在概率学中我们用偏度和峰度去刻画分布的情况:
偏度描述的是分布的对称性程度,如上面,右偏表示在u值的右侧分布占多数,左偏则反向,并且通过阴影的面积去刻画概率。而峰度是描述分布的最高值的情况,在常用情况下,减去3的原因在于正态分布的超值峰度恰好为3。
下面使用python代入公式计算和调用函数库计算进行比较:
#!/usr/bin/python
#coding:utf8
#coding=utf8
#encoding:utf8
#encoding=utf8
#_*_ coding:utf8 _*_
# -*- coding:utf-8 -*-
import numpy as np
from scipy import stats
import math
import matplotlib as mpl
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm
def calc_statistics(x):
n = x.shape[0] # 样本个数
# 手动计算
# 分别表示各个k阶矩
m = 0
m2 = 0
m3 = 0
m4 = 0
for t in x:
m += t
m2 += t*t
m3 += t**3
m4 += t**4
m /= n
m2 /= n
m3 /= n
m4 /= n
# 代入公式求个值
mu = m
sigma = np.sqrt(m2 - mu*mu)
skew = (m3 - 3*mu*m2 + 2*mu**3) / sigma**3
kurtosis = (m4 - 4*mu*m3 + 6*mu*mu*m2 - 4*mu**3*mu + mu**4) / sigma**4 - 3
print('手动计算均值、标准差、偏度、峰度:', mu, sigma, skew, kurtosis)
# 使用系统函数验证
mu = np.mean(x, axis=0)
sigma = np.std(x, axis=0)
skew = stats.skew(x)
kurtosis = stats.kurtosis(x)
return mu, sigma, skew, kurtosis
if __name__ == '__main__':
d = np.random.randn(100000)
print(d)
mu, sigma, skew, kurtosis = calc_statistics(d)
print('函数库计算均值、标准差、偏度、峰度:', mu, sigma, skew, kurtosis)
# 一维直方图
mpl.rcParams[u'font.sans-serif'] = 'SimHei'
mpl.rcParams[u'axes.unicode_minus'] = False
y1, x1, dummy = plt.hist(d, bins=50, normed=True, color='g', alpha=0.75)
t = np.arange(x1.min(), x1.max(), 0.05)
y = np.exp(-t**2 / 2) / math.sqrt(2*math.pi)
plt.plot(t, y, 'r-', lw=2)
plt.title(u'高斯分布,样本个数:%d' % d.shape[0])
plt.grid(True)
plt.show()
d = np.random.randn(100000, 2)
mu, sigma, skew, kurtosis = calc_statistics(d)
print('函数库计算均值、标准差、偏度、峰度:', mu, sigma, skew, kurtosis)
# 二维图像
N = 30
density, edges = np.histogramdd(d, bins=[N, N])
print('样本总数:', np.sum(density))
density /= density.max()
x = y = np.arange(N)
# 高斯分布
t = np.meshgrid(x, y)
fig = plt.figure(facecolor='w')
ax = fig.add_subplot(111, projection='3d')
ax.scatter(t[0], t[1], density, c='r', s=15*density, marker='o', depthshade=True)
ax.plot_surface(t[0], t[1], density, cmap=cm.Accent, rstride=2, cstride=2, alpha=0.9, lw=0.75)
ax.set_xlabel(u'X')
ax.set_ylabel(u'Y')
ax.set_zlabel(u'Z')
plt.title(u'二元高斯分布,样本个数:%d' % d.shape[0], fontsize=20)
plt.tight_layout(0.1)
plt.show()