Python习题——2018-05-23作业

Matplotlib 练习题

Exercise 11.1: Plotting a function
　　Plot the function

$$f(x)=sin^2(x-2)e^{-x^2}$$

over the interval [0, 2]. Add proper axis labels, a title, etc.

import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns

sns.set_style("darkgrid")

x = np.linspace(0, 2, 100)
y = np.power(np.sin(x - 2), 2) * np.exp(-1 * np.power(x, 2))


plt.xlim((0, 2))
plt.ylim((0, 1))
plt.xlabel('x label')
plt.ylabel('y label')
plt.title('$f(x)=sin^2(x-2)e^{-x^2}$')

plt.plot(x, y)
plt.savefig("Figure_1.png")
plt.show()

Output:

　
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

$$\hat{b}=\arg\min_b{\parallel}Xb-y\parallel_2$$

Plot the true parameters $b$ and estimated parameters $\hat{b}$. See Figure 1 for an example plot.

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import leastsq


def error(b, X, y):
    '''返回误差'''
    result = np.ravel(X @ b - y)
    return result


x = np.linspace(0, 9, 10)
X = np.random.random((20, 10))
b = np.random.random((10, 1))
z = np.random.normal(loc=0, scale=1, size=(20,1))

y = X @ b + z

b_ = np.random.random((10, 1))
b_ = leastsq(error, b_, args=(X,y))[0]

plt.plot(x, b,  '.', label='True coefficients')
plt.plot(x, b_, 'x', label='Estimated coefficients')
plt.xlim((0, 9))
plt.ylim((-1.5, 1.5))
plt.xlabel('index')
plt.ylabel('value')
plt.legend()

plt.savefig("Figure_2.png")
plt.show()

Output:

　
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.

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import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import gaussian_kde
import seaborn as sns

sns.set_style("darkgrid")

z = np.random.random(size=10000)
kde = gaussian_kde(z)
x = np.linspace(0, 0.9999, 10000)
plt.xlim((0, 1))
plt.hist(z, bins=25, normed=True, color='b')
plt.plot(x, kde(x), 'r-')
plt.savefig('Figure_3.png')
plt.show()

Output: