import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
%matplotlib inline
sns.set()

import scipy.stats as ss
import scikit_posthocs as sp

from sklearn.cluster import KMeans, AgglomerativeClustering
from sklearn.mixture import GaussianMixture
from sklearn.metrics import silhouette_score

regression_B_link = 'regression_data/inc_analysis_regression_1/Data for Regression B-Table 1.csv'

df_clusters = pd.read_csv(regression_B_link).reset_index()
df_clusters.columns = df_clusters.iloc[0].values

df_clusters = df_clusters.iloc[1:]

df_clusters.head()

	InclusionDistanceAvg	InclusionDistanceStdDev	Cluster	Epoch	NoOfCommittees	AttesterSlashings	VoluntaryExits	MissingBlocks	AttesterSlashings-t-1	VoluntaryExits-t-1	...	MissingBlocks-t-2	AttesterSlashings-t-3	VoluntaryExits-t-3	MissingBlocks-t-3	AttesterSlashings-t-4	VoluntaryExits-t-4	MissingBlocks-t-4	AttesterSlashings-t-5	VoluntaryExits-t-5	MissingBlocks-t-5
1	26.875	3.782387026	C2	5005	256	0	0	7	0	0	...	12	0	0	12	0	0	8	0	0	8
2	26.625	4.513420847	C2	5006	256	0	0	10	0	0	...	8	0	0	12	0	0	12	0	0	8
3	29.34375	3.694890762	C2	5007	256	0	0	12	0	0	...	7	0	0	8	0	0	12	0	0	12
4	27.8125	3.880783925	C2	5008	256	0	0	10	0	0	...	10	0	0	7	0	0	8	0	0	12
5	28.90625	3.062355987	C2	5009	256	0	0	10	0	0	...	12	0	0	10	0	0	7	0	0	8

5 rows × 23 columns

df_clusters['Epoch'] = pd.to_numeric(df_clusters['Epoch'])

df_clusters['Epoch'].describe()

count     7143.000000
mean      8576.000000
std       2062.150819
min       5005.000000
25%       6790.500000
50%       8576.000000
75%      10361.500000
max      12147.000000
Name: Epoch, dtype: float64

df_final = df_clusters[['InclusionDistanceAvg','InclusionDistanceStdDev','NoOfCommittees','AttesterSlashings', 'VoluntaryExits', 'MissingBlocks']]

df_final.head()

	InclusionDistanceAvg	InclusionDistanceStdDev	NoOfCommittees	AttesterSlashings	VoluntaryExits	MissingBlocks
1	26.875	3.782387026	256	0	0	7
2	26.625	4.513420847	256	0	0	10
3	29.34375	3.694890762	256	0	0	12
4	27.8125	3.880783925	256	0	0	10
5	28.90625	3.062355987	256	0	0	10

k = 5
k_means = KMeans(n_clusters=k)

k_means_fit = k_means.fit(df_final)

pd.DataFrame(k_means_fit.fit_predict(df_final))

	0
0	2
1	2
2	2
3	2
4	2
...	...
7138	4
7139	4
7140	4
7141	4
7142	2

7143 rows × 1 columns

silhouette_score(df_final,k_means_fit.fit_predict(df_final))

0.6449778138066845

r = {
    'k': [],
    'silhouette_score': []
}

for k in range(3, 30):
    k_means = KMeans(n_clusters=k) 
    k_means_fit = k_means.fit(df_final)
    r['k'].append(k)
    r['silhouette_score'].append(silhouette_score(df_final,k_means_fit.fit_predict(df_final)))
    
    print("Fitted ", k)

Fitted  3
Fitted  4
Fitted  5
Fitted  6
Fitted  7
Fitted  8
Fitted  9
Fitted  10
Fitted  11
Fitted  12
Fitted  13
Fitted  14
Fitted  15
Fitted  16
Fitted  17
Fitted  18
Fitted  19
Fitted  20
Fitted  21
Fitted  22
Fitted  23
Fitted  24
Fitted  25
Fitted  26
Fitted  27
Fitted  28
Fitted  29

r

	0	1	2	3	4	5	6
0	1.000000e+00	5.633329e-01	1.786142e-15	1.588939e-02	2.317615e-01	3.055948e-02	1.543623e-03
1	5.633329e-01	1.000000e+00	5.905219e-09	1.000000e+00	4.038892e-05	1.307735e-06	1.000000e+00
2	1.786142e-15	5.905219e-09	1.000000e+00	3.990906e-06	3.260306e-25	3.697313e-28	1.012276e-04
3	1.588939e-02	1.000000e+00	3.990906e-06	1.000000e+00	7.550611e-08	1.184910e-09	1.000000e+00
4	2.317615e-01	4.038892e-05	3.260306e-25	7.550611e-08	1.000000e+00	1.000000e+00	2.063077e-09
5	3.055948e-02	1.307735e-06	3.697313e-28	1.184910e-09	1.000000e+00	1.000000e+00	2.348468e-11
6	1.543623e-03	1.000000e+00	1.012276e-04	1.000000e+00	2.063077e-09	2.348468e-11	1.000000e+00

plt.figure(figsize=(20,15), )
cluster_scores = sns.lineplot(x=r['k'], y= r['silhouette_score'], linewidth = 5)
cluster_scores.axes.set_title("Cluster count (k) vs Silhouette Score ",fontsize=40)
cluster_scores.set_xlabel("K cluster",fontsize=20)
cluster_scores.set_ylabel("Silhouette Score",fontsize=20)
cluster_scores.tick_params(labelsize=15)
plt.savefig('cluster_scores_per_k.png')

r['silhouette_score']

[0.6256661429337039,
 0.6390691517962414,
 0.6920747182244511,
 0.7536393082219609,
 0.8257589175062543,
 0.8265823959914539,
 0.8326246614450671,
 0.774807544694714,
 0.7057761776707536,
 0.6477942741728239,
 0.5703401664958195,
 0.5087912939048966,
 0.42135348189272137,
 0.4181730869113912,
 0.3511231582400592,
 0.349618866315711,
 0.3445744012196581,
 0.32165255589997055,
 0.3210595556388046,
 0.3241353318859474,
 0.31487638976484184,
 0.31341380139291225,
 0.30378211846839887,
 0.3114911320404254,
 0.30902657280342083,
 0.30758765311812125,
 0.3084788765273082]

r['k']

[3,
 4,
 5,
 6,
 7,
 8,
 9,
 10,
 11,
 12,
 13,
 14,
 15,
 16,
 17,
 18,
 19,
 20,
 21,
 22,
 23,
 24,
 25,
 26,
 27,
 28,
 29]

# optimal number of clusters is 9

k_means = KMeans(n_clusters=9, random_state=42) 
k_means_fit = k_means.fit(df_final)

k_means_fit.labels_

array([2, 2, 2, ..., 7, 7, 8], dtype=int32)

df_labeled = df_final
df_labeled['clusters'] = k_means_fit.labels_

/opt/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:2: SettingWithCopyWarning:


A value is trying to be set on a copy of a slice from a DataFrame.
Try using .loc[row_indexer,col_indexer] = value instead

See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy

df_labeled.head()

	InclusionDistanceAvg	InclusionDistanceStdDev	NoOfCommittees	AttesterSlashings	VoluntaryExits	MissingBlocks	clusters
1	26.875	3.782387026	256	0	0	7	2
2	26.625	4.513420847	256	0	0	10	2
3	29.34375	3.694890762	256	0	0	12	2
4	27.8125	3.880783925	256	0	0	10	2
5	28.90625	3.062355987	256	0	0	10	2

df_labeled.info()

<class 'pandas.core.frame.DataFrame'>
RangeIndex: 7143 entries, 1 to 7143
Data columns (total 7 columns):
 #   Column                   Non-Null Count  Dtype 
---  ------                   --------------  ----- 
 0   InclusionDistanceAvg     7143 non-null   object
 1   InclusionDistanceStdDev  7143 non-null   object
 2   NoOfCommittees           7143 non-null   object
 3   AttesterSlashings        7143 non-null   object
 4   VoluntaryExits           7143 non-null   object
 5   MissingBlocks            7143 non-null   object
 6   clusters                 7143 non-null   int32 
dtypes: int32(1), object(6)
memory usage: 362.9+ KB

mask = (df_labeled['clusters'] != 8) & (df_labeled['clusters'] != 7)

df_labeled = df_labeled[mask]

df_labeled

	InclusionDistanceAvg	InclusionDistanceStdDev	NoOfCommittees	AttesterSlashings	VoluntaryExits	MissingBlocks	clusters
1	26.875	3.782387026	256	0	0	7	2
2	26.625	4.513420847	256	0	0	10	2
3	29.34375	3.694890762	256	0	0	12	2
4	27.8125	3.880783925	256	0	0	10	2
5	28.90625	3.062355987	256	0	0	10	2
...	...	...	...	...	...	...	...
7100	24.625	6.499379623	448	0	0	6	1
7101	23.71875	6.706926487	448	0	0	8	1
7102	20.84375	7.135144165	448	0	0	5	1
7103	24.625	6.020127531	448	0	0	5	1
7104	23.9375	5.836136039	448	0	0	8	1

7104 rows × 7 columns

df_labeled['clusters'].value_counts()

1    1081
3    1068
0    1051
5    1037
4    1029
6    1025
2     813
Name: clusters, dtype: int64

# Removed clusters 8 and 9 because there was < 30.
df_labeled['clusters'] = df_labeled['clusters'].astype(str)

/opt/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:2: SettingWithCopyWarning:


A value is trying to be set on a copy of a slice from a DataFrame.
Try using .loc[row_indexer,col_indexer] = value instead

See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy

df_labeled.head()

	InclusionDistanceAvg	InclusionDistanceStdDev	NoOfCommittees	AttesterSlashings	VoluntaryExits	MissingBlocks	clusters
1	26.875	3.782387026	256	0	0	7	2
2	26.625	4.513420847	256	0	0	10	2
3	29.34375	3.694890762	256	0	0	12	2
4	27.8125	3.880783925	256	0	0	10	2
5	28.90625	3.062355987	256	0	0	10	2

import plotly.express as px
import plotly.offline as pyo

pyo.init_notebook_mode()
    
fig = (px.scatter_3d(df_labeled, x='InclusionDistanceAvg', y='InclusionDistanceStdDev', z='MissingBlocks', 
              color='clusters', size_max=10, opacity=.3))

# tight layout
fig.update_layout(margin=dict(l=0, r=0, b=0, t=0))
fig.write_html("inc_distance_plotplot.html")

df_labeled.columns

Index(['InclusionDistanceAvg', 'InclusionDistanceStdDev', 'NoOfCommittees',
       'AttesterSlashings', 'VoluntaryExits', 'MissingBlocks', 'clusters'],
      dtype='object')

# Computing Dunn test results for all of the following

# InclusionDistanceAvg vs clusters
# InclusionDistanceStdDev vs clusters
# NoOfCommittees vs clusters
# AttesterSlashings vs clusters
# VoluntaryExits vs clusters
# MissingBlocks vs clusters

df_inc_dist_avg = df_labeled[['InclusionDistanceAvg', 'clusters']]
df_inc_dist_stdev = df_labeled[['InclusionDistanceStdDev', 'clusters']]
# df_num_committees = df_labeled[['NoOfCommittees', 'clusters']]
df_att_slashings = df_labeled[['AttesterSlashings', 'clusters']]
df_vol_exits = df_labeled[['VoluntaryExits', 'clusters']]
df_missing_blocks = df_labeled[['MissingBlocks', 'clusters']]

dfs = [df_inc_dist_avg, df_inc_dist_stdev, df_att_slashings, df_vol_exits, df_missing_blocks]

# make sure leading column is numeric

for i in range(5):
    dfs[i][dfs[i].columns[0]] = pd.to_numeric(dfs[i][dfs[i].columns[0]])

/opt/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:4: SettingWithCopyWarning:


A value is trying to be set on a copy of a slice from a DataFrame.
Try using .loc[row_indexer,col_indexer] = value instead

See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy

dunn_test_results_raw = []

for df in dfs:
    cols = df.columns
    dunn_test_results_raw.append(sp.posthoc_dunn(df, val_col=cols[0], group_col=cols[1], p_adjust = 'bonferroni'))

len(dunn_test_results_raw)

5

dunn_test_results_modified = []

for i, r in enumerate(dunn_test_results_raw):
    dunn_test_results_modified.append({ element: [] for element in r.columns })
    for column in r.columns:
        for item in r[column]:
            if item <= 0.01:
                dunn_test_results_modified[i][column].append(1)
            else:
                dunn_test_results_modified[i][column].append(0)

dunn_test_results_modified_dfs = [ pd.DataFrame(df) for df in dunn_test_results_modified ]

dunn_test_results_modified_dfs[0]

	0	1	2	3	4	5	6
0	0	1	1	0	1	1	0
1	1	0	1	0	0	1	1
2	1	1	0	1	1	1	1
3	0	0	1	0	0	1	1
4	1	0	1	0	0	1	1
5	1	1	1	1	1	0	1
6	0	1	1	1	1	1	0

# Saving output dfs
for i in range(5):
    pd.DataFrame(dunn_test_results_modified_dfs[i], index=dunn_test_results_modified_dfs[i].columns).to_csv(f'dunn_test_clusters_output_{dfs[i].columns[0]}.csv')

df_labeled.head()

	InclusionDistanceAvg	InclusionDistanceStdDev	NoOfCommittees	AttesterSlashings	VoluntaryExits	MissingBlocks	clusters
1	26.875	3.782387026	256	0	0	7	2
2	26.625	4.513420847	256	0	0	10	2
3	29.34375	3.694890762	256	0	0	12	2
4	27.8125	3.880783925	256	0	0	10	2
5	28.90625	3.062355987	256	0	0	10	2

median_per_cluster_dfs = [] 
for df in dfs:
    median_per_cluster_dfs.append(df.groupby('clusters')
                                          .median()
                                          .reset_index()
                                          .sort_values(df.columns[1], ascending=False))

avgs_per_cluster_dfs = [] 
for df in dfs:
    avgs_per_cluster_dfs.append(df.groupby('clusters')
                                        .mean()
                                        .reset_index()
                                        .sort_values(df.columns[1], ascending=False))

stdev_per_cluster_dfs = [] 
for df in dfs:
    stdev_per_cluster_dfs.append(df.groupby('clusters')
                                         .std()
                                         .reset_index()
                                         .sort_values(df.columns[1], ascending=False))
    

for df in median_per_cluster_dfs:
    df.sort_values(df.columns[1], ascending=False, inplace=True)
    
for df in avgs_per_cluster_dfs:
    df.sort_values(df.columns[1], ascending=False, inplace=True)
    
for df in stdev_per_cluster_dfs:
    df.sort_values(df.columns[1], ascending=False, inplace=True)

median_per_cluster_dfs

[   clusters  InclusionDistanceAvg
 2         2             27.593750
 5         5             26.187500
 4         4             23.625000
 1         1             23.500000
 3         3             23.265625
 0         0             22.750000
 6         6             22.687500,
    clusters  InclusionDistanceStdDev
 0         0                 7.229462
 6         6                 7.206628
 4         4                 6.728610
 3         3                 6.611863
 1         1                 6.082431
 5         5                 5.245197
 2         2                 4.433992,
    clusters  AttesterSlashings
 6         6                  0
 5         5                  0
 4         4                  0
 3         3                  0
 2         2                  0
 1         1                  0
 0         0                  0,
    clusters  VoluntaryExits
 6         6               0
 5         5               0
 4         4               0
 3         3               0
 2         2               0
 1         1               0
 0         0               0,
    clusters  MissingBlocks
 2         2              8
 6         6              7
 3         3              7
 1         1              7
 0         0              7
 5         5              6
 4         4              6]

median_per_cluster_dfs[1].columns

Index(['clusters', 'InclusionDistanceStdDev'], dtype='object')

# Box plots!
sns.set_style(style='white')

for i, df in enumerate(dfs):
    plt.figure(figsize=(25,18))
    a = sns.boxplot(x=df.columns[1], y=df.columns[0], data=df, order=median_per_cluster_dfs[i].index)
    a.set_xlabel(df.columns[1].capitalize() ,fontsize=35)
    a.set_ylabel(df.columns[0].capitalize() ,fontsize=35)
    a.tick_params(labelsize=35)
    plt.savefig(f'boxplots/boxplot_{df.columns[0]}_{df.columns[1]}.png')

# Bar plots!
sns.set_style(style='white')

for i, df in enumerate(median_per_cluster_dfs):
    plt.figure(figsize=(25,18))
    a = sns.barplot(x=df.columns[0], y=df.columns[1], data=df, order=median_per_cluster_dfs[i].index)
    a.axes.set_title(f"Median {df.columns[1]} per Cluster",fontsize=40)
    a.set_xlabel(df.columns[0].capitalize() ,fontsize=35)
    a.set_ylabel(df.columns[1].capitalize() ,fontsize=35)
    a.tick_params(labelsize=35)
    plt.savefig(f'barplots/barplot_median_{df.columns[0]}_{df.columns[1]}.png')

# Bar plots!
sns.set_style(style='white')

for i, df in enumerate(avgs_per_cluster_dfs):
    plt.figure(figsize=(25,18))
    a = sns.barplot(x=df.columns[0], y=df.columns[1], data=df, order=avgs_per_cluster_dfs[i].index)
    a.axes.set_title(f"Mean {df.columns[1]} per Cluster",fontsize=40)
    a.set_xlabel(df.columns[0].capitalize() ,fontsize=35)
    a.set_ylabel(df.columns[1].capitalize() ,fontsize=35)
    a.tick_params(labelsize=35)
    plt.savefig(f'barplots/barplot_mean_{df.columns[0]}_{df.columns[1]}.png')

# Bar plots!
sns.set_style(style='white')

for i, df in enumerate(stdev_per_cluster_dfs):
    plt.figure(figsize=(25,18))
    a = sns.barplot(x=df.columns[0], y=df.columns[1], data=df, order=stdev_per_cluster_dfs[i].index)
    a.axes.set_title(f"Standard Deviation of {df.columns[1]} per Cluster",fontsize=40)
    a.set_xlabel(df.columns[0].capitalize() ,fontsize=25)
    a.set_ylabel(df.columns[1].capitalize() ,fontsize=25)
    a.tick_params(labelsize=20)
    plt.savefig(f'barplots/barplot_stdev_{df.columns[0]}_{df.columns[1]}.png')

df_labeled.shape

(7104, 7)

df_labeled['VoluntaryExits'].value_counts()

0     7042
1        8
4        7
8        7
9        5
6        5
17       5
3        4
2        4
12       3
7        3
5        3
13       2
10       1
18       1
14       1
15       1
22       1
16       1
Name: VoluntaryExits, dtype: int64

df_labeled['AttesterSlashings'].value_counts()

0    7035
1      58
2       7
5       2
3       1
4       1
Name: AttesterSlashings, dtype: int64

df_labeled_no_zeroes = df_labeled[(df_labeled['AttesterSlashings'] != '0') & (df_labeled['VoluntaryExits'] != '0')]

df_labeled_no_zeroes.head()

	InclusionDistanceAvg	InclusionDistanceStdDev	NoOfCommittees	AttesterSlashings	VoluntaryExits	MissingBlocks	clusters

df_att_slashings_no_zeros = df_labeled[df_labeled['AttesterSlashings'] != '0'][['AttesterSlashings', 'clusters']]

df_vol_exits_no_zeros = df_labeled[df_labeled['VoluntaryExits'] != '0'][['VoluntaryExits', 'clusters']]

sns.distplot(df_att_slashings_no_zeros, kde=False)

<matplotlib.axes._subplots.AxesSubplot at 0x7f889774cf90>

sns.distplot(df_vol_exits_no_zeros, kde=False)

<matplotlib.axes._subplots.AxesSubplot at 0x7f8897b1d110>

df_labeled.head()

	InclusionDistanceAvg	InclusionDistanceStdDev	NoOfCommittees	AttesterSlashings	VoluntaryExits	MissingBlocks	clusters
1	26.875	3.782387026	256	0	0	7	2
2	26.625	4.513420847	256	0	0	10	2
3	29.34375	3.694890762	256	0	0	12	2
4	27.8125	3.880783925	256	0	0	10	2
5	28.90625	3.062355987	256	0	0	10	2

df_labeled.columns

Index(['InclusionDistanceAvg', 'InclusionDistanceStdDev', 'NoOfCommittees',
       'AttesterSlashings', 'VoluntaryExits', 'MissingBlocks', 'clusters'],
      dtype='object')

#Multi-distribution plot
plt.subplots(figsize=(12,8), dpi=100)
sns.distplot( df_labeled['InclusionDistanceAvg'].loc[df_labeled.clusters=='0'] , color="dodgerblue", label="Cluster 0", hist=False)
sns.distplot( df_labeled['InclusionDistanceAvg'].loc[df_labeled.clusters=='1'] , color="orange", label="Cluster 1", hist=False)
sns.distplot( df_labeled['InclusionDistanceAvg'].loc[df_labeled.clusters=='2'] , color="deeppink", label="Cluster 2", hist=False)
sns.distplot( df_labeled['InclusionDistanceAvg'].loc[df_labeled.clusters=='3'] , color="red", label="Cluster 3", hist=False)
sns.distplot( df_labeled['InclusionDistanceAvg'].loc[df_labeled.clusters=='4'] , color="blue", label="Cluster 4", hist=False)
sns.distplot( df_labeled['InclusionDistanceAvg'].loc[df_labeled.clusters=='5'] , color="purple", label="Cluster 5", hist=False)
sns.distplot( df_labeled['InclusionDistanceAvg'].loc[df_labeled.clusters=='6'] , color="green", label="Cluster 6", hist=False)

plt.title('Inclusion Distance Avg Across Clusters')
plt.legend();
plt.savefig('inc_dist_avg_across_cluster_distributions.png')

#Multi-distribution plot
plt.subplots(figsize=(12,8), dpi=100)
sns.distplot( df_labeled['InclusionDistanceStdDev'].loc[df_labeled.clusters=='0'] , color="dodgerblue", label="Cluster 0", hist=False)
sns.distplot( df_labeled['InclusionDistanceStdDev'].loc[df_labeled.clusters=='1'] , color="orange", label="Cluster 1", hist=False)
sns.distplot( df_labeled['InclusionDistanceStdDev'].loc[df_labeled.clusters=='2'] , color="deeppink", label="Cluster 2", hist=False)
sns.distplot( df_labeled['InclusionDistanceStdDev'].loc[df_labeled.clusters=='3'] , color="red", label="Cluster 3", hist=False)
sns.distplot( df_labeled['InclusionDistanceStdDev'].loc[df_labeled.clusters=='4'] , color="blue", label="Cluster 4", hist=False)
sns.distplot( df_labeled['InclusionDistanceStdDev'].loc[df_labeled.clusters=='5'] , color="purple", label="Cluster 5", hist=False)
sns.distplot( df_labeled['InclusionDistanceStdDev'].loc[df_labeled.clusters=='6'] , color="green", label="Cluster 6", hist=False)

plt.title('Inclusion Distance Standard Deviation Across Clusters')
plt.legend();
plt.savefig('inc_dist_std_across_cluster_distributions.png')

#Multi-distribution plot
plt.subplots(figsize=(12,8), dpi=100)
sns.distplot( df_labeled['MissingBlocks'].loc[df_labeled.clusters=='0'] , color="dodgerblue", label="Cluster 0", hist=False)
sns.distplot( df_labeled['MissingBlocks'].loc[df_labeled.clusters=='1'] , color="orange", label="Cluster 1", hist=False)
sns.distplot( df_labeled['MissingBlocks'].loc[df_labeled.clusters=='2'] , color="deeppink", label="Cluster 2", hist=False)
sns.distplot( df_labeled['MissingBlocks'].loc[df_labeled.clusters=='3'] , color="red", label="Cluster 3", hist=False)
sns.distplot( df_labeled['MissingBlocks'].loc[df_labeled.clusters=='4'] , color="blue", label="Cluster 4", hist=False)
sns.distplot( df_labeled['MissingBlocks'].loc[df_labeled.clusters=='5'] , color="purple", label="Cluster 5", hist=False)
sns.distplot( df_labeled['MissingBlocks'].loc[df_labeled.clusters=='6'] , color="green", label="Cluster 6", hist=False)

plt.title('Missing Blocks Across Clusters')
plt.legend();
plt.savefig('missing_blocks_across_cluster_distributions.png')