使用python中的对数轴缩放和拟合对数正态分布

Question

我有一个对数正态的分布式样本集。我可以使用具有线性或对数x轴的自然图来可视化样本。我可以对直方图进行拟合以获得PDF，然后将其缩放到具有线性x轴的图中的histrogram，另请参见this previously posted question。

但是，我无法使用对数x轴将PDF正确绘制到绘图中。

不幸的是，将PDF区域缩放到直方图不仅存在问题，而且PDF也向左移动，如下图所示。

我现在的问题是，我在这里做错了什么？使用CDF绘制预期的直方图，as suggested in this answer，有效。我想知道我在这段代码中做错了什么，因为在我的理解中它也应该工作。

这是python代码（我很抱歉它很长但我想发布一个“完整的独立版本”）：

import numpy as np
import matplotlib.pyplot as plt
import scipy.stats

# generate log-normal distributed set of samples
np.random.seed(42)
samples   = np.random.lognormal( mean=1, sigma=.4, size=10000 )

# make a fit to the samples
shape, loc, scale = scipy.stats.lognorm.fit( samples, floc=0 )
x_fit       = np.linspace( samples.min(), samples.max(), 100 )
samples_fit = scipy.stats.lognorm.pdf( x_fit, shape, loc=loc, scale=scale )

# plot a histrogram with linear x-axis
plt.subplot( 1, 2, 1 )
N_bins = 50
counts, bin_edges, ignored = plt.hist( samples, N_bins, histtype='stepfilled', label='histogram' )
# calculate area of histogram (area under PDF should be 1)
area_hist = .0
for ii in range( counts.size):
    area_hist += (bin_edges[ii+1]-bin_edges[ii]) * counts[ii]
# oplot fit into histogram
plt.plot( x_fit, samples_fit*area_hist, label='fitted and area-scaled PDF', linewidth=2)
plt.legend()

# make a histrogram with a log10 x-axis
plt.subplot( 1, 2, 2 )
# equally sized bins (in log10-scale)
bins_log10 = np.logspace( np.log10( samples.min()  ), np.log10( samples.max() ), N_bins )
counts, bin_edges, ignored = plt.hist( samples, bins_log10, histtype='stepfilled', label='histogram' )
# calculate area of histogram
area_hist_log = .0
for ii in range( counts.size):
    area_hist_log += (bin_edges[ii+1]-bin_edges[ii]) * counts[ii]
# get pdf-values for log10 - spaced intervals
x_fit_log       = np.logspace( np.log10( samples.min()), np.log10( samples.max()), 100 )
samples_fit_log = scipy.stats.lognorm.pdf( x_fit_log, shape, loc=loc, scale=scale )
# oplot fit into histogram
plt.plot( x_fit_log, samples_fit_log*area_hist_log, label='fitted and area-scaled PDF', linewidth=2 )

plt.xscale( 'log' )
plt.xlim( bin_edges.min(), bin_edges.max() )
plt.legend()
plt.show()

更新1：

我忘了提到我使用的版本：

python      2.7.6
numpy       1.8.2
matplotlib  1.3.1
scipy       0.13.3

更新2：

正如@Christoph和@zaxliu所指出的那样（感谢两者），问题在于缩放PDF。当我使用与直方图相同的箱子时，它可以工作，就像@ zaxliu的解决方案一样，但是当我为PDF使用更高的分辨率时仍然存在一些问题（如上例所示）。如下图所示：

右边图的代码是（我省略了导入和数据样本生成的东西，你可以在上面的例子中找到）：

# equally sized bins in log10-scale
bins_log10 = np.logspace( np.log10( samples.min()  ), np.log10( samples.max() ), N_bins )
counts, bin_edges, ignored = plt.hist( samples, bins_log10, histtype='stepfilled', label='histogram' )

# calculate length of each bin (required for scaling PDF to histogram)
bins_log_len = np.zeros( bins_log10.size )
for ii in range( counts.size):
    bins_log_len[ii] = bin_edges[ii+1]-bin_edges[ii]

# get pdf-values for same intervals as histogram
samples_fit_log = scipy.stats.lognorm.pdf( bins_log10, shape, loc=loc, scale=scale )

# oplot fitted and scaled PDF into histogram
plt.plot( bins_log10, np.multiply(samples_fit_log,bins_log_len)*sum(counts), label='PDF using histogram bins', linewidth=2 )

# make another pdf with a finer resolution
x_fit_log       = np.logspace( np.log10( samples.min()), np.log10( samples.max()), 100 )
samples_fit_log = scipy.stats.lognorm.pdf( x_fit_log, shape, loc=loc, scale=scale )
# calculate length of each bin (required for scaling PDF to histogram)
# in addition, estimate middle point for more accuracy (should in principle also be done for the other PDF)
bins_log_len       = np.diff( x_fit_log )
samples_log_center = np.zeros( x_fit_log.size-1 )
for ii in range( x_fit_log.size-1 ):
    samples_log_center[ii] = .5*(samples_fit_log[ii] + samples_fit_log[ii+1] )

# scale PDF to histogram
# NOTE: THIS IS NOT WORKING PROPERLY (SEE FIGURE)
pdf_scaled2hist = np.multiply(samples_log_center,bins_log_len)*sum(counts)

# oplot fit into histogram
plt.plot( .5*(x_fit_log[:-1]+x_fit_log[1:]), pdf_scaled2hist, label='PDF using own bins', linewidth=2 )

plt.xscale( 'log' )
plt.xlim( bin_edges.min(), bin_edges.max() )
plt.legend(loc=3)

Answer 1

另一答案