這100道練習題玩轉Numpy！

對于Numpy，我講的不多，因為和Pandas相比，他距離日常的資料處理更“遠”一些。
但是，Numpy仍然是Python做資料分析所必須要掌握的基礎庫之一，以下題是github上的開源項目，主要為了檢測你的Numpy能力，同時對你的學習作為一個補充。
1. 導入numpy庫并取别名為np (★☆☆)
(提示: import … as …)
import numpy as np
2. 列印輸出numpy的版本和配置資訊 (★☆☆)
(提示: np.version, np.show_config)
print(np.__version__)
print(np.show_config())
3. 建立一個長度為10的空向量 (★☆☆)
(提示: np.zeros)
Z = np.zeros(10)
print(Z)
4. 如何找到任何一個數組的記憶體大小？(★☆☆)
(提示: size, itemsize)
Z = np.zeros((10,10))
print("%d bytes" % (Z.size * Z.itemsize))
5. 如何從指令行得到numpy中add函數的說明文檔? (★☆☆)
(提示: np.info)
import numpy
numpy.info(numpy.add)
6. 建立一個長度為10并且除了第五個值為1的空向量 (★☆☆)
(提示: array[4])
Z = np.zeros(10)
Z[4] = 1
print(Z)
7. 建立一個值域範圍從10到49的向量(★☆☆)
(提示: np.arange)
Z = np.arange(10,50)
print(Z)
8.  反轉一個向量(第一個元素變為最後一個) (★☆☆)
(提示: array[::-1])
Z = np.arange(50)
Z = Z[::-1]
print(Z)
9. 建立一個 3x3 并且值從0到8的矩陣(★☆☆)
(提示: reshape)
Z = np.arange(9).reshape(3,3)
print(Z)
10. 找到數組[1,2,0,0,4,0]中非0元素的位置索引 (★☆☆)
(提示: np.nonzero)
nz = np.nonzero([1,2,0,0,4,0])
print(nz)
11. 建立一個 3x3 的機關矩陣 (★☆☆)
(提示: np.eye)
Z = np.eye(3)
print(Z)
12. 建立一個 3x3x3的随機數組 (★☆☆)
(提示: np.random.random)
Z = np.random.random((3,3,3))
print(Z)
13. 建立一個 10x10 的随機數組并找到它的最大值和最小值 (★☆☆)
(提示: min, max)
Z = np.random.random((10,10))
Zmin, Zmax = Z.min(), Z.max()
print(Zmin, Zmax)
14. 建立一個長度為30的随機向量并找到它的平均值 (★☆☆)
(提示: mean)
Z = np.random.random(30)
m = Z.mean()
print(m)
15. 建立一個二維數組，其中邊界值為1，其餘值為0 (★☆☆)
(提示: array[1:-1, 1:-1])
Z = np.ones((10,10))
Z[1:-1,1:-1] = 0
print(Z)
16. 對于一個存在在數組，如何添加一個用0填充的邊界? (★☆☆)
(提示: np.pad)
Z = np.ones((5,5))
Z = np.pad(Z, pad_width=1, mode='constant', constant_values=0)
print(Z)
17. 下面表達式運作的結果是什麼？(★☆☆)
(提示: NaN = not a number, inf = infinity)
(提示：NaN : 不是一個數，inf : 無窮)
# 表達式                           # 結果
0 * np.nan                        nan
np.nan == np.nan                  False
np.inf > np.nan                   False
np.nan - np.nan                   nan
0.3 == 3 * 0.1                    False
18. 建立一個 5x5的矩陣，并設定值1,2,3,4落在其對角線下方位置 (★☆☆)
(提示: np.diag)
Z = np.diag(1+np.arange(4),k=-1)
print(Z)
19. 建立一個8x8 的矩陣，并且設定成棋盤樣式 (★☆☆)
(提示: array[::2])
Z = np.zeros((8,8),dtype=int)
Z[1::2,::2] = 1
Z[::2,1::2] = 1
print(Z)
20. 考慮一個 (6,7,8) 形狀的數組，其第100個元素的索引(x,y,z)是什麼?
(提示: np.unravel_index)
print(np.unravel_index(100,(6,7,8)))
21. 用tile函數去建立一個 8x8的棋盤樣式矩陣(★☆☆)
(提示: np.tile)
Z = np.tile( np.array([[0,1],[1,0]]), (4,4))
print(Z)
22. 對一個5x5的随機矩陣做歸一化(★☆☆)
(提示: (x - min) / (max - min))
Z = np.random.random((5,5))
Zmax, Zmin = Z.max(), Z.min()
Z = (Z - Zmin)/(Zmax - Zmin)
print(Z)
23. 建立一個将顔色描述為(RGBA)四個無符号位元組的自定義dtype？(★☆☆)
(提示: np.dtype)
color = np.dtype([("r", np.ubyte, 1),
                  ("g", np.ubyte, 1),
                  ("b", np.ubyte, 1),
                  ("a", np.ubyte, 1)])
color
24. 一個5x3的矩陣與一個3x2的矩陣相乘，實矩陣乘積是什麼？(★☆☆)
(提示: np.dot | @)
Z = np.dot(np.ones((5,3)), np.ones((3,2)))
print(Z)
25. 給定一個一維數組，對其在3到8之間的所有元素取反 (★☆☆)
(提示: >, <=)
Z = np.arange(11)
Z[(3 < Z) & (Z <= 8)] *= -1
print(Z)
26. 下面腳本運作後的結果是什麼? (★☆☆)
(提示: np.sum)
# Author: Jake VanderPlas               # 結果

print(sum(range(5),-1))                 9
from numpy import *                     
print(sum(range(5),-1))                 10    #numpy.sum(a, axis=None)
27. 考慮一個整數向量Z,下清單達合法的是哪個? (★☆☆)
（提示：這裡還有“位運算符”）
Z**Z                        True
2 << Z >> 2                 False
Z <- Z                      True
1j*Z                        True  #複數           
Z/1/1                       True
Z<Z>Z                       False
28. 下面表達式的結果分别是什麼？(★☆☆)
np.array(0) / np.array(0)                           nan
np.array(0) // np.array(0)                          0
np.array([np.nan]).astype(int).astype(float)        -2.14748365e+09
29. 如何從零位開始舍入浮點數組？(★☆☆)
(提示: np.uniform, np.copysign, np.ceil, np.abs)
# Author: Charles R Harris

Z = np.random.uniform(-10,+10,10)
print (np.copysign(np.ceil(np.abs(Z)), Z))
30. 如何找出兩個數組公共的元素? (★☆☆)
(提示: np.intersect1d)
Z1 = np.random.randint(0, 10, 10)
Z2 = np.random.randint(0, 10, 10)
print (np.intersect1d(Z1, Z2))
31. 如何忽略所有的 numpy 警告(盡管不建議這麼做)? (★☆☆)
(提示: np.seterr, np.errstate)
# Suicide mode on
defaults = np.seterr(all="ignore")
Z = np.ones(1) / 0

# Back to sanity
_ = np.seterr(**defaults)

# 另一個等價的方式， 使用上下文管理器（context manager）
with np.errstate(divide='ignore'):
    Z = np.ones(1) / 0
32. 下面的表達式是否為真? (★☆☆)
(提示: 虛數)
np.sqrt(-1) == np.emath.sqrt(-1)     Faslse
33. 如何獲得昨天，今天和明天的日期? (★☆☆)
(提示: np.datetime64, np.timedelta64)
yesterday = np.datetime64('today', 'D') - np.timedelta64(1, 'D')
today = np.datetime64('today', 'D')
tomorrow = np.datetime64('today', 'D') + np.timedelta64(1, 'D')
34. 怎麼獲得所有與2016年7月的所有日期? (★★☆)
(提示: np.arange(dtype=datetime64['D']))
Z = np.arange('2016-07', '2016-08', dtype='datetime64[D]')
print (Z)
35. 如何計算 ((A+B)*(-A/2)) (不使用中間變量)? (★★☆)
(提示: np.add(out=), np.negative(out=), np.multiply(out=), np.divide(out=))
A = np.ones(3) * 1
B = np.ones(3) * 1
C = np.ones(3) * 1
np.add(A, B, out=B)
np.divide(A, 2, out=A)
np.negative(A, out=A)
np.multiply(A, B, out=A)
36. 用5種不同的方法提取随機數組中的整數部分 (★★☆)
(提示: %, np.floor, np.ceil, astype, np.trunc)
Z = np.random.uniform(0, 10, 10)
print (Z - Z % 1)
print (np.floor(Z))
print (np.cell(Z)-1)
print (Z.astype(int))
print (np.trunc(Z))
37. 建立一個5x5的矩陣且每一行的值範圍為從0到4 (★★☆)
(提示: np.arange)
Z = np.zeros((5, 5))
Z += np.arange(5)
print (Z)
38. 如何用一個生成10個整數的函數來建構數組 (★☆☆)
(提示: np.fromiter)
def generate():
    for x in range(10):
      yield x
Z = np.fromiter(generate(), dtype=float, count=-1)
print (Z)
39. 建立一個大小為10的向量， 值域為0到1，不包括0和1 (★★☆)
(提示: np.linspace)
Z = np.linspace(0, 1, 12, endpoint=True)[1: -1]
print (Z)
40. 建立一個大小為10的随機向量，并把它排序 (★★☆)
(提示: sort)
Z = np.random.random(10)
Z.sort()
print (Z)
41.  對一個小數組進行求和有沒有辦法比np.sum更快? (★★☆)
(提示: np.add.reduce)
# Author: Evgeni Burovski

Z = np.arange(10)
np.add.reduce(Z)

# np.add.reduce 是numpy.add子產品中的一個ufunc(universal function)函數,C語言實作
42. 如何判斷兩和随機數組相等 (★★☆)
(提示: np.allclose, np.array_equal)
A = np.random.randint(0, 2, 5)
B = np.random.randint(0, 2, 5)

# 假設array的形狀(shape)相同和一個誤差容限（tolerance）
equal = np.allclose(A,B)
print(equal)

# 檢查形狀和元素值，沒有誤差容限（值必須完全相等）
equal = np.array_equal(A,B)
print(equal)
43. 把數組變為隻讀 (★★☆)
(提示: flags.writeable)
Z = np.zeros(5)
Z.flags.writeable = False
Z[0] = 1
44. 将一個10x2的笛卡爾坐标矩陣轉換為極坐标 (★★☆)
(提示: np.sqrt, np.arctan2)
Z = np.random.random((10, 2))
X, Y = Z[:, 0], Z[:, 1]
R = np.sqrt(X**2 + Y**2)
T = np.arctan2(Y, X)
print (R)
print (T)
45. 建立一個大小為10的随機向量并且将該向量中最大的值替換為0(★★☆)
(提示: argmax)
Z = np.random.random(10)
Z[Z.argmax()] = 0
print (Z)
46. 建立一個結構化數組，其中x和y坐标覆寫[0, 1]x[1, 0]區域 (★★☆)
(提示: np.meshgrid)
Z = np.zeros((5, 5), [('x', float), ('y', float)])
Z['x'], Z['y'] = np.meshgrid(np.linspace(0, 1, 5), np.linspace(0, 1, 5))
print (Z)
47. 給定兩個數組X和Y，構造柯西(Cauchy)矩陣C () (★★☆)
(提示: np.subtract.outer)
# Author: Evgeni Burovski

X = np.arange(8)
Y = X + 0.5
C = 1.0 / np.subtract.outer(X, Y)
print (C)
print(np.linalg.det(C)) # 計算行列式
48. 列印每個numpy 類型的最小和最大可表示值 (★★☆)
(提示: np.iinfo, np.finfo, eps)
for dtype in [np.int8, np.int32, np.int64]:
   print(np.iinfo(dtype).min)
   print(np.iinfo(dtype).max)
for dtype in [np.float32, np.float64]:
   print(np.finfo(dtype).min)
   print(np.finfo(dtype).max)
   print(np.finfo(dtype).eps)
49. 如何列印數組中所有的值？(★★☆)
(提示: np.set_printoptions)
np.set_printoptions(threshold=np.nan)
Z = np.zeros((16,16))
print(Z)
50. 如何在數組中找到與給定标量接近的值? (★★☆)
(提示: argmin)
Z = np.arange(100)
v = np.random.uniform(0, 100)
index = (np.abs(Z-v)).argmin()
print(Z[index])
51. 建立表示位置(x, y)和顔色(r, g, b, a)的結構化數組 (★★☆)
(提示: dtype)
Z = np.zeros(10, [('position', [('x', float, 1), 
                                ('y', float, 1)]),
                  ('color',    [('r', float, 1), 
                                ('g', float, 1), 
                                ('b', float, 1)])])
print (Z)
52. 思考形狀為(100, 2)的随機向量，求出點與點之間的距離 (★★☆)
(提示: np.atleast_2d, T, np.sqrt)
Z = np.random.random((100, 2))
X, Y = np.atleast_2d(Z[:, 0], Z[:, 1])
D = np.sqrt((X-X.T)**2 + (Y-Y.T)**2)
print (D)

# 使用scipy庫可以更快
import scipy.spatial

Z = np.random.random((100,2))
D = scipy.spatial.distance.cdist(Z,Z)
print(D)
53. 如何将類型為float(32位)的數組類型轉換位integer(32位)? (★★☆)
(提示: astype(copy=False))
Z = np.arange(10, dtype=np.int32)
Z = Z.astype(np.float32, copy=False)
print(Z)
54. 如何讀取下面的檔案? (★★☆)
(提示: np.genfromtxt)
1, 2, 3, 4, 5
6,  ,  , 7, 8
 ,  , 9,10,11

# 先把上面儲存到檔案example.txt中
# 這裡不使用StringIO， 因為Python2 和Python3 在這個地方有相容性問題
Z = np.genfromtxt("example.txt", delimiter=",")  
print(Z)
55. numpy數組枚舉(enumerate)的等價操作? (★★☆)
(提示: np.ndenumerate, np.ndindex)
Z = np.arange(9).reshape(3,3)
for index, value in np.ndenumerate(Z):
    print(index, value)
for index in np.ndindex(Z.shape):
    print(index, Z[index])
56. 構造一個二維高斯矩陣(★★☆)
(提示: np.meshgrid, np.exp)
X, Y = np.meshgrid(np.linspace(-1, 1, 10), np.linspace(-1, 1, 10))
D = np.sqrt(X**2 + Y**2)
sigma, mu = 1.0, 0.0
G = np.exp(-( (D-mu)**2 / (2.0*sigma**2) ))
print (G)
57. 如何在二維數組的随機位置放置p個元素? (★★☆)
(提示: np.put, np.random.choice)
# Author: Divakar

n = 10
p = 3
Z = np.zeros((n,n))
np.put(Z, np.random.choice(range(n*n), p, replace=False),1)
print(Z)
58. 減去矩陣每一行的平均值 (★★☆)
(提示: mean(axis=,keepdims=))
# Author: Warren Weckesser

X = np.random.rand(5, 10)

# 新
Y = X - X.mean(axis=1, keepdims=True)

# 舊
Y = X - X.mean(axis=1).reshape(-1, 1)

print(Y)
59. 如何對數組通過第n列進行排序? (★★☆)
(提示: argsort)
# Author: Steve Tjoa

Z = np.random.randint(0,10,(3,3))
print(Z)
print(Z[ Z[:,1].argsort() ])
60. 如何判斷一個給定的二維數組存在空列? (★★☆)
(提示: any, ~)
# Author: Warren Weckesser

Z = np.random.randint(0,3,(3,10))
print((~Z.any(axis=0)).any())
61. 從數組中找出與給定值最接近的值 (★★☆)
(提示: np.abs, argmin, flat)
Z = np.random.uniform(0,1,10)
z = 0.5
m = Z.flat[np.abs(Z - z).argmin()]
print(m)
62. 思考形狀為(1, 3)和(3, 1)的兩個數組形狀，如何使用疊代器計算它們的和? (★★☆)
(提示: np.nditer)
A = np.arange(3).reshape(3, 1)
B = np.arange(3).reshape(1, 3)
it = np.nditer([A, B, None])
for x, y, z in it:
    z[...] = x + y
print (it.operands[2])
63. 建立一個具有name屬性的數組類 (★★☆)
(提示: class method)
class NameArray(np.ndarray):
    def __new__(cls, array, name="no name"):
        obj = np.asarray(array).view(cls)
        obj.name = name
        return obj
    def __array_finalize__(self, obj):
        if obj is None: return
        self.info = getattr(obj, 'name', "no name")

Z = NameArray(np.arange(10), "range_10")
print (Z.name)
64. 給定一個向量，如何讓在第二個向量索引的每個元素加1(注意重複索引)? (★★★)
(提示: np.bincount | np.add.at)
# Author: Brett Olsen

Z = np.ones(10)
I = np.random.randint(0,len(Z),20)
Z += np.bincount(I, minlength=len(Z))
print(Z)

# Another solution
# Author: Bartosz Telenczuk
np.add.at(Z, I, 1)
print(Z)
65. 如何根據索引清單I将向量X的元素累加到數組F? (★★★)
(提示: np.bincount)
# Author: Alan G Isaac

X = [1,2,3,4,5,6]
I = [1,3,9,3,4,1]
F = np.bincount(I,X)
print(F)
66. 思考(dtype = ubyte)的(w, h, 3)圖像，計算唯一顔色的值(★★★)
(提示: np.unique)
# Author: Nadav Horesh

w,h = 16,16
I = np.random.randint(0,2,(h,w,3)).astype(np.ubyte)
F = I[...,0]*256*256 + I[...,1]*256 +I[...,2]
n = len(np.unique(F))
print(np.unique(I))
67. 思考如何求一個四維數組最後兩個軸的資料和(★★★)
(提示: sum(axis=(-2,-1)))
A = np.random.randint(0,10,(3,4,3,4))
# 傳遞一個元組(numpy 1.7.0)
sum = A.sum(axis=(-2,-1))
print(sum)

# 将最後兩個次元壓縮為一個
# (适用于不接受軸元組參數的函數)
sum = A.reshape(A.shape[:-2] + (-1,)).sum(axis=-1)
print(sum)
68. 考慮一維向量D，如何使用相同大小的向量S來計算D的子集的均值，其描述子集索引？(★★★)
(提示: np.bincount)
# Author: Jaime Fernández del Río

D = np.random.uniform(0,1,100)
S = np.random.randint(0,10,100)
D_sums = np.bincount(S, weights=D)
D_counts = np.bincount(S)
D_means = D_sums / D_counts
print(D_means)

# Pandas solution as a reference due to more intuitive code
import pandas as pd
print(pd.Series(D).groupby(S).mean())
69. 如何獲得點積的對角線？(★★★)
(提示: np.diag)
# Author: Mathieu Blondel

A = np.random.uniform(0,1,(5,5))
B = np.random.uniform(0,1,(5,5))

# Slow version  
np.diag(np.dot(A, B))

# Fast version
np.sum(A * B.T, axis=1)

# Faster version
np.einsum("ij,ji->i", A, B)
70.考慮向量[1,2,3,4,5]，如何建立一個新的向量，在每個值之間交錯有3個連續的零？(★★★)
(提示: array[::4])
# Author: Warren Weckesser

Z = np.array([1,2,3,4,5])
nz = 3
Z0 = np.zeros(len(Z) + (len(Z)-1)*(nz))
Z0[::nz+1] = Z
print(Z0)
71. 考慮一個次元(5,5,3)的數組，如何将其與一個(5,5)的數組相乘？(★★★)
(提示: array[:, :, None])
A = np.ones((5,5,3))
B = 2*np.ones((5,5))
print(A * B[:,:,None])
72. 如何對一個數組中任意兩行做交換? (★★★)
(提示: array[[]] = array[[]])
# Author: Eelco Hoogendoorn

A = np.arange(25).reshape(5,5)
A[[0,1]] = A[[1,0]]
print(A)
73. 思考描述10個三角形（共享頂點）的一組10個三元組，找到組成所有三角形的唯一線段集 (★★★)
(提示: repeat, np.roll, np.sort, view, np.unique)
# Author: Nicolas P. Rougier

faces = np.random.randint(0,100,(10,3))
F = np.roll(faces.repeat(2,axis=1),-1,axis=1)
F = F.reshape(len(F)*3,2)
F = np.sort(F,axis=1)
G = F.view( dtype=[('p0',F.dtype),('p1',F.dtype)] )
G = np.unique(G)
print(G)
74. 給定一個二進制的數組C，如何生成一個數組A滿足np.bincount(A)==C? (★★★)
(提示: np.repeat)
# Author: Jaime Fernández del Río

C = np.bincount([1,1,2,3,4,4,6])
A = np.repeat(np.arange(len(C)), C)
print(A)
75. 如何通過滑動視窗計算一個數組的平均數? (★★★)
(提示: np.cumsum)
# Author: Jaime Fernández del Río

def moving_average(a, n=3) :
    ret = np.cumsum(a, dtype=float)
    ret[n:] = ret[n:] - ret[:-n]
    return ret[n - 1:] / n
Z = np.arange(20)
print(moving_average(Z, n=3))
76. 思考以為數組Z，建構一個二維數組，其第一行是(Z[0],Z[1],Z[2])， 然後每一行移動一位，最後一行為 (Z[-3],Z[-2],Z[-1]) (★★★)
(提示: from numpy.lib import stride_tricks)
# Author: Joe Kington / Erik Rigtorp
from numpy.lib import stride_tricks

def rolling(a, window):
    shape = (a.size - window + 1, window)
    strides = (a.itemsize, a.itemsize)
    return stride_tricks.as_strided(a, shape=shape, strides=strides)
Z = rolling(np.arange(10), 3)
print(Z)
77. 如何對布爾值取反，或改變浮點數的符号(sign)? (★★★)
(提示: np.logical_not, np.negative)
# Author: Nathaniel J. Smith

Z = np.random.randint(0,2,100)
np.logical_not(Z, out=Z)

Z = np.random.uniform(-1.0,1.0,100)
np.negative(Z, out=Z)
78. 思考兩組點集P0和P1去描述一組線(二維)和一個點p,如何計算點p到每一條線 i (P0[i],P1[i])的距離？(★★★)
def distance(P0, P1, p):
    T = P1 - P0
    L = (T**2).sum(axis=1)
    U = -((P0[:,0]-p[...,0])*T[:,0] + (P0[:,1]-p[...,1])*T[:,1]) / L
    U = U.reshape(len(U),1)
    D = P0 + U*T - p
    return np.sqrt((D**2).sum(axis=1))

P0 = np.random.uniform(-10,10,(10,2))
P1 = np.random.uniform(-10,10,(10,2))
p  = np.random.uniform(-10,10,( 1,2))
print(distance(P0, P1, p))
79. 考慮兩組點集P0和P1去描述一組線(二維)和一組點集P，如何計算每一個點 j(P[j]) 到每一條線 i (P0[i],P1[i])的距離? (★★★)
# Author: Italmassov Kuanysh

# based on distance function from previous question
P0 = np.random.uniform(-10, 10, (10,2))
P1 = np.random.uniform(-10,10,(10,2))
p = np.random.uniform(-10, 10, (10,2))
print(np.array([distance(P0,P1,p_i) for p_i in p]))
80. 思考一個任意的數組，編寫一個函數，該函數提取一個具有固定形狀的子部分，并以一個給定的元素為中心(在該部分填充值) (★★★)
(提示: minimum, maximum)
# Author: Nicolas Rougier

Z = np.random.randint(0,10,(10,10))
shape = (5,5)
fill  = 0
position = (1,1)

R = np.ones(shape, dtype=Z.dtype)*fill
P  = np.array(list(position)).astype(int)
Rs = np.array(list(R.shape)).astype(int)
Zs = np.array(list(Z.shape)).astype(int)R_start = np.zeros((len(shape),)).astype(int)
R_stop  = np.array(list(shape)).astype(int)
Z_start = (P-Rs//2)
Z_stop  = (P+Rs//2)+Rs%2

R_start = (R_start - np.minimum(Z_start,0)).tolist()
Z_start = (np.maximum(Z_start,0)).tolist()
R_stop = np.maximum(R_start, (R_stop - np.maximum(Z_stop-Zs,0))).tolist()
Z_stop = (np.minimum(Z_stop,Zs)).tolist()

r = [slice(start,stop) for start,stop in zip(R_start,R_stop)]
z = [slice(start,stop) for start,stop in zip(Z_start,Z_stop)]
R[r] = Z[z]
print(Z)
print(R)
81. 考慮一個數組Z = [1,2,3,4,5,6,7,8,9,10,11,12,13,14],如何生成一個數組R = [[1,2,3,4], [2,3,4,5], [3,4,5,6], ...,[11,12,13,14]]? (★★★)
(提示: stride_tricks.as_strided)
# Author: Stefan van der Walt

Z = np.arange(1,15,dtype=np.uint32)
R = stride_tricks.as_strided(Z,(11,4),(4,4))
print(R)
82. 計算矩陣的秩 (★★★)
(提示: np.linalg.svd)
# Author: Stefan van der Walt

Z = np.random.uniform(0,1,(10,10))
U, S, V = np.linalg.svd(Z) # Singular Value Decomposition
rank = np.sum(S > 1e-10)
print(rank)
83. 如何找出數組中出現頻率最高的值?(★★★)
(提示: np.bincount, argmax)
Z = np.random.randint(0,10,50)
print(np.bincount(Z).argmax())
84. 從一個10x10的矩陣中提取出連續的3x3區塊(★★★)
(提示: stride_tricks.as_strided)
# Author: Chris Barker

Z = np.random.randint(0,5,(10,10))
n = 3
i = 1 + (Z.shape[0]-3)
j = 1 + (Z.shape[1]-3)
C = stride_tricks.as_strided(Z, shape=(i, j, n, n), strides=Z.strides + Z.strides)
print(C)
85.建立一個滿足 Z[i,j] == Z[j,i]的二維數組子類 (★★★)
(提示: class method)
# Author: Eric O. Lebigot
# Note: only works for 2d array and value setting using indices

class Symetric(np.ndarray):
    def __setitem__(self, index, value):
        i,j = index
        super(Symetric, self).__setitem__((i,j), value)
        super(Symetric, self).__setitem__((j,i), value)

def symetric(Z):
    return np.asarray(Z + Z.T - np.diag(Z.diagonal())).view(Symetric)

S = symetric(np.random.randint(0,10,(5,5)))
S[2,3] = 42
print(S)
86. 考慮p個 nxn 矩陣和一組形狀為(n,1)的向量，如何直接計算p個矩陣的乘積(n,1)? (★★★)
(提示: np.tensordot)
# Author: Stefan van der Walt

p, n = 10, 20
M = np.ones((p,n,n))
V = np.ones((p,n,1))
S = np.tensordot(M, V, axes=[[0, 2], [0, 1]])
print(S)

# It works, because:
# M is (p,n,n)
# V is (p,n,1)
# Thus, summing over the paired axes 0 and 0 (of M and V independently),
# and 2 and 1, to remain with a (n,1) vector.
87. 對于一個16x16的數組，如何得到一個區域的和(區域大小為4x4)? (★★★)
(提示: np.add.reduceat)
# Author: Robert Kern

Z = np.ones((16,16))
k = 4
S = np.add.reduceat(np.add.reduceat(Z, np.arange(0, Z.shape[0], k), axis=0), np.arange(0, Z.shape[1], k), axis=1)
print(S)
88. 如何利用numpy數組實作Game of Life? (★★★)
(提示: Game of Life , Game of Life有哪些圖形？)
# Author: Nicolas Rougier

def iterate(Z):
    # Count neighbours
    N = (Z[0:-2,0:-2] + Z[0:-2,1:-1] + Z[0:-2,2:] +
         Z[1:-1,0:-2]                + Z[1:-1,2:] +
         Z[2:  ,0:-2] + Z[2:  ,1:-1] + Z[2:  ,2:])

    # Apply rules
    birth = (N==3) & (Z[1:-1,1:-1]==0)
    survive = ((N==2) | (N==3)) & (Z[1:-1,1:-1]==1)
    Z[...] = 0
    Z[1:-1,1:-1][birth | survive] = 1
    return Z

Z = np.random.randint(0,2,(50,50))
for i in range(100): Z = iterate(Z)
print(Z)
89. 如何找到一個數組的第n個最大值? (★★★)
(提示: np.argsort | np.argpartition)
Z = np.arange(10000)
np.random.shuffle(Z)
n = 5

# Slow
print (Z[np.argsort(Z)[-n:]])

# Fast
print (Z[np.argpartition(-Z,n)[:n]])
90. 給定任意個數向量，建立笛卡爾積(每一個元素的每一種組合) (★★★)
(提示: np.indices)
# Author: Stefan Van der Walt

def cartesian(arrays):
    arrays = [np.asarray(a) for a in arrays]
    shape = (len(x) for x in arrays)

    ix = np.indices(shape, dtype=int)
    ix = ix.reshape(len(arrays), -1).T

    for n, arr in enumerate(arrays):
        ix[:, n] = arrays[n][ix[:, n]]

    return ix

print (cartesian(([1, 2, 3], [4, 5], [6, 7])))
91. 如何從一個正常數組中建立記錄數組(record array)? (★★★)
(提示: np.core.records.fromarrays)
Z = np.array([("Hello", 2.5, 3),
              ("World", 3.6, 2)])
R = np.core.records.fromarrays(Z.T, 
                               names='col1, col2, col3',
                               formats = 'S8, f8, i8')
print(R)
92. 思考一個大向量Z, 用三種不同的方法計算它的立方 (★★★)
(提示: np.power, *, np.einsum)
# Author: Ryan G.

x = np.random.rand(5e7)

%timeit np.power(x,3)
%timeit x*x*x
%timeit np.einsum('i,i,i->i',x,x,x)
93. 考慮兩個形狀分别為(8,3) 和(2,2)的數組A和B. 如何在數組A中找到滿足包含B中元素的行？(不考慮B中每行元素順序)？(★★★)
(提示: np.where)
# Author: Gabe Schwartz

A = np.random.randint(0,5,(8,3))
B = np.random.randint(0,5,(2,2))

C = (A[..., np.newaxis, np.newaxis] == B)
rows = np.where(C.any((3,1)).all(1))[0]
print(rows)
94. 思考一個10x3的矩陣，如何分解出有不全相同值的行 (如 [2,2,3]) (★★★)
# Author: Robert Kern

Z = np.random.randint(0,5,(10,3))
print(Z)
# solution for arrays of all dtypes (including string arrays and record arrays)
E = np.all(Z[:,1:] == Z[:,:-1], axis=1)
U = Z[~E]
print(U)
# soluiton for numerical arrays only, will work for any number of columns in Z
U = Z[Z.max(axis=1) != Z.min(axis=1),:]
print(U)
95. 将一個整數向量轉換為二進制矩陣 (★★★)
(提示: np.unpackbits)
# Author: Warren Weckesser

I = np.array([0, 1, 2, 3, 15, 16, 32, 64, 128])
B = ((I.reshape(-1,1) & (2**np.arange(8))) != 0).astype(int)
print(B[:,::-1])

# Author: Daniel T. McDonald

I = np.array([0, 1, 2, 3, 15, 16, 32, 64, 128], dtype=np.uint8)
print(np.unpackbits(I[:, np.newaxis], axis=1))
96. 給定一個二維數組，如何提取出唯一的行?(★★★)
(提示: np.ascontiguousarray)
# Author: Jaime Fernández del Río

Z = np.random.randint(0,2,(6,3))
T = np.ascontiguousarray(Z).view(np.dtype((np.void, Z.dtype.itemsize * Z.shape[1])))
_, idx = np.unique(T, return_index=True)
uZ = Z[idx]
print(uZ)
97. 考慮兩個向量A和B，寫出用einsum等式對應的inner, outer, sum, mul函數 (★★★)
(提示: np.einsum)
# Author: Alex Riley
# Make sure to read: http://ajcr.net/Basic-guide-to-einsum/

A = np.random.uniform(0,1,10)
B = np.random.uniform(0,1,10)

np.einsum('i->', A)       # np.sum(A)
np.einsum('i,i->i', A, B) # A * B
np.einsum('i,i', A, B)    # np.inner(A, B)
np.einsum('i,j->ij', A, B)    # np.outer(A, B)
98. 考慮一個由兩個向量描述的路徑(X,Y)，如何用等距樣例(equidistant samples)對其進行采樣(sample)(★★★)?
(提示: np.cumsum, np.interp)
# Author: Bas Swinckels

phi = np.arange(0, 10*np.pi, 0.1)
a = 1
x = a*phi*np.cos(phi)
y = a*phi*np.sin(phi)

dr = (np.diff(x)**2 + np.diff(y)**2)**.5 # segment lengths
r = np.zeros_like(x)
r[1:] = np.cumsum(dr)                # integrate path
r_int = np.linspace(0, r.max(), 200) # regular spaced path
x_int = np.interp(r_int, r, x)       # integrate path
y_int = np.interp(r_int, r, y)
99. 給定一個整數n 和一個二維數組X，從X中選擇可以被解釋為從多n度的多項分布式的行，即這些行隻包含整數對n的和. (★★★)
(提示: np.logical_and.reduce, np.mod)
# Author: Evgeni Burovski

X = np.asarray([[1.0, 0.0, 3.0, 8.0],
                [2.0, 0.0, 1.0, 1.0],
                [1.5, 2.5, 1.0, 0.0]])
n = 4
M = np.logical_and.reduce(np.mod(X, 1) == 0, axis=-1)
M &= (X.sum(axis=-1) == n)
print(X[M])
100. 對于一個一維數組X，計算它boostrapped之後的95%置信區間的平均值. (★★★)
(提示: np.percentile)
# Author: Jessica B. Hamrick

X = np.random.randn(100) # random 1D array
N = 1000 # number of bootstrap samples
idx = np.random.randint(0, X.size, (N, X.size))
means = X[idx].mean(axis=1)
confint = np.percentile(means, [2.5, 97.5])
print(confint)