1_senator_pca - Jupyter Notebook

PCA and senate voting data In this problem, we are given the data matrix with entries in , where each row corresponds to a senator and each column to a bill. We first import this data, print some relevant values, and normalize it as necessary to ready it for further computation. To run this code, you'll need a number of standard Python libraries, all of which are installable via or . We highly recommend using a virtual environment (https://realpython.com/python-virtual-environments-a-primer/) , for this class and in general. Lastly, ensure that all data files ( senator_pca_data_matrix.csv and senator_pca_politician_labels.txt ) are located in the same folder as the notebook. Places you will need to modify this code are enclosed in a block. You should not need to modify code outside these blocks to complete the problems. Questions that you are expected to answer in text are marked in red . For solution files, solutions will be presented in blue . # In [1]: In [2]: We observe that the number of rows, , is the number of senators and is equal to 100. The number of columns, , is the number of bills and is equal to 542. X.shape: (100, 542) # import the necessary packages for data manipulation, computation and PCA import pandas as pd import numpy as np import scipy as sp from numpy import linalg as LA import matplotlib.pyplot as plt from sklearn.decomposition import PCA % matplotlib inline np.random.seed( 7 ) # import the data matrix senator_df = pd.read_csv( 'senator_pca_data_matrix.csv' ) affiliation_file = open ( 'senator_pca_politician_labels.txt' , 'r' ) affiliations = [line.split( '\n' )[ 0 ].split( ' ' )[ 1 ] for line in affiliation_file X = np.array(senator_df.values[:, 3 :].T, dtype = 'float64' ) # transpose to get s print ( 'X.shape: ' , X.shape) n = X.shape[ 0 ] # number of senators d = X.shape[ 1 ] # number of bills # this is just used for plotting, feel free to ignore assert set (affiliations) == { "Red" , "Blue" , "Yellow" } # assign a marker and hatch to each affiliation markers = [( "Red" , "o" , "/" ), ( "Blue" , "^" , "-" ), ( "Yellow" , "D" , "+" )]

In [3]: A row of consists of 542 entries -1 (senator voted against), 1 (senator voted for), or 0 (senator abstained), one for each bill. In [4]: (542,) [ 1. 1. 1. -1. -1. 1. 1. 1. 1. -1. 1. -1. -1. 1. 1. -1. 1. 1. 1. 1. 1. -1. 1. 1. 1. -1. 1. -1. 1. 1. 1. 1. 1. -1. 1. -1. -1. -1. -1. 1. 1. -1. -1. -1. -1. 1. 1. 1. -1. 1. 1. -1. 1. 1. -1. 1. 1. 1. 1. -1. 1. -1. -1. -1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. -1. 0. -1. 1. 1. 1. -1. -1. 1. 1. -1. -1. 1. 1. 1. -1. 1. -1. 1. -1. 1. 1. -1. -1. -1. 1. 1. 1. -1. -1. -1. -1. -1. -1. 1. -1. 1. 1. -1. -1. -1. 1. -1. 1. -1. 1. 0. 0. 1. 1. -1. 1. 1. -1. 1. 1. -1. 1. -1. -1. 1. 1. 1. 1. 0. -1. -1. 1. 1. -1. 1. 1. 1. 1. 1. 0. 1. 0. 1. 1. 1. 1. 1. 1. 1. -1. 1. 1. -1. 1. -1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. -1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 0. 1. -1. -1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. -1. 1. 1. 1. 1. 1. 1. 1. 1. -1. 1. 1. 0. 1. 0. -1. 1. 1. 1. 1. 1. 1. -1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. -1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. -1. 1. 1. 1. -1. 1. 1. 1. 1. 1. 1. -1. -1. -1. 1. 1. -1. 1. -1. -1. 1. 1. 1. -1. 1. 1. 1. -1. 1. -1. 1. -1. -1. 1. -1. -1. 1. 1. 1. -1. 1. 1. 1. 1. 1. -1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. -1. -1. 1. -1. 1. -1. -1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. -1. 1. 1. 1. 1. 1. -1. 1. -1. 1. 1. 1. 1. 1. 1. -1. 1. -1. -1. -1. -1. 1. 1. 1. 1. 1. 1. 1. 1. 1. -1. 1. 1. 1. 1. 1. 1. 1. -1. -1. 1. -1. 1. 1. 1. 1. 1. -1. 1. -1. 1. -1. 1. 1. -1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 0. 1. -1. 1. 1. 1. 1. 1. -1. -1. -1. 1. 1. 0. 1. 1. 1. 1. 1. 1. 1. -1. -1. 0. 0. 0. 0. 0. 0. 0. 1. 1. -1. -1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. -1. 1. -1. 1. 1. 1. 1. -1. -1. 1. 1. 1. 1. -1. -1. 1. 1. -1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. -1. 1. 1. 1. -1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. -1. -1. -1. 1. 1. 1. 1. -1. -1. 1. 1. -1. 1. 1. 1. 1. 1. 1.] (100,) [ 1. 1. 1. 1. 1. 1. 1. -1. 1. -1. 1. -1. 1. -1. -1. -1. 1. 1. -1. 1. 1. -1. 1. -1. 1. 1. 1. -1. -1. 1. 1. 1. -1. 1. 1. 1. -1. -1. -1. -1. 1. -1. -1. 1. 1. -1. -1. -1. -1. -1. 1. 1. -1. -1. 1. 1. -1. -1. -1. -1. -1. 1. 1. 1. 1. 1. -1. -1. -1. 1. -1. -1. 1. -1. -1. 1. 1. 1. -1. -1. -1. 1. 1. -1. 1. -1. 1. 1. 1. -1. -1. -1. -1. -1. 1. 1. 1. -1. -1. -1.] # print an example row of the data matrix typical_row = X[ 0 ] print (typical_row.shape) print (typical_row) # print an example column of the data matrix typical_column = X[:, 0 ] print (typical_column.shape) print (typical_column)

In [7]: Before we calculate the that maximizes variance, let's observe what the scalar projections on a random direction look like. # define score function def f (X, a): return X @ a

In [8]: Note here that projecting along the random vector does not explain much variance at all — data points are clustered together and intermixed across parties. It is clear that this variance along random direction a_rand: 9.267454390893336 # generate a random direction and normalize the vector a_rand = np.random.rand(d) a_rand = a_rand / np.linalg.norm(a_rand) # compute associated scores along a_rand scores_rand = f(X, a_rand) # visualize the scores along a_rand, coloring them by party affiliation for aff, marker, _ in markers: plt.scatter( scores_rand[np.array(affiliations) == aff], np.zeros_like(scores_rand[np.array(affiliations) == aff]), c = aff, marker = marker, edgecolors = "black" , linewidth = 0.5 , label = aff ) plt.legend() plt.title( 'projections along random direction a_rand' ) plt.xlabel( '$\\langle x_i, a\\rangle$' ) cur_axes = plt.gca() cur_axes.axes.get_yaxis().set_visible( False ) plt.show() print ( 'variance along random direction a_rand: ' , scores_rand.var())