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CLUSTER STABILITY IN
ELECTORAL DATA (2000–2016)
Formal Analysis using Hierarchical Ward Clustering and AWE Criterion

Problem Setup
Given X ∈ R3142×5 , where each row corresponds to a U
...
county and each column contains the
Democratic vote share in the elections {2000, 2004, 2008, 2012, 2016}, perform the following:
1
...

2
...

3
...

4
...


MATHEMATICAL DERIVATION
Given:
log Bk = 25(k − 1) − 2k

⇒

k
1
2
2whitegray!10 3
4
5
6

AWEk = 2 log Bk = 2(23k − 25) = 46k − 50
AWEk

∆k = AWEk − AWEk−1

0
42
88
134
180
226

–
42
46
46
46
46

Conclusion: Maximum ∆k occurs first at k = 3; hence, k ∗ = 3 is chosen via the parsimony
principle
...
THREE PERSISTENT BLOCS
The value k ∗ = 3 supports a stable trichotomy:
(i)Democratic urban clusters,

(ii)Republican rural zones,

(iii)Heterogeneous swing regions

These reflect spatial and ideological cohesion across time
...
TEMPORAL STABILITY SIGNAL
For k > 3, ∆k remains constant:
∆k = 46 for all k = 3, 4, 5, 6
This plateau in AWEk implies no significant informational gain, evidencing that the blocs are temporally resilient over five election cycles
...
STRATEGIC TARGETING
The third (swing) cluster likely holds:
Highest campaign leverage and policy reactivity
...


SUMMARY INSIGHT
Ward clustering with AWE analysis identifies three enduring U
...
electoral blocs
...
These results align strongly with spatially observable voting behavior and provide a
principled basis for both political analysis and campaign strategy
...
S
...
S
...
3

Cluster 2:

λ1 /λ4 = 1
...
6

A new county has feature vector:
0
...
65 
0
...
25
48,000


µ2 = 

 0
...
55


assigned to Cluster 2, whose statistics are:



Covariance matrix: Σ2 = diag(0
...
01, 0
...
03
−4000


d = xnew − µ2 = 
,
 0
...
03

Σ2 = diag(0
...
01, 0
...
032 (−4000)2 0
...
032
+
+
+
= 0
...
56 + 0
...
09 = 2
...
01
25002
0
...
01
MD =

√
2
...
68

3

OUTLIER DETERMINATION (THRESHOLD: 3
...
68 < 3
...


CLUSTER SHAPE INTERPRETATION
Cluster 2’s eigenvalue ratio λ1 /λ4 = 1
...
e
...
This results in a spherical cluster, sharply contrasting with
elongated ellipsoids seen in Clusters 1 (5
...
6)
...


4

SPATIAL CLUSTER CONTIGUITY IN
STATEWIDE PATTERNS
Analyzing Geopolitical Coherence in Regional k-Means Voting Blocs

Problem Setup
Suppose a U
...
state with 100 counties is studied
...
Voter turnout
2
...
Republican vote share
• Adjacency matrix A ∈ {0, 1}100×100 where Aij = 1 iff counties i and j share a border
...
Define the Cluster Contiguity
Ratio (CCRk ) as:
CCRk =

Number of adjacent pairs in the same cluster
Total number of adjacent pairs

Assume the following empirical values:
k

CCRk

2
3
4
5

0
...
78
0
...
43

SPATIAL COHERENCE VERDICT
Maximum spatial coherence: CCR2 = 0
...


INTERPRETIVE RATIONALE
SPATIAL AUTOCORRELATION
High Moran-I statistics in electoral behavior indicate that adjacent counties vote similarly
...


5

VOTING REGIONALISM
A binary partition (k = 2) often captures:
Urban/metro Democratic core

vs
...


COMPACTNESS VS INTERPRETABILITY
k
2
3
4–5

Geometric Compactness

Spatial Contiguity

Interpretability

Lowest (broad centroids)
Moderate
Highest

Highest (0
...
78)
Fragmented (≤ 0
...
Here, k = 3 offers a strong compromise—retaining over 75% spatial cohesion
while isolating a pivotal suburban swing belt
...


6



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