plot, just so we don't keep repeating them
kwargs = dict(transform=ax.transAxes, color='k', clip_on=False)
ax.plot((1-d/3,1+d/3), (-d,+d), **kwargs)
ax.plot((1-d/3,1+d/3),(1-d,1+d), **kwargs)
kwargs.update(transform=ax2.transAxes) # switch to the bottom axes
ax2.plot((-d,+d), (1-d,1+d), **kwargs)
ax2.plot((-d,+d), (-d,+d), **kwargs)
ax2.tick_params(labelright=False)
ax.tick_params(labeltop=False)
ax.tick_params(axis='x', which='major', pad=15)
ax2.tick_params(axis='x', which='major', pad=15)
ax2.set_yticks([])
f.text(0.5, -0.04, 'Tamanho lateral do triângulo ($\mu m$)', ha='center', fontsize=22)
f.text(-0.02, 0.5, 'Frequência relativa', va='center', rotation='vertical', fontsize=22)
#ax.set_xlabel('Tamanho lateral do triângulo ($\mu m$)', fontsize=22)
#ax.set_ylabel('Frequência relativa', fontsize=22)
#x_axis = np.arange(0, 29, 0.001)
#ax.plot(x_axis, norm.pdf(x_axis,2.232,1.888), linewidth=3)
f.tight_layout()
plt.show()
#plt.savefig('output.png', dpi=500, bbox_inches='tight')
嘗試使用curve_fit：
#amostra 17B (menor intervalo)
import pandas as pd
import matplotlib.pyplot as plt
import numpy as np
from scipy.stats import lognorm
import matplotlib.ticker as tkr
import scipy, pylab
import locale
import matplotlib.gridspec as gridspec
from scipy.stats import lognorm
locale.setlocale(locale.LC_NUMERIC, "de_DE")
plt.rcParams['axes.formatter.use_locale'] = True
from scipy.optimize import *
frequencia_relativa=[0.000, 0.000, 0.038, 0.097, 0.091, 0.118, 0.070, 0.124, 0.097, 0.059, 0.059, 0.048, 0.054, 0.043,
0.032, 0.005, 0.027, 0.016, 0.005, 0.000, 0.005, 0.000, 0.005, 0.000, 0.000, 0.000, 0.000, 0.000,
0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000,
0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000,
0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000,
0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.005, 0.000, 0.000]
x=[0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, 0.90, 1.00, 1.10, 1.20, 1.30, 1.40,
1.50, 1.60, 1.70, 1.80, 1.90, 2.00, 2.10, 2.20, 2.30, 2.40, 2.50, 2.60, 2.70, 2.80,
2.90, 3.00, 3.10, 3.20, 3.30, 3.40, 3.50, 3.60, 3.70, 3.80, 3.90, 4.00, 4.10, 4.20,
4.30, 4.40, 4.50, 4.60, 4.70, 4.80, 4.90, 5.00, 5.10, 5.20, 5.30, 5.40, 5.50, 5.60,
5.70, 5.80, 5.90, 6.00, 6.10, 6.20, 6.30, 6.40, 6.50, 6.60, 6.70, 6.80, 6.90, 7.00,
7.10, 7.20, 7.30, 7.40, 7.50, 7.60, 7.70, 7.80, 7.90, 8.00]
plt.rcParams["figure.figsize"] = [20,8]
f, (ax,ax2) = plt.subplots(1,2, sharex=True, sharey=True, facecolor='w')
axes = f.add_subplot(111, frameon=False)
ax.spines['top'].set_color('none')
ax2.spines['top'].set_color('none')
gs = gridspec.GridSpec(1,2,width_ratios=[3,1])
ax = plt.subplot(gs[0])
ax2 = plt.subplot(gs[1])
def f(x, mu, sigma) :
return 1/(np.sqrt(2*np.pi)*sigma*x)*np.exp(-((np.log(x)-
mu)**2)/(2*sigma**2))
params, extras = curve_fit(f, x, frequencia_relativa)
plt.plot(x, f(x ,params[0], params[1]))
print("mu=%g, sigma=%g" % (params[0], params[1]))
plt.subplots_adjust(wspace=0.04)
# hide the spines between ax and ax2
ax.spines['right'].set_visible(False)
ax2.spines['left'].set_visible(False)
d = .015 # how big to make the diagonal lines in axes coordinates
# arguments to pass plot, just so we don't keep repeating them
kwargs = dict(transform=ax.transAxes, color='k', clip_on=False)
ax.plot((1-d/3,1+d/3), (-d,+d), **kwargs)
ax.plot((1-d/3,1+d/3),(1-d,1+d), **kwargs)
kwargs.update(transform=ax2.transAxes) # switch to the bottom axes
ax2.plot((-d,+d), (1-d,1+d), **kwargs)
ax2.plot((-d,+d), (-d,+d), **kwargs)
f.tight_layout()
plt.show()
#plt.savefig('output.png', dpi=500, bbox_inches='tight')
錯誤：
import matplotlib.ticker as tkr
import scipy, pylab
import locale
import matplotlib.gridspec as gridspec
#from scipy.stats import lognorm
locale.setlocale(locale.LC_NUMERIC, "de_DE")
plt.rcParams['axes.formatter.use_locale'] = True
from scipy.optimize import curve_fit
x=np.asarray([0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, 0.90, 1.00, 1.10, 1.20, 1.30, 1.40,
1.50, 1.60, 1.70, 1.80, 1.90, 2.00, 2.10, 2.20, 2.30, 2.40, 2.50, 2.60, 2.70, 2.80,
2.90, 3.00, 3.10, 3.20, 3.30, 3.40, 3.50, 3.60, 3.70, 3.80, 3.90, 4.00, 4.10, 4.20,
4.30, 4.40, 4.50, 4.60, 4.70, 4.80, 4.90, 5.00, 5.10, 5.20, 5.30, 5.40, 5.50, 5.60,
5.70, 5.80, 5.90, 6.00, 6.10, 6.20, 6.30, 6.40, 6.50, 6.60, 6.70, 6.80, 6.90, 7.00,
7.10, 7.20, 7.30, 7.40, 7.50, 7.60, 7.70, 7.80, 7.90, 8.00], dtype=np.float64)
frequencia_relativa=np.asarray([0.000, 0.000, 0.038, 0.097, 0.091, 0.118, 0.070, 0.124, 0.097, 0.059, 0.059, 0.048, 0.054, 0.043,
0.032, 0.005, 0.027, 0.016, 0.005, 0.000, 0.005, 0.000, 0.005, 0.000, 0.000, 0.000, 0.000, 0.000,
0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000,
0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000,
0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000,
0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.005, 0.000, 0.000], dtype=np.float64)
f, (ax,ax2) = plt.subplots(1,2, sharex=True, sharey=True, facecolor='w')
def fun(y, mu, sigma):
return 1.0/(np.sqrt(2.0*np.pi)*sigma*y)*np.exp(-(np.log(y)-mu)**2/(2.0*sigma*sigma))
step = 0.1
xx = x
nrm = np.sum(frequencia_relativa*step) # normalization integral
print(nrm)
frequencia_relativa /= nrm # normalize frequences histogram
print(np.sum(frequencia_relativa*step)) # check normalizatio
params, extras = curve_fit(fun, xx, frequencia_relativa)
print(params[0])
print(params[1])
axes = f.add_subplot(111, frameon=False)
axes.plot(x, fun(x, params[0], params[1]), "b-", linewidth=3)
ax.spines['top'].set_color('none')
ax2.spines['top'].set_color('none')
gs = gridspec.GridSpec(1,2,width_ratios=[3,1])
ax = plt.subplot(gs[0])
ax2 = plt.subplot(gs[1])
ax.axvspan(0.190, 1.616, label='Média $\pm$ desvio padrão', ymin=0.0, ymax=1.0, alpha=0.2, color='Plum')
ax.yaxis.tick_left()
ax.xaxis.tick_bottom()
ax2.xaxis.tick_bottom()
ax.tick_params(labeltop='off') # don't put tick labels at the top
ax2.yaxis.tick_right()
ax.bar(xx, height=frequencia_relativa, label='Frequência relativa do tamanho lateral triangular', alpha=0.5, width=0.1, align='edge', edgecolor='black', hatch="///")
ax2.bar(xx, height=frequencia_relativa, alpha=0.5, width=0.1, align='edge', edgecolor='black', hatch="///")
#plt.plot(xx, frequencia_relativa, "ro")
ax.tick_params(axis = 'both', which = 'major', labelsize = 18)
ax.tick_params(axis = 'both', which = 'minor', labelsize = 18)
ax2.tick_params(axis = 'both', which = 'major', labelsize = 18)
ax2.tick_params(axis = 'both', which = 'minor', labelsize = 18)
ax2.xaxis.set_ticks(np.arange(7.0, 8.5, 0.5))
ax2.xaxis.set_major_formatter(tkr.FormatStrFormatter('%0.1f'))
plt.subplots_adjust(wspace=0.04)
ax.set_xlim(0,2.5)
ax.set_ylim(0,1.4)
ax2.set_xlim(7.0,8.0)
def func(x, pos): # formatter function takes tick label and tick position
s = str(x)
ind = s.index('.')
return s[:ind] + ',' + s[ind+1:] # change dot to comma
x_format = tkr.FuncFormatter(func)
ax.xaxis.set_major_formatter(x_format)
ax2.xaxis.set_major_formatter(x_format)
# hide the spines between ax and ax2
ax.spines['right'].set_visible(False)
ax2.spines['left'].set_visible(False)
d = .015 # how big to make the diagonal lines in axes coordinates
# arguments to pass plot, just so we don't keep repeating them
kwargs = dict(transform=ax.transAxes, color='k', clip_on=False)
ax.plot((1-d/3,1+d/3), (-d,+d), **kwargs)
ax.plot((1-d/3,1+d/3),(1-d,1+d), **kwargs)
kwargs.update(transform=ax2.transAxes) # switch to the bottom axes
ax2.plot((-d,+d), (1-d,1+d), **kwargs)
ax2.plot((-d,+d), (-d,+d), **kwargs)
ax2.tick_params(labelright=False)
ax.tick_params(labeltop=False)
ax.tick_params(axis='x', which='major', pad=15)
ax2.tick_params(axis='x', which='major', pad=15)
ax2.set_yticks([])
f.text(0.5, -0.04, 'Tamanho lateral do triângulo ($\mu m$)', ha='center', fontsize=22)
f.text(-0.02, 0.5, 'Frequência relativa', va='center', rotation='vertical', fontsize=22)
#ax.set_xlabel('Tamanho lateral do triângulo ($\mu m$)', fontsize=22)
#ax.set_ylabel('Frequência relativa', fontsize=22)
#x_axis = np.arange(0, 29, 0.001)
#ax.plot(x_axis, norm.pdf(x_axis,2.232,1.888), linewidth=3)
ax.axvline(0.903, color='k', linestyle='-', linewidth=1.3)
ax.axvline(0.190, color='k', linestyle='--', linewidth=1)
ax.axvline(1.616, color='k', linestyle='--', linewidth=1)
f.legend(loc=9,
bbox_to_anchor=(.79,.99),
labelspacing=1.5,
numpoints=1,
columnspacing=0.2,
ncol=1, fontsize=18)
ax.text(0.903*0.70, 1.4*0.92, '$\mu$ = (0,90 $\pm$ 0,71) $\mu m$', fontsize=20)
f.tight_layout()
plt.show()
您正在嘗試同時制作精美的圖形和擬合。 你幫助你適應，圖表是次要問題。
首先，使用 NumPy arrays 作為數據，幫助很大。 其次，您的直方圖 function 是非規范化的。
因此，如果在您的第一個程序中，我將規范化 freqs 數組
x=np.asarray([0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, 0.90, 1.00, 1.10, 1.20, 1.30, 1.40,
1.50, 1.60, 1.70, 1.80, 1.90, 2.00, 2.10, 2.20, 2.30, 2.40, 2.50, 2.60, 2.70, 2.80,
2.90, 3.00, 3.10, 3.20, 3.30, 3.40, 3.50, 3.60, 3.70, 3.80, 3.90, 4.00, 4.10, 4.20,
4.30, 4.40, 4.50, 4.60, 4.70, 4.80, 4.90, 5.00, 5.10, 5.20, 5.30, 5.40, 5.50, 5.60,
5.70, 5.80, 5.90, 6.00, 6.10, 6.20, 6.30, 6.40, 6.50, 6.60, 6.70, 6.80, 6.90, 7.00,
7.10, 7.20, 7.30, 7.40, 7.50, 7.60, 7.70, 7.80, 7.90, 8.00], dtype=np.float64)
frequencia_relativa=np.asarray([0.000, 0.000, 0.038, 0.097, 0.091, 0.118, 0.070, 0.124, 0.097, 0.059, 0.059, 0.048, 0.054, 0.043,
0.032, 0.005, 0.027, 0.016, 0.005, 0.000, 0.005, 0.000, 0.005, 0.000, 0.000, 0.000, 0.000, 0.000,
0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000,
0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000,
0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000,
0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.005, 0.000, 0.000], dtype=np.float64)
step = 0.1
nrm = np.sum(frequencia_relativa*step) # normalization integral
print(nrm)
frequencia_relativa /= nrm
print(np.sum(frequencia_relativa*step))
並將 Y 限制設置為 1.4，我將在下面得到圖表
然后，在擬合部分我會做類似的變換，將X軸移動步長的一半，使直方圖值在bin的中間，擬合開始工作，代碼，Python 3.9.1 Win 10 x64 . 我刪除了與適合無關的所有內容，以便它適合您，並繪制適合的 function 與輸入數據。
我也不太了解歸一化積分的部分（直方圖中所有條形的總和為 1，因為它是相對頻率）並且我不了解步進和移位的選擇。 你能更好地解釋這部分嗎？
您要擬合的 function 是對數范數分布的兩個參數 PDF。 它的條件是0 ∫ ∞ PDF(x,μ,σ)=1。 您必須以相同的方式調整輸入數據。 對於直方圖，積分是 bin 的總和乘以步長。 Step 顯然是 0.1，所以我計算這個總和，檢查它不是 1，然后將頻率除以歸一化值，使得積分等於 1。您可以嘗試擬合的不是 2 參數，而是 3 參數曲線，第三參數是歸一化值，但非線性擬合中的更多參數意味着您可能會遇到更多問題。
Wrt shift，必須做出一個假設，即 bin 描述的值。 我假設 bin 的值應該是 bin 中間的值。 同樣，這是一個假設，我不知道您的數據是如何制作的，也許直方圖值實際上是 bin 左側的值。 就是這樣，您只需刪除班次並重新運行代碼。
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
x=np.asarray([0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, 0.90, 1.00, 1.10, 1.20, 1.30, 1.40,
1.50, 1.60, 1.70, 1.80, 1.90, 2.00, 2.10, 2.20, 2.30, 2.40, 2.50, 2.60, 2.70, 2.80,
2.90, 3.00, 3.10, 3.20, 3.30, 3.40, 3.50, 3.60, 3.70, 3.80, 3.90, 4.00, 4.10, 4.20,
4.30, 4.40, 4.50, 4.60, 4.70, 4.80, 4.90, 5.00, 5.10, 5.20, 5.30, 5.40, 5.50, 5.60,
5.70, 5.80, 5.90, 6.00, 6.10, 6.20, 6.30, 6.40, 6.50, 6.60, 6.70, 6.80, 6.90, 7.00,
7.10, 7.20, 7.30, 7.40, 7.50, 7.60, 7.70, 7.80, 7.90, 8.00], dtype=np.float64)
frequencia_relativa=np.asarray([0.000, 0.000, 0.038, 0.097, 0.091, 0.118, 0.070, 0.124, 0.097, 0.059, 0.059, 0.048, 0.054, 0.043,
0.032, 0.005, 0.027, 0.016, 0.005, 0.000, 0.005, 0.000, 0.005, 0.000, 0.000, 0.000, 0.000, 0.000,
0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000,
0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000,
0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000,
0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.005, 0.000, 0.000], dtype=np.float64)
def f(y, mu, sigma):
return 1/(np.sqrt(2.0*np.pi)*sigma*y)*np.exp(-(np.log(y)-mu)**2/(2.0*sigma*sigma))
step = 0.1
nrm = np.sum(frequencia_relativa*step)
frequencia_relativa /= nrm
xx = x - 0.5*step
params, extras = curve_fit(f, xx, frequencia_relativa)
mu = params[0]
sigma = params[1]
print((mu,sigma))
# calculate mean value, https://en.wikipedia.org/wiki/Log-normal_distribution
print(np.exp(mu + sigma*sigma/2.0))
# calculate stddev as sq.root of variance
z=np.sqrt((np.exp(sigma*sigma)-1)*np.exp(mu+mu+sigma*sigma))
print(z)
xxx=np.linspace(0.001,8,1000)
plt.plot(xxx, f(xxx, mu, sigma), "b-")
plt.plot(xx, frequencia_relativa, "ro")
plt.show()
我得到的 lognorm 曲線看起來不錯 wrt 輸入數據。 兩條曲線的大部分數據都在 [0...2] 區間，峰值在 ~(0.8, 1.2)。 這是最簡單的圖表，它將擬合曲線（藍色）與頻率直方圖箱（紅點）的中心重疊。 現在您可以嘗試將其放入您的精美圖表中，祝您好運。
僅供參考，適合 3 參數對數范數曲線的代碼適用於非規范化數據。 似乎也有效
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
x=np.asarray([0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, 0.90, 1.00, 1.10, 1.20, 1.30, 1.40,
1.50, 1.60, 1.70, 1.80, 1.90, 2.00, 2.10, 2.20, 2.30, 2.40, 2.50, 2.60, 2.70, 2.80,
2.90, 3.00, 3.10, 3.20, 3.30, 3.40, 3.50, 3.60, 3.70, 3.80, 3.90, 4.00, 4.10, 4.20,
4.30, 4.40, 4.50, 4.60, 4.70, 4.80, 4.90, 5.00, 5.10, 5.20, 5.30, 5.40, 5.50, 5.60,
5.70, 5.80, 5.90, 6.00, 6.10, 6.20, 6.30, 6.40, 6.50, 6.60, 6.70, 6.80, 6.90, 7.00,
7.10, 7.20, 7.30, 7.40, 7.50, 7.60, 7.70, 7.80, 7.90, 8.00], dtype=np.float64)
frequencia_relativa=np.asarray([0.000, 0.000, 0.038, 0.097, 0.091, 0.118, 0.070, 0.124, 0.097, 0.059, 0.059, 0.048, 0.054, 0.043,
0.032, 0.005, 0.027, 0.016, 0.005, 0.000, 0.005, 0.000, 0.005, 0.000, 0.000, 0.000, 0.000, 0.000,
0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000,
0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000,
0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000,
0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.005, 0.000, 0.000], dtype=np.float64)
def f(y, mu, sigma, N):
return N/(np.sqrt(2.0*np.pi)*sigma*y)*np.exp(-(np.log(y)-mu)**2/(2.0*sigma*sigma))
step = 0.1
xx = x - 0.5*step
params, extras = curve_fit(f, xx, frequencia_relativa)
print(params)
plt.plot(xx, f(xx, params[0], params[1], params[2]), "b-")
plt.plot(xx, frequencia_relativa, "ro")
plt.show()
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