Fitting Log-normal distribution (Python plot)
Question
I am trying to fit a log-normal distribution to the histogram data. I've tried to follow examples of other questions here on the Stack Exchange but I'm not getting the fit, because in this case I have a broken axis. I already put the broken axis on that plot, I tried to prevent the numbers from overlapping on the axes, I removed the numbers from the repeated axes, I reduced the size of the second subplot, but I'm not able to fit the log-normal. How can I fit the log-normal distribution for this data set?
Code:
#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')
Attempt with 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')
Error:
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()
1 answers
solution1
1 ACCPTED 2021-02-20 22:17:48
You're trying at the same time to do fancy graphs and fit. you help you with fit, graphs are secondary problem.
First, use NumPy arrays for data, helps a lot. Second, your histogram function is denormalized.
So if in the first of your programs I'll normalize freqs array
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))
and set Y limit to 1.4, I'll get graph below
Then, in the fitting part I'll do similar transformation, and shift X axis by half the step size, so that histogram value is in the middle of the bin, fit is starting to work, code, Python 3.9.1 Win 10 x64. I removed everything not related to fit, just so it works for you, and plotted fitted function vs input data.
I also didn't quite understand the part of normalizing the integral (the sum of all the bars in the histogram gives 1 because it's the relative frequency) and I didn't understand the choice of step and shift. Could you explain this part better, please?
Your function to fit is two-parameters PDF of log-norm distribution. It conditioned such that the 0 ∫ ∞ PDF(x,μ,σ)=1. You have to condition your input data in the same way. For histogram, integral is the sum of bins multiplied by steps. Step is obviously 0.1, so I compute this sum, check it is not 1, and then divide frequencies by normalization value, such that integral is equal to 1. You could try to fit not 2-parametric, but 3-parametric curve, third parameter being normalization value, but more parameters in non-linear fit means more problems you could get.
Wrt shift, one has to make an assumption, what value of the bin describes. I assumed that value of the bin should be the value in the middle of the bin. Again, this is an assumption, I don't know how your data were made, maybe histogram value is really value at the left side of the bin. It that is so, you just remove the shift and rerun the code.
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()
and I'm getting lognorm curve which looks ok wrt input data. Both curves have majority of data in the [0...2] interval with peak value at ~(0.8, 1.2). Here is simplest graph which overlaps fitted curve (blue) with centers of the frequency histogram bins (red dots). Now you could try to put it into your fancy graphs, good luck.
And just for reference, code which fits 3-parameters log-norm curve to apply to denormalized data. Seems to work as well
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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