Numpy基础 --数组和矢量计算 利用Python进行数据分析读书笔记

Numpy 数组和矢量计算

代码下载

import numpy as np

#ndarray对象  数组 NumPy数组

创建ndarray

data1=[6,7.5,8,0,1]

arr1=np.array(data1)

arr1

array([ 6. ,  7.5,  8. ,  0. ,  1. ])

data2=[[1,2,3,4],[5,6,7,8]]

arr2=np.array(data2)

arr2

array([[1, 2, 3, 4],       [5, 6, 7, 8]])

arr2.ndim

2

arr2.shape

(2, 4)

arr1.dtype

dtype('float64')

arr2.dtype

dtype('int32')

np.zeros(10)

array([ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.])

np.zeros((3,6))

array([[ 0.,  0.,  0.,  0.,  0.,  0.],       [ 0.,  0.,  0.,  0.,  0.,  0.],       [ 0.,  0.,  0.,  0.,  0.,  0.]])

np.empty((2,3,2))

array([[[  1.37556714e-311,   0.00000000e+000],        [  0.00000000e+000,   0.00000000e+000],        [  0.00000000e+000,   0.00000000e+000]],       [[  0.00000000e+000,   0.00000000e+000],        [  0.00000000e+000,   0.00000000e+000],        [  0.00000000e+000,   0.00000000e+000]]])

np.arange(15)

array([ 0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14])

arr3=np.array([1,2,3],dtype=np.float64)

arr4=np.array([1,2,3],dtype=np.int32)

arr3.dtype

dtype('float64')

arr4.dtype

dtype('int32')

arr5=np.array([1,2,3,4,5])

arr5.dtype

dtype('int32')

float_arr5=arr5.astype(np.float64)

float_arr5.dtype

dtype('float64')

arr6=np.array([3.7,1.2,3.5,6.4,-0.5,0.9])

arr6

array([ 3.7,  1.2,  3.5,  6.4, -0.5,  0.9])

arr6.astype(np.int32)

array([3, 1, 3, 6, 0, 0])

numeric_strings=np.array(['1.25','3.44','5.64'],dtype=np.string_)

numeric_strings.astype(float)

array([ 1.25,  3.44,  5.64])

int_array=np.arange(10)

calibers=np.array([.22,.270,.357,.380,.44,.50],dtype=np.float64)

int_array.astype(calibers.dtype)

array([ 0.,  1.,  2.,  3.,  4.,  5.,  6.,  7.,  8.,  9.])

empty_uint32=np.empty(8,dtype='u4')

empty_uint32

array([0, 0, 1, 0, 2, 0, 3, 0], dtype=uint32)

#调用astype就会创建一个新的数组

数组和标量之间的运算

#数组很重要，因为它使你不用编写循环即可对数据执行批量运算，这通常就叫做矢量化（vectorization

arr=np.array([[1.,2.,3.],[4.,5.,6.]])

arr

array([[ 1.,  2.,  3.],       [ 4.,  5.,  6.]])

arr*arr

array([[  1.,   4.,   9.],       [ 16.,  25.,  36.]])

1/arr

array([[ 1.        ,  0.5       ,  0.33333333],       [ 0.25      ,  0.2       ,  0.16666667]])

arr*0.5

array([[ 0.5,  1. ,  1.5],       [ 2. ,  2.5,  3. ]])

基本的索引和切片

arr=np.arange(10)arr

array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])

arr[5]

5

arr[5:8]

array([5, 6, 7])

arr[5:8]=12arr

array([ 0,  1,  2,  3,  4, 12, 12, 12,  8,  9])

arr_slice=arr[5:8]arr_slice[1]=12345arr

array([    0,     1,     2,     3,     4,    12, 12345,    12,     8,     9])

arr_slice[:]=64arr

array([ 0,  1,  2,  3,  4, 64, 64, 64,  8,  9])

#数组切片是原始数组的视图，视图上的任何修改都会直接反映到源数组上arr2d=np.array([[1,2,3],[4,5,6],[7,8,9]])arr2d[2]

array([7, 8, 9])

arr2d[0][2]

3

arr2d[0,2]

3

arr3d=np.array([[[1,2,3],[4,5,6]],[[7,8,9],[10,11,12]]])arr3d

array([[[ 1,  2,  3],        [ 4,  5,  6]],       [[ 7,  8,  9],        [10, 11, 12]]])

arr3d[0]

array([[1, 2, 3],       [4, 5, 6]])

old_values=arr3d[0].copy()arr3d[0]=42arr3d

array([[[42, 42, 42],        [42, 42, 42]],       [[ 7,  8,  9],        [10, 11, 12]]])

arr3d[0]=old_valuesarr3d

array([[[ 1,  2,  3],        [ 4,  5,  6]],       [[ 7,  8,  9],        [10, 11, 12]]])

arr3d[1,0]

array([7, 8, 9])

arr[1:6]

array([ 1,  2,  3,  4, 64])

arr2d

array([[1, 2, 3],       [4, 5, 6],       [7, 8, 9]])

arr2d[:2]

array([[1, 2, 3],       [4, 5, 6]])

arr2d[:2,1:]

array([[2, 3],       [5, 6]])

arr2d[1,:2]

array([4, 5])

arr2d[2,:1]

array([7])

arr2d[:,:1]

array([[1],       [4],       [7]])

布尔型索引

names=np.array(['Bob','Joe','Will','Bob','Will','Joe','Joe'])data=np.random.randn(7,4)

names

array(['Bob', 'Joe', 'Will', 'Bob', 'Will', 'Joe', 'Joe'],       dtype='<U4')

data

array([[  1.60975139e-03,  -3.23542576e-01,   1.76647590e+00,          1.00434873e-01],       [ -1.00265678e+00,  -2.06101922e-01,  -1.98938974e+00,          1.03029242e-01],       [ -4.59143820e-01,   6.32877040e-01,   6.65959171e-02,         -9.06221248e-01],       [  1.69835755e-01,  -3.53395803e-01,   1.05681390e+00,         -4.89362964e-01],       [ -1.63716077e+00,   3.09182690e+00,  -2.81776081e-01,          6.14541313e-01],       [  8.23892259e-01,  -6.11722686e-01,   6.27307169e-01,         -3.55724014e-02],       [  1.71960690e+00,   2.35358233e-01,  -1.58146445e+00,          1.11900395e+00]])

names=='Bob'

array([ True, False, False,  True, False, False, False], dtype=bool)

data[names=='Bob']

array([[  1.60975139e-03,  -3.23542576e-01,   1.76647590e+00,          1.00434873e-01],       [  1.69835755e-01,  -3.53395803e-01,   1.05681390e+00,         -4.89362964e-01]])

data[names=='Bob',2:]

array([[ 1.7664759 ,  0.10043487],       [ 1.0568139 , -0.48936296]])

data[names=='Bob',3]

array([ 0.10043487, -0.48936296])

names!='Bob'

array([False,  True,  True, False,  True,  True,  True], dtype=bool)

data[~(names=='Bob')]

array([[-1.00265678, -0.20610192, -1.98938974,  0.10302924],       [-0.45914382,  0.63287704,  0.06659592, -0.90622125],       [-1.63716077,  3.0918269 , -0.28177608,  0.61454131],       [ 0.82389226, -0.61172269,  0.62730717, -0.0355724 ],       [ 1.7196069 ,  0.23535823, -1.58146445,  1.11900395]])

mask=(names=='Bob')|(names=='Will')mask

array([ True, False,  True,  True,  True, False, False], dtype=bool)

data[mask]

array([[  1.60975139e-03,  -3.23542576e-01,   1.76647590e+00,          1.00434873e-01],       [ -4.59143820e-01,   6.32877040e-01,   6.65959171e-02,         -9.06221248e-01],       [  1.69835755e-01,  -3.53395803e-01,   1.05681390e+00,         -4.89362964e-01],       [ -1.63716077e+00,   3.09182690e+00,  -2.81776081e-01,          6.14541313e-01]])

data[data<0]=0data

array([[  1.60975139e-03,   0.00000000e+00,   1.76647590e+00,          1.00434873e-01],       [  0.00000000e+00,   0.00000000e+00,   0.00000000e+00,          1.03029242e-01],       [  0.00000000e+00,   6.32877040e-01,   6.65959171e-02,          0.00000000e+00],       [  1.69835755e-01,   0.00000000e+00,   1.05681390e+00,          0.00000000e+00],       [  0.00000000e+00,   3.09182690e+00,   0.00000000e+00,          6.14541313e-01],       [  8.23892259e-01,   0.00000000e+00,   6.27307169e-01,          0.00000000e+00],       [  1.71960690e+00,   2.35358233e-01,   0.00000000e+00,          1.11900395e+00]])

data[names!='Joe']=7data

array([[ 7.        ,  7.        ,  7.        ,  7.        ],       [ 0.        ,  0.        ,  0.        ,  0.10302924],       [ 7.        ,  7.        ,  7.        ,  7.        ],       [ 7.        ,  7.        ,  7.        ,  7.        ],       [ 7.        ,  7.        ,  7.        ,  7.        ],       [ 0.82389226,  0.        ,  0.62730717,  0.        ],       [ 1.7196069 ,  0.23535823,  0.        ,  1.11900395]])

花式索引

指的是利用整数数组进行索引。

arr=np.empty((8,4))for i in range(8):    arr[i]=iarr

array([[ 0.,  0.,  0.,  0.],       [ 1.,  1.,  1.,  1.],       [ 2.,  2.,  2.,  2.],       [ 3.,  3.,  3.,  3.],       [ 4.,  4.,  4.,  4.],       [ 5.,  5.,  5.,  5.],       [ 6.,  6.,  6.,  6.],       [ 7.,  7.,  7.,  7.]])

arr[[4,3,0,6]]

array([[ 4.,  4.,  4.,  4.],       [ 3.,  3.,  3.,  3.],       [ 0.,  0.,  0.,  0.],       [ 6.,  6.,  6.,  6.]])

arr[[-3,-5,-7]]

array([[ 5.,  5.,  5.,  5.],       [ 3.,  3.,  3.,  3.],       [ 1.,  1.,  1.,  1.]])

arr=np.arange(32).reshape((8,4))arr

array([[ 0,  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, 26, 27],       [28, 29, 30, 31]])

arr[[1,5,7,2],[0,3,1,2]]

array([ 4, 23, 29, 10])

arr[[1,5,7,2]][:,[0,3,1,2]]

array([[ 4,  7,  5,  6],       [20, 23, 21, 22],       [28, 31, 29, 30],       [ 8, 11,  9, 10]])

arr[np.ix_([1,5,7,2],[0,3,1,2])]

array([[ 4,  7,  5,  6],       [20, 23, 21, 22],       [28, 31, 29, 30],       [ 8, 11,  9, 10]])

记住，花式索引跟切片不一样，它总是将数据复制到新数组中。

数组转置和轴对称

转置（transpose)是重塑的一种特殊形式，它返回的是源数据的视图（不会进行任何复制操作）。

arr=np.arange(15).reshape((3,5))arr

array([[ 0,  1,  2,  3,  4],       [ 5,  6,  7,  8,  9],       [10, 11, 12, 13, 14]])

arr.T

array([[ 0,  5, 10],       [ 1,  6, 11],       [ 2,  7, 12],       [ 3,  8, 13],       [ 4,  9, 14]])

arr=np.random.randn(6,3)np.dot(arr.T,arr)

array([[ 8.84595216,  2.30542093,  3.92854057],       [ 2.30542093,  2.28401128,  1.73860755],       [ 3.92854057,  1.73860755,  9.77924613]])

arr=np.arange(16).reshape((2,2,4))arr

array([[[ 0,  1,  2,  3],        [ 4,  5,  6,  7]],       [[ 8,  9, 10, 11],        [12, 13, 14, 15]]])

arr.transpose((1,0,2))

array([[[ 0,  1,  2,  3],        [ 8,  9, 10, 11]],       [[ 4,  5,  6,  7],        [12, 13, 14, 15]]])

arr

array([[[ 0,  1,  2,  3],        [ 4,  5,  6,  7]],       [[ 8,  9, 10, 11],        [12, 13, 14, 15]]])

arr.swapaxes(1,2)#也是返回源数据的视图,不会进行任何复制操作

array([[[ 0,  4],        [ 1,  5],        [ 2,  6],        [ 3,  7]],       [[ 8, 12],        [ 9, 13],        [10, 14],        [11, 15]]])

通用函数：快速的元素级数据函数

arr=np.arange(10)np.sqrt(arr)

array([ 0.        ,  1.        ,  1.41421356,  1.73205081,  2.        ,        2.23606798,  2.44948974,  2.64575131,  2.82842712,  3.        ])

np.exp(arr)

array([  1.00000000e+00,   2.71828183e+00,   7.38905610e+00,         2.00855369e+01,   5.45981500e+01,   1.48413159e+02,         4.03428793e+02,   1.09663316e+03,   2.98095799e+03,         8.10308393e+03])

x=np.random.randn(8)y=np.random.randn(8)x

array([-1.55455343,  0.58957206,  1.12291564,  0.84985964,  1.81809564,        0.96211051,  0.03536402, -0.29113791])

y

array([ 1.35585258, -0.18208383, -0.96881932, -0.97084842,  0.15031288,       -0.21753205, -0.12555617,  1.07649061])

np.maximum(x,y)

array([ 1.35585258,  0.58957206,  1.12291564,  0.84985964,  1.81809564,        0.96211051,  0.03536402,  1.07649061])

arr=np.random.randn(7)*5np.modf(arr)

(array([-0.99321578,  0.62223866,  0.32422504, -0.20182624, -0.74306072,        -0.10960894,  0.95203083]), array([-2.,  2.,  2., -4., -1., -6.,  5.]))

np.fabs(arr)

array([ 2.99321578,  2.62223866,  2.32422504,  4.20182624,  1.74306072,        6.10960894,  5.95203083])

利用数组进行数据处理

points=np.arange(-5,5,0.01)#1000个间隔相等的点

xs,ys=np.meshgrid(points,points)ys

array([[-5.  , -5.  , -5.  , ..., -5.  , -5.  , -5.  ],       [-4.99, -4.99, -4.99, ..., -4.99, -4.99, -4.99],       [-4.98, -4.98, -4.98, ..., -4.98, -4.98, -4.98],       ...,        [ 4.97,  4.97,  4.97, ...,  4.97,  4.97,  4.97],       [ 4.98,  4.98,  4.98, ...,  4.98,  4.98,  4.98],       [ 4.99,  4.99,  4.99, ...,  4.99,  4.99,  4.99]])

import matplotlib.pyplot as  pltz=np.sqrt(xs**2+ys**2)z

array([[ 7.07106781,  7.06400028,  7.05693985, ...,  7.04988652,         7.05693985,  7.06400028],       [ 7.06400028,  7.05692568,  7.04985815, ...,  7.04279774,         7.04985815,  7.05692568],       [ 7.05693985,  7.04985815,  7.04278354, ...,  7.03571603,         7.04278354,  7.04985815],       ...,        [ 7.04988652,  7.04279774,  7.03571603, ...,  7.0286414 ,         7.03571603,  7.04279774],       [ 7.05693985,  7.04985815,  7.04278354, ...,  7.03571603,         7.04278354,  7.04985815],       [ 7.06400028,  7.05692568,  7.04985815, ...,  7.04279774,         7.04985815,  7.05692568]])

plt.imshow(z,cmap=plt.cm.gray)plt.colorbar()

<matplotlib.colorbar.Colorbar at 0x28840dd65c0>

plt.title('Image plot of $\sqrt（x^2+y^2)$ for a grid of values')

<matplotlib.text.Text at 0x28841041dd8>

将条件逻辑表述为数组运算

xarr=np.array([1.1,1.2,1.3,1.4,1.5])yarr=np.array([2.1,2.2,2.3,2.4,2.5])cond=np.array([True,False,True,True,False])

result=[(x if c else y) for x,y,c in zip(xarr,yarr,cond)]result

[1.1000000000000001, 2.2000000000000002, 1.3, 1.3999999999999999, 2.5]

result=np.where(cond,xarr,yarr)result

array([ 1.1,  2.2,  1.3,  1.4,  2.5])

arr=np.random.randn(4,4)arr

array([[-1.124892  ,  0.16102557, -0.84624401, -1.61350592],       [ 0.93525737, -1.97957635, -2.53954932,  0.79295019],       [-1.40451591,  0.31596234, -1.43060903, -1.61587221],       [-1.00342438,  0.88479574,  1.52961242,  0.72461918]])

np.where(arr>0,2,-2)

array([[-2,  2, -2, -2],       [ 2, -2, -2,  2],       [-2,  2, -2, -2],       [-2,  2,  2,  2]])

np.where(arr>0,2,arr)

array([[-1.124892  ,  2.        , -0.84624401, -1.61350592],       [ 2.        , -1.97957635, -2.53954932,  2.        ],       [-1.40451591,  2.        , -1.43060903, -1.61587221],       [-1.00342438,  2.        ,  2.        ,  2.        ]])

数学和统计方法

arr=np.random.randn(5,4)#正态分布的数据arr.mean()

0.22588105368397526

np.mean(arr)

0.22588105368397526

arr.sum()

4.5176210736795053

arr.mean(axis=1)

array([ 0.67230987,  0.01274547,  0.18780888,  0.54016805, -0.283627  ])

arr.sum(0)

array([ 0.06725924,  0.51484031,  1.7496478 ,  2.18587372])

arr=np.array([[0,1,2],[3,4,5],[6,7,8]])arr

array([[0, 1, 2],       [3, 4, 5],       [6, 7, 8]])

arr.cumsum(0)

array([[ 0,  1,  2],       [ 3,  5,  7],       [ 9, 12, 15]], dtype=int32)

arr.cumprod(1)

array([[  0,   0,   0],       [  3,  12,  60],       [  6,  42, 336]], dtype=int32)

用于布尔型数组的方法

arr=np.random.randn(100)(arr>0).sum() #正值的数量

49

bools=np.array([False,False,True,False])

bools.any()#测试数组中是否存在一个或多个True

True

bools.all()#测试数组中所有值是否都是True

False

排序

arr=np.random.randn(8)arr

array([ 0.19051791, -0.9561823 , -0.88527884,  1.72500065,  0.7121868 ,       -0.98016434, -0.62017177,  1.56115109])

arr.sort()arr

array([-0.98016434, -0.9561823 , -0.88527884, -0.62017177,  0.19051791,        0.7121868 ,  1.56115109,  1.72500065])

arr=np.random.randn(5,3)arr

array([[ 2.18671772, -0.52656283,  0.9128075 ],       [-0.60204952,  0.71479588, -0.03902287],       [-0.63784626, -1.89380845, -0.28438434],       [ 1.22924442,  0.16689474, -0.63089802],       [ 0.72705863,  2.18074376,  0.47051067]])

arr.sort(1)#这里属于就地排序，会改变原始数组arr

array([[-0.52656283,  0.9128075 ,  2.18671772],       [-0.60204952, -0.03902287,  0.71479588],       [-1.89380845, -0.63784626, -0.28438434],       [-0.63089802,  0.16689474,  1.22924442],       [ 0.47051067,  0.72705863,  2.18074376]])

唯一化以及其他的集合逻辑

names=np.array(['Bob','Joe','Will','Bob','Will','Joe','Joe'])np.unique(names)

array(['Bob', 'Joe', 'Will'],       dtype='<U4')

ints=np.array([3,3,4,2,1,3,2,4])np.unique(ints)

array([1, 2, 3, 4])

sorted(set(names))

['Bob', 'Joe', 'Will']

#np.in1d用于测试一个数组中的值在另一个数组中的成员资格，返回一个布尔型数组values=np.array([6,0,0,3,2,5,6])

np.in1d(values,[2,3,6])

array([ True, False, False,  True,  True, False,  True], dtype=bool)

用于数组的文件输入输出

arr=np.arange(10)np.save('ch04/some_array',arr)

np.load('ch04/some_array.npy')

array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])

#将多个数组保存到一个压缩文件中np.savez('ch04/array_archive.npz',a=arr,b=arr)

#加载.npz文件时，你会得到一个类似字典的对象arch=np.load('ch04/array_archive.npz')arch['b']

array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])

存取文本文件

arr=np.loadtxt('ch04/array_ex.txt',delimiter=',')arr

array([[ 0.580052,  0.18673 ,  1.040717,  1.134411],       [ 0.194163, -0.636917, -0.938659,  0.124094],       [-0.12641 ,  0.268607, -0.695724,  0.047428],       [-1.484413,  0.004176, -0.744203,  0.005487],       [ 2.302869,  0.200131,  1.670238, -1.88109 ],       [-0.19323 ,  1.047233,  0.482803,  0.960334]])

np.savetxt('ch04/array_txt.txt',arr,delimiter=' ',newline='\n')

线性代数

x=np.array([[1.,2.,3.],[4.,5.,6.]])y=np.array([[6.,23.],[-1,7],[8,9]])x

array([[ 1.,  2.,  3.],       [ 4.,  5.,  6.]])

y

array([[  6.,  23.],       [ -1.,   7.],       [  8.,   9.]])

x.dot(y)#相当于np.dot(x,y)

array([[  28.,   64.],       [  67.,  181.]])

#一个二维数组跟一个大小合适的一维数组的矩阵点积运算之后将会得到一个一维数组np.dot(x,np.ones(3))

array([  6.,  15.])

#numpy.linalg中有一组标准的矩阵分解运算以及诸如求逆和行列式之类的东西。from numpy.linalg import inv,qrX=np.random.randn(5,5)mat=X.T.dot(X)mat

array([[ 3.52812683,  0.50014532, -1.33983697,  1.65988419, -0.76535951],       [ 0.50014532,  4.08419311, -2.5690617 , -0.16615284, -3.74006228],       [-1.33983697, -2.5690617 ,  2.72421214, -0.13432057,  4.16986366],       [ 1.65988419, -0.16615284, -0.13432057,  3.2039997 , -0.87473058],       [-0.76535951, -3.74006228,  4.16986366, -0.87473058,  8.06038483]])

inv(mat)

array([[  54.04030339,   25.14468473,  138.84119709,  -36.99114311,         -59.04224289],       [  25.14468473,   12.31566469,   65.26559276,  -17.16636841,         -27.52455358],       [ 138.84119709,   65.26559276,  359.29868506,  -95.18079197,        -152.73752953],       [ -36.99114311,  -17.16636841,  -95.18079197,   25.66540608,          40.54724899],       [ -59.04224289,  -27.52455358, -152.73752953,   40.54724899,          65.1619639 ]])

mat.dot(inv(mat))

array([[  1.00000000e+00,  -1.42108547e-14,  -2.84217094e-14,          1.42108547e-14,  -1.42108547e-14],       [ -2.84217094e-14,   1.00000000e+00,   1.13686838e-13,         -5.68434189e-14,  -8.52651283e-14],       [  2.84217094e-14,   0.00000000e+00,   1.00000000e+00,          0.00000000e+00,   5.68434189e-14],       [ -7.10542736e-15,  -7.10542736e-15,  -5.68434189e-14,          1.00000000e+00,  -7.10542736e-15],       [  1.13686838e-13,   2.84217094e-14,   0.00000000e+00,         -5.68434189e-14,   1.00000000e+00]])

q,r=qr(mat)r

array([[ -4.22302950e+00,  -2.32915443e+00,   3.09645494e+00,         -2.82756748e+00,   4.20997326e+00],       [  0.00000000e+00,  -5.66758935e+00,   4.67966804e+00,          5.91077981e-01,   8.21617203e+00],       [  0.00000000e+00,   0.00000000e+00,  -1.31721365e+00,          1.90947603e-01,  -3.21644932e+00],       [  0.00000000e+00,   0.00000000e+00,   0.00000000e+00,         -2.33466484e+00,   1.45658073e+00],       [  0.00000000e+00,   0.00000000e+00,   0.00000000e+00,          0.00000000e+00,   5.46664408e-03]])

随机数生成

samples=np.random.randn(4,4)samples

array([[ 1.5404748 , -0.92115435,  1.00509721,  0.43422671],       [ 0.69277073,  0.18068919,  0.60346547, -0.35861855],       [ 1.05033574,  1.16613186, -1.0336046 , -0.71084958],       [-0.06515771,  1.3693006 ,  1.40907517, -0.94190917]])

#范例：随机漫步nsteps=1000draws=np.random.randint(0,2,size=nsteps)steps=np.where(draws>0,1,-1)walk=steps.cumsum()

walk.min()

-25

walk.max()

7

(np.abs(walk)>=10).argmax()

65

###一次模拟多个随机漫步nwalks=5000nsteps=1000draws=np.random.randint(0,2,size=(nwalks,nsteps))#0或1steps=np.where(draws>0,1,-1)walks=steps.cumsum(1)#沿着第二个轴方向累加walks

array([[  1,   2,   3, ..., -28, -29, -30],       [ -1,   0,   1, ...,  10,  11,  12],       [  1,   2,   3, ...,  46,  47,  48],       ...,        [ -1,  -2,  -3, ...,   0,   1,   0],       [ -1,   0,   1, ...,  -8,  -7,  -8],       [  1,   2,   3, ...,  44,  43,  42]], dtype=int32)

walks.max()

146

walks.min()

-114

hits30=(np.abs(walks)>=30).any(1)hits30

array([ True, False,  True, ..., False,  True,  True], dtype=bool)

hits30.sum()#到达30或-30的数量

3417

crossing_times=(np.abs(walks[hits30])>=30).argmax(1)

crossing_times.mean()

500.91747146619844