Numpy组队学习下-统计相关

统计相关

次序统计

计算最小值

  • numpy.amin(a[, axis=None, out=None, keepdims=np._NoValue, initial=np._NoValue, where=np._NoValue])Return the minimum of an array or minimum along an axis.

【例】计算最小值

import numpy as np

x = np.array([[11, 12, 13, 14, 15],
              [16, 17, 18, 19, 20],
              [21, 22, 23, 24, 25],
              [26, 27, 28, 29, 30],
              [31, 32, 33, 34, 35]])
y = np.amin(x)
print(y)  # 11

y = np.amin(x, axis=0)
print(y)  # [11 12 13 14 15]

y = np.amin(x, axis=1)
print(y)  # [11 16 21 26 31]

计算最大值

  • numpy.amax(a[, axis=None, out=None, keepdims=np._NoValue, initial=np._NoValue, where=np._NoValue])Return the maximum of an array or maximum along an axis.

【例】计算最大值

import numpy as np

x = np.array([[11, 12, 13, 14, 15],
              [16, 17, 18, 19, 20],
              [21, 22, 23, 24, 25],
              [26, 27, 28, 29, 30],
              [31, 32, 33, 34, 35]])
y = np.amax(x)
print(y)  # 35

y = np.amax(x, axis=0)
print(y)  # [31 32 33 34 35]

y = np.amax(x, axis=1)
print(y)  # [15 20 25 30 35]

计算极差

  • numpy.ptp(a, axis=None, out=None, keepdims=np._NoValue) Range of values (maximum - minimum) along an axis. The name of the function comes from the acronym for ‘peak to peak’.

【例】计算极差

import numpy as np

np.random.seed(20200623)
x = np.random.randint(0, 20, size=[4, 5])
print(x)
# [[10  2  1  1 16]
#  [18 11 10 14 10]
#  [11  1  9 18  8]
#  [16  2  0 15 16]]

print(np.ptp(x))  # 18
print(np.ptp(x, axis=0))  # [ 8 10 10 17  8]
print(np.ptp(x, axis=1))  # [15  8 17 16]

计算分位数

  • numpy.percentile(a, q, axis=None, out=None, overwrite_input=False, interpolation='linear', keepdims=False) Compute the q-th percentile of the data along the specified axis. Returns the q-th percentile(s) of the array elements.
    • a:array,用来算分位数的对象,可以是多维的数组。
    • q:介于0-100的float,用来计算是几分位的参数,如四分之一位就是25,如要算两个位置的数就[25,75]。
    • axis:坐标轴的方向,一维的就不用考虑了,多维的就用这个调整计算的维度方向,取值范围0/1。

【例】计算分位数

import numpy as np

np.random.seed(20200623)
x = np.random.randint(0, 20, size=[4, 5])
print(x)
# [[10  2  1  1 16]
#  [18 11 10 14 10]
#  [11  1  9 18  8]
#  [16  2  0 15 16]]

print(np.percentile(x, [25, 50]))  
# [ 2. 10.]

print(np.percentile(x, [25, 50], axis=0))
# [[10.75  1.75  0.75 10.75  9.5 ]
#  [13.5   2.    5.   14.5  13.  ]]

print(np.percentile(x, [25, 50], axis=1))
# [[ 1. 10.  8.  2.]
#  [ 2. 11.  9. 15.]]

均值与方差

计算中位数

  • numpy.median(a, axis=None, out=None, overwrite_input=False, keepdims=False) Compute the median along the specified axis. Returns the median of the array elements.

【例】计算中位数

import numpy as np

np.random.seed(20200623)
x = np.random.randint(0, 20, size=[4, 5])
print(x)
# [[10  2  1  1 16]
#  [18 11 10 14 10]
#  [11  1  9 18  8]
#  [16  2  0 15 16]]
print(np.percentile(x, 50))
print(np.median(x))
# 10.0

print(np.percentile(x, 50, axis=0))
print(np.median(x, axis=0))
# [13.5  2.   5.  14.5 13. ]

print(np.percentile(x, 50, axis=1))
print(np.median(x, axis=1))
# [ 2. 11.  9. 15.]

计算平均值

  • numpy.mean(a[, axis=None, dtype=None, out=None, keepdims=np._NoValue)])Compute the arithmetic mean along the specified axis.

【例】计算平均值(沿轴的元素的总和除以元素的数量)。

import numpy as np

x = np.array([[11, 12, 13, 14, 15],
              [16, 17, 18, 19, 20],
              [21, 22, 23, 24, 25],
              [26, 27, 28, 29, 30],
              [31, 32, 33, 34, 35]])
y = np.mean(x)
print(y)  # 23.0

y = np.mean(x, axis=0)
print(y)  # [21. 22. 23. 24. 25.]

y = np.mean(x, axis=1)
print(y)  # [13. 18. 23. 28. 33.]

计算加权平均值

  • numpy.average(a[, axis=None, weights=None, returned=False])Compute the weighted average along the specified axis.

meanaverage都是计算均值的函数,在不指定权重的时候averagemean是一样的。指定权重后,average可以计算加权平均值。

【例】计算加权平均值(将各数值乘以相应的权数,然后加总求和得到总体值,再除以总的单位数。)

import numpy as np

x = np.array([[11, 12, 13, 14, 15],
              [16, 17, 18, 19, 20],
              [21, 22, 23, 24, 25],
              [26, 27, 28, 29, 30],
              [31, 32, 33, 34, 35]])
y = np.average(x)
print(y)  # 23.0

y = np.average(x, axis=0)
print(y)  # [21. 22. 23. 24. 25.]

y = np.average(x, axis=1)
print(y)  # [13. 18. 23. 28. 33.]


y = np.arange(1, 26).reshape([5, 5])
print(y)
# [[ 1  2  3  4  5]
#  [ 6  7  8  9 10]
#  [11 12 13 14 15]
#  [16 17 18 19 20]
#  [21 22 23 24 25]]

z = np.average(x, weights=y)
print(z)  # 27.0

z = np.average(x, axis=0, weights=y)
print(z)
# [25.54545455 26.16666667 26.84615385 27.57142857 28.33333333]

z = np.average(x, axis=1, weights=y)
print(z)
# [13.66666667 18.25       23.15384615 28.11111111 33.08695652]

计算方差

  • numpy.var(a[, axis=None, dtype=None, out=None, ddof=0, keepdims=np._NoValue])Compute the variance along the specified axis.
    • ddof=0:是“Delta Degrees of Freedom”,表示自由度的个数。

要注意方差和样本方差的无偏估计,方差公式中分母上是n;样本方差无偏估计公式中分母上是n-1n为样本个数)。

证明的链接

【例】计算方差

import numpy as np

x = np.array([[11, 12, 13, 14, 15],
              [16, 17, 18, 19, 20],
              [21, 22, 23, 24, 25],
              [26, 27, 28, 29, 30],
              [31, 32, 33, 34, 35]])
y = np.var(x)
print(y)  # 52.0
y = np.mean((x - np.mean(x)) ** 2)
print(y)  # 52.0

y = np.var(x, ddof=1)
print(y)  # 54.166666666666664
y = np.sum((x - np.mean(x)) ** 2) / (x.size - 1)
print(y)  # 54.166666666666664

y = np.var(x, axis=0)
print(y)  # [50. 50. 50. 50. 50.]

y = np.var(x, axis=1)
print(y)  # [2. 2. 2. 2. 2.]

计算标准差

  • numpy.std(a[, axis=None, dtype=None, out=None, ddof=0, keepdims=np._NoValue])Compute the standard deviation along the specified axis.

标准差是一组数据平均值分散程度的一种度量,是方差的算术平方根。

【例】计算标准差

import numpy as np

x = np.array([[11, 12, 13, 14, 15],
              [16, 17, 18, 19, 20],
              [21, 22, 23, 24, 25],
              [26, 27, 28, 29, 30],
              [31, 32, 33, 34, 35]])
y = np.std(x)
print(y)  # 7.211102550927978
y = np.sqrt(np.var(x))
print(y)  # 7.211102550927978

y = np.std(x, axis=0)
print(y)
# [7.07106781 7.07106781 7.07106781 7.07106781 7.07106781]

y = np.std(x, axis=1)
print(y)
# [1.41421356 1.41421356 1.41421356 1.41421356 1.41421356]

相关

计算协方差矩阵

  • numpy.cov(m, y=None, rowvar=True, bias=False, ddof=None, fweights=None,aweights=None) Estimate a covariance matrix, given data and weights.

【例】计算协方差矩阵

import numpy as np

x = [1, 2, 3, 4, 6]
y = [0, 2, 5, 6, 7]
print(np.cov(x))  # 3.7   #样本方差
print(np.cov(y))  # 8.5   #样本方差
print(np.cov(x, y))
# [[3.7  5.25]
#  [5.25 8.5 ]]

print(np.var(x))  # 2.96    #方差
print(np.var(x, ddof=1))  # 3.7    #样本方差
print(np.var(y))  # 6.8    #方差
print(np.var(y, ddof=1))  # 8.5    #样本方差

z = np.mean((x - np.mean(x)) * (y - np.mean(y)))    #协方差
print(z)  # 4.2

z = np.sum((x - np.mean(x)) * (y - np.mean(y))) / (len(x) - 1)   #样本协方差
print(z)  # 5.25

z = np.dot(x - np.mean(x), y - np.mean(y)) / (len(x) - 1)     #样本协方差     
print(z)  # 5.25

计算相关系数

  • numpy.corrcoef(x, y=None, rowvar=True, bias=np._NoValue, ddof=np._NoValue) Return Pearson product-moment correlation coefficients.

理解了np.cov()函数之后,很容易理解np.correlate(),二者参数几乎一模一样。

np.cov()描述的是两个向量协同变化的程度,它的取值可能非常大,也可能非常小,这就导致没法直观地衡量二者协同变化的程度。相关系数实际上是正则化的协方差,n个变量的相关系数形成一个n维方阵。

【例】计算相关系数

import numpy as np

np.random.seed(20200623)
x, y = np.random.randint(0, 20, size=(2, 4))

print(x)  # [10  2  1  1]
print(y)  # [16 18 11 10]

z = np.corrcoef(x, y)
print(z)
# [[1.         0.48510096]
#  [0.48510096 1.        ]]

a = np.dot(x - np.mean(x), y - np.mean(y))
b = np.sqrt(np.dot(x - np.mean(x), x - np.mean(x)))
c = np.sqrt(np.dot(y - np.mean(y), y - np.mean(y)))
print(a / (b * c))  # 0.4851009629263671

直方图

  • numpy.digitize(x, bins, right=False)Return the indices of the bins to which each value in input array belongs.
    • x:numpy数组
    • bins:一维单调数组,必须是升序或者降序
    • right:间隔是否包含最右
    • 返回值:x在bins中的位置。

【例】

import numpy as np

x = np.array([0.2, 6.4, 3.0, 1.6])
bins = np.array([0.0, 1.0, 2.5, 4.0, 10.0])
inds = np.digitize(x, bins)
print(inds)  # [1 4 3 2]
for n in range(x.size):
    print(bins[inds[n] - 1], "<=", x[n], "<", bins[inds[n]])

# 0.0 <= 0.2 < 1.0
# 4.0 <= 6.4 < 10.0
# 2.5 <= 3.0 < 4.0
# 1.0 <= 1.6 < 2.5

【例】

import numpy as np

x = np.array([1.2, 10.0, 12.4, 15.5, 20.])
bins = np.array([0, 5, 10, 15, 20])
inds = np.digitize(x, bins, right=True)
print(inds)  # [1 2 3 4 4]

inds = np.digitize(x, bins, right=False)
print(inds)  # [1 3 3 4 5]

我想问自由度在计算方差的时候起到了什么作用。