2016-05-16-scratch.sagews

Problem 3: (numpy)

1. Generate a random vector of integers between 0 and 10 (inclusive) of length 100. (hint: np.bincount)

2. Which number appears most often?

# Solution

Z = np.random.randint(0, 10, size = 100)
np.bincount(Z).argmax()

3

%md
### Problem 4 (numpy)

For two different types of random matrices with size 1000x1000 (e.g. rand, randn):
 - Multiply the matrix with its own transpose: $B = A^T * A$
 - Then compute the eigenvalues of that resulting matrix $B$ and plot a histogram via matplotlib.

Guess what each the distributions of eigenvalues looks like. (Compare with the distribution of the random sampling routines.)

* random matrices: http://docs.scipy.org/doc/numpy/reference/routines.random.html
* linear algebra: http://docs.scipy.org/doc/numpy/reference/routines.linalg.html

  e.g. `np.random.beta(.1, 5, size=(1000, 1000))`

Problem 4 (numpy)

1. For different types of random matrices with size 1000x1000 (e.g. rand, randn)

2. Multiply the matrix with its own transpose: $B = A^T * A$

3. Then compute the eigenvalues of that resulting matrix $B$ and plot a histogram via matplotlib!

Guess, what each the distributions of eigenvalues looks like! (Compare with the distribution of the random sampling routines)

• random matrices: http://docs.scipy.org/doc/numpy/reference/routines.random.html

• linear algebra: http://docs.scipy.org/doc/numpy/reference/routines.linalg.html

  e.g. np.random.beta(.1, 5, size=(1000, 1000))

# Solution
%python
A = np.random.beta(.1, 5, size=(1000, 1000))
A = A.T * A
Aev = np.linalg.eigvals(A)
plt.hist(Aev)
plt.show()

(array([ 17., 57., 135., 226., 270., 168., 89., 32., 5., 1.]), array([-0.31629052, -0.24057083, -0.16485113, -0.08913143, -0.01341173, 0.06230796, 0.13802766, 0.21374736, 0.28946706, 0.36518675, 0.44090645]), <a list of 10 Patch objects>)

%md
### Problem 5: (scipy)

1. create two random vectors with normal distribution
1. use a T-test to check, if they significantly differ

* http://docs.scipy.org/doc/scipy/reference/tutorial/stats.html

Problem 5: (scipy)

1. create two random vectors with normal distribution

2. use a T-test to check, if they significantly differ

• http://docs.scipy.org/doc/scipy/reference/tutorial/stats.html

# Solution

Problem 6: (scipy)

The family of BFGS optimization algorithms is very well known.

Optimize a two dimensional function over a constraint box using it!

1. choose one from this list of classical optimization functions: https://en.wikipedia.org/wiki/Test_functions_for_optimization

2. express the mathematical equation in terms of numpy vectors

3. minimize in the box $[-10, 10]^2$ using the l-bfgs-b algorithm from scipy.

   question: what happens, when you do not specify the gradients?

• theory: http://docs.scipy.org/doc/scipy/reference/tutorial/optimize.html#broyden-fletcher-goldfarb-shanno-algorithm-method-bfgs

• minimize: http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize.html#scipy.optimize.minimize

• l-bfgs-b: http://docs.scipy.org/doc/scipy/reference/optimize.minimize-lbfgsb.html#optimize-minimize-lbfgsb

# Solution