class: center, middle, title-slide .title[ # Hierarchical clustering ] .author[ ### Claus O. Wilke ] .date[ ### last updated: 2023-04-10 ] --- ## Example data: State level statistics for US .tiny-font[ ```r US_state_stats ``` ``` # A tibble: 50 × 20 state homeo…¹ multi…² income med_i…³ poverty fed_s…⁴ smoke murder robbery <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> 1 Alabama 71.1 15.5 22984 42081 17.1 11.7 24.8 8.2 141. 2 Alaska 64.7 24.6 30726 66521 9.5 16.8 25 4.8 80.9 3 Arizona 67.4 20.7 25680 50448 15.3 9.85 20.4 7.5 144. 4 Arkansas 67.7 15.2 21274 39267 18 9.61 23.5 6.7 91.1 5 Californ… 57.4 30.7 29188 60883 13.7 8.89 15.2 6.9 176. 6 Colorado 67.6 25.6 30151 56456 12.2 9.15 19.9 3.7 84.6 7 Connecti… 69.2 34.6 36775 67740 9.2 14.8 16.5 2.9 113 8 Delaware 73.6 17.7 29007 57599 11 8.89 20.7 4.4 155. 9 Florida 69.7 30 26551 47661 13.8 9.62 21.6 5 169. 10 Georgia 67.2 20.5 25134 49347 15.7 8.88 22.2 6.2 155. # … with 40 more rows, 10 more variables: agg_assault <dbl>, larceny <dbl>, # motor_theft <dbl>, soc_sec <dbl>, nuclear <dbl>, coal <dbl>, # tr_deaths <dbl>, tr_deaths_no_alc <dbl>, unempl <dbl>, popdens2010 <dbl>, # and abbreviated variable names ¹​homeownership, ²​multiunit, ³​med_income, # ⁴​fed_spend ``` ] --- ## PCA reveals broad trends .center[ <!-- --> ] --- class: middle, center ## Can we get a more detailed picture? --- ## Hierarchical clustering .center[ <!-- --> ] -- This visualization is called a dendrogram --- ## Cut the dendrogram to obtain defined clusters <!-- --> --- ## Cut the dendrogram to obtain defined clusters <!-- --> --- ## Cut the dendrogram to obtain defined clusters <!-- --> --- ## Cut the dendrogram to obtain defined clusters <!-- --> --- ## Cut the dendrogram to obtain defined clusters <!-- --> --- ## Display clusters in PCA space .center[ <!-- --> ] --- ## Hierarchical clustering can be useful with heatmaps .pull-left[ Arbitrary ordering: <!-- --> ] -- .pull-right[ Ordering by clustering similarity: <!-- --> ] [//]: # "segment ends here" --- class: center middle ## Hierarchical clustering via UPGMA --- ## Hierarchical clustering methods require two ingredients -- 1\. Distance metric Examples: Euclidean distance, maximum distance <br> -- 2\. Linkage criteria Examples: - Unweighted average linkage clustering (unweighted pair group method with arithmetic mean, UPGMA) - Complete linkage clustering - Minimum energy clustering (See: https://en.wikipedia.org/wiki/Hierarchical_clustering) --- ## Hierarchical clustering via UPGMA .pull-left.move-up-1em[ <!-- --> ] -- .pull-right[ Distance matrix: <table class=" lightable-paper" style="font-family: sans-serif; width: auto !important; margin-left: auto; margin-right: auto;"> <thead> <tr> <th style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> </th> <th style="text-align:right;font-weight: bold;color: black !important;background-color: white !important;"> A </th> <th style="text-align:right;font-weight: bold;color: black !important;background-color: white !important;"> B </th> <th style="text-align:right;font-weight: bold;color: black !important;background-color: white !important;"> C </th> <th style="text-align:right;font-weight: bold;color: black !important;background-color: white !important;"> D </th> <th style="text-align:right;font-weight: bold;color: black !important;background-color: white !important;"> E </th> </tr> </thead> <tbody> <tr> <td style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> A </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.00 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 2.79 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.840 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.56 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.610 </td> </tr> <tr> <td style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> B </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 2.79 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.00 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 4.020 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 1.34 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.330 </td> </tr> <tr> <td style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> C </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.84 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 4.02 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.000 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.57 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.791 </td> </tr> <tr> <td style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> D </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.56 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 1.34 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.570 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.00 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 2.820 </td> </tr> <tr> <td style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> E </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.61 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.33 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.791 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 2.82 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.000 </td> </tr> </tbody> </table> ] --- ## Hierarchical clustering via UPGMA .pull-left.move-up-1em[ <!-- --> ] .pull-right[ Distance matrix: <table class=" lightable-paper" style="font-family: sans-serif; width: auto !important; margin-left: auto; margin-right: auto;"> <thead> <tr> <th style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> </th> <th style="text-align:right;font-weight: bold;color: black !important;background-color: white !important;"> A </th> <th style="text-align:right;font-weight: bold;color: black !important;background-color: white !important;"> B </th> <th style="text-align:right;font-weight: bold;color: black !important;background-color: white !important;"> C </th> <th style="text-align:right;font-weight: bold;color: black !important;background-color: white !important;"> D </th> <th style="text-align:right;font-weight: bold;color: black !important;background-color: white !important;"> E </th> </tr> </thead> <tbody> <tr> <td style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> A </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.00 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 2.79 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.840 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.56 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.610 </td> </tr> <tr> <td style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> B </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 2.79 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.00 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 4.020 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 1.34 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.330 </td> </tr> <tr> <td style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> C </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.84 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 4.02 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.000 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.57 </td> <td style="text-align:right;color: black !important;background-color: yellow !important;"> 0.791 </td> </tr> <tr> <td style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> D </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.56 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 1.34 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.570 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.00 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 2.820 </td> </tr> <tr> <td style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> E </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.61 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.33 </td> <td style="text-align:right;color: black !important;background-color: yellow !important;"> 0.791 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 2.82 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.000 </td> </tr> </tbody> </table> Join points with minimum distance ] --- ## Hierarchical clustering via UPGMA .pull-left.move-up-1em[ <!-- --> ] .pull-right[ Update distances by averaging: `$$d_{A:(C, E)} = (d_{A:C} + d_{A:E})/2,$$` similarly for points B and D ] --- ## Hierarchical clustering via UPGMA .pull-left.move-up-1em[  ] .pull-right[ Updated distance matrix: <table class=" lightable-paper" style="font-family: sans-serif; width: auto !important; margin-left: auto; margin-right: auto;"> <thead> <tr> <th style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> </th> <th style="text-align:right;font-weight: bold;color: black !important;background-color: white !important;"> A </th> <th style="text-align:right;font-weight: bold;color: black !important;background-color: white !important;"> B </th> <th style="text-align:right;font-weight: bold;color: black !important;background-color: white !important;"> D </th> <th style="text-align:right;font-weight: bold;color: black !important;background-color: white !important;"> (C, E) </th> </tr> </thead> <tbody> <tr> <td style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> A </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.00 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 2.79 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.56 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.73 </td> </tr> <tr> <td style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> B </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 2.79 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.00 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 1.34 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.68 </td> </tr> <tr> <td style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> D </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.56 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 1.34 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.00 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.20 </td> </tr> <tr> <td style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> (C, E) </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.73 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.68 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.20 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.00 </td> </tr> </tbody> </table> ] --- ## Hierarchical clustering via UPGMA .pull-left.move-up-1em[ <!-- --> ] .pull-right[ Updated distance matrix: <table class=" lightable-paper" style="font-family: sans-serif; width: auto !important; margin-left: auto; margin-right: auto;"> <thead> <tr> <th style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> </th> <th style="text-align:right;font-weight: bold;color: black !important;background-color: white !important;"> A </th> <th style="text-align:right;font-weight: bold;color: black !important;background-color: white !important;"> B </th> <th style="text-align:right;font-weight: bold;color: black !important;background-color: white !important;"> D </th> <th style="text-align:right;font-weight: bold;color: black !important;background-color: white !important;"> (C, E) </th> </tr> </thead> <tbody> <tr> <td style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> A </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.00 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 2.79 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.56 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.73 </td> </tr> <tr> <td style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> B </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 2.79 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.00 </td> <td style="text-align:right;color: black !important;background-color: yellow !important;"> 1.34 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.68 </td> </tr> <tr> <td style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> D </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.56 </td> <td style="text-align:right;color: black !important;background-color: yellow !important;"> 1.34 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.00 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.20 </td> </tr> <tr> <td style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> (C, E) </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.73 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.68 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.20 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.00 </td> </tr> </tbody> </table> Continue joining points with minimum distance ] --- ## Hierarchical clustering via UPGMA .pull-left.move-up-1em[ <!-- --> ] .pull-right[ Updated distance matrix: <table class=" lightable-paper" style="font-family: sans-serif; width: auto !important; margin-left: auto; margin-right: auto;"> <thead> <tr> <th style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> </th> <th style="text-align:right;font-weight: bold;color: black !important;background-color: white !important;"> A </th> <th style="text-align:right;font-weight: bold;color: black !important;background-color: white !important;"> (C, E) </th> <th style="text-align:right;font-weight: bold;color: black !important;background-color: white !important;"> (B, D) </th> </tr> </thead> <tbody> <tr> <td style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> A </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.00 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.73 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.17 </td> </tr> <tr> <td style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> (C, E) </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.73 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.00 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.44 </td> </tr> <tr> <td style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> (B, D) </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.17 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.44 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.00 </td> </tr> </tbody> </table> ] --- ## Hierarchical clustering via UPGMA .pull-left.move-up-1em[ <!-- --> ] .pull-right[ Updated distance matrix: <table class=" lightable-paper" style="font-family: sans-serif; width: auto !important; margin-left: auto; margin-right: auto;"> <thead> <tr> <th style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> </th> <th style="text-align:right;font-weight: bold;color: black !important;background-color: white !important;"> A </th> <th style="text-align:right;font-weight: bold;color: black !important;background-color: white !important;"> (C, E) </th> <th style="text-align:right;font-weight: bold;color: black !important;background-color: white !important;"> (B, D) </th> </tr> </thead> <tbody> <tr> <td style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> A </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.00 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.73 </td> <td style="text-align:right;color: black !important;background-color: yellow !important;"> 3.17 </td> </tr> <tr> <td style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> (C, E) </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.73 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.00 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.44 </td> </tr> <tr> <td style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> (B, D) </td> <td style="text-align:right;color: black !important;background-color: yellow !important;"> 3.17 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.44 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.00 </td> </tr> </tbody> </table> ] --- ## Hierarchical clustering via UPGMA .pull-left.move-up-1em[  ] .pull-right[ In general, distances are weighted by # of data points added to the new cluster: `$$\begin{eqnarray} d_{(C, E): (A, (B, D))} &=& \\ \qquad\qquad(d_{(C, E): A} &+& 2 d_{(C, E): (B, D)})/(1+2) \end{eqnarray}$$` ] --- ## Hierarchical clustering via UPGMA .pull-left.move-up-1em[ <!-- --> ] .pull-right[ Distance matrix: <table class=" lightable-paper" style="font-family: sans-serif; width: auto !important; margin-left: auto; margin-right: auto;"> <thead> <tr> <th style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> </th> <th style="text-align:right;font-weight: bold;color: black !important;background-color: white !important;"> (C, E) </th> <th style="text-align:right;font-weight: bold;color: black !important;background-color: white !important;"> (A, (B, D)) </th> </tr> </thead> <tbody> <tr> <td style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> (C, E) </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.00 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.53 </td> </tr> <tr> <td style="text-align:left;font-weight: bold;color: black !important;background-color: white !important;"> (A, (B, D)) </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 3.53 </td> <td style="text-align:right;color: black !important;background-color: white !important;"> 0.00 </td> </tr> </tbody> </table> ] --- ## Hierarchical clustering via UPGMA .pull-left.move-up-1em[ <!-- --> ] .pull-right[ Final clustering: <br><br> ((A, (B, D)), (C, E)) ] --- ## Hierarchical clustering via UPGMA .pull-left.move-up-1em[  ] .pull-right.move-up-1em[ Final clustering: <!-- --> ] [//]: # "segment ends here" --- class: center middle ## Doing hierarchical clustering in R --- ## Doing hierarchical clustering in R We'll cluster the US state stats dataset .tiny-font[ ```r US_state_stats <- read_csv("https://wilkelab.org/SDS375/datasets/US_state_stats.csv") %>% filter(state != "District of Columbia") US_state_stats ``` ``` # A tibble: 50 × 20 state homeo…¹ multi…² income med_i…³ poverty fed_s…⁴ smoke murder robbery <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> 1 Alabama 71.1 15.5 22984 42081 17.1 11.7 24.8 8.2 141. 2 Alaska 64.7 24.6 30726 66521 9.5 16.8 25 4.8 80.9 3 Arizona 67.4 20.7 25680 50448 15.3 9.85 20.4 7.5 144. 4 Arkansas 67.7 15.2 21274 39267 18 9.61 23.5 6.7 91.1 5 Californ… 57.4 30.7 29188 60883 13.7 8.89 15.2 6.9 176. 6 Colorado 67.6 25.6 30151 56456 12.2 9.15 19.9 3.7 84.6 7 Connecti… 69.2 34.6 36775 67740 9.2 14.8 16.5 2.9 113 8 Delaware 73.6 17.7 29007 57599 11 8.89 20.7 4.4 155. 9 Florida 69.7 30 26551 47661 13.8 9.62 21.6 5 169. 10 Georgia 67.2 20.5 25134 49347 15.7 8.88 22.2 6.2 155. # … with 40 more rows, 10 more variables: agg_assault <dbl>, larceny <dbl>, # motor_theft <dbl>, soc_sec <dbl>, nuclear <dbl>, coal <dbl>, # tr_deaths <dbl>, tr_deaths_no_alc <dbl>, unempl <dbl>, popdens2010 <dbl>, # and abbreviated variable names ¹​homeownership, ²​multiunit, ³​med_income, # ⁴​fed_spend ``` ] --- ## Step 1: Calculate the distance matrix .tiny-font[ ```r US_state_stats %>% column_to_rownames(var = "state") %>% scale() %>% dist(method = "euclidean") ``` ``` Alabama Alaska Arizona Arkansas California Colorado Connecticut Delaware Florida Georgia Hawaii Idaho Illinois Indiana Iowa Kansas Kentucky Louisiana Alaska 10.598996 Arizona 3.330008 9.367964 Arkansas 2.101623 10.258034 3.460202 California 7.387598 10.417133 4.574415 7.213409 Colorado 5.853785 8.784506 3.584944 5.667544 4.905273 Connecticut 8.163739 9.632134 6.647566 8.581965 6.413761 6.054845 Delaware 6.852026 6.873437 6.099149 6.443272 8.224788 5.836395 7.471686 Florida 3.817288 10.483883 3.058667 4.111477 5.074512 4.373974 6.059186 6.305386 Georgia 3.718662 9.954441 1.731379 3.841420 4.757818 3.278357 7.153667 6.451118 3.147390 Hawaii 8.937729 6.700074 7.068674 8.508732 7.035466 6.118096 6.293809 6.878239 7.514385 7.792204 Idaho 5.597221 9.004979 4.419521 4.548114 6.607359 4.469112 7.676599 5.801041 5.045858 4.799015 6.969588 Illinois 5.254690 10.316944 3.192744 5.609923 4.396773 3.944139 5.238648 6.674979 3.160311 3.311714 7.447009 6.015498 Indiana 3.604745 10.172037 3.853444 3.882761 7.044651 4.062005 7.864817 6.213848 3.787589 3.140930 8.375538 5.298522 4.811957 Iowa 5.007138 9.958779 4.840562 4.587101 7.634487 4.002106 6.966781 5.674885 4.667862 4.859556 7.354684 3.935105 5.395395 3.885737 Kansas 4.408601 7.409098 3.060602 4.442553 6.598661 3.512435 7.294319 3.735978 4.258347 3.615391 6.183604 3.905887 4.316269 4.282286 2.020848 Kentucky 2.896963 8.930716 5.257548 3.631240 9.364228 6.754618 10.274920 6.475033 5.154735 4.953492 9.129675 6.531273 7.275413 3.384898 5.400969 5.740308 Louisiana 2.187868 9.890864 2.803839 2.411572 6.355365 5.410132 7.975901 6.748584 3.939756 3.518586 8.236106 4.961559 5.256562 4.220968 5.384255 4.020284 3.898974 Maine 5.449459 9.500632 5.356080 4.816331 7.595522 5.251686 6.878787 5.995987 4.764959 5.823169 6.944840 3.188682 6.143050 5.141585 3.249026 4.222520 5.968985 5.401479 Maryland 7.611163 9.273961 6.327444 8.668010 6.722458 6.408493 4.868335 7.423132 6.638708 6.740797 7.515811 8.851332 5.771450 7.475384 8.018422 6.383880 9.115635 7.319302 Massachusetts 7.963086 9.781969 6.354021 8.072120 5.455518 5.195148 3.060985 7.225595 5.278116 6.487500 5.717072 7.309502 5.055135 7.032772 6.492189 6.865981 9.469819 7.491380 Michigan 3.219770 10.438840 2.876205 3.585488 6.172830 4.156116 6.718661 6.002538 2.794426 2.920050 8.228620 4.716282 3.390219 2.702043 3.778241 3.360651 4.829221 3.688944 Minnesota 5.699186 9.792369 4.346035 5.729161 6.486491 2.933653 5.706504 5.782694 4.478219 4.323841 7.140860 4.643832 3.857913 4.315508 2.716098 2.943799 7.133466 5.836457 Mississippi 2.883395 11.407078 4.665463 2.644596 8.218904 7.060859 9.761691 7.962489 5.091162 4.947830 9.825048 5.547383 7.061814 5.136325 6.216101 6.501396 3.987288 3.266231 Missouri 2.193270 9.917924 2.932619 2.820666 6.655760 4.228391 7.353772 5.963989 3.232413 2.995409 7.834503 5.026190 4.551890 2.291596 3.673264 2.990748 3.026295 2.921983 Montana 3.778946 8.061485 3.997077 3.462787 7.773931 5.249206 9.107011 4.420878 4.673883 4.419328 7.284882 3.631654 6.333142 4.746282 3.332278 3.110260 4.808803 3.374796 Nebraska 5.010139 8.818975 4.374697 4.393229 7.323869 4.062795 6.927510 5.277823 5.158065 4.691573 6.423243 4.068942 5.195917 4.537399 2.470144 2.287313 6.083651 5.127234 Nevada 6.867679 8.696989 4.599803 6.614218 5.019683 5.218190 8.358046 6.695426 5.604532 4.833401 6.778081 6.817587 5.516908 6.030305 7.702788 6.141385 7.205939 6.315798 New Hampshire 7.032307 9.342326 5.896818 6.871723 7.463344 5.057318 5.090292 6.242540 5.863305 6.372527 6.696511 5.337879 5.194277 6.441029 4.299915 4.980274 8.770819 7.124445 New Jersey 8.480882 10.991190 6.720639 8.755648 5.736034 6.305088 3.633544 7.650389 5.777136 6.795657 7.618642 8.219887 4.804280 7.707355 7.699712 8.015335 10.731633 8.247333 New Mexico 3.680324 8.522460 4.141634 3.438805 7.732973 5.405001 8.705953 5.658454 5.487654 4.472108 7.876981 5.083431 6.601013 4.562994 5.203417 4.088725 3.360258 3.383514 New York 7.740170 10.726266 5.889495 7.622262 4.069967 5.830359 5.910359 8.683771 5.156224 5.945851 6.716974 7.941413 4.649067 7.279079 7.735557 7.558194 9.310169 7.104194 North Carolina 2.964659 10.446575 2.011604 3.162292 5.346357 4.219042 7.172742 6.558336 2.854521 1.673388 8.316637 4.971980 3.299835 3.014498 4.783395 3.728261 4.552803 3.294185 North Dakota 6.830398 8.494124 6.726308 6.215588 9.102334 5.825150 8.078210 6.274746 7.143179 7.022232 6.067419 5.591146 7.800522 6.406683 4.333533 4.269167 6.384704 6.832059 Ohio 3.756038 10.503530 3.207573 3.964614 5.829037 3.634401 6.763715 6.086862 2.761796 2.645838 7.765809 5.294823 3.414807 2.337553 3.693684 2.922134 4.539927 4.122743 Oklahoma 2.746771 9.590296 3.485447 2.160842 7.093379 4.597272 8.206370 6.013341 4.026087 3.514846 7.809859 4.144994 5.635490 2.949381 3.685416 3.896042 3.236769 2.712148 Oregon 5.548912 9.264492 3.668999 4.628232 4.605672 3.465647 6.494161 6.242370 3.774078 4.028477 5.799009 3.010644 4.661539 4.918500 4.211692 3.995912 6.544470 4.859990 Pennsylvania 3.278011 10.332521 3.342653 4.088292 6.229582 4.611578 5.783043 6.249101 2.736122 3.577485 7.875235 5.678008 3.172094 3.276966 4.092980 3.516435 4.990771 4.012989 Rhode Island 8.299825 8.727605 6.894639 7.765106 6.657274 6.679551 6.042031 6.508415 6.088086 7.287467 5.160020 6.871825 6.752070 7.613243 7.566176 7.738592 8.938709 7.772664 South Carolina 2.215120 10.099229 2.490228 2.549883 6.425645 5.464897 7.510363 6.526626 3.601202 3.138396 8.511209 5.060045 4.305294 4.303923 5.276383 4.459678 4.725989 2.765308 South Dakota 5.342727 8.707087 5.423795 4.524657 8.151475 5.319894 7.848930 5.668781 5.828860 5.846903 6.674934 3.605722 6.993915 5.527714 3.431093 3.665688 5.645236 5.154360 Tennessee 1.592758 10.217475 2.929816 2.455450 6.825626 5.211316 7.910098 6.451256 3.357974 2.869454 8.663065 5.604460 4.496692 2.836403 4.945482 4.346147 2.903620 2.663005 Texas 4.732795 10.096352 2.507271 4.552391 4.110426 3.320189 7.219854 7.019202 3.748023 2.073875 7.473824 4.796109 3.747254 4.304938 5.192502 3.882935 6.240198 3.758654 Utah 7.816158 9.971541 5.886282 7.294373 7.035596 4.025850 7.845243 6.827859 6.908662 5.766022 7.201289 4.877627 6.305243 6.466673 4.757319 4.528537 9.209289 7.334168 Vermont 7.853224 9.342110 7.331872 7.357606 9.369040 7.662961 7.198676 7.277386 7.861043 8.097009 7.691579 6.377467 7.270864 8.433320 6.365156 6.540232 9.630963 7.996153 Virginia 5.753364 8.798401 4.670117 6.490933 6.171084 4.301018 4.253637 6.930069 5.205486 5.131602 6.528350 6.267057 4.758796 5.509324 5.179441 4.622325 7.411418 5.481934 Washington 6.100880 8.812749 3.607467 5.795693 3.680065 2.581905 4.948654 6.302339 3.986082 4.020214 5.250266 4.144214 3.791803 5.258658 4.655437 4.116854 7.604610 5.325361 West Virginia 3.873600 10.991010 5.946337 3.513818 9.654607 6.783430 9.605647 6.816279 5.733335 5.935861 9.565350 5.812334 7.473856 4.004706 4.372867 5.827139 3.110384 4.962556 Wisconsin 4.364111 9.825504 3.273309 4.235680 5.727754 2.696602 5.899727 5.849888 3.071053 3.246268 6.758584 4.164685 3.394139 3.129319 2.282848 2.033971 5.447424 4.520653 Wyoming 8.354493 7.492609 8.226198 7.759220 10.664883 7.652842 9.921622 5.697209 8.976007 8.516690 7.911546 6.819268 9.534238 8.124581 6.861121 5.593932 7.161100 8.439725 Maine Maryland Massachusetts Michigan Minnesota Mississippi Missouri Montana Nebraska Nevada New Hampshire New Jersey New Mexico New York North Carolina North Dakota Alaska Arizona Arkansas California Colorado Connecticut Delaware Florida Georgia Hawaii Idaho Illinois Indiana Iowa Kansas Kentucky Louisiana Maine Maryland 8.589605 Massachusetts 6.619401 5.444320 Michigan 4.510982 6.891263 6.464100 Minnesota 4.485471 6.695074 5.548868 3.581946 Mississippi 6.008202 9.723329 9.418208 4.948954 7.356641 Missouri 4.892887 6.787230 6.796929 2.684878 4.399936 4.127669 Montana 4.160841 8.207678 8.403983 4.667686 5.307321 4.753047 3.053530 Nebraska 4.054463 7.708930 6.601193 4.418092 3.198633 6.438225 4.032172 3.688675 Nevada 7.824494 7.645061 7.495758 6.191369 7.070806 7.638030 6.029728 6.659527 7.046413 New Hampshire 4.561159 7.429595 5.714043 5.619372 3.356284 8.495515 6.212605 6.589170 3.879067 8.137473 New Jersey 7.951568 5.737290 3.395446 6.730956 6.213064 10.001374 7.660068 9.696761 7.648360 7.682244 6.126948 New Mexico 5.873785 7.862505 8.285417 4.873973 6.239279 4.073579 3.355320 2.922670 4.978582 6.621719 7.716956 9.420024 New York 7.723006 7.091664 4.143738 6.772569 6.857682 9.003420 6.987451 8.708403 7.313364 6.295759 7.330703 5.408281 8.515242 North Carolina 5.487191 7.071247 6.794340 2.102464 4.565711 4.446084 2.746438 4.567294 4.785123 5.333058 6.327946 6.862369 4.664005 6.268931 North Dakota 5.008931 8.852050 7.533362 6.931520 5.893309 7.612123 5.711863 3.859775 3.748656 8.534916 5.974491 9.465422 5.503519 8.669587 7.282022 Ohio 5.087764 6.627267 5.791114 2.109893 3.650569 5.593289 2.440079 4.327799 4.194522 5.893166 5.992299 6.502035 4.923876 5.875571 2.489281 6.361795 Oklahoma 4.358549 8.039996 7.394851 3.590942 4.839118 3.673270 2.266944 2.940617 3.679251 6.424026 6.381849 8.500283 3.190292 7.414962 3.526575 5.343542 Oregon 3.656905 7.934215 5.557330 4.272458 4.331402 6.002440 4.659856 4.650299 4.258624 5.426520 5.288891 6.817479 5.634277 5.463209 4.215200 5.934161 Pennsylvania 4.838128 5.912135 5.665029 2.080653 3.777518 5.651725 2.761281 5.021885 4.355587 6.503932 5.146335 6.006059 5.459081 6.083498 2.754122 6.788586 Rhode Island 6.528749 8.114487 5.008645 7.227053 7.547637 8.840137 7.557499 8.276331 7.313252 6.409380 7.182960 5.772403 8.093078 5.986392 7.275119 7.663675 South Carolina 5.630992 7.487290 7.603678 3.235657 5.423490 3.523234 3.158677 4.523471 4.874313 6.177323 6.430322 7.649252 4.164027 7.175516 2.329532 7.236684 South Dakota 3.438765 8.644534 7.405576 5.657342 5.175878 5.748209 4.624357 1.983415 3.157721 7.953556 5.346449 8.968003 4.399006 8.359152 5.988650 2.701569 Tennessee 5.609207 7.334145 7.474428 2.804910 5.237431 3.764199 2.195797 4.277971 4.776192 6.037960 6.787385 7.877657 4.008972 6.990206 2.214183 6.978381 Texas 5.934945 6.905895 6.151740 3.844443 4.498445 5.802564 4.034896 4.834120 4.890119 5.384088 6.552911 6.946721 5.132624 5.322347 3.176829 7.118197 Utah 6.366016 8.354401 7.296731 6.164955 4.413720 8.625013 6.365106 6.177063 4.851634 7.792762 6.146926 8.237101 6.680453 8.446089 6.541281 6.005682 Vermont 5.750459 9.430157 8.260979 7.389970 6.408482 8.907407 7.833774 7.430777 5.091342 9.671883 4.209895 8.633394 8.253892 9.087153 7.741963 6.307128 Virginia 5.805975 4.003073 4.769055 5.014803 4.261891 7.689765 4.790444 6.459736 5.068947 7.419712 4.989951 6.063187 6.111441 6.426794 5.365508 6.397805 Washington 4.621560 6.256025 4.237212 4.434985 3.653237 7.171936 4.981488 5.769384 4.569590 5.395136 4.714175 5.548317 6.173554 4.770796 4.514208 6.387711 West Virginia 4.895459 9.802177 9.108152 4.655579 6.370725 4.226412 3.790895 4.718712 5.231977 8.486123 7.616860 10.111724 4.390298 9.593673 5.287663 5.860449 Wisconsin 3.920615 6.744556 5.216107 2.734616 2.202186 5.931485 2.848045 3.955807 2.896716 6.147798 4.157304 6.216018 5.146218 5.846711 3.303504 5.291888 Wyoming 7.126880 10.105004 9.906165 8.743662 7.897871 8.711600 7.551584 4.333896 6.103995 9.153755 7.491362 10.901153 6.510596 11.108905 8.808637 4.795353 Ohio Oklahoma Oregon Pennsylvania Rhode Island South Carolina South Dakota Tennessee Texas Utah Vermont Virginia Washington West Virginia Wisconsin Alaska Arizona Arkansas California Colorado Connecticut Delaware Florida Georgia Hawaii Idaho Illinois Indiana Iowa Kansas Kentucky Louisiana Maine Maryland Massachusetts Michigan Minnesota Mississippi Missouri Montana Nebraska Nevada New Hampshire New Jersey New Mexico New York North Carolina North Dakota Ohio Oklahoma 3.450637 Oregon 4.221053 4.206247 Pennsylvania 2.405404 3.980486 4.781168 Rhode Island 6.996460 7.656694 5.560826 7.069158 South Carolina 4.078721 3.618968 5.026658 3.454359 7.704205 South Dakota 5.575686 3.772286 4.674182 5.793156 7.443807 5.658435 Tennessee 3.006111 2.665695 5.200718 2.852044 7.860988 2.250386 5.683295 Texas 3.311304 3.957736 3.692048 4.403557 7.459784 4.340904 5.933274 4.070610 Utah 5.820170 6.516477 5.388702 6.975968 8.438162 7.360012 5.741575 7.592907 5.444679 Vermont 7.981582 7.654685 6.981145 7.030258 8.167112 7.041482 5.854341 7.898382 8.341997 7.926106 Virginia 4.993860 5.631299 5.561860 4.104011 7.531600 5.724971 5.947832 5.660883 5.197778 6.452475 7.215077 Washington 4.321383 5.108779 2.237714 4.559541 5.678181 5.464591 5.505772 5.650595 3.492007 5.017698 7.085641 4.083678 West Virginia 4.933222 3.494353 6.332121 5.195085 8.738638 5.204569 4.787221 4.310947 6.850242 8.065666 8.282668 7.472877 7.462814 Wisconsin 2.327254 3.507086 3.276736 2.778869 6.699085 4.264763 4.428909 3.954716 3.628860 4.813705 6.653113 4.176491 3.228251 5.200544 Wyoming 8.644524 7.238965 8.071010 8.835251 9.108380 8.328574 4.973078 8.477997 9.144074 7.732709 7.294573 8.519401 8.512051 7.556514 7.685097 ``` ] --- ## Step 1: Calculate the distance matrix .tiny-font[ ```r US_state_stats %>% column_to_rownames(var = "state") %>% scale() %>% dist(method = "manhattan") ``` ``` Alabama Alaska Arizona Arkansas California Colorado Connecticut Delaware Florida Georgia Hawaii Idaho Illinois Indiana Iowa Kansas Kentucky Louisiana Alaska 39.364617 Arizona 11.457393 34.256754 Arkansas 7.563727 36.979698 12.282109 California 27.092569 35.778165 17.267495 27.473571 Colorado 20.486385 26.830814 12.303813 18.723663 16.577681 Connecticut 27.881993 32.490872 23.810292 29.803273 23.519703 20.333207 Delaware 25.565431 24.114268 22.085864 24.654841 28.860191 18.919099 28.283941 Florida 14.757703 37.487025 10.813581 15.947303 14.771631 15.949390 21.862262 21.536576 Georgia 12.722919 35.663923 6.033199 13.493929 15.433219 12.168010 26.520525 22.328604 9.602667 Hawaii 34.819691 19.783897 27.766370 32.305331 24.016901 21.547728 20.223625 23.485511 28.690091 30.706527 Idaho 19.745449 32.227050 15.412817 15.999750 23.900019 15.496762 26.810291 20.777305 18.505853 17.569117 23.408972 Illinois 19.534776 36.554042 11.861539 21.097936 12.472733 12.717012 17.823019 22.194732 9.976896 11.117761 25.943819 22.116396 Indiana 12.753238 35.519382 12.812347 14.848398 22.206667 13.352968 26.441818 20.677876 12.183197 10.113156 30.437677 16.207176 15.523156 Iowa 16.887848 36.070057 15.710044 15.935214 26.912765 12.974164 25.517710 18.651812 15.708193 15.867194 25.775804 13.555946 19.622490 12.111341 Kansas 16.132276 28.312776 10.999284 15.758134 24.228312 12.783725 26.664013 13.944676 15.248182 12.667386 22.191425 12.943138 16.005273 12.645178 6.528961 Kentucky 8.879573 33.818930 18.687634 11.872833 35.856603 23.981172 37.865451 24.475962 18.640994 17.745640 34.148013 20.581355 27.759876 10.986555 18.385613 19.250124 Louisiana 7.341806 35.810798 10.924207 8.575925 24.205810 18.792486 26.200688 24.897868 14.158552 12.542016 29.682396 17.218745 20.217441 13.946551 17.005256 13.780842 11.500321 Maine 18.893906 31.188934 18.303720 17.954955 26.770330 16.574771 25.198845 19.631394 17.253112 20.260276 23.406068 9.825994 22.515036 16.250223 9.054923 13.491801 20.787123 17.698318 Maryland 25.805656 31.675335 20.806442 29.704523 24.095626 19.793805 15.526273 27.350394 23.930061 22.707238 26.258090 31.258360 19.061221 26.788525 27.085683 21.811227 34.558648 24.019984 Massachusetts 27.610370 32.348280 22.706743 28.991138 20.095868 17.844923 9.836393 25.796140 19.106424 23.719136 17.807773 26.258625 18.528311 23.938428 22.671947 24.176227 35.593178 23.416700 Michigan 10.892408 37.731162 9.333866 12.990808 18.845509 14.659201 23.999365 19.623196 9.285337 6.945999 30.860859 17.966644 11.158765 8.069259 11.947356 12.242973 17.086716 13.017955 Minnesota 19.445605 32.475859 14.420244 20.003122 20.599158 9.133611 19.435055 18.127856 17.212266 14.384316 22.538527 17.623839 12.072838 14.771491 9.745534 11.476417 27.329071 19.707720 Mississippi 9.663576 41.516351 15.960822 8.415174 29.470958 23.447245 32.729832 30.242143 17.246017 17.527327 36.513600 19.940493 24.719654 17.346186 19.728993 22.334799 14.005749 11.439921 Missouri 7.610217 35.417698 9.961130 10.138051 22.859271 14.467240 25.367467 21.763993 10.972625 9.934061 30.042068 17.101701 17.060413 7.803763 11.519870 10.530092 10.061856 8.091016 Montana 13.043659 29.681100 14.858367 13.219795 30.821872 18.277643 34.740921 15.034760 17.402251 16.386288 26.661423 11.731440 23.457669 15.508538 12.219795 10.143727 15.291925 11.828956 Nebraska 18.737310 32.137251 13.935147 15.665617 27.906480 13.838232 25.181656 17.958883 18.467426 15.358309 23.227217 12.914711 18.901255 15.574059 8.283421 7.150246 22.523107 17.457847 Nevada 25.071960 30.052598 16.832981 23.195984 17.162031 17.188030 32.418899 22.901218 19.861440 17.464555 23.546802 23.620484 18.225749 22.043616 24.762569 20.244312 27.873980 22.828637 New Hampshire 25.637810 30.952351 21.461364 24.533148 25.554741 16.771438 17.519240 20.889699 21.502428 23.563682 21.808457 17.975251 17.969381 22.780764 13.282445 19.341475 34.108060 24.302570 New Jersey 29.229522 37.110486 20.867874 30.068037 18.720608 20.180116 10.704776 27.749816 17.831686 20.929297 25.755317 28.711458 13.270208 24.967964 27.643298 27.550045 41.034094 27.931479 New Mexico 12.685293 31.647812 15.669371 11.617146 31.227012 19.638299 31.876644 21.272159 21.565973 17.598827 28.172359 15.988255 25.428154 16.280465 18.731333 14.175000 9.304608 11.454328 New York 26.114656 37.852758 18.181192 26.783061 12.968452 19.077331 20.286075 27.630758 14.139010 18.674509 24.293757 27.347706 12.921183 22.303422 26.024397 24.813468 34.887015 24.230301 North Carolina 10.319309 38.802747 7.446743 11.907696 17.557676 15.648799 25.558962 22.449534 9.897016 4.868379 32.148962 18.056033 10.884715 9.480474 16.099142 12.500428 16.711862 11.406332 North Dakota 26.061847 29.624019 22.675114 24.010357 34.721456 19.589040 31.217967 21.930020 24.312128 24.777472 22.020599 18.003735 28.687232 22.217109 14.389066 14.677949 20.608835 23.732389 Ohio 13.826927 36.220795 11.794776 15.322582 17.309009 12.460479 23.316813 19.716962 7.436731 8.085043 29.193779 18.706337 11.021343 7.764234 12.622645 11.068050 16.217131 13.993672 Oklahoma 8.929367 33.166327 12.491627 7.080033 26.986146 15.193265 28.506319 21.671704 15.195802 13.047307 28.282550 14.175486 20.920568 10.631406 13.021217 12.579024 10.516689 8.034751 Oregon 20.609313 30.077416 11.721054 17.447993 16.885635 10.594128 21.472858 22.143437 14.021381 14.083179 19.588485 9.833415 16.409481 14.170722 12.942133 12.391392 21.916091 16.839859 Pennsylvania 10.622794 35.821054 11.706567 14.798249 19.463068 15.562500 21.086788 20.360240 9.355917 10.615641 29.680309 21.651206 10.029327 10.968759 14.606625 12.341548 18.968970 15.124029 Rhode Island 32.574009 29.404593 26.342309 28.871952 23.870786 21.575907 22.988741 21.949430 22.771182 28.842765 16.554458 22.700590 24.092097 28.247248 25.561136 26.981077 33.852692 29.478300 South Carolina 7.534312 38.926594 9.025351 9.182206 23.984869 19.746607 25.090109 24.506851 13.059331 10.801379 32.799809 17.540498 15.024322 13.249993 18.192465 15.527733 13.546455 7.710250 South Dakota 18.426645 31.431828 18.285225 16.001491 31.778541 18.799516 29.442488 20.436812 20.137226 21.171254 24.509916 10.754237 25.537500 18.932092 12.213913 13.109386 17.378240 16.386444 Tennessee 5.779919 38.584488 9.344479 9.345081 23.533637 18.465746 26.990937 23.338196 12.108336 9.307380 33.710207 19.067733 15.956930 9.262523 16.440284 13.619979 9.678722 8.508175 Texas 16.139287 32.808115 9.153110 16.353664 13.120956 11.597821 26.650051 23.620983 11.708841 6.976128 27.715606 18.155539 12.106739 13.872454 16.821420 14.259404 22.383038 13.597655 Utah 26.260382 32.311477 21.223510 24.655275 25.258503 12.561220 27.417314 23.393697 25.550868 20.756311 24.448452 14.706314 22.678405 18.392145 13.307426 15.572565 28.263004 25.332988 Vermont 28.777699 31.700599 24.958772 27.095827 33.149379 24.024744 27.053885 23.658541 26.765388 29.616863 24.792779 18.914221 25.568846 28.418282 17.566895 19.346604 33.946720 27.623194 Virginia 19.980949 27.270728 14.400683 21.668542 19.218570 13.698095 15.551851 23.509069 17.543482 15.395190 21.843737 23.801562 12.962813 17.677008 17.885015 16.284319 28.253853 18.770132 Washington 22.185323 26.463377 12.501963 20.806337 13.112197 7.064249 16.019791 22.210963 14.974729 14.868716 17.297056 14.635503 12.965558 16.412173 16.550914 14.672008 27.581817 18.355271 West Virginia 13.819041 38.774625 21.780926 11.528138 36.257929 22.314531 35.763154 24.985774 22.396856 21.603417 34.105318 17.104790 29.709254 14.629234 15.803289 21.185233 10.987548 16.239855 Wisconsin 15.455751 32.496025 11.453120 15.346039 19.430383 7.884163 19.637135 17.978207 10.239927 10.439779 23.813409 16.070823 11.643856 9.960649 8.265174 7.044122 19.823646 14.339788 Wyoming 29.906309 27.173530 28.517161 28.801932 40.629967 24.873327 37.049912 19.816190 31.764923 30.393671 27.741781 22.234256 33.339692 27.328553 21.455659 16.974509 23.731847 29.182222 Maine Maryland Massachusetts Michigan Minnesota Mississippi Missouri Montana Nebraska Nevada New Hampshire New Jersey New Mexico New York North Carolina North Dakota Alaska Arizona Arkansas California Colorado Connecticut Delaware Florida Georgia Hawaii Idaho Illinois Indiana Iowa Kansas Kentucky Louisiana Maine Maryland 30.761983 Massachusetts 23.673133 18.172812 Michigan 15.372217 23.593521 22.387806 Minnesota 15.867013 19.750511 17.626432 11.557404 Mississippi 20.157039 33.626080 31.821963 16.215734 24.547154 Missouri 15.135665 23.887302 22.766769 8.863121 15.870148 12.623194 Montana 13.097015 29.363592 30.401189 16.045604 19.392116 17.669828 10.707897 Nebraska 12.969741 26.692684 24.738214 14.884091 12.135819 22.638419 14.889073 12.634011 Nevada 27.260706 26.608474 28.298399 22.327457 23.193390 27.304145 21.705181 24.553351 22.629561 New Hampshire 13.491797 23.730018 20.213063 19.890094 12.076524 28.743098 23.038951 23.515789 14.249492 28.861021 New Jersey 29.334700 18.139708 11.151831 20.932440 19.801818 33.821907 26.755159 35.999363 26.767042 28.573222 20.112728 New Mexico 19.865650 28.943640 30.601543 18.907161 23.012443 15.115886 11.884397 9.623528 16.850745 24.134972 29.028684 35.749268 New York 26.117693 24.806937 15.036843 18.240629 20.031358 30.670356 23.188818 30.670798 26.066276 23.642492 24.274177 16.069327 32.126386 North Carolina 20.007664 24.188508 24.404811 6.196769 15.385799 15.667506 9.426212 15.332328 14.644977 19.506845 22.866755 20.392445 17.183687 19.163133 North Dakota 15.670782 32.391528 28.148771 25.368076 22.295672 28.822664 19.326822 13.637233 13.003647 27.942718 21.223498 36.068846 18.968782 33.570273 25.987557 Ohio 16.858559 22.435097 20.133078 6.549243 13.748388 19.599339 7.675325 14.546511 15.082362 21.733593 21.307301 20.127389 18.065514 16.510663 8.205989 21.902712 Oklahoma 16.148164 27.825917 25.634278 13.205045 17.165680 12.114288 8.812459 8.048941 13.561660 22.310308 22.878414 29.549027 11.049805 25.847849 12.043752 19.444345 Oregon 11.547196 27.348602 18.282478 14.015917 14.019834 20.424974 15.707416 16.046703 12.922771 18.650262 17.055011 21.491776 20.289379 18.359011 15.201312 18.783142 Pennsylvania 15.790440 20.929248 20.607652 7.511174 13.816418 18.161597 9.604403 18.815490 16.035231 22.514320 19.591486 19.571947 21.482173 17.812933 9.400328 25.309376 Rhode Island 20.023439 32.257531 17.739055 27.134583 26.205180 31.655257 28.125811 27.794018 23.864827 21.130403 25.102546 22.101111 30.014082 21.496894 28.779428 24.377074 South Carolina 19.983025 26.618771 27.081046 10.496299 19.884757 11.387501 9.407850 13.529259 17.005252 22.188035 22.116064 25.021297 13.283755 23.666508 8.224572 26.588417 South Dakota 11.338985 31.209689 26.741990 19.973828 19.381215 20.485043 14.796577 6.452830 10.406530 27.268253 18.940603 33.924858 15.114362 30.753969 20.933066 9.063993 Tennessee 19.996080 24.459514 26.681235 9.295897 18.274519 13.412056 7.191153 12.720936 16.484459 22.621774 24.497506 26.197992 14.591292 22.555724 7.536444 25.273351 Texas 19.401428 23.217318 20.545190 11.496219 15.212010 20.639393 13.346891 17.611304 18.099673 18.387170 23.076380 21.673437 19.417936 17.217764 10.607533 25.243526 Utah 17.847726 28.262351 25.874627 21.554323 12.975953 28.642052 20.604086 20.643193 14.897366 25.459721 19.026230 29.940862 22.102110 30.982791 23.901114 20.562986 Vermont 15.174179 33.390229 29.301329 25.321936 20.217056 31.553956 26.101607 22.970440 16.182841 33.251526 14.279325 31.772850 29.596370 32.123540 28.867406 17.051873 Virginia 21.417589 12.309755 15.702151 15.089570 10.392017 25.490009 16.915196 23.814041 17.765089 25.070717 16.138210 18.442732 22.272700 19.651659 15.921588 25.146148 Washington 16.160035 20.368978 13.191440 15.954762 10.824922 23.867231 16.921238 21.710647 17.286058 19.080272 16.709648 16.702402 23.386570 14.796973 17.306217 22.708232 West Virginia 15.695032 36.503979 33.703987 18.633802 23.585395 14.556596 13.858266 16.615720 19.101353 30.742534 26.926197 38.404246 14.460035 34.453129 20.818540 21.703957 Wisconsin 13.658604 21.062285 16.898714 9.542290 7.794500 19.606658 10.214511 14.078979 9.638931 20.772246 14.426283 19.631344 18.809586 18.189315 11.128356 18.298274 Wyoming 23.358541 37.018216 34.966270 31.018062 24.654408 30.299377 24.946730 14.636990 19.824625 32.127388 26.132282 40.851635 19.980524 39.945219 31.444499 16.295284 Ohio Oklahoma Oregon Pennsylvania Rhode Island South Carolina South Dakota Tennessee Texas Utah Vermont Virginia Washington West Virginia Wisconsin Alaska Arizona Arkansas California Colorado Connecticut Delaware Florida Georgia Hawaii Idaho Illinois Indiana Iowa Kansas Kentucky Louisiana Maine Maryland Massachusetts Michigan Minnesota Mississippi Missouri Montana Nebraska Nevada New Hampshire New Jersey New Mexico New York North Carolina North Dakota Ohio Oklahoma 13.192733 Oregon 13.804684 14.460831 Pennsylvania 7.703123 15.367328 17.442336 Rhode Island 26.637750 27.820854 17.992930 26.225444 South Carolina 14.102342 9.890319 17.500843 13.029383 29.416749 South Dakota 19.629029 12.427434 15.363569 21.531977 23.655525 17.862611 Tennessee 10.562274 8.320692 18.777025 10.487609 31.254816 6.561510 18.955424 Texas 9.617867 14.888464 12.572941 12.845620 28.050352 15.391440 21.492927 14.036198 Utah 19.457312 19.992631 16.236775 23.546790 29.039621 26.397215 18.648339 25.608327 19.685801 Vermont 26.663653 26.033323 20.144664 23.134844 24.666394 25.446111 16.016361 28.546519 28.645105 23.935294 Virginia 15.422349 19.279284 19.893824 12.800078 28.712510 19.784178 22.391629 19.051815 15.704802 21.262725 25.094335 Washington 15.130598 17.846874 8.440218 16.310862 18.038310 19.784162 20.709173 19.883098 12.571810 16.828013 22.159450 13.641454 West Virginia 19.313608 11.168484 21.990419 20.567401 31.415898 18.177472 17.246612 14.878690 23.865663 22.410871 29.028447 28.749971 26.168082 Wisconsin 7.971613 12.889222 11.039865 9.867973 23.284892 15.468640 16.634515 13.320727 11.235804 15.952210 20.450072 12.313105 10.541120 19.852494 Wyoming 29.363848 24.440022 27.382921 30.496770 31.239854 29.665212 16.379129 30.899715 32.785761 24.033473 21.141481 28.721895 28.481907 26.311153 24.315956 ``` ] --- ## Step 1: Calculate the distance matrix .tiny-font[ ```r US_state_stats %>% column_to_rownames(var = "state") %>% scale() %>% dist(method = "maximum") ``` ``` Alabama Alaska Arizona Arkansas California Colorado Connecticut Delaware Florida Georgia Hawaii Idaho Illinois Indiana Iowa Kansas Kentucky Louisiana Alaska 5.0230030 Arizona 1.8008087 5.0313330 Arkansas 0.9729239 4.3482712 1.8791047 California 3.3092126 5.3811939 2.4154837 2.7012130 Colorado 2.7795090 5.0146730 1.5829457 2.8578051 2.4637933 Connecticut 3.6364712 5.1312933 2.9255781 4.0873715 2.8502707 2.6410745 Delaware 3.5402592 3.6328111 3.5485892 2.8655274 3.9130835 3.5319291 3.6485494 Florida 1.7813269 5.2395836 1.6050686 1.8181819 2.9710449 2.5837690 2.6959090 3.7568397 Georgia 2.4271769 5.2312535 0.8018184 2.5054729 2.3671740 1.3229045 3.0695498 3.7485097 2.2314368 Hawaii 3.6822218 4.5481918 3.0120101 3.4456765 4.0055004 3.2800948 3.5008704 3.4541417 3.9739610 3.7531854 Idaho 2.4160750 4.4149114 2.1887403 1.8399567 3.2850578 2.3458157 3.7593481 2.9321676 2.6237046 2.3696855 3.1381676 Illinois 2.2162604 5.2395836 1.5110566 2.1867323 2.8502707 2.5343795 2.9061736 3.7568397 1.8440448 1.5356266 3.9660762 2.8377070 Indiana 1.5564577 5.2229235 2.1840278 1.5241176 3.8299617 2.3422906 3.4184782 3.7401797 2.1801508 1.6142814 3.7374157 3.0913080 2.5343795 Iowa 2.8742961 4.8480725 2.5827009 2.2494491 3.8164642 2.5446210 3.0165492 3.3653287 2.2705136 2.1922888 3.6743370 2.5413004 2.4845160 2.1840278 Kansas 2.2156803 2.9542017 1.4290311 1.8877119 2.8985804 1.6833646 2.8657218 1.6313258 1.8137702 1.6488820 2.6781328 2.4617812 1.9164620 3.0069948 0.9178838 Kentucky 1.5414323 3.6016173 2.6271638 1.6459340 4.2730978 2.7854267 3.8616143 2.5322061 2.2473329 2.2705848 3.6716989 3.4184782 2.7854267 1.6165849 2.6271638 3.4501308 Louisiana 1.0503158 4.6564820 0.9997552 1.3330069 2.6087224 2.5827009 3.6074659 3.1737382 2.0411668 1.5659206 3.6033734 3.1242349 1.8673220 2.2961162 3.5824560 1.9214210 2.1934040 Maine 2.8326397 5.0729831 2.5410444 2.2077927 3.7923094 3.3275812 3.0033650 3.5902393 2.5227929 2.9752491 3.5481795 2.3097329 2.7367953 3.0648016 2.5147940 2.4352748 2.9620893 3.5407996 Maryland 3.4064270 4.8564026 3.1247996 3.7419898 3.5670360 3.4444384 2.9159526 3.5653120 3.2288227 3.5686674 3.3325172 4.1425999 3.7366825 3.3075913 3.7894089 2.8867025 3.4666471 3.2437733 Massachusetts 3.2186734 5.0646531 2.9995576 3.3466821 2.2991351 3.0292250 1.9575245 3.5819093 2.0301622 2.6044228 3.8004944 3.2923835 2.3299369 2.9120370 3.0061926 3.0814208 3.3551731 2.9992655 Michigan 2.0809162 5.1479533 1.6425289 1.8271459 4.0580126 2.3488809 3.0692861 3.6652095 1.4742016 1.9965487 3.7768399 1.9695183 1.8427520 1.6459340 1.9995103 1.4839938 2.0890700 1.6410477 Minnesota 2.4993879 5.2062635 2.2077927 2.4944764 4.0580126 1.5942192 2.9786360 3.7235196 2.0336212 1.8687330 3.7453005 2.0277390 1.8687330 2.3106381 1.2918845 1.1594322 2.7537741 3.2075478 Mississippi 1.3820748 5.0396630 2.1993423 1.5564577 3.3497676 3.2820955 4.4293701 3.5569192 2.4944764 2.3880289 3.9102604 2.5618947 3.5189477 2.8591450 3.2360775 2.9664044 1.7958081 1.7425559 Missouri 1.0449834 4.9480328 1.5141777 1.3483656 3.0435094 2.3880289 3.1776604 3.4652890 1.8852672 2.0356968 3.5718340 2.7964244 1.9858944 1.2344505 2.3327621 1.7725443 1.6775865 2.0012326 Montana 1.7725443 3.0926932 1.6575024 1.3610608 2.8019611 2.2670144 3.4118121 1.7763919 1.8942885 1.7616607 2.8132680 2.0840651 2.5343795 2.5638587 1.7933099 1.7594739 3.0069948 1.4996450 Nebraska 2.3744185 4.5815119 2.5832485 2.0899667 3.7924287 2.0885839 3.0444998 3.0987680 2.9679877 2.8030995 2.7912345 2.1985094 2.8030995 2.4183603 1.1726234 1.1542174 2.5832485 3.0825784 Nevada 2.7833497 4.3399412 2.1131380 2.5832485 3.1066282 2.4733230 3.4354174 3.2609030 3.0750889 2.8543133 3.2428013 3.0638885 3.0672040 2.8385436 3.8473914 3.4076895 2.5749199 2.9130249 New Hampshire 3.1349501 4.9980129 2.5410444 3.4383324 3.6715352 2.6667725 2.7090692 3.5152691 2.4705972 2.6630221 3.5402946 2.6667725 2.6845996 2.6667725 2.1750272 1.9217134 3.3372050 3.5407996 New Jersey 4.2173266 5.2812337 4.3632746 4.3641692 3.6628521 4.3929420 2.6596375 3.7984898 3.2359428 3.9336543 3.8714580 4.5061457 3.6936539 3.8856315 4.3699096 4.4451378 4.1578889 4.1769078 New Mexico 1.7576345 3.2653684 1.7659646 1.7156437 2.9468901 2.0899667 3.6412175 2.4944764 1.9742151 2.0509729 2.9664044 2.7931111 2.5343795 1.9575551 2.5410444 2.0225485 2.2789855 1.9979193 New York 4.3120396 5.3562039 3.6732189 4.3488946 2.4447176 3.0712533 3.3816771 4.4444900 3.5024513 3.6977889 4.0764640 4.3857497 3.3816771 3.9372384 4.3478706 4.0786243 4.0417693 4.0294843 North Carolina 1.2857694 5.2062635 0.8752176 1.2918845 2.5845675 1.7211085 3.1721230 3.7235196 1.6093367 1.2135885 3.8872277 2.2078787 1.9778871 1.7211085 2.5282858 2.0885839 1.8991546 1.5939193 North Dakota 2.9576090 4.6481520 3.0779131 2.4733230 4.2870933 2.6046466 2.8931450 3.1654082 3.4626523 3.2977641 2.6705190 3.1111878 3.2977641 2.9130249 2.1310745 2.1705389 3.0779131 3.6657690 Ohio 2.0639982 5.3562039 1.5771303 1.8102279 2.8527505 2.1531408 3.0750872 3.8734600 1.9482198 1.8008087 3.9108823 2.8593771 1.9480678 1.5826288 2.1609026 1.4243659 2.0257649 2.0641853 Oklahoma 1.5414323 4.6314920 1.4876711 1.2293632 3.1336051 2.0356968 3.6074659 3.1487481 1.8687330 1.7038448 3.3352887 2.2789855 2.5343795 1.3650792 1.6662586 2.3106381 1.4839938 1.9161974 Oregon 2.4993879 5.0646531 2.2077927 1.8745409 1.9578539 2.1437045 2.7961091 3.5819093 1.7624753 1.8819539 3.0672040 1.7391482 2.5343795 2.8891968 2.3391891 2.2596699 3.2285628 3.2075478 Pennsylvania 1.8609820 5.2978937 1.9574007 1.8877119 3.2850578 2.9361011 2.5645942 3.8151499 1.2482765 2.5837690 4.0055004 2.3662058 1.6442166 1.9386112 2.0130147 1.8675020 1.9386112 1.9214210 Rhode Island 3.5380808 4.2982911 3.6840288 3.6849234 2.9836063 3.7136962 2.4208998 2.8155473 2.8779678 3.2544085 3.0883497 3.8268999 3.0144081 3.2063857 3.6906638 3.7658920 3.4786431 3.4976620 South Carolina 1.1726234 4.4149114 1.2388198 1.4846925 3.0193546 2.7140557 3.5154401 2.9321676 2.0237210 1.6643686 3.5008704 2.7140557 2.1993423 2.7140557 2.5410444 1.7116518 2.7140557 1.9669811 South Dakota 2.4577315 4.3482712 2.4183603 1.8942885 3.6275405 2.2889319 3.3395626 2.8655274 2.8030995 2.6382113 3.1066282 2.0336212 2.8377070 2.3739432 1.8153598 1.8548241 2.8170793 3.1658914 Tennessee 0.6444247 4.7481123 2.0257649 1.3257712 3.6716989 2.1840278 3.4418721 3.2653684 1.6459340 1.8008087 3.6349127 2.8170793 2.1840278 1.5508890 2.4577315 2.8487319 1.5508890 1.3294082 Texas 2.7795090 5.3478738 1.0213171 2.8578051 2.0885839 1.5506205 3.1391625 3.8651300 2.5837690 1.0992547 4.0291549 2.4010029 2.0804608 2.3106381 2.5446210 1.6833646 2.7537741 1.9182527 Utah 4.2097927 5.0230030 2.8170793 3.7983092 3.3333675 2.6588164 3.5956001 3.5402592 3.3275812 3.3868257 3.1775918 3.0913080 2.6588164 5.0011071 3.2884332 2.4271769 5.4442431 3.5134360 Vermont 3.2755405 4.7756032 3.4333978 3.5462400 4.1136384 4.7756032 3.1571474 4.7756032 4.0852685 3.7259162 4.7756032 4.7756032 3.6307196 4.7756032 4.2838579 3.7731994 4.7756032 4.0285286 Virginia 2.4156125 5.2978937 2.8179051 2.9264437 3.2601415 3.1375439 2.0509251 3.8151499 2.9219282 3.2617730 3.9503065 3.2883654 3.4297881 3.0006969 2.9763605 2.5798081 2.5638587 2.6293130 Washington 2.3854405 5.1979334 1.7495715 2.2305061 1.7874579 2.0343656 2.4393689 3.7151896 1.7225126 1.7726150 3.2327857 1.9024828 2.1466573 3.0702999 2.2298503 2.1503311 3.5134360 2.7493267 West Virginia 1.6841817 5.1283899 3.1709892 1.6666837 4.1546319 4.1496896 4.0984462 3.0487879 2.2928029 3.7973574 3.6956900 3.2039602 2.8578051 2.3488809 1.9624597 2.7854267 1.5267726 2.4087684 Wisconsin 1.9578539 5.2562436 1.6662586 2.1573850 2.9227352 1.9965487 2.6766601 3.7734998 1.5171555 1.6442166 3.8399187 2.2033439 1.7311579 2.0574175 0.9164422 0.9950861 2.5005535 2.6660138 Wyoming 4.1475733 3.4005216 4.2422668 3.7519651 4.6768045 4.1580948 4.1938679 2.5884568 4.5821110 4.4316537 3.3666856 3.4342159 4.5526509 4.3390645 4.0244718 3.4333978 3.7096482 4.1517819 Maine Maryland Massachusetts Michigan Minnesota Mississippi Missouri Montana Nebraska Nevada New Hampshire New Jersey New Mexico New York North Carolina North Dakota Alaska Arizona Arkansas California Colorado Connecticut Delaware Florida Georgia Hawaii Idaho Illinois Indiana Iowa Kansas Kentucky Louisiana Maine Maryland 4.0416882 Massachusetts 3.0501224 2.9992655 Michigan 1.9578539 3.4044967 2.9115481 Minnesota 2.0748448 3.8091449 2.9599580 1.7038448 Mississippi 3.1636694 4.2473518 4.1618324 3.3836036 3.5731689 Missouri 2.7699180 2.9078528 2.8813937 1.6241297 1.9578539 2.4270582 Montana 2.1485883 3.1928546 3.1890410 2.3685225 2.5207847 2.2585124 1.6180381 Nebraska 2.1370779 3.4379578 3.1240256 2.6382113 1.3526709 3.1686593 1.8687330 1.7425559 Nevada 3.1401288 4.2659263 3.1208605 3.4058320 3.6275405 3.1349501 2.6931740 3.3527268 4.5619070 New Hampshire 2.6667725 3.9894925 2.6518749 2.3596399 1.4876713 4.5170249 2.2911056 2.6667725 1.6617153 3.9023541 New Jersey 4.4138394 3.4901464 1.6832819 3.9092571 4.3236750 4.3365924 4.2451107 4.5527580 4.4877426 4.4845775 4.0155919 New Mexico 2.7666047 3.3034958 3.2678135 2.0809162 2.6293130 1.7742946 1.6826644 1.3234250 2.2248033 2.8030995 3.5731689 4.5137749 New York 4.3237158 3.3333675 2.1256256 4.5894190 4.5894190 4.5700249 3.8083540 4.2137595 3.8452091 3.1775918 4.2029416 3.0038970 4.3611796 North Carolina 2.2077927 3.5133130 3.0466832 1.4734450 2.3084348 2.5884568 1.3740684 2.0336212 3.2428013 2.9883556 2.5956039 3.8276012 1.9956185 4.1400494 North Dakota 3.0846814 4.3374639 3.1777438 3.1328759 2.6046466 3.5176150 2.6835753 1.7038448 1.8867862 5.0565716 2.6667725 4.5414608 2.6243573 3.0499696 3.7374659 Ohio 2.8328707 3.4608698 2.3343978 1.2077418 1.2370384 3.3666856 1.6072117 2.3516045 1.9786584 3.0120101 2.3609862 3.4974257 2.0908354 3.3906636 1.2641429 2.8808970 Oklahoma 1.6997224 3.2993427 3.2432434 1.6918018 1.8877119 1.9236957 1.0701956 1.8675020 1.6643686 3.4626523 2.8315678 4.3693346 1.4474456 4.3366096 2.1435466 2.4073249 Oregon 2.2463998 3.6201592 3.0622123 2.5121030 2.5121030 3.1636694 2.5943132 2.1485883 2.6382113 2.6243573 2.6667725 4.4259293 2.5909999 3.3538086 1.8745409 3.1328759 Pennsylvania 2.2652940 2.5888574 2.5798527 0.8156393 1.6833646 3.1636694 1.4041955 2.1485883 1.9786584 3.1066282 2.2130983 3.4914475 2.0325253 3.8164642 1.3701805 2.8808970 Rhode Island 3.7345936 3.2117763 2.7045012 3.2300113 3.6444292 3.9080622 3.5658649 3.8735122 3.8473914 3.8053317 3.3363461 2.7754648 3.8345290 2.9804707 3.1483554 4.3420560 South Carolina 2.7140557 3.1848650 2.9852582 2.0639982 2.2162604 2.0048077 2.1655706 2.7140557 2.8030995 2.7140557 2.8989861 3.9894733 2.7140557 4.0786243 1.2688514 3.2977641 South Dakota 1.7087198 4.1425999 3.1739443 2.5257179 2.2889319 2.8580622 2.3678606 1.0442920 1.6643686 4.3970188 2.6667725 4.5376613 2.1244797 3.9434892 3.0779131 1.6168899 Tennessee 2.4862559 3.2593947 2.8992631 1.4876711 2.1523752 1.9286541 1.0761876 2.4055958 2.1985094 2.7360406 2.9326953 3.9893437 1.6775865 3.9926292 1.2977556 2.7820316 Texas 3.3275812 3.6961715 2.8464051 2.3488809 2.2705547 2.9606532 2.3880289 2.1139928 1.7616607 3.1302828 3.0338227 4.2101221 2.1784770 3.2555285 1.5659206 2.9992901 Utah 4.0713935 3.9366101 3.0862485 3.3551731 2.6904690 3.8616143 3.7666566 2.8578051 3.1019525 3.6716989 2.8487319 4.4499655 3.1652576 3.8647739 3.5450886 2.9752491 Vermont 4.7756032 4.2626501 4.3122278 3.5517909 3.5084134 4.0663552 4.2271181 4.7756032 3.1112346 4.7756032 2.1088307 4.3187628 4.7756032 3.9130835 3.5123265 4.7756032 Virginia 2.3488809 2.7402750 2.4938576 3.0976022 3.5022505 3.6742964 2.1727685 2.5342250 3.1184066 3.9590319 3.2326836 3.8027927 2.7304404 3.5872238 3.2064185 2.7625040 Washington 2.4663249 2.9357244 2.8275194 2.2705547 2.2705547 3.6881279 2.4849743 2.6730468 2.3084348 2.3339136 2.2790502 4.1912365 2.4816610 3.0343982 1.6632761 2.8030995 West Virginia 3.1774538 3.8477624 3.6363639 1.8008087 2.8969531 2.0111725 1.7616607 2.3422906 2.3880289 3.6799134 3.2360775 4.2835532 2.8186571 4.7297300 2.5837690 2.2534721 Wisconsin 2.1768375 3.4033435 2.8128920 1.1352773 1.1352773 3.2360775 1.4163198 1.8779000 1.5939193 2.9679877 1.7400218 4.1766090 2.2922216 3.4541417 1.6488820 2.6835753 Wyoming 3.2784978 4.3127608 4.4853133 4.5263471 4.2885613 3.9918552 4.2475275 2.3283962 3.2469333 3.8908488 3.6299157 4.5565611 3.3816085 4.6452400 4.4495402 2.3831186 Ohio Oklahoma Oregon Pennsylvania Rhode Island South Carolina South Dakota Tennessee Texas Utah Vermont Virginia Washington West Virginia Wisconsin Alaska Arizona Arkansas California Colorado Connecticut Delaware Florida Georgia Hawaii Idaho Illinois Indiana Iowa Kansas Kentucky Louisiana Maine Maryland Massachusetts Michigan Minnesota Mississippi Missouri Montana Nebraska Nevada New Hampshire New Jersey New Mexico New York North Carolina North Dakota Ohio Oklahoma 1.6748838 Oregon 2.6572658 2.0890700 Pennsylvania 1.3522995 1.9386112 1.9386112 Rhode Island 2.8891968 3.6900888 3.7466835 2.9095072 South Carolina 2.1277440 2.7140557 2.7140557 1.8440640 3.3102274 South Dakota 2.5651823 2.0916102 2.4733230 2.5651823 3.8584155 2.7140557 Tennessee 1.4380315 1.5508890 2.6271638 1.3820748 3.3100979 1.3294082 2.5871676 Texas 2.1531408 2.0356968 1.7616607 2.9361011 3.5308763 2.2601370 2.6835753 2.1531408 Utah 3.4184782 4.3047504 2.8891968 3.8616143 3.7707197 3.5134360 3.0143971 4.8428442 2.6904690 Vermont 4.1892915 4.7756032 4.7756032 3.5912552 4.7756032 3.3150048 4.7756032 3.3939335 4.3216845 4.7756032 Virginia 3.1539754 2.6582456 3.3132648 2.2819629 3.1235469 2.5409257 2.4467893 2.5979446 3.3892770 3.6297157 3.4728622 Washington 2.5479270 2.3739432 0.9520729 2.0748448 3.5119906 2.3685225 2.4467893 2.9120370 1.5843296 2.7798579 4.3878810 2.6288299 West Virginia 2.0617307 2.1139928 3.0018489 1.9386112 3.6043074 2.7140557 2.1523752 2.1348047 4.1496896 4.8935018 4.7756032 3.1709892 3.2884332 Wisconsin 1.4075445 1.5506205 2.0012326 1.2596566 3.4973632 1.7873050 2.3678606 1.8991546 1.9965487 2.9436896 3.8488525 3.0964490 1.8918938 2.1531408 Wyoming 4.5084606 3.9886987 3.8761187 4.5873718 3.8773153 4.0581406 2.9944175 4.2612054 4.6063105 3.6193943 4.7756032 4.4632181 4.2380582 3.7119834 4.3622118 ``` ] --- ## Step 1: Calculate the distance matrix .tiny-font[ ```r US_state_stats %>% column_to_rownames(var = "state") %>% scale() %>% dist(method = "euclidean") ``` ``` Alabama Alaska Arizona Arkansas California Colorado Connecticut Delaware Florida Georgia Hawaii Idaho Illinois Indiana Iowa Kansas Kentucky Louisiana Alaska 10.598996 Arizona 3.330008 9.367964 Arkansas 2.101623 10.258034 3.460202 California 7.387598 10.417133 4.574415 7.213409 Colorado 5.853785 8.784506 3.584944 5.667544 4.905273 Connecticut 8.163739 9.632134 6.647566 8.581965 6.413761 6.054845 Delaware 6.852026 6.873437 6.099149 6.443272 8.224788 5.836395 7.471686 Florida 3.817288 10.483883 3.058667 4.111477 5.074512 4.373974 6.059186 6.305386 Georgia 3.718662 9.954441 1.731379 3.841420 4.757818 3.278357 7.153667 6.451118 3.147390 Hawaii 8.937729 6.700074 7.068674 8.508732 7.035466 6.118096 6.293809 6.878239 7.514385 7.792204 Idaho 5.597221 9.004979 4.419521 4.548114 6.607359 4.469112 7.676599 5.801041 5.045858 4.799015 6.969588 Illinois 5.254690 10.316944 3.192744 5.609923 4.396773 3.944139 5.238648 6.674979 3.160311 3.311714 7.447009 6.015498 Indiana 3.604745 10.172037 3.853444 3.882761 7.044651 4.062005 7.864817 6.213848 3.787589 3.140930 8.375538 5.298522 4.811957 Iowa 5.007138 9.958779 4.840562 4.587101 7.634487 4.002106 6.966781 5.674885 4.667862 4.859556 7.354684 3.935105 5.395395 3.885737 Kansas 4.408601 7.409098 3.060602 4.442553 6.598661 3.512435 7.294319 3.735978 4.258347 3.615391 6.183604 3.905887 4.316269 4.282286 2.020848 Kentucky 2.896963 8.930716 5.257548 3.631240 9.364228 6.754618 10.274920 6.475033 5.154735 4.953492 9.129675 6.531273 7.275413 3.384898 5.400969 5.740308 Louisiana 2.187868 9.890864 2.803839 2.411572 6.355365 5.410132 7.975901 6.748584 3.939756 3.518586 8.236106 4.961559 5.256562 4.220968 5.384255 4.020284 3.898974 Maine 5.449459 9.500632 5.356080 4.816331 7.595522 5.251686 6.878787 5.995987 4.764959 5.823169 6.944840 3.188682 6.143050 5.141585 3.249026 4.222520 5.968985 5.401479 Maryland 7.611163 9.273961 6.327444 8.668010 6.722458 6.408493 4.868335 7.423132 6.638708 6.740797 7.515811 8.851332 5.771450 7.475384 8.018422 6.383880 9.115635 7.319302 Massachusetts 7.963086 9.781969 6.354021 8.072120 5.455518 5.195148 3.060985 7.225595 5.278116 6.487500 5.717072 7.309502 5.055135 7.032772 6.492189 6.865981 9.469819 7.491380 Michigan 3.219770 10.438840 2.876205 3.585488 6.172830 4.156116 6.718661 6.002538 2.794426 2.920050 8.228620 4.716282 3.390219 2.702043 3.778241 3.360651 4.829221 3.688944 Minnesota 5.699186 9.792369 4.346035 5.729161 6.486491 2.933653 5.706504 5.782694 4.478219 4.323841 7.140860 4.643832 3.857913 4.315508 2.716098 2.943799 7.133466 5.836457 Mississippi 2.883395 11.407078 4.665463 2.644596 8.218904 7.060859 9.761691 7.962489 5.091162 4.947830 9.825048 5.547383 7.061814 5.136325 6.216101 6.501396 3.987288 3.266231 Missouri 2.193270 9.917924 2.932619 2.820666 6.655760 4.228391 7.353772 5.963989 3.232413 2.995409 7.834503 5.026190 4.551890 2.291596 3.673264 2.990748 3.026295 2.921983 Montana 3.778946 8.061485 3.997077 3.462787 7.773931 5.249206 9.107011 4.420878 4.673883 4.419328 7.284882 3.631654 6.333142 4.746282 3.332278 3.110260 4.808803 3.374796 Nebraska 5.010139 8.818975 4.374697 4.393229 7.323869 4.062795 6.927510 5.277823 5.158065 4.691573 6.423243 4.068942 5.195917 4.537399 2.470144 2.287313 6.083651 5.127234 Nevada 6.867679 8.696989 4.599803 6.614218 5.019683 5.218190 8.358046 6.695426 5.604532 4.833401 6.778081 6.817587 5.516908 6.030305 7.702788 6.141385 7.205939 6.315798 New Hampshire 7.032307 9.342326 5.896818 6.871723 7.463344 5.057318 5.090292 6.242540 5.863305 6.372527 6.696511 5.337879 5.194277 6.441029 4.299915 4.980274 8.770819 7.124445 New Jersey 8.480882 10.991190 6.720639 8.755648 5.736034 6.305088 3.633544 7.650389 5.777136 6.795657 7.618642 8.219887 4.804280 7.707355 7.699712 8.015335 10.731633 8.247333 New Mexico 3.680324 8.522460 4.141634 3.438805 7.732973 5.405001 8.705953 5.658454 5.487654 4.472108 7.876981 5.083431 6.601013 4.562994 5.203417 4.088725 3.360258 3.383514 New York 7.740170 10.726266 5.889495 7.622262 4.069967 5.830359 5.910359 8.683771 5.156224 5.945851 6.716974 7.941413 4.649067 7.279079 7.735557 7.558194 9.310169 7.104194 North Carolina 2.964659 10.446575 2.011604 3.162292 5.346357 4.219042 7.172742 6.558336 2.854521 1.673388 8.316637 4.971980 3.299835 3.014498 4.783395 3.728261 4.552803 3.294185 North Dakota 6.830398 8.494124 6.726308 6.215588 9.102334 5.825150 8.078210 6.274746 7.143179 7.022232 6.067419 5.591146 7.800522 6.406683 4.333533 4.269167 6.384704 6.832059 Ohio 3.756038 10.503530 3.207573 3.964614 5.829037 3.634401 6.763715 6.086862 2.761796 2.645838 7.765809 5.294823 3.414807 2.337553 3.693684 2.922134 4.539927 4.122743 Oklahoma 2.746771 9.590296 3.485447 2.160842 7.093379 4.597272 8.206370 6.013341 4.026087 3.514846 7.809859 4.144994 5.635490 2.949381 3.685416 3.896042 3.236769 2.712148 Oregon 5.548912 9.264492 3.668999 4.628232 4.605672 3.465647 6.494161 6.242370 3.774078 4.028477 5.799009 3.010644 4.661539 4.918500 4.211692 3.995912 6.544470 4.859990 Pennsylvania 3.278011 10.332521 3.342653 4.088292 6.229582 4.611578 5.783043 6.249101 2.736122 3.577485 7.875235 5.678008 3.172094 3.276966 4.092980 3.516435 4.990771 4.012989 Rhode Island 8.299825 8.727605 6.894639 7.765106 6.657274 6.679551 6.042031 6.508415 6.088086 7.287467 5.160020 6.871825 6.752070 7.613243 7.566176 7.738592 8.938709 7.772664 South Carolina 2.215120 10.099229 2.490228 2.549883 6.425645 5.464897 7.510363 6.526626 3.601202 3.138396 8.511209 5.060045 4.305294 4.303923 5.276383 4.459678 4.725989 2.765308 South Dakota 5.342727 8.707087 5.423795 4.524657 8.151475 5.319894 7.848930 5.668781 5.828860 5.846903 6.674934 3.605722 6.993915 5.527714 3.431093 3.665688 5.645236 5.154360 Tennessee 1.592758 10.217475 2.929816 2.455450 6.825626 5.211316 7.910098 6.451256 3.357974 2.869454 8.663065 5.604460 4.496692 2.836403 4.945482 4.346147 2.903620 2.663005 Texas 4.732795 10.096352 2.507271 4.552391 4.110426 3.320189 7.219854 7.019202 3.748023 2.073875 7.473824 4.796109 3.747254 4.304938 5.192502 3.882935 6.240198 3.758654 Utah 7.816158 9.971541 5.886282 7.294373 7.035596 4.025850 7.845243 6.827859 6.908662 5.766022 7.201289 4.877627 6.305243 6.466673 4.757319 4.528537 9.209289 7.334168 Vermont 7.853224 9.342110 7.331872 7.357606 9.369040 7.662961 7.198676 7.277386 7.861043 8.097009 7.691579 6.377467 7.270864 8.433320 6.365156 6.540232 9.630963 7.996153 Virginia 5.753364 8.798401 4.670117 6.490933 6.171084 4.301018 4.253637 6.930069 5.205486 5.131602 6.528350 6.267057 4.758796 5.509324 5.179441 4.622325 7.411418 5.481934 Washington 6.100880 8.812749 3.607467 5.795693 3.680065 2.581905 4.948654 6.302339 3.986082 4.020214 5.250266 4.144214 3.791803 5.258658 4.655437 4.116854 7.604610 5.325361 West Virginia 3.873600 10.991010 5.946337 3.513818 9.654607 6.783430 9.605647 6.816279 5.733335 5.935861 9.565350 5.812334 7.473856 4.004706 4.372867 5.827139 3.110384 4.962556 Wisconsin 4.364111 9.825504 3.273309 4.235680 5.727754 2.696602 5.899727 5.849888 3.071053 3.246268 6.758584 4.164685 3.394139 3.129319 2.282848 2.033971 5.447424 4.520653 Wyoming 8.354493 7.492609 8.226198 7.759220 10.664883 7.652842 9.921622 5.697209 8.976007 8.516690 7.911546 6.819268 9.534238 8.124581 6.861121 5.593932 7.161100 8.439725 Maine Maryland Massachusetts Michigan Minnesota Mississippi Missouri Montana Nebraska Nevada New Hampshire New Jersey New Mexico New York North Carolina North Dakota Alaska Arizona Arkansas California Colorado Connecticut Delaware Florida Georgia Hawaii Idaho Illinois Indiana Iowa Kansas Kentucky Louisiana Maine Maryland 8.589605 Massachusetts 6.619401 5.444320 Michigan 4.510982 6.891263 6.464100 Minnesota 4.485471 6.695074 5.548868 3.581946 Mississippi 6.008202 9.723329 9.418208 4.948954 7.356641 Missouri 4.892887 6.787230 6.796929 2.684878 4.399936 4.127669 Montana 4.160841 8.207678 8.403983 4.667686 5.307321 4.753047 3.053530 Nebraska 4.054463 7.708930 6.601193 4.418092 3.198633 6.438225 4.032172 3.688675 Nevada 7.824494 7.645061 7.495758 6.191369 7.070806 7.638030 6.029728 6.659527 7.046413 New Hampshire 4.561159 7.429595 5.714043 5.619372 3.356284 8.495515 6.212605 6.589170 3.879067 8.137473 New Jersey 7.951568 5.737290 3.395446 6.730956 6.213064 10.001374 7.660068 9.696761 7.648360 7.682244 6.126948 New Mexico 5.873785 7.862505 8.285417 4.873973 6.239279 4.073579 3.355320 2.922670 4.978582 6.621719 7.716956 9.420024 New York 7.723006 7.091664 4.143738 6.772569 6.857682 9.003420 6.987451 8.708403 7.313364 6.295759 7.330703 5.408281 8.515242 North Carolina 5.487191 7.071247 6.794340 2.102464 4.565711 4.446084 2.746438 4.567294 4.785123 5.333058 6.327946 6.862369 4.664005 6.268931 North Dakota 5.008931 8.852050 7.533362 6.931520 5.893309 7.612123 5.711863 3.859775 3.748656 8.534916 5.974491 9.465422 5.503519 8.669587 7.282022 Ohio 5.087764 6.627267 5.791114 2.109893 3.650569 5.593289 2.440079 4.327799 4.194522 5.893166 5.992299 6.502035 4.923876 5.875571 2.489281 6.361795 Oklahoma 4.358549 8.039996 7.394851 3.590942 4.839118 3.673270 2.266944 2.940617 3.679251 6.424026 6.381849 8.500283 3.190292 7.414962 3.526575 5.343542 Oregon 3.656905 7.934215 5.557330 4.272458 4.331402 6.002440 4.659856 4.650299 4.258624 5.426520 5.288891 6.817479 5.634277 5.463209 4.215200 5.934161 Pennsylvania 4.838128 5.912135 5.665029 2.080653 3.777518 5.651725 2.761281 5.021885 4.355587 6.503932 5.146335 6.006059 5.459081 6.083498 2.754122 6.788586 Rhode Island 6.528749 8.114487 5.008645 7.227053 7.547637 8.840137 7.557499 8.276331 7.313252 6.409380 7.182960 5.772403 8.093078 5.986392 7.275119 7.663675 South Carolina 5.630992 7.487290 7.603678 3.235657 5.423490 3.523234 3.158677 4.523471 4.874313 6.177323 6.430322 7.649252 4.164027 7.175516 2.329532 7.236684 South Dakota 3.438765 8.644534 7.405576 5.657342 5.175878 5.748209 4.624357 1.983415 3.157721 7.953556 5.346449 8.968003 4.399006 8.359152 5.988650 2.701569 Tennessee 5.609207 7.334145 7.474428 2.804910 5.237431 3.764199 2.195797 4.277971 4.776192 6.037960 6.787385 7.877657 4.008972 6.990206 2.214183 6.978381 Texas 5.934945 6.905895 6.151740 3.844443 4.498445 5.802564 4.034896 4.834120 4.890119 5.384088 6.552911 6.946721 5.132624 5.322347 3.176829 7.118197 Utah 6.366016 8.354401 7.296731 6.164955 4.413720 8.625013 6.365106 6.177063 4.851634 7.792762 6.146926 8.237101 6.680453 8.446089 6.541281 6.005682 Vermont 5.750459 9.430157 8.260979 7.389970 6.408482 8.907407 7.833774 7.430777 5.091342 9.671883 4.209895 8.633394 8.253892 9.087153 7.741963 6.307128 Virginia 5.805975 4.003073 4.769055 5.014803 4.261891 7.689765 4.790444 6.459736 5.068947 7.419712 4.989951 6.063187 6.111441 6.426794 5.365508 6.397805 Washington 4.621560 6.256025 4.237212 4.434985 3.653237 7.171936 4.981488 5.769384 4.569590 5.395136 4.714175 5.548317 6.173554 4.770796 4.514208 6.387711 West Virginia 4.895459 9.802177 9.108152 4.655579 6.370725 4.226412 3.790895 4.718712 5.231977 8.486123 7.616860 10.111724 4.390298 9.593673 5.287663 5.860449 Wisconsin 3.920615 6.744556 5.216107 2.734616 2.202186 5.931485 2.848045 3.955807 2.896716 6.147798 4.157304 6.216018 5.146218 5.846711 3.303504 5.291888 Wyoming 7.126880 10.105004 9.906165 8.743662 7.897871 8.711600 7.551584 4.333896 6.103995 9.153755 7.491362 10.901153 6.510596 11.108905 8.808637 4.795353 Ohio Oklahoma Oregon Pennsylvania Rhode Island South Carolina South Dakota Tennessee Texas Utah Vermont Virginia Washington West Virginia Wisconsin Alaska Arizona Arkansas California Colorado Connecticut Delaware Florida Georgia Hawaii Idaho Illinois Indiana Iowa Kansas Kentucky Louisiana Maine Maryland Massachusetts Michigan Minnesota Mississippi Missouri Montana Nebraska Nevada New Hampshire New Jersey New Mexico New York North Carolina North Dakota Ohio Oklahoma 3.450637 Oregon 4.221053 4.206247 Pennsylvania 2.405404 3.980486 4.781168 Rhode Island 6.996460 7.656694 5.560826 7.069158 South Carolina 4.078721 3.618968 5.026658 3.454359 7.704205 South Dakota 5.575686 3.772286 4.674182 5.793156 7.443807 5.658435 Tennessee 3.006111 2.665695 5.200718 2.852044 7.860988 2.250386 5.683295 Texas 3.311304 3.957736 3.692048 4.403557 7.459784 4.340904 5.933274 4.070610 Utah 5.820170 6.516477 5.388702 6.975968 8.438162 7.360012 5.741575 7.592907 5.444679 Vermont 7.981582 7.654685 6.981145 7.030258 8.167112 7.041482 5.854341 7.898382 8.341997 7.926106 Virginia 4.993860 5.631299 5.561860 4.104011 7.531600 5.724971 5.947832 5.660883 5.197778 6.452475 7.215077 Washington 4.321383 5.108779 2.237714 4.559541 5.678181 5.464591 5.505772 5.650595 3.492007 5.017698 7.085641 4.083678 West Virginia 4.933222 3.494353 6.332121 5.195085 8.738638 5.204569 4.787221 4.310947 6.850242 8.065666 8.282668 7.472877 7.462814 Wisconsin 2.327254 3.507086 3.276736 2.778869 6.699085 4.264763 4.428909 3.954716 3.628860 4.813705 6.653113 4.176491 3.228251 5.200544 Wyoming 8.644524 7.238965 8.071010 8.835251 9.108380 8.328574 4.973078 8.477997 9.144074 7.732709 7.294573 8.519401 8.512051 7.556514 7.685097 ``` ] --- ## Step 2: Cluster and plot .tiny-font.pull-left[ ```r dist_out <- US_state_stats %>% column_to_rownames(var = "state") %>% scale() %>% dist(method = "euclidean") hc_out <- hclust( dist_out, method = "average" ) library(ggdendro) ggdendrogram(hc_out, rotate = TRUE) ``` `method = "average"`: UPGMA ] .pull-right.width-40.move-up-6em[ <!-- --> ] --- ## Step 2: Cluster and plot .tiny-font.pull-left[ ```r dist_out <- US_state_stats %>% column_to_rownames(var = "state") %>% scale() %>% dist(method = "euclidean") hc_out <- hclust( dist_out, method = "ward.D2" ) library(ggdendro) ggdendrogram(hc_out, rotate = TRUE) ``` `method = "ward.D2"`: Ward's minimum variance method ] .pull-right.width-40.move-up-6em[ <!-- --> ] --- ## Step 2: Cluster and plot .tiny-font.pull-left[ ```r dist_out <- US_state_stats %>% column_to_rownames(var = "state") %>% scale() %>% dist(method = "euclidean") hc_out <- hclust( dist_out, method = "complete" ) library(ggdendro) ggdendrogram(hc_out, rotate = TRUE) ``` `method = "complete"`: complete linkage method ] .pull-right.width-40.move-up-6em[ <!-- --> ] --- ## We can also plot manually .tiny-font.pull-left[ ```r ddata <- dendro_data( hc_out, type = "rectangle" ) segments <- segment(ddata) labels <- label(ddata) ggplot() + geom_segment( data = segments, aes(x, y, xend = xend, yend = yend) ) + geom_text( data = labels, aes(x, y - .5, label = label), hjust = 1, size = 8/.pt ) + coord_flip(ylim = c(-5, 12)) ``` ] .pull-right.width-45.move-up-3em[ <!-- --> ] --- ## Assign clusters by cutting the dendrogram .tiny-font[ ```r # cut dendrogram at height 5 cutree(hc_out, h = 5) ``` ``` Alabama Alaska Arizona Arkansas California Colorado 1 2 1 1 3 4 Connecticut Delaware Florida Georgia Hawaii Idaho 5 6 4 1 7 8 Illinois Indiana Iowa Kansas Kentucky Louisiana 4 1 8 8 9 1 Maine Maryland Massachusetts Michigan Minnesota Mississippi 8 10 5 4 4 9 Missouri Montana Nebraska Nevada New Hampshire New Jersey 1 8 8 11 12 5 New Mexico New York North Carolina North Dakota Ohio Oklahoma 9 3 1 13 1 1 Oregon Pennsylvania Rhode Island South Carolina South Dakota Tennessee 4 4 14 1 8 1 Texas Utah Vermont Virginia Washington West Virginia 4 15 12 10 4 9 Wisconsin Wyoming 4 13 ``` ] --- ## Assign clusters by cutting the dendrogram .tiny-font[ ```r # cut dendrogram at height 8 cutree(hc_out, h = 8) ``` ``` Alabama Alaska Arizona Arkansas California Colorado 1 2 1 1 3 4 Connecticut Delaware Florida Georgia Hawaii Idaho 5 4 4 1 3 4 Illinois Indiana Iowa Kansas Kentucky Louisiana 4 1 4 4 1 1 Maine Maryland Massachusetts Michigan Minnesota Mississippi 4 5 5 4 4 1 Missouri Montana Nebraska Nevada New Hampshire New Jersey 1 4 4 3 6 5 New Mexico New York North Carolina North Dakota Ohio Oklahoma 1 3 1 6 1 1 Oregon Pennsylvania Rhode Island South Carolina South Dakota Tennessee 4 4 3 1 4 1 Texas Utah Vermont Virginia Washington West Virginia 4 4 6 5 4 1 Wisconsin Wyoming 4 6 ``` ] --- ## Assign clusters by cutting the dendrogram .tiny-font[ ```r # cut dendrogram so there are 4 clusters cutree(hc_out, k = 4) ``` ``` Alabama Alaska Arizona Arkansas California Colorado 1 2 1 1 3 1 Connecticut Delaware Florida Georgia Hawaii Idaho 3 1 1 1 3 1 Illinois Indiana Iowa Kansas Kentucky Louisiana 1 1 1 1 1 1 Maine Maryland Massachusetts Michigan Minnesota Mississippi 1 3 3 1 1 1 Missouri Montana Nebraska Nevada New Hampshire New Jersey 1 1 1 3 4 3 New Mexico New York North Carolina North Dakota Ohio Oklahoma 1 3 1 4 1 1 Oregon Pennsylvania Rhode Island South Carolina South Dakota Tennessee 1 1 3 1 1 1 Texas Utah Vermont Virginia Washington West Virginia 1 1 4 3 1 1 Wisconsin Wyoming 1 4 ``` ] --- ## Add cluster info into scatterplot .pull-left.tiny-font[ ```r # cut dendrogram so there are 4 clusters cluster <- cutree(hc_out, k = 4) US_state_stats %>% left_join( tibble( state = names(cluster), cluster = factor(cluster) ) ) %>% ggplot(aes(income, tr_deaths)) + geom_point(aes(color = cluster)) ``` ] .pull-right[ <!-- --> ] [//]: # "segment ends here" --- ## Further reading - Wikipedia: [Hierarchical clustering](https://en.wikipedia.org/wiki/Hierarchical_clustering) - **ggdendro** documentation: [Introduction to ggdendro](https://cran.r-project.org/web/packages/ggdendro/vignettes/ggdendro.html) - **ggdendro** documentation: [reference documentation (pdf)](https://cran.r-project.org/web/packages/ggdendro/ggdendro.pdf)