Principal Component Analysis II¶

Data 100, Fall 2023

Updated by Dominic Liu

Acknowledgments Page

In [1]:
import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt
import numpy as np
import yaml
from datetime import datetime
from ds100_utils import *

np.random.seed(23) #kallisti
plt.rcParams['figure.dpi'] = 150

PCA is a Linear Transformation¶

In [2]:
df = pd.read_csv("data/2d.csv")
df
Out[2]:
x y
0 2.311043 5.436627
1 2.951447 6.093710
2 2.628517 6.776799
3 2.041157 5.335430
4 3.916969 8.948526
... ... ...
95 3.639231 8.331902
96 2.765474 5.621709
97 2.745027 7.134981
98 3.945360 8.198725
99 2.743246 6.145312

100 rows × 2 columns

Let's visualize the dataset first.

In [3]:
fig = plt.figure()
ax = fig.add_subplot(1, 1, 1)

# Move left y-axis and bottom x-axis to centre, passing through (0,0)
ax.spines['left'].set_position('zero')
ax.spines['bottom'].set_position('zero')

# Eliminate upper and right axes
ax.spines['right'].set_color('none')
ax.spines['top'].set_color('none')

# Show ticks in the left and lower axes only
ax.xaxis.set_ticks_position('bottom')
ax.yaxis.set_ticks_position('left')
sns.scatterplot(x="x", y="y", data = df);
plt.axis("square")
ax.set_ylabel("")
ax.set_xlabel("");
plt.xlim(-5, 5)
plt.ylim(-5, 10);

Now let's perform PCA on this 2D dataset to see what transformations are involved.

Step 1 of PCA is to center the data matrix.

In [4]:
centered_df = df - np.mean(df, axis = 0)

fig = plt.figure()
ax = fig.add_subplot(1, 1, 1)

# Move left y-axis and bottom x-axis to centre, passing through (0,0)
ax.spines['left'].set_position('zero')
ax.spines['bottom'].set_position('zero')

# Eliminate upper and right axes
ax.spines['right'].set_color('none')
ax.spines['top'].set_color('none')

# Show ticks in the left and lower axes only
ax.xaxis.set_ticks_position('bottom')
ax.yaxis.set_ticks_position('left')
sns.scatterplot(x="x", y="y", data = centered_df);
plt.axis("square")
ax.set_ylabel("")
ax.set_xlabel("");
plt.xlim(-5, 5)
plt.ylim(-5, 5);

Step 2 is to obtain the SVD of the centered data.

In [5]:
U, S, Vt = np.linalg.svd(centered_df, full_matrices = False)

We mentioned that $V$ transforms $X$ to get the principal components. What does that tranformation look like?

In [6]:
PCs = centered_df @ Vt.T

fig = plt.figure()
ax = fig.add_subplot(1, 1, 1)

# Move left y-axis and bottom x-axis to centre, passing through (0,0)
ax.spines['left'].set_position('zero')
ax.spines['bottom'].set_position('zero')

# Eliminate upper and right axes
ax.spines['right'].set_color('none')
ax.spines['top'].set_color('none')

# Show ticks in the left and lower axes only
ax.xaxis.set_ticks_position('bottom')
ax.yaxis.set_ticks_position('left')
sns.scatterplot(x=0, y=1, data = PCs);
plt.axis("square")
ax.set_ylabel("")
ax.set_xlabel("");
plt.xlim(-5, 5)
plt.ylim(-5, 5);

Turns out $V$ simply rotates the centered data matrix $X$ such that the direction with the most variation (i.e. the direction that's the most spread-out) is aligned with the x-axis!




Congressional Vote Records¶

Let's examine how the House of Representatives (of the 116th Congress, 1st session) voted in the month of September 2019.

From the U.S. Senate website:

Roll call votes occur when a representative or senator votes "yea" or "nay," so that the names of members voting on each side are recorded. A voice vote is a vote in which those in favor or against a measure say "yea" or "nay," respectively, without the names or tallies of members voting on each side being recorded.

The data, compiled from ProPublica source, is a "skinny" table of data where each record is a single vote by a member across any roll call in the 116th Congress, 1st session, as downloaded in February 2020. The member of the House, whom we'll call legislator, is denoted by their bioguide alphanumeric ID in http://bioguide.congress.gov/.

In [7]:
# February 2019 House of Representatives roll call votes
# Downloaded using https://github.com/eyeseast/propublica-congress
votes = pd.read_csv('data/votes.csv')
votes = votes.astype({"roll call": str}) 
votes
Out[7]:
chamber session roll call member vote
0 House 1 555 A000374 Not Voting
1 House 1 555 A000370 Yes
2 House 1 555 A000055 No
3 House 1 555 A000371 Yes
4 House 1 555 A000372 No
... ... ... ... ... ...
17823 House 1 515 Y000062 Yes
17824 House 1 515 Y000065 No
17825 House 1 515 Y000033 Yes
17826 House 1 515 Z000017 Yes
17827 House 1 515 P000197 Speaker

17828 rows × 5 columns

Suppose we pivot this table to group each legislator and their voting pattern across every (roll call) vote in this month. We mark 1 if the legislator voted Yes (yea), and 0 otherwise (No/nay, no vote, speaker, etc.).

In [8]:
def was_yes(s):
    return 1 if s.iloc[0] == "Yes" else 0
    
vote_pivot = votes.pivot_table(index='member', 
                                columns='roll call', 
                                values='vote', 
                                aggfunc=was_yes, 
                                fill_value=0)
print(vote_pivot.shape)
vote_pivot.head()    

(441, 41)
Out[8]:
roll call 515 516 517 518 519 520 521 522 523 524 ... 546 547 548 549 550 551 552 553 554 555
member
A000055 1 0 0 0 1 1 0 1 1 1 ... 0 0 1 0 0 1 0 0 1 0
A000367 0 0 0 0 0 0 0 0 0 0 ... 0 1 1 1 1 0 1 1 0 1
A000369 1 1 0 0 1 1 0 1 1 1 ... 0 0 1 0 0 1 0 0 1 0
A000370 1 1 1 1 1 0 1 0 0 0 ... 1 1 1 1 1 0 1 1 1 1
A000371 1 1 1 1 1 0 1 0 0 0 ... 1 1 1 1 1 0 1 1 1 1

5 rows × 41 columns

How do we analyze this data?

While we could consider loading information about the legislator, such as their party, and see how this relates to their voting pattern, it turns out that we can do a lot with PCA to cluster legislators by how they vote.

PCA¶

In [9]:
vote_pivot_centered = vote_pivot - np.mean(vote_pivot, axis = 0)
vote_pivot_centered.head(5)
Out[9]:
roll call 515 516 517 518 519 520 521 522 523 524 ... 546 547 548 549 550 551 552 553 554 555
member
A000055 0.129252 -0.668934 -0.526077 -0.52381 0.049887 0.587302 -0.562358 0.634921 0.594104 0.560091 ... -0.521542 -0.526077 0.045351 -0.521542 -0.519274 0.54195 -0.521542 -0.535147 0.086168 -0.503401
A000367 -0.870748 -0.668934 -0.526077 -0.52381 -0.950113 -0.412698 -0.562358 -0.365079 -0.405896 -0.439909 ... -0.521542 0.473923 0.045351 0.478458 0.480726 -0.45805 0.478458 0.464853 -0.913832 0.496599
A000369 0.129252 0.331066 -0.526077 -0.52381 0.049887 0.587302 -0.562358 0.634921 0.594104 0.560091 ... -0.521542 -0.526077 0.045351 -0.521542 -0.519274 0.54195 -0.521542 -0.535147 0.086168 -0.503401
A000370 0.129252 0.331066 0.473923 0.47619 0.049887 -0.412698 0.437642 -0.365079 -0.405896 -0.439909 ... 0.478458 0.473923 0.045351 0.478458 0.480726 -0.45805 0.478458 0.464853 0.086168 0.496599
A000371 0.129252 0.331066 0.473923 0.47619 0.049887 -0.412698 0.437642 -0.365079 -0.405896 -0.439909 ... 0.478458 0.473923 0.045351 0.478458 0.480726 -0.45805 0.478458 0.464853 0.086168 0.496599

5 rows × 41 columns







SLIDO QUESTION










In [10]:
vote_pivot_centered.shape
Out[10]:
(441, 41)
In [11]:
u, s, vt = np.linalg.svd(vote_pivot_centered, full_matrices = False)

PCA plot¶

In [12]:
pcs = u * s
sns.scatterplot(x=pcs[:, 0], y=pcs[:, 1]);
plt.xlabel("PC1");
plt.ylabel("PC2");

Component Scores¶

If the first two singular values are large and all others are small, then two dimensions are enough to describe most of what distinguishes one observation from another. If not, then a PCA scatter plot is omitting lots of information.

An equivalent way to evaluate this is to determine the variance ratios, i.e., the fraction of the variance each PC contributes to total variance.













SLIDO QUESTION
















In [13]:
np.round(s**2 / sum(s**2), 2)
Out[13]:
array([0.8 , 0.05, 0.02, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01,
       0.01, 0.  , 0.  , 0.  , 0.  , 0.  , 0.  , 0.  , 0.  , 0.  , 0.  ,
       0.  , 0.  , 0.  , 0.  , 0.  , 0.  , 0.  , 0.  , 0.  , 0.  , 0.  ,
       0.  , 0.  , 0.  , 0.  , 0.  , 0.  , 0.  , 0.  ])

Scree plot¶

A scree plot (and where its "elbow" is located) is a visual way of checking the distribution of variance.

In [14]:
plt.plot(s**2 / sum(s**2), marker='.');
plt.xlabel("Principal Component $i$");
plt.ylabel("Variance Ratio");

Looks reasonable! Let's plot PC2 vs. PC1 again

In [15]:
sns.scatterplot(x=pcs[:, 0], y=pcs[:, 1]);
plt.xlabel("PC1");
plt.ylabel("PC2");

Baesd on the plot above, it looks like there are two clusters of datapoints. What do you think this corresponds to?

Incorporating Member Information¶

Suppose we load in more member information, from https://github.com/unitedstates/congress-legislators. This includes each legislator's political party.

In [16]:
# You can get current information about legislators with this code. In our case, we'll use
# a static copy of the 2019 membership roster to properly match our voting data.

# base_url = 'https://raw.githubusercontent.com/unitedstates/congress-legislators/main/'
# legislators_path = 'legislators-current.yaml'
# f = fetch_and_cache(base_url + legislators_path, legislators_path)

# Use 2019 data copy
legislators_data = yaml.safe_load(open('data/legislators-2019.yaml'))

def to_date(s):
    return datetime.strptime(s, '%Y-%m-%d')

legs = pd.DataFrame(
    columns=['leg_id', 'first', 'last', 'gender', 'state', 'chamber', 'party', 'birthday'],
    data=[[x['id']['bioguide'], 
           x['name']['first'],
           x['name']['last'],
           x['bio']['gender'],
           x['terms'][-1]['state'],
           x['terms'][-1]['type'],
           x['terms'][-1]['party'],
           to_date(x['bio']['birthday'])] for x in legislators_data])

legs.set_index("leg_id")
legs.sort_index()
Out[16]:
leg_id first last gender state chamber party birthday
0 B000944 Sherrod Brown M OH sen Democrat 1952-11-09
1 C000127 Maria Cantwell F WA sen Democrat 1958-10-13
2 C000141 Benjamin Cardin M MD sen Democrat 1943-10-05
3 C000174 Thomas Carper M DE sen Democrat 1947-01-23
4 C001070 Robert Casey M PA sen Democrat 1960-04-13
... ... ... ... ... ... ... ... ...
534 M001197 Martha McSally F AZ sen Republican 1966-03-22
535 G000592 Jared Golden M ME rep Democrat 1982-07-25
536 K000395 Fred Keller M PA rep Republican 1965-10-23
537 B001311 Dan Bishop M NC rep Republican 1964-07-01
538 M001210 Gregory Murphy M NC rep Republican 1963-03-05

539 rows × 8 columns



Let's check out how party affiliations relate to the PC1, PC2 transformation from earlier:

First, let's see which party has negative PC1 scores.

In [17]:
vote2d = pd.DataFrame({
    'member': vote_pivot.index,
    'pc1': pcs[:, 0],
    'pc2': pcs[:, 1]
}).merge(legs, left_on='member', right_on='leg_id')

vote2d[vote2d['pc1'] < 0]['party'].value_counts()
Out[17]:
party
Democrat    231
Name: count, dtype: int64

Hm let's keep checking things out.

We didn't cover the query syntax that we use below, but if you're curious, check out the documentation.

In [18]:
#top right only
vote2d.query('pc2 > -2 and pc1 > 0')['party'].value_counts()
Out[18]:
party
Republican    194
Name: count, dtype: int64



Interesting. Let's use these party labels to color our principal components:

In [19]:
cp = sns.color_palette()
party_cp = [cp[i] for i in [0, 3, 1]]
party_hue = ["Democrat", "Republican", "Independent"]

sns.scatterplot(x="pc1", y="pc2",
                hue="party", palette=party_cp,  hue_order=party_hue,
                data = vote2d);

There seems to be a bunch of overplotting, so let's jitter a bit.

In [20]:
vote2d['pc1_jittered'] = vote2d['pc1'] + np.random.normal(loc = 0, scale = 0.1, size = vote2d.shape[0])
vote2d['pc2_jittered'] = vote2d['pc2'] + np.random.normal(loc = 0, scale = 0.1, size = vote2d.shape[0])
In [21]:
sns.scatterplot(x="pc1_jittered", y="pc2_jittered", 
                hue="party", palette=party_cp,  hue_order=party_hue,
                data = vote2d);
In [22]:
vote2d[vote2d['pc2'] < -1]
Out[22]:
member pc1 pc2 leg_id first last gender state chamber party birthday pc1_jittered pc2_jittered
1 A000367 0.188870 -2.433565 A000367 Justin Amash M MI rep Independent 1980-04-18 0.191451 -2.431195
6 A000374 1.247134 -3.533196 A000374 Ralph Abraham M LA rep Republican 1954-09-16 1.210379 -3.521461
47 B001311 1.695651 -2.093912 B001311 Dan Bishop M NC rep Republican 1964-07-01 1.686247 -1.982446
50 C000537 0.699636 -3.394179 C000537 James Clyburn M SC rep Democrat 1940-07-21 0.687526 -3.212611
52 C000984 0.531789 -3.099044 C000984 Elijah Cummings M MD rep Democrat 1951-01-18 0.583003 -3.131361
69 C001087 2.755060 -1.378193 C001087 Eric Crawford M AR rep Republican 1966-01-22 2.756083 -1.270555
150 G000582 2.262007 -2.632452 G000582 Jenniffer González-Colón F PR rep Republican 1976-08-05 2.259351 -2.656279
179 H001077 2.509474 -1.349023 H001077 Clay Higgins M LA rep Republican 1961-08-24 2.446616 -1.346705
196 J000299 2.908823 -1.094618 J000299 Mike Johnson M LA rep Republican 1972-01-30 3.061062 -1.116434
272 M001200 1.247134 -3.533196 M001200 A. McEachin M VA rep Democrat 1961-10-10 1.221263 -3.559693
282 M001210 1.695651 -2.093912 M001210 Gregory Murphy M NC rep Republican 1963-03-05 1.577282 -1.919511
285 N000147 1.247134 -3.533196 N000147 Eleanor Norton F DC rep Democrat 1937-06-13 1.385627 -3.546085
298 P000197 1.247134 -3.533196 P000197 Nancy Pelosi F CA rep Democrat 1940-03-26 1.217323 -3.454539
310 P000610 1.247134 -3.533196 P000610 Stacey Plaskett F VI rep Democrat 1966-05-13 1.347833 -3.381807
323 R000577 -2.069671 -1.344435 R000577 Tim Ryan M OH rep Democrat 1973-07-16 -1.992928 -1.423723
330 R000600 1.247134 -3.533196 R000600 Aumua Amata F AS rep Republican 1947-12-29 1.255358 -3.499536
360 S001177 1.247134 -3.533196 S001177 Gregorio Sablan M MP rep Democrat 1955-01-19 1.214066 -3.646465
374 S001204 1.247134 -3.533196 S001204 Michael San Nicolas M GU rep Democrat 1981-01-30 1.395281 -3.476336
In [23]:
df = votes[votes['member'].isin(vote2d[vote2d['pc2'] < -1]['member'])]
df.groupby(['member', 'vote']).size()
Out[23]:
member   vote      
A000367  No            31
         Yes           10
A000374  Not Voting    41
B001311  No            17
         Yes            7
C000537  No             1
         Not Voting    37
         Yes            3
C000984  No             4
         Not Voting    32
         Yes            5
C001087  No             6
         Not Voting    23
         Yes           12
G000582  Not Voting     1
         Yes            6
H001077  No            14
         Not Voting    15
         Yes           12
J000299  No            20
         Not Voting     7
         Yes           14
M001200  Not Voting    41
M001210  No            16
         Not Voting     1
         Yes            7
N000147  No             6
         Not Voting     1
P000197  Speaker       41
P000610  No             7
R000577  No            11
         Not Voting     8
         Yes           22
R000600  Not Voting     7
S001177  No             7
S001204  No             6
         Not Voting     1
dtype: int64

We can check each of our legislators in the legs DataFrame, as below.

In [24]:
legs.query("leg_id == 'P000197'")
Out[24]:
leg_id first last gender state chamber party birthday
181 P000197 Nancy Pelosi F CA rep Democrat 1940-03-26

Analysis: Regular Voters¶

Let's now look at regulars, whom we define as legislators who have voted above a certain threshold.

First, let's recompute the pivot table where we only consider Yes/No votes, and ignore records with "No Vote" or other entries.

In [25]:
bool_votes = votes["vote"].isin(["Yes", "No"])
num_yes_or_no_votes_per_member = votes[bool_votes].groupby("member").size()
num_yes_or_no_votes_per_member
Out[25]:
member
A000055    40
A000367    41
A000369    41
A000370    41
A000371    41
           ..
W000827    33
Y000033    41
Y000062    41
Y000065    36
Z000017    41
Length: 437, dtype: int64
In [26]:
vote_pivot_with_yes_no_count = (
    vote_pivot.merge(num_yes_or_no_votes_per_member.to_frame(),
                     left_index = True, right_index = True, how="outer")
    .fillna(0)
    .rename(columns = {0: 'yes_no_count'})
)
vote_pivot_with_yes_no_count.head(5)
Out[26]:
515 516 517 518 519 520 521 522 523 524 ... 547 548 549 550 551 552 553 554 555 yes_no_count
member
A000055 1 0 0 0 1 1 0 1 1 1 ... 0 1 0 0 1 0 0 1 0 40.0
A000367 0 0 0 0 0 0 0 0 0 0 ... 1 1 1 1 0 1 1 0 1 41.0
A000369 1 1 0 0 1 1 0 1 1 1 ... 0 1 0 0 1 0 0 1 0 41.0
A000370 1 1 1 1 1 0 1 0 0 0 ... 1 1 1 1 0 1 1 1 1 41.0
A000371 1 1 1 1 1 0 1 0 0 0 ... 1 1 1 1 0 1 1 1 1 41.0

5 rows × 42 columns

In [27]:
thresh = 30
keep_legs = vote_pivot_with_yes_no_count["yes_no_count"] >= thresh
regulars = vote_pivot_with_yes_no_count[keep_legs]
regulars = regulars.drop(columns='yes_no_count')
regulars.shape
Out[27]:
(425, 41)



Regulars: PCA¶

Let's do some PCA on these regular voters. Remember: 1. Center data, then 2. Compute SVD.

In [28]:
regulars_centered = regulars - np.mean(regulars, axis = 0)
regulars_centered.head(5)
Out[28]:
515 516 517 518 519 520 521 522 523 524 ... 546 547 548 549 550 551 552 553 554 555
member
A000055 0.101176 -0.691765 -0.543529 -0.541176 0.023529 0.581176 -0.581176 0.625882 0.588235 0.550588 ... -0.541176 -0.545882 0.014118 -0.538824 -0.536471 0.529412 -0.538824 -0.555294 0.056471 -0.522353
A000367 -0.898824 -0.691765 -0.543529 -0.541176 -0.976471 -0.418824 -0.581176 -0.374118 -0.411765 -0.449412 ... -0.541176 0.454118 0.014118 0.461176 0.463529 -0.470588 0.461176 0.444706 -0.943529 0.477647
A000369 0.101176 0.308235 -0.543529 -0.541176 0.023529 0.581176 -0.581176 0.625882 0.588235 0.550588 ... -0.541176 -0.545882 0.014118 -0.538824 -0.536471 0.529412 -0.538824 -0.555294 0.056471 -0.522353
A000370 0.101176 0.308235 0.456471 0.458824 0.023529 -0.418824 0.418824 -0.374118 -0.411765 -0.449412 ... 0.458824 0.454118 0.014118 0.461176 0.463529 -0.470588 0.461176 0.444706 0.056471 0.477647
A000371 0.101176 0.308235 0.456471 0.458824 0.023529 -0.418824 0.418824 -0.374118 -0.411765 -0.449412 ... 0.458824 0.454118 0.014118 0.461176 0.463529 -0.470588 0.461176 0.444706 0.056471 0.477647

5 rows × 41 columns

In [29]:
u, s, vt = np.linalg.svd(regulars_centered, full_matrices = False)

Check component scores to verify that using first two PCs is okay:

In [30]:
np.round(s**2 / sum(s**2), 2)
Out[30]:
array([0.84, 0.02, 0.02, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01,
       0.01, 0.  , 0.  , 0.  , 0.  , 0.  , 0.  , 0.  , 0.  , 0.  , 0.  ,
       0.  , 0.  , 0.  , 0.  , 0.  , 0.  , 0.  , 0.  , 0.  , 0.  , 0.  ,
       0.  , 0.  , 0.  , 0.  , 0.  , 0.  , 0.  , 0.  ])



Now, plot first two PCs.

In [31]:
pcs_reg = u * s

vote2d_reg = pd.DataFrame({
    'member': regulars_centered.index,
    'pc1': pcs_reg[:, 0],
    'pc2': pcs_reg[:, 1],
}).merge(legs, left_on='member', right_on='leg_id')

sns.scatterplot(x="pc1", y="pc2", data=vote2d_reg);
In [32]:
vote2d_reg['pc1_jittered'] = vote2d_reg['pc1'] + np.random.normal(loc = 0, scale = 0.1, size = vote2d_reg.shape[0])
vote2d_reg['pc2_jittered'] = vote2d_reg['pc2'] + np.random.normal(loc = 0, scale = 0.1, size = vote2d_reg.shape[0])

sns.scatterplot(x="pc1_jittered", y="pc2_jittered",
                hue="party", palette=party_cp,  hue_order=party_hue,
                data = vote2d_reg);

Exploring $V^{\top}$¶

We can also look at Vt directly to try to gain insight into why each component is as it is.

In [33]:
num_votes = vt.shape[1]
votes_cols = regulars.columns

def plot_pc(k):
    plt.bar(votes_cols, vt[k, :], alpha=0.7)
    plt.xticks(votes_cols, rotation=90);

with plt.rc_context({"figure.figsize": (12, 4)}):
    k = 1
    plot_pc(k)
    plt.title(k)

Using SVD for PCA may lead to hard to interpret $V^T$ matrices for two reasons:

  1. Each row tends to include many many attributes. What does it mean that the first PC is driven by this specific set of 30 votes?
  2. Later rows (i.e. less important PCs) will be driven more by the orthogonality constraint than the data itself.

There exists other methods for doing PCA, e.g. Sparse PCA, that will try to lead to an interpretable version of $V^T$.

In [34]:
vote2d_reg.shape
Out[34]:
(424, 13)

Biplot¶

In [35]:
import random

# roll_calls = sorted([517, 520, 526, 527, 555, 553]) # features to plot on biplot
roll_calls = [515, 516, 517, 520, 526, 527, 553]


plt.figure(figsize = (7, 7))
# first plot each datapoint in terms of the first two principal components
sns.scatterplot(x="pc1_jittered", y="pc2_jittered",
                hue="party", palette=party_cp,  hue_order=party_hue,
                data = vote2d_reg);

# next, plot the loadings for PC1 and PC2
cp = sns.color_palette()[1:] # skip blue

directions_df= pd.DataFrame(data=vt[:2,:].T, index=regulars_centered.columns, columns=["dir1", "dir2"])
dir1, dir2 = directions_df["dir1"], directions_df["dir2"]
for i, feature in enumerate(roll_calls):
    feature = str(feature)
    plt.arrow(0, 0,
              dir1.loc[feature], dir2.loc[feature],
              head_width=0.2, head_length=0.2, color=cp[i], label=feature)
plt.legend(bbox_to_anchor=(1.1, 1.05));
plt.savefig("biplot.png", dpi = 200)

Each roll call from the 116th Congress - 1st Session: https://clerk.house.gov/evs/2019/ROLL_500.asp

  • 555: Raising a question of the privileges of the House (H.Res.590)
  • 553: [https://www.congress.gov/bill/116th-congress/senate-joint-resolution/54/actions]
  • 527: On Agreeing to the Amendment H.R.1146 - Arctic Cultural and Coastal Plain Protection Act