Exercise 11.1: Plotting a function
Plot the function
over the interval . Add proper axis labels, a title, etc.
import numpy
import matplotlib.pyplot as plt
x = numpy.linspace(0, 2, 100)
y = numpy.square(numpy.sin(x - 2)) * numpy.exp(-x * x)
plt.plot(x, y)
plt.title('f(x)')
plt.xlabel('x')
plt.ylabel('y')
plt.show()
Exercise 11.2: Data
Create a data matrix
with
observations of
variables. Generate a vector
with parameters Then generate the response vector
where
is a vector with standard normally distributed variables.
Now (by only using
and
), find an estimator for
, by solving
Plot the true parameters and estimated parameters . See Figure for an example plot.
import matplotlib.pyplot as plt
import numpy
import random
X = numpy.random.randn(20, 10)
z = numpy.random.normal(size = (20, 1))
b1 = numpy.random.rand(10, 1)
y = numpy.dot(X, b1) + z
x = numpy.linspace(-1, 1, 10)
b2 = numpy.array(numpy.linalg.lstsq(X, y, rcond = -1)[0])
plt.scatter(x, b1, c = 'r', marker = 'x', label = "b")
plt.scatter(x, b2, c = 'b', marker = 'o', label = 'b~')
plt.xlabel('index')
plt.ylabel('value')
plt.legend()
plt.show()
Exercise 11.3: Histogram and density estimation
Generate a vector
of 10000 observations from your favorite exotic distribution. Then make a plot that shows a histogram of
(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.
import numpy
import matplotlib.pyplot as plt
from scipy import stats
y = numpy.random.normal(size = 10000)
kernel = stats.gaussian_kde(y)
x = numpy.linspace(-10, 10, 1000)
plt.hist(y, 25, rwidth = 0.8, density = True)
plt.plot(x, kernel.evaluate(x))
plt.show()