(Week 12)Python-Matplotlib_exercises-优快云博客

本文通过实例演示了如何使用Python进行函数绘图、最小二乘法求解参数估计及密度估计。具体包括正弦平方指数函数的绘制、通过随机生成的数据集估算参数，并对比真实值，最后展示了如何对大量数据进行密度估计。

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Exercise 11.1: Plotting a function

Plot the function $y=sin^2(x-2)e^{-x^2}$ over the interval [0, 2]. Add proper axis labels, a title, etc.

编写代码如下：

import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
sns.set(color_codes=True)   #show grid

def plot_a_fun():
    x = np.linspace(0, 2, 100)
    print(x)
    y1 = np.sin(x - 2) ** 2     
    y2 = np.exp(-(x ** 2))
    y = y1 * y2             #get y value
    plt.plot(x, y, label='$y=sin^2(x-2)e^{-x^2}$')
    plt.xlabel('x axis')
    plt.ylabel('y axis')
    plt.legend()            #show legend
    plt.show()

plot_a_fun()

运行结果如下：

Exercise 11.2: Data

Create a data matrix X with 20 observations of 10 variables. Generate a vector b with parameters Then generate the response vector $y=Xb+z$ where z is a vector with standard normally distributed variables. Now (by only using y and X), find an estimator for b, by solving

b^= a r g m i n b | | X b - y | | 2

$\hat{b} = \mathop{argmin}\limits_{b}||Xb - y||^2$

Plot the true parameters b and estimated parameters b. See Figure 1 for an example plot.

编写代码如下：

def least_square():
    X = np.random.random((20, 10)) * 10         #create X, b, z
    b = np.random.random(10) * 6 - 3
    z = np.random.randn(20)
    y = X.dot(b) + z                           #create y    
    b1 = np.linalg.lstsq(X, y, rcond=-1)[0]      #get least square solution of Xb = y
    index = np.linspace(0, 9, 10)               
    plt.plot(index, b, 'r*', label="True coefficients")
    plt.plot(index, b1, 'bo', label='Estimated coefficients')
    plt.hlines(0, 0, 9, colors='k', linestyles='solid') #show middle line
    plt.tight_layout()
    plt.legend()
    plt.savefig('p2.png')
    plt.show()

least_square()

运行结果如下：

Exercise 11.3: Histogram and density estimation

Generate a vector z of 10000 observations from your favorite exotic distribution. Then make a plot that shows a histogram of z (with 25 bins), along with an estimate for the density, using a Gaussian kernel density estimator (see scipy.stats). See Figure 2 for an example plot.

编写代码如下：

def density_estimation():
    x = np.random.normal(60, 10, 10000)
    sns.distplot(x, bins=25, kde=True)  #kde=True -- using Gaussian kernel density estimator
    plt.savefig('p3.png')
    plt.show()

density_estimation()

运行结果如下：