Task03 统计相关-次序统计

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.
mean和average都是计算均值的函数，在不指定权重的时候average和mean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-1（n为样本个数）。

证明的链接 7

【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
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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]
练习
（1） y=X β

对于简单线性回归，向量计法等同于
在这里插入图片描述
给定X跟y我们可以使用 NumPy 库解出β值，
【知识点：统计相关】

from numpy . linalg import inv
from numpy import dot, transpose
X = [[1, 6, 2] , [1, 8, 1] , [1, 10, 0] , [1 , 14, 2] , [1, 18, 0]]
y = [[7] , [9] , [13] , [17.5] , [18]]
（2）计算给定数组中每行的最大值。

a = np.random.randint(1, 10, [5, 3])

如何在二维numpy数组的每一行中找到最大值？

【知识点：统计相关】

（3）计算数组的元素最大值与最小值之差（极值）

【知识点：统计相关】

数组为：
A=[[3 7 5]
[8 4 3]
[2 4 9]]

（4）计算s的均值，方差，标准差，协方差

【知识点：统计相关】

s=[9.7, 10, 10.3, 9.7,10,10.3,9.7,10,10.3]