使用 scipy.stats 拟合非标准化分布

Question

I'm tryng to fit a histogram but the fit only works with normalised data, ie with option normed=True in the histogram. Is there a way of doing this with scipy stats (or other method)? Here is a MWE using a uniform distribution:

import matplotlib.pyplot as plt
import numpy as np
import random
from scipy.stats import uniform

data = []
for i in range(1000):
    data.append(random.uniform(-1,1))

loc, scale = uniform.fit(data)

x = np.linspace(-1,1, 1000)
y = uniform.pdf(x, loc, scale)

plt.hist(data, bins=100, normed=False)
plt.plot(x, y, 'r-')
plt.show()

I also tried defining my own function (below) but I'm getting a bad fit.

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

data = []
for i in range(1000):
    data.append(random.uniform(-1,1))

def unif(x,avg,sig):
    return avg*x + sig

y, base = np.histogram(data,bins=100)
x = [0.5 * (base[i] + base[i+1]) for i in xrange(len(base)-1)]

popt, pcov = optimize.curve_fit(unif, x, y)
x_fit = np.linspace(x[0], x[-1], 100)
y_fit = unif(x_fit, *popt)

plt.hist(data, bins=100, normed=False)
plt.plot(x_fit, y_fit, 'r-')
plt.show()

Answer 1

Note that it is generally a bad idea to fit a distribution to the histogram. Compared to the raw data the histogram contains less information so the fit will most likely be worse. Thus, the first MWE in the question actually contains the best approach. Simply normalize the histogram and it will match the distribution of the data: plt.hist(data, bins=100, normed=True) .

However, it seems you actually want to work with the unnormalized histogram. In that case take the normalization that the histogram would normally use and apply it inverted to the fitted distribution. The documentation describes the normalization as

n/(len(x)`dbin)

which is verbose for saying dividing by the number of observations times the bin width .

Multiplying the distribution by this value results in the expected counts per bin:

loc, scale = uniform.fit(data)

x = np.linspace(-1,1, 1000)
y = uniform.pdf(x, loc, scale)

n_bins = 100      
bin_width = np.ptp(data) / n_bins

plt.hist(data, bins=n_bins, normed=False)
plt.plot(x, y * len(data) * bin_width, 'r-')

---

The second MWE is interesting because you describe the line aa bad fit , but actually it is a very good fit :). You simply overfit the histogram because although you expect a horizontal line (one degree of freedom) you fit an arbitrary line (two degrees of freedom).

So if you want a horizontal line fit a horizontal line and don't be surprised to get something else if you fit something else...

def unif(x, sig):
    return 0 * x + sig  # slope is zero -> horizontal line

However, there is a much simpler way of obtaining the height of the unnormalized uniform distribution. Just average the histogram over all bins:

y, base = np.histogram(data,bins=100)
y_hat = np.mean(y)
print(y_hat)
# 10.0

Or, even simpler use the theoretical value of len(data) / n_bins == 10 .