matrix
Matrix is the object numpy.matrix class type, the class inherits from numpy.ndarray, any operation against a multi-dimensional array, also valid for the matrix, but as a sub-category matrix and combined with its own characteristics, to do the necessary expansion, such as: multiplication, inversion and so on.
- Create a matrix object can be in one of three ways:
numpy.matrix(任何可被解释为矩阵的二维容器,copy=是否复制数据(缺省值为True，即复制数据))- If a copy of the value True (the default), the matrix obtained shared object with the same data source container parameter, otherwise, each with a separate copy of the data.
numpy.mat(任何可被解释为矩阵的二维容器)Equivalent tonumpy.matrix(..., copy=False)- Matrix object created by this function must share data with parameter source container, you can not have a separate copy of the data.
numpy.bmat(拼块规则)- Comprising a large number of small matrix data matrix, the rules specified by the parameter splicing
The above described functions may be acceptable matrix form of character string: data items separated by white space, the data lines separated by a semicolon. For example:
'123; 456'
/ 123 456 /
#mat.py
import numpy as np
a = np.array([
[1, 2, 6],
[3, 5, 7],
[4, 8, 9]])
print(a, type(a))
b = np.matrix(a)
print(b, type(b))
b += 10
print(b)
print(a)
c = np.matrix(a, copy=False)
print(c, type(c))
c += 10
print(c)
print(a)
d = np.mat(a)
print(d, type(d))
d -= 10
print(d)
print(a)
e = np.mat('1 2 6; 3 5 7; 4 8 9')
print(e)
f = np.bmat('b e')
print(f)
g = np.bmat('b e; e b')
# 多維數組的乘法：對應元素相乘
h = a * a
print('h等於\n', h)
# 矩陣相乘：乘積矩陣的第i行第j列元素等於被乘矩陣的第i行於乘數矩陣的第j列的點積
i = e * e
print('i等於\n', i)
j = e.I
print('e的逆矩陣為\n', j)
print(j * e)
k = a.dot(a) # 數組使用矩陣乘法
l = np.linalg.inv(a) # 數組使用逆矩陣[[1 2 6]
[3 5 7]
[4 8 9]] <class 'numpy.ndarray'>
[[1 2 6]
[3 5 7]
[4 8 9]] <class 'numpy.matrixlib.defmatrix.matrix'>
[[11 12 16]
[13 15 17]
[14 18 19]]
[[1 2 6]
[3 5 7]
[4 8 9]]
[[1 2 6]
[3 5 7]
[4 8 9]] <class 'numpy.matrixlib.defmatrix.matrix'>
[[11 12 16]
[13 15 17]
[14 18 19]]
[[11 12 16]
[13 15 17]
[14 18 19]]
[[11 12 16]
[13 15 17]
[14 18 19]] <class 'numpy.matrixlib.defmatrix.matrix'>
[[1 2 6]
[3 5 7]
[4 8 9]]
[[1 2 6]
[3 5 7]
[4 8 9]]
[[1 2 6]
[3 5 7]
[4 8 9]]
[[11 12 16 1 2 6]
[13 15 17 3 5 7]
[14 18 19 4 8 9]]
h等於
[[ 1 4 36]
[ 9 25 49]
[16 64 81]]
i等於
[[ 31 60 74]
[ 46 87 116]
[ 64 120 161]]
e的逆矩陣為
[[-0.73333333 2. -1.06666667]
[ 0.06666667 -1. 0.73333333]
[ 0.26666667 0. -0.06666667]]
[[ 1.00000000e+00 1.77635684e-15 3.55271368e-15]
[ 0.00000000e+00 1.00000000e+00 -1.11022302e-16]
[ 0.00000000e+00 0.00000000e+00 1.00000000e+00]]General function
1. frompyfunc-> ufunc Object
def 标量函数(标量参数1, 标量参数2, ...):
...
return 标量返回值1, 标量返回值2, ...
矢量参数1
矢量参数2
...
numpy.frompyfunc(标量函数, 参数个数, 返回值个数)
->矢量函数 # numpy.ufunc类类型的对象，可调用对象
矢量函数(矢量参数1, 矢量参数2, ...)
->矢量返回值1, 矢量返回值2# ufunc.py
import numpy as np
def foo(x, y):
return x + y, x - y, x * y
def hum(x):
def fun(y):
return x + y, x - y, x * y
return np.frompyfunc(fun, 1, 1)
x, y = 1, 4
print(foo(x, y))
X, Y = np.array([1, 2, 3]), np.array([4, 5, 6])
bar = np.frompyfunc(foo, 2, 3)
print(bar(X, Y))
print(np.frompyfunc(foo, 2, 3)(X, Y))
print(hum(100)(X))(5, -3, 4)
(array([5, 7, 9], dtype=object), array([-3, -3, -3], dtype=object), array([4, 10, 18], dtype=object))
(array([5, 7, 9], dtype=object), array([-3, -3, -3], dtype=object), array([4, 10, 18], dtype=object))
[(101, 99, 100) (102, 98, 200) (103, 97, 300)]2. Adder generic function: add
add.reduce() - 累加和
add.accumulate() - 累加和过程
add.reduceat() - 分段累加和
add.outer() - 外和# add.py
a = np.arange(1, 7)
print(a)
b = a + a
print(b)
b = np.add(a, a)
print(b)
c = np.add.reduce(a)
print(c)
d = np.add.accumulate(a)
print(d)
e = np.add.reduceat(a, [0, 2, 4]) # 按照下標分段
print(e)
f = np.add.outer([10, 20, 30], a)
print(f)
g = np.outer([10, 20, 30], a) # 不加add的話是外積
print(g)[1 2 3 4 5 6]
[ 2 4 6 8 10 12]
[ 2 4 6 8 10 12]
21
[ 1 3 6 10 15 21]
[ 3 7 11]
[[11 12 13 14 15 16]
[21 22 23 24 25 26]
[31 32 33 34 35 36]]
[[ 10 20 30 40 50 60]
[ 20 40 60 80 100 120]
[ 30 60 90 120 150 180]]3. The general function division
[5 5 -5 -5]<真除>[2 -2 2 -2] = [2.5 -2.5 -2.5 2.5]
numpy.true_divide()
numpy.divide()
/
[5 5 -5 -5]<地板除>[2 -2 2 -2] = [2 -3 -3 2]
numpy.floor_divide()
//
[5 5 -5 -5]<天花板除>[2 -2 2 -2] = [3 -2 -2 3]
天花板取整(真除的结果)：numpy.ceil()
[5 5 -5 -5]<截断除>[2 -2 2 -2] = [2 -2 -2 2]
截断取整(真除的结果)：numpy.trunc()# div.py
a = np.array([5, 5, -5, -5])
b = np.array([2, -2, 2, -2])
print(a, b)
# c = np.true_divide(a, b)
# c = np.divide(a, b)
c = a / b
print(c)
# d = np.floor_divide(a, b)
d = a // b
print(d)
e = np.ceil(a / b).astype(int)
print(e)
f = np.trunc(a / b).astype(int)
print(f)[ 5 5 -5 -5] [ 2 -2 2 -2]
[ 2.5 -2.5 -2.5 2.5]
[ 2 -3 -3 2]
[ 3 -2 -2 3]
[ 2 -2 -2 2]4. The general function modulo
被除数/除数=商...余数
除数 * 商 + 余数 = 被除数
[5 5 -5 -5]<地板除>[2 -2 2 -2] = [2 -3 -3 2]...[1 -1 1 -1]
numpy.remainder()
numpy.mod()
%
[5 5 -5 -5]<截断除>[2 -2 2 -2] = [2 -2 -2 2]...[1 1 -1 -1]
numpy.fmod()Numpy Python language for the scalar operator overloading be defined by the generic function to support an array of vector calculation.
# fib.py
import numpy as np
n = 35
# 递归方法
# def fibo(n):
# return 1 if n < 3 else fibo(n - 1) + fibo(n - 2)
# print(fibo(n))
# 循环换位方法
fn_1, fn_2 = 0, 1
for i in range(n):
fn = fn_1 + fn_2
fn_1, fn_2 = fn, fn_1
print(fn)
# 矩阵乘方
print(int((np.mat('1. 1.; 1. 0.') ** (n - 1))[0, 0]))
# 斐波那契公式
r = np.sqrt(5)
print(int((((1 + r) / 2) ** n -
((1 - r) / 2) ** n) / r))9227465
9227465
92274655. Numpy all trigonometric function are common in
x = Asin(at+pi/2)
y = Bsin(bt)# lissa.py
import numpy as np
import matplotlib.pyplot as mp
t = np.linspace(0, 2 * np.pi, 201)
A, a, B, b = 10, 9, 5, 8
x = A * np.sin(a * t + np.pi / 2)
y = B * np.sin(b * t)
mp.figure('Lissajous', facecolor='lightgray')
mp.title('Lissajous', fontsize=20)
mp.xlabel('x', fontsize=14)
mp.ylabel('y', fontsize=14)
mp.tick_params(labelsize=10)
mp.grid(linestyle=':')
mp.plot(x, y, c='orangered', label='Lissajous')
mp.legend()
mp.show()
y = 4/(1pi) sin(1x) 2 x 1 - 1
y = 4/(3pi) sin(3x) 2 x 2 - 1
y = 4/(5pi) sin(5x) 2 x 3 - 1import numpy as np
import matplotlib.pyplot as mp
def squarewave(n):
k = np.arange(1, n + 1)
def fun(x):
return np.sum(4 / ((2 * k - 1) * np.pi) *
np.sin((2 * k - 1) * x))
return np.frompyfunc(fun, 1, 1)
x = np.linspace(0, 2 * np.pi, 201)
y1 = squarewave(1)(x)
y2 = squarewave(2)(x)
y3 = squarewave(3)(x)
y4 = squarewave(10)(x)
y5 = squarewave(100)(x)
y6 = squarewave(1000)(x)
mp.figure('Squarewave', facecolor='lightgray')
mp.title('Squarewave', fontsize=20)
mp.xlabel('x', fontsize=14)
mp.ylabel('y', fontsize=14)
mp.tick_params(labelsize=10)
mp.grid(linestyle=':')
mp.plot(x, y1, label='n=1')
mp.plot(x, y2, label='n=2')
mp.plot(x, y3, label='n=3')
mp.plot(x, y4, label='n=10')
mp.plot(x, y5, label='n=100')
mp.plot(x, y6, label='n=1000')
mp.legend()
mp.show()
7. The general function Bitwise
位异或：^/__xor__/bitwise_xor
0 ^ 0 = 0
0 ^ 1 = 1
1 ^ 0 = 1
1 ^ 1 = 0
if a^b<0 then a和b异号
位与：&/__and__/bitwise_and
0 & 0 = 0
0 & 1 = 0
1 & 0 = 0
1 & 1 = 1
# 判断是否为2的幂
1 2^0 00001 0 00000
2 2^1 00010 1 00001
4 2^2 00100 3 00011
8 2^3 01000 7 00111
16 2^4 10000 15 01111
...
if a & (a-1) == 0 then a是2的幂
位或：|/__or__/bitwise_or
0 | 0 = 0
0 | 1 = 1
1 | 0 = 1
1 | 1 = 1
位反：~/__not__/bitwise_not
~0 = 1
~1 = 0
移位：<</__lshift__/left_shift>>/__rshift__/right_shift
左移1位相当于乘2，右移1位相当于除2。import numpy as np
a = np.array([0, -1, 2, -3, 4, -5])
b = np.array([0, 1, 2, 3, 4, 5])
print(a, b)
# c = a ^ b
# c = a.__xor__(b)
c = np.bitwise_xor(a, b)
print(c)
print(np.where(c < 0)[0])
d = np.arange(1, 21)
print(d)
# e = d & (d - 1)
# e = d.__and__(d - 1)
e = np.bitwise_and(d, d - 1)
print(e)
print(d[e == 0])
# f = d << 1
# f = d.__lshift__(1)
f = np.left_shift(d, 1)
print(f)
# g = d >> 1
# g = d.__rshift__(1)
g = np.right_shift(d, 1)
print(g)
[ 0 -1 2 -3 4 -5] [0 1 2 3 4 5]
[ 0 -2 0 -2 0 -2]
[1 3 5]
[ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20]
[ 0 0 2 0 4 4 6 0 8 8 10 8 12 12 14 0 16 16 18 16]
[ 1 2 4 8 16]
[ 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40]
[ 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10]Linear algebra module (linalg)
1. inverse and generalized inverse matrix
If the product of a matrix A and the other B is a square matrix, then A and B to mutually inverse.
np.linalg.inv (A) -> A ^ -1The above definition of the inverse matrix is extended to non-square, then called the generalized inverse matrix.
np.linalg.pinv (A) -> A ^ -1
np.matrix.I -> INV / Pinv
import numpy as np
A = np.mat('1 2 3; 8 9 4; 7 6 5')
print(A)
B = np.linalg.inv(A)
print(B)
print(A * B)
print(A.I)
C = np.mat('11 12 13 14; 20 21 22 15; 19 18 17 16')
print(C)
# D = np.linalg.inv(C)
D = np.linalg.pinv(C)
print(D)
print(C * D)
print(C.I)
E = np.mat('1 2 3; 4 5 6; 7 8 9')
print(E)
F = np.linalg.inv(E)
print(F)
print(E * F)[[1 2 3]
[8 9 4]
[7 6 5]]
[[-0.4375 -0.16666667 0.39583333]
[ 0.25 0.33333333 -0.41666667]
[ 0.3125 -0.16666667 0.14583333]]
[[ 1.00000000e+00 2.77555756e-17 -5.55111512e-17]
[ 0.00000000e+00 1.00000000e+00 2.22044605e-16]
[ 0.00000000e+00 1.94289029e-16 1.00000000e+00]]
[[-0.4375 -0.16666667 0.39583333]
[ 0.25 0.33333333 -0.41666667]
[ 0.3125 -0.16666667 0.14583333]]
[[11 12 13 14]
[20 21 22 15]
[19 18 17 16]]
[[-0.18055556 -0.08333333 0.23611111]
[-0.04305556 0.04166667 -0.00138889]
[ 0.09444444 0.16666667 -0.23888889]
[ 0.1625 -0.125 0.0375 ]]
[[ 1.00000000e+00 2.22044605e-16 -3.33066907e-16]
[ 1.77635684e-15 1.00000000e+00 -1.88737914e-15]
[ 1.77635684e-15 8.88178420e-16 1.00000000e+00]]
[[-0.18055556 -0.08333333 0.23611111]
[-0.04305556 0.04166667 -0.00138889]
[ 0.09444444 0.16666667 -0.23888889]
[ 0.1625 -0.125 0.0375 ]]
[[1 2 3]
[4 5 6]
[7 8 9]]
[[ 3.15251974e+15 -6.30503948e+15 3.15251974e+15]
[-6.30503948e+15 1.26100790e+16 -6.30503948e+15]
[ 3.15251974e+15 -6.30503948e+15 3.15251974e+15]]
[[ 0. 1. -0.5]
[ 0. 2. -1. ]
[ 0. 3. 2.5]]2. The solution of linear equations
$$
\left( \begin{array}{l}
x - 2y + z = 0\
2y - 8z - 8 = 0\
- 4x + 5y + 9z + 9 = 0
\end{array} \right.
$$
$$
\left( \begin{array}{l}
1x + - 2y + 1z = 0\
0x + 2y + - 8z = 8\
- 4x + 5y + 9z = - 9
\end{array} \right.
$$
$$
\left( {\begin{array}{{20}{c}}
1&{ - 2}&1\
0&2&{ - 8}\
{ - 4}&5&9
\end{array}} \right) \times \left( {\begin{array}{{20}{c}}
x\
y\
z
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
0\
8\
{ - 9}
\end{array}} \right)
$$
a x b
= np.linalg.lstsq(a, b)[0] -> 拟合
= np.linalg.solve(a, b) -> 解import numpy as np
a = np.mat('1 -2 1; 0 2 -8; -4 5 9')
b = np.mat('0; 8; -9')
x = np.linalg.solve(a, b)
print(x)
x = np.linalg.lstsq(a, b)[0]
print(x)[[29.]
[16.]
[ 3.]]
[[29.]
[16.]
[ 3.]]
/Users/haoen110/miniconda3/lib/python3.7/site-packages/ipykernel_launcher.py:6: FutureWarning: `rcond` parameter will change to the default of machine precision times ``max(M, N)`` where M and N are the input matrix dimensions.
To use the future default and silence this warning we advise to pass `rcond=None`, to keep using the old, explicitly pass `rcond=-1`.
3. The eigenvalues and eigenvectors
- For n-order square matrix A, if a number of non-zero column vector of dimension n x exists, so that Ax = ax, a feature value is called a matrix A, the matrix A is x belongs to an eigenvalue of a
np.linalg .eig (A) -> array eigenvalues, eigenvectors array
[A1 A2]
[[X11 X12]
[X21 X22]]
import numpy as np
A = np.mat('3 -2; 1 0')
print(A)
eigvals, eigvecs = np.linalg.eig(A)
print(eigvals)
print(eigvecs)
print(A * eigvecs[:, 0])
print(eigvals[0] * eigvecs[:, 0])
print(A * eigvecs[:, 1])
print(eigvals[1] * eigvecs[:, 1])[[ 3 -2]
[ 1 0]]
[2. 1.]
[[0.89442719 0.70710678]
[0.4472136 0.70710678]]
[[1.78885438]
[0.89442719]]
[[1.78885438]
[0.89442719]]
[[0.70710678]
[0.70710678]]
[[0.70710678]
[0.70710678]]4. Singular Value Decomposition
- On the main diagonal elements of the singular value matrix M is referred to, the other elements are 0
|
M S X X = the U-V
| |
orthogonal matrix
Uxu vxv = E = T ^ T ^
np.linalg.svd (M, full_matrices = False) -> U, singular value, V
import numpy as np
M = np.mat('4 11 14; 8 7 -2')
print(M)
U, sv, V = np.linalg.svd(M, full_matrices=False)
print(U * U.T)
print(V * V.T)
print(sv)
S = np.diag(sv)
print(S)
print(U * S * V)[[ 4 11 14]
[ 8 7 -2]]
[[1.0000000e+00 3.2123061e-17]
[3.2123061e-17 1.0000000e+00]]
[[ 1.00000000e+00 -6.16790569e-18]
[-6.16790569e-18 1.00000000e+00]]
[18.97366596 9.48683298]
[[18.97366596 0. ]
[ 0. 9.48683298]]
[[ 4. 11. 14.]
[ 8. 7. -2.]]5. determinant
np.det(方阵)->行列式的值import numpy as np
A = np.mat('2 1; 3 4')
print(A)
print(np.linalg.det(A))
B = np.mat('3 2 1; 4 9 8; 5 6 7')
print(B)
print(np.linalg.det(B))[[2 1]
[3 4]]
5.000000000000001
[[3 2 1]
[4 9 8]
[5 6 7]]
47.999999999999986Fast Fourier Transform block (FFT)
- Primitive: y = f (x) - the temporal / spatial domain function
A series of superimposed sine function
Y = A 1 sin (WlX fai1 +) + A 2 sin (W2X fai2 +) + ... + Ansin (Fain WNx +) + R & lt
N-> OO: the R-> 0
[X1, X2, .. ., xn] -> [y1 , y2, ..., yn]w1->A1, fai1 \ w2->A2, fai2 | A,fai = f'(w) - 频率域函数 ... | wn->An, fain / f(x) -傅里叶变换-> f'(w) 时空域 频率域 f(x) <-反傅里叶变换- f'(w) 时空域 频率域 np.fft.fftfreq(采样数, 采样周期)->频率序列 np.fft.fft(原函数值序列) -> 目标函数值序列(复数) 复数的模反映了振幅A，辐角反映了初相位fai np.fft.ifft(目标函数值序列(复数))->原函数值序列
import numpy as np
import numpy.fft as nf
import matplotlib.pyplot as mp
times = np.linspace(0, 2 * np.pi, 201)
sigs1 = 4 / (1 * np.pi) * np.sin(1 * times)
sigs2 = 4 / (3 * np.pi) * np.sin(3 * times)
sigs3 = 4 / (5 * np.pi) * np.sin(5 * times)
sigs4 = 4 / (7 * np.pi) * np.sin(7 * times)
sigs5 = 4 / (9 * np.pi) * np.sin(9 * times)
sigs6 = sigs1 + sigs2 + sigs3 + sigs4 + sigs5
freqs = nf.fftfreq(times.size, times[1] - times[0])
ffts = nf.fft(sigs6)
pows = np.abs(ffts)
sigs7 = nf.ifft(ffts).real
mp.subplot(121)
mp.title('Time Domain', fontsize=16)
mp.xlabel('Time', fontsize=12)
mp.ylabel('Signal', fontsize=12)
mp.tick_params(labelsize=10)
mp.grid(linestyle=':')
mp.plot(times, sigs1, label='{:.4f}'.format(
1 / (2 * np.pi)))
mp.plot(times, sigs2, label='{:.4f}'.format(
3 / (2 * np.pi)))
mp.plot(times, sigs3, label='{:.4f}'.format(
5 / (2 * np.pi)))
mp.plot(times, sigs4, label='{:.4f}'.format(
7 / (2 * np.pi)))
mp.plot(times, sigs5, label='{:.4f}'.format(
9 / (2 * np.pi)))
mp.plot(times, sigs6, label='{:.4f}'.format(
1 / (2 * np.pi)))
mp.plot(times, sigs7, label='{:.4f}'.format(
1 / (2 * np.pi)), alpha=0.5, linewidth=6)
mp.legend()
mp.subplot(122)
mp.title('Frequency Domain', fontsize=16)
mp.xlabel('Frequency', fontsize=12)
mp.ylabel('Power', fontsize=12)
mp.tick_params(labelsize=10)
mp.grid(linestyle=':')
mp.plot(freqs[freqs >= 0], pows[freqs >= 0],
c='orangered', label='Frequency Spectrum')
mp.legend()
mp.tight_layout()
- Frequency filtering Fourier transform
____________________IFFT_____________________
| |
V |
energetic signal FFT frequency domain filter |
> signal with noise -----> noisy spectrum ----------> High spectral
low energy noise /
import numpy as np
import numpy.fft as nf
import scipy.io.wavfile as wf
import matplotlib.pyplot as mp
sample_rate, noised_sigs = wf.read(
'../data/noised.wav')
noised_sigs = noised_sigs / 2 ** 15
times = np.arange(len(noised_sigs)) / sample_rate
freqs = nf.fftfreq(times.size, 1 / sample_rate)
noised_ffts = nf.fft(noised_sigs)
noised_pows = np.abs(noised_ffts)
fund_freq = np.abs(freqs[noised_pows.argmax()])
print(fund_freq)
noised_indices = np.where(np.abs(freqs) != fund_freq)
filter_ffts = noised_ffts.copy()
filter_ffts[noised_indices] = 0
filter_pows = np.abs(filter_ffts)
filter_sigs = nf.ifft(filter_ffts).real
wf.write('../data/filter.wav', sample_rate,
(filter_sigs * 2 ** 15).astype(np.int16))
mp.figure('Filter', facecolor='lightgray')
mp.subplot(221)
mp.title('Time Domain', fontsize=16)
mp.ylabel('Signal', fontsize=12)
mp.tick_params(labelsize=10)
mp.grid(linestyle=':')
mp.plot(times[:178], noised_sigs[:178],
c='orangered', label='Noised')
mp.legend()
mp.subplot(222)
mp.title('Frequency Domain', fontsize=16)
mp.ylabel('Power', fontsize=12)
mp.tick_params(labelsize=10)
mp.grid(linestyle=':')
mp.semilogy(freqs[freqs >= 0],
noised_pows[freqs >= 0], c='limegreen',
label='Noised')
mp.legend()
mp.subplot(223)
mp.xlabel('Time', fontsize=12)
mp.ylabel('Signal', fontsize=12)
mp.tick_params(labelsize=10)
mp.grid(linestyle=':')
mp.plot(times[:178], filter_sigs[:178],
c='hotpink', label='Filter')
mp.legend()
mp.subplot(224)
mp.xlabel('Frequency', fontsize=12)
mp.ylabel('Power', fontsize=12)
mp.tick_params(labelsize=10)
mp.grid(linestyle=':')
mp.plot(freqs[freqs >= 0], filter_pows[freqs >= 0],
c='dodgerblue', label='Filter')
mp.legend()
mp.tight_layout()
mp.show()1000.0
Random Number Module (Random)
Generating a random number sequence specific to obey the laws of statistics.
1. Binomial
np.random.binomial(n, p, size)Size generating random numbers, each random number from n number of successful attempts in which the probability of success of each attempt p.
# 猜硬币游戏：初始筹码1000，每轮猜9次，猜对5次及5次以上为赢，筹码加1，否则为输，筹码减1，求10000轮的过程中手中筹码的变化。
# 代码：bi.py
import numpy as np
import matplotlib.pyplot as mp
outcomes = np.random.binomial(9, 0.5, 10000)
chips = [1000]
for outcome in outcomes:
if outcome >= 5:
chips.append(chips[-1] + 1)
else:
chips.append(chips[-1] - 1)
chips = np.array(chips)
mp.figure('Binomial Distribution',
facecolor='lightgray')
mp.title('Binomial Distribution', fontsize=20)
mp.xlabel('Round', fontsize=14)
mp.ylabel('Chip', fontsize=14)
mp.tick_params(labelsize=12)
mp.grid(linestyle=':')
o, h, l, c = 0, chips.argmax(), chips.argmin(), \
chips.size - 1
if chips[o] < chips[c]:
color = 'orangered'
elif chips[c] < chips[o]:
color = 'limegreen'
else:
color = 'dodgerblue'
mp.plot(chips, c=color, label='Chip')
mp.axhline(y=chips[o], linestyle='--',
color='deepskyblue', linewidth=1)
mp.axhline(y=chips[h], linestyle='--',
color='crimson', linewidth=1)
mp.axhline(y=chips[l], linestyle='--',
color='seagreen', linewidth=1)
mp.axhline(y=chips[c], linestyle='--',
color='orange', linewidth=1)
mp.legend()
mp.show()
2. hypergeometric distribution
np.random.hypergeometric(ngood, nbad, nsample, size)Size generating random numbers, each random number from the random number sample in nsample good samples, samples of the total samples ngood good and bad samples composed nbad
# 模球游戏：将25个好球和1个坏球放在一起，每次模3个球，全为好球加1分，只要摸到了坏球减6分，求100轮的过程中分值的变化。
# 代码：hyper.py
import numpy as np
import matplotlib.pyplot as mp
outcomes = np.random.hypergeometric(25, 1, 3, 100)
scores = [0]
for outcome in outcomes:
if outcome == 3:
scores.append(scores[-1] + 1)
else:
scores.append(scores[-1] - 6)
scores = np.array(scores)
mp.figure('Hypergeometric Distribution',
facecolor='lightgray')
mp.title('Hypergeometric Distribution', fontsize=20)
mp.xlabel('Round', fontsize=14)
mp.ylabel('Score', fontsize=14)
mp.tick_params(labelsize=12)
mp.grid(linestyle=':')
o, h, l, c = 0, scores.argmax(), scores.argmin(), \
scores.size - 1
if scores[o] < scores[c]:
color = 'orangered'
elif scores[c] < scores[o]:
color = 'limegreen'
else:
color = 'dodgerblue'
mp.plot(scores, c=color, label='Score')
mp.axhline(y=scores[o], linestyle='--',
color='deepskyblue', linewidth=1)
mp.axhline(y=scores[h], linestyle='--',
color='crimson', linewidth=1)
mp.axhline(y=scores[l], linestyle='--',
color='seagreen', linewidth=1)
mp.axhline(y=scores[c], linestyle='--',
color='orange', linewidth=1)
mp.legend()
mp.show()
3. The standard normal distribution
np.random.norm(size)`Size generating random numbers, subject to standard normal (mean = 0, standard deviation = 1) distribution.
Probability Density: $ f (x) = \ frac {{{e ^ {- \ frac {{{{(x - \ mu)} ^ 2}}} {{2 {\ sigma ^ 2}}}}}} } {{\ sigma \ sqrt {2 \ pi}}} $
# norm.py
import numpy as np
import matplotlib.pyplot as mp
samples = np.random.normal(size=10000)
mp.figure('Normal Distribution', facecolor='lightgray')
mp.title('Normal Distribution', fontsize=20)
mp.xlabel('Sample', fontsize=14)
mp.ylabel('Occurrence', fontsize=14)
mp.tick_params(labelsize=12)
mp.grid(axis='y', linestyle=':')
bins = mp.hist(samples, 100, normed=True,
edgecolor='steelblue',
facecolor='deepskyblue',
label='Normal')[1]
probs = np.exp(-bins ** 2 / 2) / np.sqrt(2 * np.pi)
mp.plot(bins, probs, 'o-', c='orangered',
label='Probability')
mp.legend()
mp.show()
Miscellaneous
1. Sort
Sort indirect joint
numpy.lexsort((参考序列, 待排序列))-> ordered index
[Zhao Wang Wu six Zhangsanlisi]
[70,608,070] <- [30203020]numpy.sort_complex(复数数组)- In ascending order of the real part, the same elements of the real part, the imaginary part of the reference ascending, returns the result directly sorted array.
numpy.searchsorted(有序序列, 待插序列)-> Location Array- It represents an element in the sequence to be inserted is inserted into a position at which the ordered sequence, the result is still orderly
numpy.insert(被插序列, 位置序列, 待插序列)-> element in the sequence to be inserted- Position according to the position in the sequence, is inserted into the insertion sequence, the inserted result returned
# sort.py
import numpy as np
ages = np.array([30, 20, 30, 20])
scores = np.array([70, 60, 80, 70])
names = np.array(['zhangsan', 'lisi', 'wangwu', 'zhaoliu'])
# 按照成绩的升序打印姓名，成绩相同的按照年龄的升序排列
print(np.take(names, np.lexsort((ages, scores))))
compleies = scores + ages * 1j
print(compleies)
sorted_compleies = np.sort_complex(compleies)
print(sorted_compleies)
# 0 1 2 3 4 5 6
a = np.array([1, 2, 4, 5, 6, 8, 9])
b = np.array([7, 3])
c = np.searchsorted(a, b)
print(c)
d = np.insert(a, c, b)
print(d)
['lisi' 'zhaoliu' 'zhangsan' 'wangwu']
[70.+30.j 60.+20.j 80.+30.j 70.+20.j]
[60.+20.j 70.+20.j 70.+30.j 80.+30.j]
[5 2]
[1 2 3 4 5 6 7 8 9]2. Interpolation
import scipy.interpolate as si
si.interp1d(离散水平坐标, 离散垂直坐标, kind=插值算法(缺省为线性插值)) -> 插值器
插值器(水平坐标)->垂直坐标import numpy as np
import scipy.interpolate as si
import matplotlib.pyplot as mp
min_x, max_x = -2.5, 2.5
con_x = np.linspace(min_x, max_x, 1001)
con_y = np.sinc(con_x)
dis_x = np.linspace(min_x, max_x, 11)
dis_y = np.sinc(dis_x)
# 线性插值
linear = si.interp1d(dis_x, dis_y)
lin_x = np.linspace(min_x, max_x, 51)
lin_y = linear(lin_x)
# 三次样条插值
cubic = si.interp1d(dis_x, dis_y, kind='cubic')
cub_x = np.linspace(min_x, max_x, 51)
cub_y = cubic(cub_x)
mp.figure('Interpolation', facecolor='lightgray')
mp.subplot(221)
mp.title('Continuous', fontsize=16)
mp.ylabel('y', fontsize=12)
mp.tick_params(labelsize=10)
mp.grid(linestyle=':')
mp.plot(con_x, con_y, c='hotpink',
label='Continuous')
mp.legend()
mp.subplot(222)
mp.title('Discrete', fontsize=16)
mp.tick_params(labelsize=10)
mp.grid(linestyle=':')
mp.scatter(dis_x, dis_y, c='orangered', s=80,
label='Discrete')
mp.legend()
mp.subplot(223)
mp.title('Linear', fontsize=16)
mp.xlabel('x', fontsize=12)
mp.ylabel('y', fontsize=12)
mp.tick_params(labelsize=10)
mp.grid(linestyle=':')
mp.plot(lin_x, lin_y, 'o-', c='limegreen',
label='Linear')
mp.scatter(dis_x, dis_y, c='orangered', s=80,
zorder=3)
mp.legend()
mp.subplot(224)
mp.title('Cubic', fontsize=16)
mp.xlabel('x', fontsize=12)
mp.tick_params(labelsize=10)
mp.grid(linestyle=':')
mp.plot(cub_x, cub_y, 'o-', c='dodgerblue',
label='Cubic')
mp.scatter(dis_x, dis_y, c='orangered', s=80,
zorder=3)
mp.legend()
mp.tight_layout()
mp.show()
3. Points
import scipy.integrate as si
si.quad(积分函数, 积分下限, 积分上限)->积分值, 最大误差import numpy as np
import scipy.integrate as si
import matplotlib.pyplot as mp
import matplotlib.patches as mc
# 定义函数
def f(x):
return 2 * x ** 2 + 3 * x + 4
# 确定x范围
a, b = -5, 5
x1 = np.linspace(a, b, 1001)
y1 = f(x1)
# 用scipy来计算积分
area = si.quad(f, a, b)[0]
print(area)
# 手动分割小梯形来计算面积，n越大，越接近真实值
n = 10
x2 = np.linspace(a, b, n + 1)
y2 = f(x2)
area = 0
for i in range(n):
area += (y2[i] + y2[i + 1]) * (x2[i + 1] - x2[i]) / 2
print(area)
# 绘制函数图
mp.figure('Integral', dpi=120, facecolor='lightgray')
mp.title('Integral', fontsize=20)
mp.xlabel('x', fontsize=14)
mp.ylabel('y', fontsize=14)
mp.tick_params(labelsize=10)
mp.grid(linestyle=':')
mp.plot(x1, y1, c='orangered', linewidth=6,
label=r'$y=2x^2+3x+4$', zorder=0)
for i in range(n):
mp.gca().add_patch(mc.Polygon([
[x2[i], 0], [x2[i], y2[i]],
[x2[i + 1], y2[i + 1]], [x2[i + 1], 0]],
fc='deepskyblue', ec='dodgerblue',
alpha=0.5))
mp.legend()
mp.show()206.66666666666669
210.0
4. Image
scipy.ndimage中提供了一些简单的图像处理，如高斯模糊、任意角度旋转、边缘识别等功能。import numpy as np
import scipy.misc as sm # 杂项工具
import scipy.ndimage as sn
import matplotlib.pyplot as mp
original = sm.imread('../data/lily.jpg', True) # True就是黑白
print(original.shape, original.dtype)
median = sn.median_filter(original, (31, 31)) # 中值滤波，消除细节干扰、模糊化
rotate = sn.rotate(original, 45)
prewitt = sn.prewitt(original)
mp.figure('Image', dpi=120, facecolor='lightgray')
mp.subplot(221)
mp.title('Original', fontsize=16)
mp.axis('off')
mp.imshow(original, cmap='gray')
mp.subplot(222)
mp.title('Median', fontsize=16)
mp.axis('off')
mp.imshow(median, cmap='gray')
mp.subplot(223)
mp.title('Rotate', fontsize=16)
mp.axis('off')
mp.imshow(rotate, cmap='gray')
mp.subplot(224)
mp.title('Prewitt', fontsize=16)
mp.axis('off')
mp.imshow(prewitt, cmap='gray')
mp.tight_layout()
mp.show()/Users/haoen110/miniconda3/lib/python3.7/site-packages/ipykernel_launcher.py:6: DeprecationWarning: `imread` is deprecated!
`imread` is deprecated in SciPy 1.0.0, and will be removed in 1.2.0.
Use ``imageio.imread`` instead.
(517, 690) float32
5. Financial
import numpy as np
# 终值 = fv(利率, 期数, 每期支付, 现值)
# 将1000元以1%的年利率存入银行5年，每年加存100元，
# 到期后本息合计多少钱？
fv = np.fv(0.01, 5, -100, -1000) # 凡是掏钱，都是负数
print(round(fv, 2))
# 现值 = pv(利率, 期数, 每期支付, 终值)
# 将多少钱以1%的年利率存入银行5年，每年加存100元，
# 到期后本息合计fv元？
pv = np.pv(0.01, 5, -100, fv)
print(pv)
# 净现值 = npv(利率, 现金流)
# 将1000元以1%的年利率存入银行5年，每年加存100元，
# 相当于一次性存入多少钱？
npv = np.npv(0.01, [-1000, -100, -100, -100, -100, -100])
print(round(npv, 2))
fv = np.fv(0.01, 5, 0, npv)
print(round(fv, 2))
# 内部收益率 = irr(现金流)
# 将1000元存入银行5年，以后逐年提现100元、200元、
# 300元、400元、500元，银行利率达到多少，可在最后
# 一次提现后偿清全部本息，即净现值为0元？
irr = np.irr([-1000, 100, 200, 300, 400, 500])
print(round(irr, 2))
npv = np.npv(irr, [-1000, 100, 200, 300, 400, 500])
print(npv)
# 每期支付 = pmt(利率, 期数, 现值)
# 以1%的年利率从银行贷款1000元，分5年还清，
# 平均每年还多少钱？
pmt = np.pmt(0.01, 5, 1000)
print(round(pmt, 2))
# 期数 = nper(利率, 每期支付, 现值)
# 以1%的年利率从银行贷款1000元，平均每年还pmt元，
# 多少年还清？
nper = np.nper(0.01, pmt, 1000)
print(int(nper))
# 利率 = rate(期数, 每期支付, 现值, 终值)
# 从银行贷款1000元，平均每年还pmt元，nper年还清，
# 年利率多少？
rate = np.rate(nper, pmt, 1000, 0)
print(round(rate, 2))1561.11
-1000.0
-1485.34
1561.11
0.12
0.0
-206.04
5
0.01