SciPy
SciPy 是一款方便、易于使用、专为科学和工程设计的Python工具包，它包括统计、优化、整合、线性代数模块、傅里叶变换、信号和图像处理、常微分方程求解器等等。它和Numpy联系很密切，Scipy一般都是操控Numpy数组来进行科学计算，所以可以说是基于Numpy之上了。
安装方法
pip install numpy scipy matplotlib iPython jupyter pandas sympy nose
Scipy子模块
| 模块名 | 功能 |
|---|---|
| scipy.cluster | 向量量化 |
| scipy.constants | 数学常量 |
| scipy.fftpack | 快速傅里叶变换 |
| scipy.integrate | 积分 |
| scipy.interpolate | 插值 |
| scipy.io | 插值 |
| scipy.interpolate | 数据输入输出 |
| scipy.linalg | 线性代数 |
| scipy.ndimage | N维图像 |
| scipy.odr | 正交距离回归 |
| scipy.optimize | 优化算法 |
| scipy.signal | 信号处理 |
| scipy.sparse | 稀疏矩阵 |
| scipy.spatial | 空间数据结构和算法 |
| scipy.special | 特殊数学函数 |
| scipy.stats | 统计函数 |
文件输入和输出：scipy.io
>>> from scipy import io as spio
>>> a = np.ones((3, 3))
>>> spio.savemat('file.mat', {'a': a}) # 保存字典到file.mat
>>> data = spio.loadmat('file.mat', struct_as_record=True)
>>> data['a']
array([[ 1., 1., 1.],
[ 1., 1., 1.],
[ 1., 1., 1.]])
线性代数操作：scipy.linalg
假如我们要计算一个方阵的行列式，我们需要调用det()函数：
>>> import numpy as np
>>> from scipy import linalg
>>> arr = np.array([[1,2],[3,4]])
>>> linalg.det(arr)
-2.0
>>> arr = np.array([[3,2],[6,4]])
>>> linalg.det(arr)
0.0
比如求一个矩阵的转置：
>>> arr = np.array([[1,2],[3,4]])
>>> iarr = linalg.inv(arr)
>>> iarr
array([[-2. , 1. ],
[ 1.5, -0.5]])
快速傅里叶变换：scipy.fftpack
首先我们用numpy初始化正弦信号：
>>> import numpy as np
>>> time_step = 0.02
>>> period = 5.
>>> time_vec = np.arange(0, 20, time_step)
>>> sig = np.sin(2 * np.pi / period * time_vec) + 0.5 * np.random.randn(time_vec.size)
>>> time_vec
array([ 0. , 0.02, 0.04, 0.06, 0.08, 0.1 , 0.12, 0.14,
0.16, 0.18, 0.2 , 0.22, 0.24, 0.26, 0.28, 0.3 ,
0.32, 0.34, 0.36, 0.38, 0.4 , 0.42, 0.44, 0.46,
0.48, 0.5 , 0.52, 0.54, 0.56, 0.58, 0.6 , 0.62,
0.64, 0.66, 0.68, 0.7 , 0.72, 0.74, 0.76, 0.78,
0.8 , 0.82, 0.84, 0.86, 0.88, 0.9 , 0.92, 0.94,
0.96, 0.98, 1. , 1.02, 1.04, 1.06, 1.08, 1.1 ,
1.12, 1.14, 1.16, 1.18, 1.2 , 1.22, 1.24, 1.26,
1.28, 1.3 , 1.32, 1.34, 1.36, 1.38, 1.4 , 1.42,
1.44, 1.46, 1.48, 1.5 , 1.52, 1.54, 1.56, 1.58,
1.6 , 1.62, 1.64, 1.66, 1.68, 1.7 , 1.72, 1.74,
1.76, 1.78, 1.8 , 1.82, 1.84, 1.86, 1.88, 1.9 ,
1.92, 1.94, 1.96, 1.98, 2. , 2.02, 2.04, 2.06,
2.08, 2.1 , 2.12, 2.14, 2.16, 2.18, 2.2 , 2.22,
2.24, 2.26, 2.28, 2.3 , 2.32, 2.34, 2.36, 2.38,
2.4 , 2.42, 2.44, 2.46, 2.48, 2.5 , 2.52, 2.54,
2.56, 2.58, 2.6 , 2.62, 2.64, 2.66, 2.68, 2.7 ,
2.72, 2.74, 2.76, 2.78, 2.8 , 2.82, 2.84, 2.86,
2.88, 2.9 , 2.92, 2.94, 2.96, 2.98, 3. , 3.02,
3.04, 3.06, 3.08, 3.1 , 3.12, 3.14, 3.16, 3.18,
3.2 , 3.22, 3.24, 3.26, 3.28, 3.3 , 3.32, 3.34,
3.36, 3.38, 3.4 , 3.42, 3.44, 3.46, 3.48, 3.5 ,
3.52, 3.54, 3.56, 3.58, 3.6 , 3.62, 3.64, 3.66,
3.68, 3.7 , 3.72, 3.74, 3.76, 3.78, 3.8 , 3.82,
3.84, 3.86, 3.88, 3.9 , 3.92, 3.94, 3.96, 3.98,
4. , 4.02, 4.04, 4.06, 4.08, 4.1 , 4.12, 4.14,
4.16, 4.18, 4.2 , 4.22, 4.24, 4.26, 4.28, 4.3 ,
4.32, 4.34, 4.36, 4.38, 4.4 , 4.42, 4.44, 4.46,
4.48, 4.5 , 4.52, 4.54, 4.56, 4.58, 4.6 , 4.62,
4.64, 4.66, 4.68, 4.7 , 4.72, 4.74, 4.76, 4.78,
4.8 , 4.82, 4.84, 4.86, 4.88, 4.9 , 4.92, 4.94,
4.96, 4.98, 5. , 5.02, 5.04, 5.06, 5.08, 5.1 ,
5.12, 5.14, 5.16, 5.18, 5.2 , 5.22, 5.24, 5.26,
5.28, 5.3 , 5.32, 5.34, 5.36, 5.38, 5.4 , 5.42,
5.44, 5.46, 5.48, 5.5 , 5.52, 5.54, 5.56, 5.58,
5.6 , 5.62, 5.64, 5.66, 5.68, 5.7 , 5.72, 5.74,
5.76, 5.78, 5.8 , 5.82, 5.84, 5.86, 5.88, 5.9 ,
5.92, 5.94, 5.96, 5.98, 6. , 6.02, 6.04, 6.06,
6.08, 6.1 , 6.12, 6.14, 6.16, 6.18, 6.2 , 6.22,
6.24, 6.26, 6.28, 6.3 , 6.32, 6.34, 6.36, 6.38,
6.4 , 6.42, 6.44, 6.46, 6.48, 6.5 , 6.52, 6.54,
6.56, 6.58, 6.6 , 6.62, 6.64, 6.66, 6.68, 6.7 ,
6.72, 6.74, 6.76, 6.78, 6.8 , 6.82, 6.84, 6.86,
6.88, 6.9 , 6.92, 6.94, 6.96, 6.98, 7. , 7.02,
7.04, 7.06, 7.08, 7.1 , 7.12, 7.14, 7.16, 7.18,
7.2 , 7.22, 7.24, 7.26, 7.28, 7.3 , 7.32, 7.34,
7.36, 7.38, 7.4 , 7.42, 7.44, 7.46, 7.48, 7.5 ,
7.52, 7.54, 7.56, 7.58, 7.6 , 7.62, 7.64, 7.66,
7.68, 7.7 , 7.72, 7.74, 7.76, 7.78, 7.8 , 7.82,
7.84, 7.86, 7.88, 7.9 , 7.92, 7.94, 7.96, 7.98,
8. , 8.02, 8.04, 8.06, 8.08, 8.1 , 8.12, 8.14,
8.16, 8.18, 8.2 , 8.22, 8.24, 8.26, 8.28, 8.3 ,
8.32, 8.34, 8.36, 8.38, 8.4 , 8.42, 8.44, 8.46,
8.48, 8.5 , 8.52, 8.54, 8.56, 8.58, 8.6 , 8.62,
8.64, 8.66, 8.68, 8.7 , 8.72, 8.74, 8.76, 8.78,
8.8 , 8.82, 8.84, 8.86, 8.88, 8.9 , 8.92, 8.94,
8.96, 8.98, 9. , 9.02, 9.04, 9.06, 9.08, 9.1 ,
9.12, 9.14, 9.16, 9.18, 9.2 , 9.22, 9.24, 9.26,
9.28, 9.3 , 9.32, 9.34, 9.36, 9.38, 9.4 , 9.42,
9.44, 9.46, 9.48, 9.5 , 9.52, 9.54, 9.56, 9.58,
9.6 , 9.62, 9.64, 9.66, 9.68, 9.7 , 9.72, 9.74,
9.76, 9.78, 9.8 , 9.82, 9.84, 9.86, 9.88, 9.9 ,
9.92, 9.94, 9.96, 9.98, 10. , 10.02, 10.04, 10.06,
10.08, 10.1 , 10.12, 10.14, 10.16, 10.18, 10.2 , 10.22,
10.24, 10.26, 10.28, 10.3 , 10.32, 10.34, 10.36, 10.38,
10.4 , 10.42, 10.44, 10.46, 10.48, 10.5 , 10.52, 10.54,
10.56, 10.58, 10.6 , 10.62, 10.64, 10.66, 10.68, 10.7 ,
10.72, 10.74, 10.76, 10.78, 10.8 , 10.82, 10.84, 10.86,
10.88, 10.9 , 10.92, 10.94, 10.96, 10.98, 11. , 11.02,
11.04, 11.06, 11.08, 11.1 , 11.12, 11.14, 11.16, 11.18,
11.2 , 11.22, 11.24, 11.26, 11.28, 11.3 , 11.32, 11.34,
11.36, 11.38, 11.4 , 11.42, 11.44, 11.46, 11.48, 11.5 ,
11.52, 11.54, 11.56, 11.58, 11.6 , 11.62, 11.64, 11.66,
11.68, 11.7 , 11.72, 11.74, 11.76, 11.78, 11.8 , 11.82,
11.84, 11.86, 11.88, 11.9 , 11.92, 11.94, 11.96, 11.98,
12. , 12.02, 12.04, 12.06, 12.08, 12.1 , 12.12, 12.14,
12.16, 12.18, 12.2 , 12.22, 12.24, 12.26, 12.28, 12.3 ,
12.32, 12.34, 12.36, 12.38, 12.4 , 12.42, 12.44, 12.46,
12.48, 12.5 , 12.52, 12.54, 12.56, 12.58, 12.6 , 12.62,
12.64, 12.66, 12.68, 12.7 , 12.72, 12.74, 12.76, 12.78,
12.8 , 12.82, 12.84, 12.86, 12.88, 12.9 , 12.92, 12.94,
12.96, 12.98, 13. , 13.02, 13.04, 13.06, 13.08, 13.1 ,
13.12, 13.14, 13.16, 13.18, 13.2 , 13.22, 13.24, 13.26,
13.28, 13.3 , 13.32, 13.34, 13.36, 13.38, 13.4 , 13.42,
13.44, 13.46, 13.48, 13.5 , 13.52, 13.54, 13.56, 13.58,
13.6 , 13.62, 13.64, 13.66, 13.68, 13.7 , 13.72, 13.74,
13.76, 13.78, 13.8 , 13.82, 13.84, 13.86, 13.88, 13.9 ,
13.92, 13.94, 13.96, 13.98, 14. , 14.02, 14.04, 14.06,
14.08, 14.1 , 14.12, 14.14, 14.16, 14.18, 14.2 , 14.22,
14.24, 14.26, 14.28, 14.3 , 14.32, 14.34, 14.36, 14.38,
14.4 , 14.42, 14.44, 14.46, 14.48, 14.5 , 14.52, 14.54,
14.56, 14.58, 14.6 , 14.62, 14.64, 14.66, 14.68, 14.7 ,
14.72, 14.74, 14.76, 14.78, 14.8 , 14.82, 14.84, 14.86,
14.88, 14.9 , 14.92, 14.94, 14.96, 14.98, 15. , 15.02,
15.04, 15.06, 15.08, 15.1 , 15.12, 15.14, 15.16, 15.18,
15.2 , 15.22, 15.24, 15.26, 15.28, 15.3 , 15.32, 15.34,
15.36, 15.38, 15.4 , 15.42, 15.44, 15.46, 15.48, 15.5 ,
15.52, 15.54, 15.56, 15.58, 15.6 , 15.62, 15.64, 15.66,
15.68, 15.7 , 15.72, 15.74, 15.76, 15.78, 15.8 , 15.82,
15.84, 15.86, 15.88, 15.9 , 15.92, 15.94, 15.96, 15.98,
16. , 16.02, 16.04, 16.06, 16.08, 16.1 , 16.12, 16.14,
16.16, 16.18, 16.2 , 16.22, 16.24, 16.26, 16.28, 16.3 ,
16.32, 16.34, 16.36, 16.38, 16.4 , 16.42, 16.44, 16.46,
16.48, 16.5 , 16.52, 16.54, 16.56, 16.58, 16.6 , 16.62,
16.64, 16.66, 16.68, 16.7 , 16.72, 16.74, 16.76, 16.78,
16.8 , 16.82, 16.84, 16.86, 16.88, 16.9 , 16.92, 16.94,
16.96, 16.98, 17. , 17.02, 17.04, 17.06, 17.08, 17.1 ,
17.12, 17.14, 17.16, 17.18, 17.2 , 17.22, 17.24, 17.26,
17.28, 17.3 , 17.32, 17.34, 17.36, 17.38, 17.4 , 17.42,
17.44, 17.46, 17.48, 17.5 , 17.52, 17.54, 17.56, 17.58,
17.6 , 17.62, 17.64, 17.66, 17.68, 17.7 , 17.72, 17.74,
17.76, 17.78, 17.8 , 17.82, 17.84, 17.86, 17.88, 17.9 ,
17.92, 17.94, 17.96, 17.98, 18. , 18.02, 18.04, 18.06,
18.08, 18.1 , 18.12, 18.14, 18.16, 18.18, 18.2 , 18.22,
18.24, 18.26, 18.28, 18.3 , 18.32, 18.34, 18.36, 18.38,
18.4 , 18.42, 18.44, 18.46, 18.48, 18.5 , 18.52, 18.54,
18.56, 18.58, 18.6 , 18.62, 18.64, 18.66, 18.68, 18.7 ,
18.72, 18.74, 18.76, 18.78, 18.8 , 18.82, 18.84, 18.86,
18.88, 18.9 , 18.92, 18.94, 18.96, 18.98, 19. , 19.02,
19.04, 19.06, 19.08, 19.1 , 19.12, 19.14, 19.16, 19.18,
19.2 , 19.22, 19.24, 19.26, 19.28, 19.3 , 19.32, 19.34,
19.36, 19.38, 19.4 , 19.42, 19.44, 19.46, 19.48, 19.5 ,
19.52, 19.54, 19.56, 19.58, 19.6 , 19.62, 19.64, 19.66,
19.68, 19.7 , 19.72, 19.74, 19.76, 19.78, 19.8 , 19.82,
19.84, 19.86, 19.88, 19.9 , 19.92, 19.94, 19.96, 19.98])
>>> sig
array([ -2.93061365e-01, 5.30383599e-01, -7.86370317e-04,
1.74994340e-02, -4.00626581e-01, 3.98398431e-01,
3.42639567e-02, -6.53931036e-02, -1.60064023e-03,
-8.97586494e-01, 1.20580358e-01, -1.22774216e-01,
3.41717628e-01, 3.29396885e-01, 2.71381465e-01,
-2.73727232e-01, 5.84773311e-01, -3.96603527e-01,
3.36738802e-01, 5.33249224e-01, 8.09188994e-01,
6.16943664e-01, 9.41794048e-01, 1.57217093e+00,
5.04685129e-01, 8.07153869e-01, 1.69752076e+00,
5.71163945e-01, 8.62580691e-01, 5.74879584e-02,
7.94747235e-01, 5.03786115e-01, 8.99139226e-01,
6.52174251e-01, 8.07423267e-01, 1.16583534e-01,
9.60585385e-01, 1.36413672e+00, -3.24389404e-02,
-2.92449427e-01, 8.61060088e-01, 1.55747607e+00,
8.83334027e-01, 9.82984623e-01, 6.85542375e-01,
1.94259820e-01, 1.10254845e-01, 7.34512698e-01,
1.39866177e+00, 4.64662031e-01, 9.95219483e-01,
7.07519649e-01, 1.23160486e+00, 2.36767343e-01,
4.75380302e-01, 1.11754658e+00, 1.19468090e+00,
9.32997486e-01, 1.01038191e+00, 1.15856960e+00,
1.99480246e+00, 1.34766857e+00, 9.43238862e-01,
4.84596263e-01, 9.26676312e-01, 1.26340466e+00,
5.12428418e-01, 6.27235106e-01, 1.17578533e+00,
1.55764087e+00, 8.93571365e-01, 3.10470631e-01,
4.95802522e-01, 1.16142975e+00, 1.27039352e+00,
1.31633868e+00, 1.18593599e+00, 3.07263457e-01,
9.11270627e-01, 5.94357333e-01, 3.54458727e-01,
-8.94667775e-02, 1.23122235e+00, 8.56676212e-01,
2.74555416e-01, 1.58927970e-01, 1.52997615e+00,
1.01712751e+00, 1.28028751e-02, 1.98422137e+00,
4.51400974e-01, 7.96112447e-01, 8.19000835e-01,
1.90891585e+00, 9.04393382e-01, 2.03910499e-01,
7.68422390e-01, -1.82638197e-01, 6.58540255e-01,
5.50297803e-01, 9.66490279e-01, 5.21687006e-01,
7.15277298e-01, 7.07519489e-01, 1.53324824e-01,
9.69425488e-02, -2.29997637e-03, 1.08177960e-02,
6.96045685e-01, 4.64431255e-01, 6.29775702e-01,
-4.16012523e-01, -1.30210340e-03, -2.09754408e-01,
1.55144428e-01, 1.54505989e-01, 6.54218639e-01,
7.53668750e-01, 1.31010811e-01, 1.14102603e-01,
-2.71547410e-01, 7.27897650e-01, 4.10680433e-01,
-1.57141119e-01, 7.32861087e-01, 1.56043065e-01,
6.83736437e-01, 1.35222381e-01, -7.12529673e-02,
-2.80339432e-02, 5.26768559e-01, -7.16710284e-01,
-2.59366660e-01, -3.96207291e-01, -9.72489776e-01,
-2.36314201e-01, -5.95575022e-01, -7.95455490e-01,
-7.00804281e-01, 1.23742557e-01, -5.14575141e-01,
1.60481984e-01, -1.19447383e+00, -1.26820616e-01,
-9.85165338e-01, -5.33921036e-01, -1.32061199e+00,
-4.09534800e-01, -7.24462364e-01, -6.69272312e-01,
-3.73749627e-01, -3.28131803e-01, -7.32815690e-02,
-6.57413726e-01, -1.39742590e+00, -1.34532818e+00,
-4.43693186e-02, -3.07525352e-01, -2.67572179e-01,
-1.34921678e+00, -5.80931986e-01, -2.60574257e-01,
-5.36740956e-01, -1.67078264e+00, -1.47498820e+00,
-6.85598652e-01, -1.39832957e+00, -1.41001620e+00,
-1.42059291e-01, -8.44455893e-01, -7.61067108e-01,
-7.83881581e-01, -9.31292278e-01, -3.95430909e-01,
-1.05723303e+00, -1.36372162e+00, -9.44235485e-01,
-9.80532721e-01, -2.76094282e-01, -9.44703328e-01,
-9.66490088e-01, -6.89958892e-01, -1.31514928e+00,
-5.00789754e-01, -9.09349476e-01, -7.15899428e-01,
-9.04283401e-01, -6.65353370e-01, -5.57884960e-01,
-8.31588604e-01, -8.69471764e-01, -1.50629799e+00,
-5.74604102e-01, -4.25208267e-01, -1.12595627e+00,
-2.03770880e-01, -1.20871569e+00, -5.66338769e-01,
-1.11128101e+00, -9.00090785e-01, -1.30618012e+00,
-9.93153914e-01, -4.79937889e-01, -6.05025564e-01,
-1.20848766e+00, 1.14094247e-01, -1.07667907e-01,
-1.72077243e+00, 7.98020330e-02, -4.07745726e-01,
-1.17493353e+00, -8.59395460e-01, -1.45554672e+00,
-4.41884867e-01, -7.05359919e-01, -4.47513566e-01,
-1.10823838e+00, -2.00488234e-01, -5.22921471e-01,
-1.03825872e+00, -5.21339369e-04, -5.64949626e-01,
-3.49073503e-01, -9.16730497e-01, -1.83943501e+00,
-2.62080910e-01, -6.71249545e-01, -3.08455576e-01,
-5.67330456e-02, -5.57832898e-01, -8.31544496e-01,
-3.72480493e-01, -4.84647989e-01, -9.67571869e-01,
-4.67879634e-01, -6.93935391e-01, -4.16381765e-01,
5.90474384e-01, -9.52995472e-01, -3.71775445e-01,
-2.83067691e-01, -7.41163157e-01, -1.10789452e+00,
2.86401105e-01, 2.00736389e-01, -7.55584226e-02,
-3.10052458e-01, -2.96762745e-01, -1.28275058e-01,
6.84287131e-01, -4.67745200e-01, -1.17043956e+00,
6.52014728e-01, 6.45613600e-01, 5.47059315e-01,
7.93297338e-01, -4.89486230e-01, 3.35264385e-01,
4.49610772e-01, 1.76029656e-01, -1.23878916e-01,
1.10948065e+00, -3.03361658e-01, 9.72584916e-02,
-2.71865649e-01, -3.77260872e-01, 7.29011555e-01,
-6.21677362e-01, 1.19646906e+00, 1.09444477e+00,
1.36376464e+00, 2.43205908e-02, 6.58978509e-01,
6.38967098e-01, 6.46744052e-01, 2.26331531e-01,
-1.16949298e-01, -3.21334006e-01, 1.40535651e+00,
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8.55870285e-01, 1.57705360e+00, 9.27119940e-01,
1.20263930e+00, 1.98014967e+00, 1.61704210e+00,
6.91991248e-01, 4.29015928e-01, 1.61845142e+00,
5.95862527e-01, 1.20783676e+00, 8.00959630e-01,
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5.35211574e-01, 5.97611913e-01, 9.54356802e-01,
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4.14232758e-01, 1.95546238e-02, 5.65787515e-01,
1.02266564e+00, 1.09555792e+00, 1.78059208e-01,
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-2.86262286e-01, -5.88550031e-01, 1.36685859e-01,
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9.25672220e-02, -2.01203440e-02, -5.58450651e-01,
-6.60814093e-02, 5.87296449e-01, 1.80829630e-02,
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-1.09024460e-01, -2.81376585e-01, -3.24903951e-01,
-6.01183552e-01, 1.23278244e-01, 1.29714399e-01,
-5.14855614e-01])
如果我们要计算该信号的采样频率，可以用scipy.fftpack.fftreq()函数
计算它的快速傅里叶变换，使用scipy.fftpack.fft():
>>> from scipy import fftpack
>>> sample_freq = fftpack.fftfreq(sig.size, d=time_step)
>>> sig_fft = fftpack.fft(sig)
>>> sig_fft
array([ -2.15395509 +0.00000000e+00j, -6.84339997 +3.72950387e-01j,
-1.35865175 +2.23642454e+00j, -14.22631157 +1.40328658e+01j,
-1.41924893 -4.98010803e+02j, -3.81579545 +1.78527435e+01j,
3.02203464 -1.37534067e+01j, -12.48741665 -9.35088951e+00j,
-5.29439657 -2.84245298e+00j, 17.15426594 -1.66581323e-02j,
9.95052908 +9.31427197e+00j, 5.98106833 -5.42015882e+00j,
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-5.23001175 +4.86218848e+00j, -6.10597712 +1.12132247e+01j,
10.16949218 +7.41068976e+00j, 5.38999278 +2.06692234e+01j,
-9.62034449 -6.48986336e+00j, 18.15508153 -3.47809813e+00j,
-14.63687090 +2.46882484e+00j, 5.21885389 +2.41996035e+00j,
13.48961390 +1.68130269e+01j, -10.66025119 +4.37643147e+00j,
-12.43390627 +2.10406611e+01j, 0.58507188 +6.98480969e+00j,
-1.46803834 -1.60060763e+01j, -24.29452647 -5.24676682e+00j,
-7.54205380 -3.21887194e+00j, 5.26387189 +5.87977505e+00j,
9.97773909 -1.12743939e+01j, -3.72800953 +1.00824348e+01j,
18.66733138 -1.20931219e+01j, 0.50515517 -1.43795885e+01j,
-0.39428169 +1.13438538e+01j, 11.67640153 +3.28776107e+00j,
4.32988759 -1.38839278e+00j, 17.58160580 +4.10860999e+00j,
-5.42306064 +9.27021008e+00j, -1.45474552 +1.54220167e+01j,
1.48481138 +6.82391210e+00j, -4.42054209 -1.98925835e+01j,
-13.42460404 +8.05574615e-01j, -13.47652545 -4.56139815e+00j,
-8.51838961 -7.57773439e+00j, -10.22408704 -7.53333610e+00j,
-1.43247318 -1.23658590e+00j, -4.57380298 +5.31640857e-01j,
-2.53856737 -1.04138895e+01j, 11.59059296 -1.27590241e+01j,
15.24450891 -3.29444574e+00j, -10.81927864 -2.38331421e+00j,
14.85505773 +3.74696613e+00j, -2.14299713 -1.87982717e+01j,
8.80037448 +9.09984010e-01j, 3.45058499 +9.45392579e+00j,
-11.95068650 -2.50702822e+01j, -17.76819393 -1.57704148e+00j,
-14.66290261 +4.27838327e+00j, 17.95480670 -7.08575108e-01j,
-2.77229909 -2.65764710e+00j, 9.64742719 +4.06928418e+00j,
15.72094430 +1.74336858e+01j, -9.88021679 -9.88978646e+00j,
18.93804440 -3.44990274e+00j, 5.13840297 +5.53326826e+00j,
15.71422456 +7.48998739e+00j, 9.32464975 +7.28098566e-01j,
3.36492721 +7.98976393e+00j, 14.24697873 +8.14464536e+00j,
-2.54008261 +1.92598459e+01j, 8.19113247 -1.45283653e+01j,
-5.60651149 +9.74808530e+00j, 15.49493435 +1.35103238e+00j,
9.17529075 -4.19436604e+00j, 0.98895437 +1.73509646e+01j,
17.84469300 -1.48124772e+00j, -12.87864403 +4.87054691e+00j,
32.24238221 -5.57547662e+00j, -5.87143614 +1.18266090e+01j,
8.81357518 -9.64433869e+00j, -17.77990626 +7.75607121e+00j,
-3.34180041 +8.26505111e-01j, -0.85835968 -5.19893971e+00j,
20.63197209 +1.04995824e+01j, 8.30585259 +3.40695090e+00j,
16.34265089 -1.73518718e+00j, 4.42381960 -1.38600407e+01j,
-1.26911958 -2.36113726e+01j, 7.17224579 +1.49502551e+00j,
-6.63167813 +9.02120461e+00j, -10.68073483 -4.83099355e+00j,
-18.62549910 +4.63085334e+00j, 12.53425237 -7.20152587e+00j,
-10.57760888 +6.66335300e+00j, 2.07013917 +2.47164868e+00j,
-1.93591302 +6.81772531e+00j, -18.18424132 +8.89215838e+00j,
-20.35766472 +1.77115265e+01j, -3.92038028 -2.93027748e+00j,
-27.29102845 +5.18345072e+00j, -8.34503927 -2.83642270e+00j,
-0.97972716 +5.83870961e+00j, -11.11711153 -9.00672680e+00j,
-14.33051218 -7.96662353e+00j, -19.97601778 -1.77650459e+01j,
8.19048556 -8.81449084e+00j, 5.98106833 +5.42015882e+00j,
9.95052908 -9.31427197e+00j, 17.15426594 +1.66581323e-02j,
-5.29439657 +2.84245298e+00j, -12.48741665 +9.35088951e+00j,
3.02203464 +1.37534067e+01j, -3.81579545 -1.78527435e+01j,
-1.41924893 +4.98010803e+02j, -14.22631157 -1.40328658e+01j,
-1.35865175 -2.23642454e+00j, -6.84339997 -3.72950387e-01j])
>>> sample_freq
array([ 0. , 0.05, 0.1 , 0.15, 0.2 , 0.25, 0.3 , 0.35,
0.4 , 0.45, 0.5 , 0.55, 0.6 , 0.65, 0.7 , 0.75,
0.8 , 0.85, 0.9 , 0.95, 1. , 1.05, 1.1 , 1.15,
1.2 , 1.25, 1.3 , 1.35, 1.4 , 1.45, 1.5 , 1.55,
1.6 , 1.65, 1.7 , 1.75, 1.8 , 1.85, 1.9 , 1.95,
2. , 2.05, 2.1 , 2.15, 2.2 , 2.25, 2.3 , 2.35,
2.4 , 2.45, 2.5 , 2.55, 2.6 , 2.65, 2.7 , 2.75,
2.8 , 2.85, 2.9 , 2.95, 3. , 3.05, 3.1 , 3.15,
3.2 , 3.25, 3.3 , 3.35, 3.4 , 3.45, 3.5 , 3.55,
3.6 , 3.65, 3.7 , 3.75, 3.8 , 3.85, 3.9 , 3.95,
4. , 4.05, 4.1 , 4.15, 4.2 , 4.25, 4.3 , 4.35,
4.4 , 4.45, 4.5 , 4.55, 4.6 , 4.65, 4.7 , 4.75,
4.8 , 4.85, 4.9 , 4.95, 5. , 5.05, 5.1 , 5.15,
5.2 , 5.25, 5.3 , 5.35, 5.4 , 5.45, 5.5 , 5.55,
5.6 , 5.65, 5.7 , 5.75, 5.8 , 5.85, 5.9 , 5.95,
6. , 6.05, 6.1 , 6.15, 6.2 , 6.25, 6.3 , 6.35,
6.4 , 6.45, 6.5 , 6.55, 6.6 , 6.65, 6.7 , 6.75,
6.8 , 6.85, 6.9 , 6.95, 7. , 7.05, 7.1 , 7.15,
7.2 , 7.25, 7.3 , 7.35, 7.4 , 7.45, 7.5 , 7.55,
7.6 , 7.65, 7.7 , 7.75, 7.8 , 7.85, 7.9 , 7.95,
8. , 8.05, 8.1 , 8.15, 8.2 , 8.25, 8.3 , 8.35,
8.4 , 8.45, 8.5 , 8.55, 8.6 , 8.65, 8.7 , 8.75,
8.8 , 8.85, 8.9 , 8.95, 9. , 9.05, 9.1 , 9.15,
9.2 , 9.25, 9.3 , 9.35, 9.4 , 9.45, 9.5 , 9.55,
9.6 , 9.65, 9.7 , 9.75, 9.8 , 9.85, 9.9 , 9.95,
10. , 10.05, 10.1 , 10.15, 10.2 , 10.25, 10.3 , 10.35,
10.4 , 10.45, 10.5 , 10.55, 10.6 , 10.65, 10.7 , 10.75,
10.8 , 10.85, 10.9 , 10.95, 11. , 11.05, 11.1 , 11.15,
11.2 , 11.25, 11.3 , 11.35, 11.4 , 11.45, 11.5 , 11.55,
11.6 , 11.65, 11.7 , 11.75, 11.8 , 11.85, 11.9 , 11.95,
12. , 12.05, 12.1 , 12.15, 12.2 , 12.25, 12.3 , 12.35,
12.4 , 12.45, 12.5 , 12.55, 12.6 , 12.65, 12.7 , 12.75,
12.8 , 12.85, 12.9 , 12.95, 13. , 13.05, 13.1 , 13.15,
13.2 , 13.25, 13.3 , 13.35, 13.4 , 13.45, 13.5 , 13.55,
13.6 , 13.65, 13.7 , 13.75, 13.8 , 13.85, 13.9 , 13.95,
14. , 14.05, 14.1 , 14.15, 14.2 , 14.25, 14.3 , 14.35,
14.4 , 14.45, 14.5 , 14.55, 14.6 , 14.65, 14.7 , 14.75,
14.8 , 14.85, 14.9 , 14.95, 15. , 15.05, 15.1 , 15.15,
15.2 , 15.25, 15.3 , 15.35, 15.4 , 15.45, 15.5 , 15.55,
15.6 , 15.65, 15.7 , 15.75, 15.8 , 15.85, 15.9 , 15.95,
16. , 16.05, 16.1 , 16.15, 16.2 , 16.25, 16.3 , 16.35,
16.4 , 16.45, 16.5 , 16.55, 16.6 , 16.65, 16.7 , 16.75,
16.8 , 16.85, 16.9 , 16.95, 17. , 17.05, 17.1 , 17.15,
17.2 , 17.25, 17.3 , 17.35, 17.4 , 17.45, 17.5 , 17.55,
17.6 , 17.65, 17.7 , 17.75, 17.8 , 17.85, 17.9 , 17.95,
18. , 18.05, 18.1 , 18.15, 18.2 , 18.25, 18.3 , 18.35,
18.4 , 18.45, 18.5 , 18.55, 18.6 , 18.65, 18.7 , 18.75,
18.8 , 18.85, 18.9 , 18.95, 19. , 19.05, 19.1 , 19.15,
19.2 , 19.25, 19.3 , 19.35, 19.4 , 19.45, 19.5 , 19.55,
19.6 , 19.65, 19.7 , 19.75, 19.8 , 19.85, 19.9 , 19.95,
20. , 20.05, 20.1 , 20.15, 20.2 , 20.25, 20.3 , 20.35,
20.4 , 20.45, 20.5 , 20.55, 20.6 , 20.65, 20.7 , 20.75,
20.8 , 20.85, 20.9 , 20.95, 21. , 21.05, 21.1 , 21.15,
21.2 , 21.25, 21.3 , 21.35, 21.4 , 21.45, 21.5 , 21.55,
21.6 , 21.65, 21.7 , 21.75, 21.8 , 21.85, 21.9 , 21.95,
22. , 22.05, 22.1 , 22.15, 22.2 , 22.25, 22.3 , 22.35,
22.4 , 22.45, 22.5 , 22.55, 22.6 , 22.65, 22.7 , 22.75,
22.8 , 22.85, 22.9 , 22.95, 23. , 23.05, 23.1 , 23.15,
23.2 , 23.25, 23.3 , 23.35, 23.4 , 23.45, 23.5 , 23.55,
23.6 , 23.65, 23.7 , 23.75, 23.8 , 23.85, 23.9 , 23.95,
24. , 24.05, 24.1 , 24.15, 24.2 , 24.25, 24.3 , 24.35,
24.4 , 24.45, 24.5 , 24.55, 24.6 , 24.65, 24.7 , 24.75,
24.8 , 24.85, 24.9 , 24.95, -25. , -24.95, -24.9 , -24.85,
-24.8 , -24.75, -24.7 , -24.65, -24.6 , -24.55, -24.5 , -24.45,
-24.4 , -24.35, -24.3 , -24.25, -24.2 , -24.15, -24.1 , -24.05,
-24. , -23.95, -23.9 , -23.85, -23.8 , -23.75, -23.7 , -23.65,
-23.6 , -23.55, -23.5 , -23.45, -23.4 , -23.35, -23.3 , -23.25,
-23.2 , -23.15, -23.1 , -23.05, -23. , -22.95, -22.9 , -22.85,
-22.8 , -22.75, -22.7 , -22.65, -22.6 , -22.55, -22.5 , -22.45,
-22.4 , -22.35, -22.3 , -22.25, -22.2 , -22.15, -22.1 , -22.05,
-22. , -21.95, -21.9 , -21.85, -21.8 , -21.75, -21.7 , -21.65,
-21.6 , -21.55, -21.5 , -21.45, -21.4 , -21.35, -21.3 , -21.25,
-21.2 , -21.15, -21.1 , -21.05, -21. , -20.95, -20.9 , -20.85,
-20.8 , -20.75, -20.7 , -20.65, -20.6 , -20.55, -20.5 , -20.45,
-20.4 , -20.35, -20.3 , -20.25, -20.2 , -20.15, -20.1 , -20.05,
-20. , -19.95, -19.9 , -19.85, -19.8 , -19.75, -19.7 , -19.65,
-19.6 , -19.55, -19.5 , -19.45, -19.4 , -19.35, -19.3 , -19.25,
-19.2 , -19.15, -19.1 , -19.05, -19. , -18.95, -18.9 , -18.85,
-18.8 , -18.75, -18.7 , -18.65, -18.6 , -18.55, -18.5 , -18.45,
-18.4 , -18.35, -18.3 , -18.25, -18.2 , -18.15, -18.1 , -18.05,
-18. , -17.95, -17.9 , -17.85, -17.8 , -17.75, -17.7 , -17.65,
-17.6 , -17.55, -17.5 , -17.45, -17.4 , -17.35, -17.3 , -17.25,
-17.2 , -17.15, -17.1 , -17.05, -17. , -16.95, -16.9 , -16.85,
-16.8 , -16.75, -16.7 , -16.65, -16.6 , -16.55, -16.5 , -16.45,
-16.4 , -16.35, -16.3 , -16.25, -16.2 , -16.15, -16.1 , -16.05,
-16. , -15.95, -15.9 , -15.85, -15.8 , -15.75, -15.7 , -15.65,
-15.6 , -15.55, -15.5 , -15.45, -15.4 , -15.35, -15.3 , -15.25,
-15.2 , -15.15, -15.1 , -15.05, -15. , -14.95, -14.9 , -14.85,
-14.8 , -14.75, -14.7 , -14.65, -14.6 , -14.55, -14.5 , -14.45,
-14.4 , -14.35, -14.3 , -14.25, -14.2 , -14.15, -14.1 , -14.05,
-14. , -13.95, -13.9 , -13.85, -13.8 , -13.75, -13.7 , -13.65,
-13.6 , -13.55, -13.5 , -13.45, -13.4 , -13.35, -13.3 , -13.25,
-13.2 , -13.15, -13.1 , -13.05, -13. , -12.95, -12.9 , -12.85,
-12.8 , -12.75, -12.7 , -12.65, -12.6 , -12.55, -12.5 , -12.45,
-12.4 , -12.35, -12.3 , -12.25, -12.2 , -12.15, -12.1 , -12.05,
-12. , -11.95, -11.9 , -11.85, -11.8 , -11.75, -11.7 , -11.65,
-11.6 , -11.55, -11.5 , -11.45, -11.4 , -11.35, -11.3 , -11.25,
-11.2 , -11.15, -11.1 , -11.05, -11. , -10.95, -10.9 , -10.85,
-10.8 , -10.75, -10.7 , -10.65, -10.6 , -10.55, -10.5 , -10.45,
-10.4 , -10.35, -10.3 , -10.25, -10.2 , -10.15, -10.1 , -10.05,
-10. , -9.95, -9.9 , -9.85, -9.8 , -9.75, -9.7 , -9.65,
-9.6 , -9.55, -9.5 , -9.45, -9.4 , -9.35, -9.3 , -9.25,
-9.2 , -9.15, -9.1 , -9.05, -9. , -8.95, -8.9 , -8.85,
-8.8 , -8.75, -8.7 , -8.65, -8.6 , -8.55, -8.5 , -8.45,
-8.4 , -8.35, -8.3 , -8.25, -8.2 , -8.15, -8.1 , -8.05,
-8. , -7.95, -7.9 , -7.85, -7.8 , -7.75, -7.7 , -7.65,
-7.6 , -7.55, -7.5 , -7.45, -7.4 , -7.35, -7.3 , -7.25,
-7.2 , -7.15, -7.1 , -7.05, -7. , -6.95, -6.9 , -6.85,
-6.8 , -6.75, -6.7 , -6.65, -6.6 , -6.55, -6.5 , -6.45,
-6.4 , -6.35, -6.3 , -6.25, -6.2 , -6.15, -6.1 , -6.05,
-6. , -5.95, -5.9 , -5.85, -5.8 , -5.75, -5.7 , -5.65,
-5.6 , -5.55, -5.5 , -5.45, -5.4 , -5.35, -5.3 , -5.25,
-5.2 , -5.15, -5.1 , -5.05, -5. , -4.95, -4.9 , -4.85,
-4.8 , -4.75, -4.7 , -4.65, -4.6 , -4.55, -4.5 , -4.45,
-4.4 , -4.35, -4.3 , -4.25, -4.2 , -4.15, -4.1 , -4.05,
-4. , -3.95, -3.9 , -3.85, -3.8 , -3.75, -3.7 , -3.65,
-3.6 , -3.55, -3.5 , -3.45, -3.4 , -3.35, -3.3 , -3.25,
-3.2 , -3.15, -3.1 , -3.05, -3. , -2.95, -2.9 , -2.85,
-2.8 , -2.75, -2.7 , -2.65, -2.6 , -2.55, -2.5 , -2.45,
-2.4 , -2.35, -2.3 , -2.25, -2.2 , -2.15, -2.1 , -2.05,
-2. , -1.95, -1.9 , -1.85, -1.8 , -1.75, -1.7 , -1.65,
-1.6 , -1.55, -1.5 , -1.45, -1.4 , -1.35, -1.3 , -1.25,
-1.2 , -1.15, -1.1 , -1.05, -1. , -0.95, -0.9 , -0.85,
-0.8 , -0.75, -0.7 , -0.65, -0.6 , -0.55, -0.5 , -0.45,
-0.4 , -0.35, -0.3 , -0.25, -0.2 , -0.15, -0.1 , -0.05])
Numpy中也有用于计算快速傅里叶变换的模块：numpy.fft
但是scipy.fftpack是我们的首选，因为应用了更多底层的工具，工作效率要高一些。
优化器：scipy.optimize
scipy.optimize通常用来最小化一个函数值
构建一个函数并绘制函数图：
>>> def f(x):
... return x**2+10*np.sin(x)
>>> x = np.arange(-10, 10, 0.1)
>>> import matplotlib.pyplot as plt
>>> plt.plot(x, f(x))
[<matplotlib.lines.Line2D object at 0x7fc8817a1dd0>]
>>> plt.show()
如果我们要找出这个函数的最小值，也就是曲线的最低点。就可以用到BFGS优化算法(Broyden–Fletcher–Goldfarb–Shanno algorithm)：
>>> from scipy import optimize
>>> optimize.fmin_bfgs(f,0)
Optimization terminated successfully.
Current function value: -7.945823
Iterations: 5
Function evaluations: 18
Gradient evaluations: 6
array([-1.30644012])
可以得到最低点的值为-1.30644012，optimize.fmin_bfgs(f, 0)第二个参数0表示从0点的位置最小化，找到最低点（该点刚好为全局最低点）。假如我从3点的位置开始梯度下降，那么得到的将会是局部最低点 3.83746709：
>>> optimize.fmin_bfgs(f,3,disp=0)
array([ 3.83746709])
假如你无法选出the global minimum的邻近点作为初始点的话可以使用scipy.optimize.basinhopping()，具体就不展开描述。
统计工具：scipy.stats
首先我们随机生成1000个服从正态分布的数：
>>> from scipy import stats
>>> a = np.random.normal(size=1000)
>>> loc, std = stats.norm.fit(a)
>>> loc
0.023660388470055038
>>> std
0.98751169104611047
>>> np.median(a)
0.03376630034524767
>>> plt.plot(a)
[<matplotlib.lines.Line2D object at 0x7fc87bb79a50>]
>>> plt.show()
还有像scipy的其他模块（计算积分、信号处理、图像处理的模块）就不一一介绍了。其实机器学习最基础的部分还是属于一些统计算法和优化算法。
参考链接
请戳这里