第十三周python作业：Scipy练习

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本文通过三个实例展示了如何使用Python及Scipy库进行最小二乘法求解过拟合问题、寻找函数最大值以及计算矩阵中各行列间的欧氏距离。这些示例涵盖了从数据生成到结果解释的完整过程。

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Exercise 10.1: Least squares

Generate matrix A ∈ Rm×n with m > n. Also generate some vector b ∈ Rm.

Now find x = arg minx ∥Ax − b∥2.

Print the norm of the residual.

代码：

import numpy as np
from scipy.linalg import lstsq

A = np.mat(np.random.randint(0,10,(20,10)))
b = np.mat(np.random.randint(0,10,(20,1)))
ret = lstsq(A,b)
x = np.mat(ret[0])
norm_re = np.linalg.norm(A*x-b,ord=2)
print("x= \n",x)
print("norm = ",norm_re,"\n")

样例输出：

x= 
 [[ 0.27747234]
 [ 0.01934983]
 [-0.17288909]
 [ 0.40072252]
 [ 0.25301335]
 [ 0.34450555]
 [ 0.03587142]
 [-0.16029266]
 [-0.18460015]
 [-0.14021874]]
norm =  5.7700277909560445

Exercise 10.2: Optimization

Find the maximum of the function f(x) = sin2(x − 2)e−x2

代码：

from scipy.optimize import fmin
import numpy as np

def f(x):
	return (-1)*(np.sin(x-2)**2)*(np.e**(-1*x**2))

res = fmin(f,1)
res_ = -1*res[0]
print(res_)

样例输出：

Optimization terminated successfully.
         Current function value: -0.911685
         Iterations: 16
         Function evaluations: 32
-0.2162109374999993

Exercise 10.3: Pairwise distances

Let X be a matrix with n rows and m columns. How can you compute the pairwise distances betweenevery two rows?

As an example application, consider n cities, and we are given their coordinates in two columns. Nowwe want a nice table that tells us for each two cities, how far they are apart.

Again, make sure you make use of Scipy’s functionality instead of writing your own routine.

代码：

from scipy.spatial.distance import pdist
import numpy as np
X = np.mat(np.random.randint(0,10,(4,3)))
dis = pdist(X,'euclidean')
print(dis)

样例输出：

[ 7.34846923 11.04536102  9.43398113  7.07106781  8.30662386  8.18535277]