Numpy and Seaborn

Q1. (15 pts) Exercise 1. Write a 12 by 12 times table matrix shwon below. Do this

• using nested for loops

• using numpy fromfunction array constructor

• using numpy broadcasting

  array([[  1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11,  12],
     [  2,   4,   6,   8,  10,  12,  14,  16,  18,  20,  22,  24],
     [  3,   6,   9,  12,  15,  18,  21,  24,  27,  30,  33,  36],
     [  4,   8,  12,  16,  20,  24,  28,  32,  36,  40,  44,  48],
     [  5,  10,  15,  20,  25,  30,  35,  40,  45,  50,  55,  60],
     [  6,  12,  18,  24,  30,  36,  42,  48,  54,  60,  66,  72],
     [  7,  14,  21,  28,  35,  42,  49,  56,  63,  70,  77,  84],
     [  8,  16,  24,  32,  40,  48,  56,  64,  72,  80,  88,  96],
     [  9,  18,  27,  36,  45,  54,  63,  72,  81,  90,  99, 108],
     [ 10,  20,  30,  40,  50,  60,  70,  80,  90, 100, 110, 120],
     [ 11,  22,  33,  44,  55,  66,  77,  88,  99, 110, 121, 132],
     [ 12,  24,  36,  48,  60,  72,  84,  96, 108, 120, 132, 144]])
  

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Q2. (25 points) In this exercise, we will practice using Pandas dataframes to explore and summarize a data set heart.

This data contains the survival time after receiving a heart transplant, the age of the patient and whether or not the survival time was censored

• Number of Observations - 69
• Number of Variables - 3

Variable name definitions:

-  survival - Days after surgery until death
-  censors - indicates if an observation is censored. 1 is uncensored
-  age - age at the time of surgery

Answer the following questions with respect to the heart data set:

• How many patients were censored?
• What is the correlation coefficient between age and survival for uncensored patients?
• What is the average age for censored and uncensored patients?
• What is the average survival time for censored and uncensored patients under the age of 45?
• What is the survival time of the youngest and oldest uncensored patient?

In [2]:

import statsmodels.api as sm
heart = sm.datasets.heart.load_pandas().data
heart.head(n=6)

Out[2]:

	survival	censors	age
0	15	1	54.3
1	3	1	40.4
2	624	1	51.0
3	46	1	42.5
4	127	1	48.0
5	64	1	54.6

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Q3. (40 pts)

Given matrix \(M\)

[[7, 8, 8],
 [1, 3, 8],
 [9, 2, 1]]

• (10 pts) Normalize the given matrix \(M\) so that all rows sum to 1.0. This cna then be considered as a transition matrix \(P\) for a Markov chain. Find the stationary distribution of this matrix in the following ways using numpy and numpy.linalg (or scipy.linalg):

• (10 pts) By raising the matrix \(P\) to some large power unitl it doesn’t change with higher powers (see np.linalg.matrix_power) and then calculating \(vP\)

• (10 pts) From the equation for stationarity \(wP = w\), we can see that \(w\) must be a left eigenvector of \(P\) with eigenvalue \(1\) (Note: np.linalg.eig returns the right eigenvectors, but the left eighenvector of a matrix is the right eigenvector of the transposed matrix). Use this to find \(w\) using np.linalg.eig.

• (10 pts) Suppose \(w = (w_1, w_2, w_3)\). Then from \(wP = w\), we have:

  \[ \begin{align}\begin{aligned}:nowrap:\\\begin{split} \begin{align} w_1 P_{11} + w_2 P_{21} + w_3 P_{31} &= w_1 \\ w_1 P_{12} + w_2 P_{22} + w_3 P_{32} &= w_2 \\ w_1 P_{13} + w_2 P_{23} + w_3 P_{331} &= w_3 \\ \end{align}\end{split}\\This is a singular system, but we also know that\end{aligned}\end{align} \]

  \(w_1 + w_2 + w_3 = 1\). Use these facts to set up a linear system of equations that can be solved with np.linalg.solve to find \(w\).

In [34]:

M = np.array([7,8,8,1,3,8,9,2,1.0]).reshape(3,3)
M

Out[34]:

array([[ 7.,  8.,  8.],
       [ 1.,  3.,  8.],
       [ 9.,  2.,  1.]])

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Q4. Write code to replicate the following plot using seaborn.

In [41]:

from IPython.display import Image
Image('../images/hw2_q4.png')

Out[41]:

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