ISLR Home

Question

In the chapter, we mentioned the use of correlation-based distance and Euclidean distance as dissimilarity measures for hierarchical clustering. It turns out that these two measures are almost equivalent: if each observation has been centered to have mean zero and standard deviation one, and if we let rij denote the correlation between the ith and jth observations, then the quantity $$1 âˆ’ r_{ij}$$ is proportional to the squared Euclidean distance between the ith and jth observations.

On the USArrests data, show that this proportionality holds.

Hint: The Euclidean distance can be calculated using the dist() function, and correlations can be calculated using the cor() function.

library(ISLR)
set.seed(1)
names(USArrests)
## [1] "Murder"   "Assault"  "UrbanPop" "Rape"

Scale

dsc <-  scale(USArrests)
head(dsc)
##                Murder   Assault   UrbanPop         Rape
## Alabama    1.24256408 0.7828393 -0.5209066 -0.003416473
## Alaska     0.50786248 1.1068225 -1.2117642  2.484202941
## Arizona    0.07163341 1.4788032  0.9989801  1.042878388
## Arkansas   0.23234938 0.2308680 -1.0735927 -0.184916602
## California 0.27826823 1.2628144  1.7589234  2.067820292
## Colorado   0.02571456 0.3988593  0.8608085  1.864967207
a <- dist(dsc)^2
b <- as.dist(1 - cor(t(dsc)))
summary(b / a)
##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max.
## 0.000086 0.069135 0.133943 0.234193 0.262589 4.887686

a Result

a
##                    Alabama      Alaska     Arizona    Arkansas  California
## Arizona         5.26023278  7.29347205
## Arkansas        1.66361028  7.98649428  7.38621016
## California     10.64788918  9.07540626  1.71736889 14.16499276
## Colorado        7.02815781  5.41268921  1.86330880  8.01485074  1.65796150
## Connecticut    10.33813789 22.46677045 10.64623933  6.79978363 16.53552607
## Delaware        4.07754307 13.11427242  3.64569237  3.24116621  9.44815517
## Florida         5.28142566  8.98059543  3.06037530 11.37171123  4.10064060
## Georgia         1.28014540  7.94923506  7.76846344  4.89188867 11.41127921
## Hawaii         11.48213585 20.52211419 10.64143234  8.83462477 13.38760961
## Idaho           8.49525655 16.46781442 12.39749070  3.12838990 20.14163739
## Illinois        3.50999964 10.67369811  1.17191703  6.06460748  3.65477638
## Indiana         4.31036167 11.32723119  6.97355332  2.08817050 11.60170337
## Iowa           12.16541263 22.32742956 16.93940863  5.88191549 24.70944059
## Kansas          5.26293868 13.54841569  7.70775193  2.47068453 13.01169377
## Kentucky        3.41358096 12.56057601 11.26789198  1.12319808 18.03175051
## Louisiana       0.59632751  8.78021688  4.91886698  4.10235707  9.10628726
## Maine          12.14600188 23.35074194 18.45725130  5.57993812 27.77273418
## Maryland        1.66318693  5.18818595  1.46830307  4.23628759  4.97851292
## Massachusetts   8.92504280 19.12306325  6.33140382  7.22602687 10.34040454
## Michigan        3.53995592  4.47531360  1.42585234  6.70553617  2.29423689
## Minnesota      10.44216437 19.59532401 12.52342409  5.42936227 18.59604712
## Mississippi     1.64657827 10.59784130 11.93799654  3.73209520 19.53705043
## Missouri        2.66005864  6.43158652  2.54681087  2.79474818  5.24032265
## Montana         5.43695119 13.37802891 11.06950744  1.51045714 18.05754995
## Nebraska        7.08899655 15.31697104 10.01202932  2.97234668 16.15818272
## Nevada          9.62507498  5.49582762  3.70958842 13.75429806  1.43239266
## New Hampshire  12.68771958 23.66889235 18.00339737  6.22495555 26.28704345
## New Jersey      7.27932787 17.46557261  4.73314255  7.50676610  7.54217630
## New Mexico      2.55807090  4.23572974  1.07678966  5.37460577  3.24367338
## New York        4.29470910 10.82658009  1.15030330  7.55074861  2.81805702
## North Carolina  2.57399095 10.49958985  9.90924086  4.29232957 18.32059908
## North Dakota   16.49577270 27.15478586 24.32446969  8.26935844 34.41077602
## Ohio            5.15222767 12.89050377  5.56285484  3.84830773  9.08023330
## Oklahoma        3.83019123 11.16673433  5.12948980  2.02338516  9.91538983
## Oregon          5.61959184  7.28497679  4.00346648  3.41422652  7.06178489
## Pennsylvania    6.33093011 17.00699373  8.51992298  3.89667037 13.79718497
## Rhode Island   11.52690580 25.63353589  9.34534128  9.53786230 15.11965101
## South Carolina  0.83868385  6.57437373  7.83554381  2.91528136 14.09774107
## South Dakota    9.50833401 18.03462912 16.82648492  3.50618977 25.53195296
## Tennessee       0.70685875  5.45808328  5.28533038  2.03190371  9.07170233
## Texas           2.71037787  9.94008765  2.70555598  5.52510645  4.70809967
## Utah            9.55181256 15.58739966  6.37281277  6.78594964  9.42592130
## Vermont        15.83365636 23.72457189 26.01373812  7.53117389 36.38925099
## Virginia        2.20811667  9.29766981  5.33912675  0.99421537 10.34247768
## Washington      7.01286989 10.70287758  4.57922199  4.54261108  7.69880444
## West Virginia   9.76154469 20.25410214 20.22677731  3.98069981 30.12205684
## Wisconsin      12.28315357 23.72814438 15.54398956  6.81337957 22.42492239
## Wyoming         3.34561684 12.24541987  7.24849420  0.98260421 13.87023590
##                   Colorado Connecticut    Delaware     Florida     Georgia
## Arizona
## Arkansas
## California
## Connecticut    11.07553068
## Delaware        6.52672513  3.08651312
## Florida         5.98223097 19.98152656  9.37227420
## Georgia         8.20771199 15.79126676  8.90348920  4.75805123
## Hawaii          7.97131856  1.91636702  6.12503464 19.00640730 14.52007651
## Idaho          12.08791906  2.67460320  4.15447951 22.08983753 14.44434344
## Illinois        3.20349937  7.50790667  2.42883468  3.13710100  5.35264212
## Indiana         5.59588463  2.60754615  2.88094279 13.06878725  7.24909451
## Iowa           15.52903598  2.39323648  6.79572222 27.75473724 18.07770851
## Kansas          6.90232753  1.50808805  2.40586899 14.76446625  9.04293557
## Kentucky       10.41611905  5.45053733  5.06922482 15.58274268  5.95760124
## Louisiana       7.04729559 12.48167150  5.41353118  3.07289585  0.73831819
## Maine          18.24438016  3.53144577  7.05476523 29.10257068 18.77854388
## Maryland        3.86813010 12.22779824  3.85134128  2.06071870  3.38143975
## Massachusetts   7.03458569  0.89646785  2.06857093 14.25295574 13.47382394
## Michigan        1.52846446 13.71803806  6.33291945  1.78409971  3.68083002
## Minnesota      11.07814842  0.96900264  4.68849362 22.69117192 15.69890364
## Mississippi    14.81560328 17.44117384  9.30898393  9.53986573  2.50544394
## Missouri        1.72328786  5.94541768  2.79668143  6.34156581  4.41920649
## Montana        10.14125584  3.45377251  4.12368175 18.22989072  9.58972949
## Nebraska        9.02077965  1.46820464  3.28096358 18.56135781 11.76182001
## Nevada          1.95680782 21.03887110 12.90310820  3.80265151  8.42336888
## New Hampshire  16.91371460  2.61436548  7.15253998 28.92081279 18.85934822
## New Jersey      5.39626853  2.59494163  2.49915588 10.17648033 10.23318484
## New Mexico      2.39241657 13.12878022  4.96026386  1.68111907  3.61584816
## New York        3.01487151  9.14407633  3.60700083  2.47463407  5.58589499
## North Carolina 14.93766452 17.55157860  7.54891363  8.99651331  5.45283520
## North Dakota   23.05346036  6.30001869 11.29984311 36.42920793 23.61644843
## Ohio            4.48941342  2.08619625  2.50858566 11.49027253  7.86421495
## Oklahoma        4.95684195  2.10558179  1.39309130 11.25835446  7.35576590
## Oregon          2.35069605  4.73365045  3.14806170 11.15541196  8.99932660
## Pennsylvania    8.14629147  0.76064399  2.52646255 15.51561791 10.12383891
## Rhode Island   12.11787531  1.15694003  2.63428960 17.90548280 17.49922478
## South Carolina 10.32409956 16.10252721  7.31143611  6.39883155  1.95162981
## South Dakota   15.73496135  5.01644621  7.14108735 26.02544595 14.99993185
## Tennessee       4.82773647 10.43443400  5.38011250  5.75629754  1.02459987
## Texas           3.22101946  8.25670065  4.01255601  3.43656855  3.08900282
## Utah            5.04506762  1.64503880  3.26920062 15.42488523 13.82649431
## Vermont        23.93629683 10.28238080 13.79725222 36.92557268 22.17612925
## Virginia        5.11779268  3.71602886  1.98478227  9.93232210  4.95042892
## Washington      3.20335541  2.87759861  2.67328071 12.65272280 10.90087257
## West Virginia  19.54557221  7.35363691  9.23041666 28.09431110 14.85742544
## Wisconsin      14.32389159  1.07217680  5.95872868 26.09715038 17.87734606
## Wyoming         7.95888274  2.63042110  1.58413047 13.19564549  7.46877639
##                     Hawaii       Idaho    Illinois     Indiana        Iowa
## Arizona
## Arkansas
## California
## Connecticut
## Delaware
## Florida
## Georgia
## Hawaii
## Idaho           5.59705761
## Illinois        7.46915588 10.71183870
## Indiana         2.39034065  2.22706397  4.85192310
## Iowa            4.65030909  0.73701571 13.97269651  3.16361273
## Kansas          2.14586355  1.46485465  5.39563441  0.18384472  2.16068385
## Kentucky        6.35208601  2.74407044  8.11046765  1.39017124  3.77387850
## Louisiana      12.73573209 12.44944547  2.73412117  6.22879147 16.28898461
## Maine           7.37695910  0.72014100 15.47795616  4.42225495  0.41694885
## Maryland       13.06726342 11.56991907  1.80364809  6.46729543 16.51808558
## Massachusetts   1.76270117  4.92885269  4.03245856  2.76081309  5.52733614
## Michigan       11.64411364 14.26973368  1.94856185  6.82174551 18.70444536
## Minnesota       2.15321861  1.02514329  9.95957129  1.73840680  0.58436614
## Mississippi    20.04999676 12.96212148  9.52924636  9.52282094 17.30832209
## Missouri        4.76657180  6.09950785  1.83683080  1.48935924  8.64274489
## Montana         5.05746152  0.68673312  8.79658837  1.00669736  1.53848336
## Nebraska        2.76512238  0.56475433  7.81884414  0.73452252  0.96468138
## Nevada         16.22637449 22.37311541  5.70815885 12.44581973 27.27692155
## New Hampshire   5.34164971  0.85554409 14.81528146  3.71669649  0.04237581
## New Jersey      2.26517546  7.52144902  2.12074421  3.11110766  8.48147900
## New Mexico     12.60682268 12.87623881  1.79379836  6.71327724 17.75054399
## New York        8.44239678 12.89550998  0.12265318  6.06564329 16.33096303
## North Carolina 22.40177844 12.90935568  9.07474138 11.18069645 18.46762384
## North Dakota   10.22394613  2.00068487 21.28864087  7.06918142  1.10973048
## Ohio            1.32119223  3.86017452  3.28514929  0.48669039  4.67002547
## Oklahoma        2.75086636  2.30071586  3.40028454  0.28124557  3.76028167
## Oregon          4.01275463  3.90348789  4.30290039  1.38787597  6.08228793
## Pennsylvania    1.46876356  2.30185532  5.35190622  0.70776879  2.46769371
## Rhode Island    4.23988480  6.04801132  6.27891398  5.44550641  6.47890146
## South Carolina 18.08904131 11.93699647  6.84538136  8.10531922 16.82256808
## South Dakota    7.86515909  0.65129587 14.44357863  3.38992001  0.97746951
## Tennessee       9.45401067  8.54649832  3.79406253  3.27621430 11.68021486
## Texas           6.70897865 11.00393558  0.67919876  4.01431766 13.66222215
## Utah            1.14698097  4.10818703  5.18547204  1.96551040  4.70112123
## Vermont        13.84461097  3.16749661 23.64332485  8.21853263  2.99235525
## Virginia        4.19399195  2.88975830  3.46889129  0.37543145  4.71106314
## Washington      2.38792134  3.55772058  4.24894202  1.30091037  5.06377441
## West Virginia  10.77931961  2.07315085 16.93164668  5.05285785  2.32772694
## Wisconsin       2.77787893  1.46540730 12.10388410  2.82445977  0.39917991
## Wyoming         4.78883792  1.36606915  5.12731438  0.79188339  2.95657109
##                     Kansas    Kentucky   Louisiana       Maine    Maryland
## Arizona
## Arkansas
## California
## Connecticut
## Delaware
## Florida
## Georgia
## Hawaii
## Idaho
## Illinois
## Indiana
## Iowa
## Kansas
## Kentucky        1.69525087
## Louisiana       7.44423549  5.86703556
## Maine           3.20902545  3.97039706 16.72967417
## Maryland        7.50811673  7.96903502  1.62285603 17.02311911
## Massachusetts   2.05733156  6.90872372  9.93797170  7.24701609  8.84657503
## Michigan        8.42213816  9.71163371  2.78155634 20.55118995  1.16661334
## Minnesota       0.94982027  3.73789651 13.62050882  1.68489790 13.28524413
## Mississippi    10.68226853  5.71156636  2.64676430 16.01167509  5.28643104
## Missouri        2.30818648  3.87191576  3.37169373 10.27584240  2.46670731
## Montana         0.84097447  0.72653492  8.66993669  1.76124712  9.15029570
## Nebraska        0.27868813  2.02192165 10.05291547  1.74739492  9.80257562
## Nevada         14.73924785 17.34246547  8.07166192 30.41868695  5.08562807
## New Hampshire   2.58697994  4.03751069 16.94814276  0.24959724 17.35867602
## New Jersey      2.91420212  7.24407975  7.19654672 10.58360912  6.89769103
## New Mexico      8.04083877  9.00439974  2.22357482 18.88818342  0.28664166
## New York        6.81002042  9.74274230  3.06078390 18.14411008  2.06271914
## North Carolina 11.82509412  8.29329403  3.94761968 16.50931251  4.21988369
## North Dakota    5.74599510  5.99395875 22.06918056  0.53371919 22.48943802
## Ohio            0.61105486  3.14236613  6.24848456  6.47995306  6.28088518
## Oklahoma        0.27026773  2.13846254  5.53922852  4.80917223  5.05932453
## Oregon          1.80281363  4.57475500  7.55732668  7.73583684  5.04735958
## Pennsylvania    0.29777100  2.54214242  8.08881680  3.68546728  8.75304098
## Rhode Island    4.03488407  9.27678579 12.70783246  7.46987885 11.81925689
## South Carolina  9.36931258  5.84014168  1.72972674 16.01246261  2.61326028
## South Dakota    2.78926951  2.28462927 14.03068398  0.61042821 14.43384182
## Tennessee       4.63672693  3.05898057  1.27656874 12.54609633  2.31113906
## Texas           5.00787825  6.39980673  1.77563217 15.51280350  2.38142544
## Utah            1.62603388  6.48300467 11.18299783  6.87388079  9.20394303
## Vermont         7.45201413  5.70652130 21.89859376  2.03167395 22.49659518
## Virginia        0.69755060  1.19216350  3.82362003  5.51059209  4.04992277
## Washington      1.34075972  5.12127859  8.82424005  6.91298000  6.46934722
## West Virginia   4.51336172  2.32144809 14.39991155  1.39667951 16.20188227
## Wisconsin       1.75375638  4.38911390 15.64929007  1.31924283 16.07327247
## Wyoming         0.57588794  1.14370352  5.68206261  3.12054491  5.77983837
##                Massachusetts    Michigan   Minnesota Mississippi    Missouri
## Arizona
## Arkansas
## California
## Connecticut
## Delaware
## Florida
## Georgia
## Hawaii
## Idaho
## Illinois
## Indiana
## Iowa
## Kansas
## Kentucky
## Louisiana
## Maine
## Maryland
## Massachusetts
## Michigan          9.34947755
## Minnesota         2.75136684 14.43627683
## Mississippi      16.98861527  8.83446266 16.86547798
## Missouri          3.92457148  1.97932272  5.80270038  8.23285169
## Montana           5.25282682 11.12149651  1.60342316  9.00915525  4.12644338
## Nebraska          2.84387944 11.20842641  0.37007934 12.43942222  3.86193175
## Nevada           14.86568907  1.58997401 21.52135471 16.86317230  5.51733243
## New Hampshire     6.03858077 19.93280673  0.86104433 17.55269491  9.53927066
## New Jersey        0.63642765  6.59382282  4.95247236 15.12768401  2.91711193
## New Mexico        9.16557542  0.33437010 13.97090623  7.05292839  2.12548917
## New York          5.05325081  1.66344295 11.82781864 10.66402702  2.33624013
## North Carolina   16.62470243  9.13975089 17.82496458  1.39875350  9.13541486
## North Dakota     11.18695298 26.18567330  3.26370639 20.02837994 14.29692398
## Ohio              1.33817699  5.98284529  2.31535568 11.66153346  1.28310556
## Oklahoma          1.79110916  5.92277037  2.01595534  9.28185692  1.19413618
## Oregon            3.50036112  4.67695720  3.71336738 12.27365305  0.99484090
## Pennsylvania      1.28547593  9.64011931  1.02143624 12.23023026  3.16611074
## Rhode Island      0.89131353 13.92819815  4.12520152 19.25015710  7.54875763
## South Carolina   14.67688662  5.39789163 15.59035972  0.61868820  5.68631756
## South Dakota      8.36663493 17.42622012  2.24709607 12.49989199  8.33040932
## Tennessee         8.80833741  2.55047150  9.62441307  3.33777169  1.54104270
## Texas             5.10487854  1.66116548  9.88364417  8.08363073  1.35819703
## Utah              0.81284812  8.66797295  2.00991494 18.12304796  3.05512270
## Vermont          15.42731858 26.26642254  5.76956751 17.67921601 15.05703387
## Virginia          3.42390423  5.03551681  3.21564976  6.44299359  0.95791432
## Washington        1.81521558  6.10841843  2.54575379 14.54690406  1.56318816
## West Virginia    11.41274347 19.95261145  4.32295006 11.07633557 10.58724291
## Wisconsin         3.56556516 17.66885220  0.24411820 18.47965902  7.93635363
## Wyoming           3.31297531  8.02281441  2.20402750  7.18951936  2.58368988
## Arizona
## Arkansas
## California
## Connecticut
## Delaware
## Florida
## Georgia
## Hawaii
## Idaho
## Illinois
## Indiana
## Iowa
## Kansas
## Kentucky
## Louisiana
## Maine
## Maryland
## Massachusetts
## Michigan
## Minnesota
## Mississippi
## Missouri
## Montana
## New Hampshire   1.77675677  1.27706268 29.02130660
## New Jersey      6.71454089  4.51408534 11.19853602    9.16224324
## New Mexico     10.18622848 10.55865777  2.97039667   18.80159747  6.86863818
## New York       10.66849454  9.56376651  4.69768691   17.29429939  2.67150641
## North Carolina 10.37436884 13.32263556 17.45020020   18.62536291 15.53843682
## North Dakota    3.34566441  3.62473636 36.62603176    0.85227873 15.33991605
## Ohio            2.70153151  1.60136636 10.84431791    5.33401871  1.23206075
## Oklahoma        1.49458332  0.93601923 11.63403150    4.28369861  2.16418915
## Oregon          3.25608376  2.47367139  8.16889579    6.95756193  3.89622373
## Pennsylvania    1.75468314  0.71962275 16.31569768    2.82980876  2.02096318
## Rhode Island    7.22995543  4.53859248 21.33084709    6.66321240  2.15161305
## South Carolina  8.42410114 11.23404454 11.85266577   17.30303273 12.83554289
## South Dakota    0.78449080  1.58543833 26.43684464    0.97507942 11.10103940
## Tennessee       5.05981058  6.51619229  7.11089831   12.45979177  6.93780191
## Texas           8.00832900  7.60036039  5.18276782   14.57579392  2.63300107
## Utah            4.39169383  2.12372682 12.86583362    5.43117088  1.72708040
## Vermont         3.70999623  5.26807477 36.52733134    2.79428918 19.36476679
## Virginia        1.33556897  1.55557312 10.89122085    5.23406772  3.33266975
## Washington      3.40132841  1.92099188  9.96715739    5.86365595  2.48363073
## West Virginia   1.62771492  3.28255887 30.20952623    2.14590987 13.98918631
## Wisconsin       2.22497950  0.94476001 25.75738752    0.51203017  6.08668685
## Wyoming         0.66423665  0.85899577 15.36852949    3.22229569  4.15023576
##                 New Mexico    New York North Carolina North Dakota        Ohio
## Arizona
## Arkansas
## California
## Connecticut
## Delaware
## Florida
## Georgia
## Hawaii
## Idaho
## Illinois
## Indiana
## Iowa
## Kansas
## Kentucky
## Louisiana
## Maine
## Maryland
## Massachusetts
## Michigan
## Minnesota
## Mississippi
## Missouri
## Montana
## New Hampshire
## New Jersey
## New Mexico
## New York        1.77531525
## North Carolina  6.42538035 10.34522795
## North Dakota   24.45353876 24.32984439    21.01011081
## Ohio            6.23046747  4.17589027    13.10852182   9.88994245
## Oklahoma        5.48789273  4.56553323     9.83866030   7.97875506  0.42041000
## Oregon          4.64537433  5.16549156    12.31672445  10.79915712  1.53948704
## Pennsylvania    9.37579177  6.73370021    13.54531184   6.53065040  0.60548603
## Rhode Island   12.98374651  7.66323173    17.79640789  11.58878201  3.86505637
## South Carolina  3.84016655  7.70375205     1.09753128  20.34386320  9.79056869
## South Dakota   15.97561960 16.91240404    13.65700621   1.06604462  5.95079431
## Tennessee       2.41128232  4.34636244     5.46374409  16.50240219  4.06740325
## Texas           2.07885671  0.71532645     9.52181377  20.93009633  2.79274563
## Utah            8.90652167  6.16379144    18.21652638  10.07314108  1.03108243
## Vermont        24.42022638 26.73382392    19.25457345   0.96527811 12.12840457
## Virginia        4.55462257  4.59639579     7.57214052   8.66917419  0.95538670
## Washington      6.14844421  5.17469914    14.48235119  10.06533325  0.94575873
## West Virginia  18.21695490 19.60683686    12.94420646   1.61717215  8.20843742
## Wisconsin      17.07983618 14.20315445    19.73976543   2.62969665  3.47815074
## Wyoming         6.90991748  6.70314952     7.56312758   5.84225475  1.71255957
##                   Oklahoma      Oregon Pennsylvania Rhode Island South Carolina
## Arizona
## Arkansas
## California
## Connecticut
## Delaware
## Florida
## Georgia
## Hawaii
## Idaho
## Illinois
## Indiana
## Iowa
## Kansas
## Kentucky
## Louisiana
## Maine
## Maryland
## Massachusetts
## Michigan
## Minnesota
## Mississippi
## Missouri
## Montana
## New Hampshire
## New Jersey
## New Mexico
## New York
## North Carolina
## North Dakota
## Ohio
## Oklahoma
## Oregon          1.15220513
## Pennsylvania    0.66916008  2.99073183
## Rhode Island    3.89932107  7.08697374   2.67952517
## South Carolina  7.54652030  9.08085251  11.17540941  17.52370125
## South Dakota    4.13735377  6.24443631   3.91672364   9.77357160    12.40165106
## Tennessee       3.42260964  4.12364431   5.92587190  12.77623630     2.06644070
## Texas           3.35353317  4.43228795   5.04483272   8.27022360     6.01832571
## Utah            1.53089038  1.64484513   1.56977787   3.08558905    15.14168381
## Vermont         9.61639102 11.73938689   9.16307553  16.89552415    18.20641622
## Virginia        0.31880185  1.60387778   1.38514914   5.91955374     5.12412866
## Washington      0.91901465  0.35228302   1.95813100   4.72780874    11.42596446
## West Virginia   6.06722404  9.21113720   5.66405621  12.53220774    12.00738896
## Wisconsin       3.21012760  5.80265478   1.48953685   4.31788699    17.80078206
## Wyoming         0.54264812  2.56416650   1.14160775   4.72054126     6.27956291
##                South Dakota   Tennessee       Texas        Utah     Vermont
## Arizona
## Arkansas
## California
## Connecticut
## Delaware
## Florida
## Georgia
## Hawaii
## Idaho
## Illinois
## Indiana
## Iowa
## Kansas
## Kentucky
## Louisiana
## Maine
## Maryland
## Massachusetts
## Michigan
## Minnesota
## Mississippi
## Missouri
## Montana
## New Hampshire
## New Jersey
## New Mexico
## New York
## North Carolina
## North Dakota
## Ohio
## Oklahoma
## Oregon
## Pennsylvania
## Rhode Island
## South Carolina
## South Dakota
## Tennessee        9.35744330
## Texas           13.76487084  2.16467664
## Utah             7.19493825  8.22434163  5.77913639
## Vermont          1.17865202 15.48951472 22.50976330 13.35613043
## Virginia         4.12776320  1.82641456  2.86341401  3.10184851  9.38707664
## Washington       6.29198248  5.66896306  4.68087805  0.48172861 12.11340609
## West Virginia    0.50535205 10.05362413 15.67097415 10.67991506  1.07755896
## Wisconsin        2.38312536 11.73554965 11.92978072  3.26968915  5.53126287
## Wyoming          2.44759295  3.72438609  5.09165889  3.44178057  6.91655040
##                   Virginia  Washington West Virginia   Wisconsin
## Arizona
## Arkansas
## California
## Connecticut
## Delaware
## Florida
## Georgia
## Hawaii
## Idaho
## Illinois
## Indiana
## Iowa
## Kansas
## Kentucky
## Louisiana
## Maine
## Maryland
## Massachusetts
## Michigan
## Minnesota
## Mississippi
## Missouri
## Montana
## New Hampshire
## New Jersey
## New Mexico
## New York
## North Carolina
## North Dakota
## Ohio
## Oklahoma
## Oregon
## Pennsylvania
## Rhode Island
## South Carolina
## South Dakota
## Tennessee
## Texas
## Utah
## Vermont
## Virginia
## Washington      1.90696640
## West Virginia   5.45372431  9.51509813
## Wisconsin       4.52263896  4.25919728    4.12451010
## Wyoming         0.49537795  2.53750448    3.54252609  3.04375685

b Result

b
##                     Alabama       Alaska      Arizona     Arkansas   California
## Arizona        1.4465947843 0.8307245898
## Arkansas       0.0877416809 0.3672785641 1.1766554172
## California     1.8659223310 0.8137128067 0.2949017541 1.5863126412
## Colorado       1.6872310193 0.5155556175 0.5983808782 1.3709254337 0.1146234010
## Connecticut    1.7135870980 1.8617870018 0.7759582405 1.8986422868 0.6631595707
## Delaware       1.1428182422 1.4788738310 0.3411540927 1.1920258659 0.8973252909
## Florida        0.1049203486 0.7861031722 1.0409926090 0.1421425885 1.7009808084
## Georgia        0.1162516972 0.7367030469 1.7839695560 0.2511757376 1.8595181408
## Hawaii         1.8059006680 1.5764752423 1.1458813737 1.9369776851 0.5651593270
## Idaho          1.4792354141 0.7916559296 0.0025732601 1.1919825459 0.2520373794
## Illinois       1.3090956230 1.8653950071 0.6962514485 1.5310645251 0.9943388791
## Indiana        1.4749888379 1.1435786713 1.5288959172 1.5522560010 0.7788542333
## Iowa           1.8953868930 1.6097240352 0.9237936588 1.9927501530 0.4446590775
## Kansas         1.7692452854 1.6312115837 1.1868843924 1.9304960880 0.6286302776
## Kentucky       0.2262067640 0.8211189668 1.8896787427 0.3945142447 1.8241158706
## Louisiana      0.0416804313 0.9708133042 1.6141501191 0.2417304309 1.9726883264
## Maine          1.2447409234 1.7558367789 0.5636322340 1.4147766173 0.9736656266
## Maryland       0.3068186638 0.4639642310 0.6726108873 0.1481740524 1.3073045972
## Massachusetts  1.7894923852 1.7874177991 0.6759197083 1.9234913599 0.5431469651
## Michigan       0.7852281487 0.0160233723 0.9351396709 0.4522302697 0.7845906291
## Minnesota      1.9380385065 1.5554053936 0.8202405883 1.9948508668 0.3603658226
## Mississippi    0.0001410904 0.6988312003 1.4366445350 0.0809945041 1.8574210543
## Missouri       1.4280619231 0.4597251487 1.0825496071 1.1987780947 0.4392700914
## Montana        0.3613201986 0.4043314953 1.6866753522 0.3341386073 1.4929667476
## Nebraska       1.9552044704 1.4469332669 0.8259917570 1.9724706420 0.3045480232
## Nevada         1.3546910596 0.2635366264 0.8480253841 1.0278904693 0.3667780030
## New Hampshire  1.8052455037 1.7834106100 0.9335432054 1.9763608584 0.5930225147
## New Jersey     1.7034480814 1.8819483401 0.9259932394 1.9203821726 0.7177102454
## New Mexico     0.4866250619 0.1544579029 0.6233621173 0.1892735525 1.0182846624
## New York       1.4405383983 1.9111958371 0.7494785897 1.6714293586 0.9119916203
## North Carolina 0.1005419555 0.6815996338 1.0190868258 0.0978489483 1.6583662185
## North Dakota   1.8325523371 0.9297233018 0.1500384685 1.5752656513 0.0376465024
## Ohio           1.7629487564 1.6040313401 1.2086276023 1.9187361114 0.6289392120
## Oklahoma       1.9163174185 1.6423424756 0.7596105442 1.9969853196 0.3978409346
## Oregon         1.6997566344 0.5157436131 0.4662059483 1.3612944186 0.0704774712
## Pennsylvania   1.5772254713 1.8951088612 1.2252417444 1.8607800842 0.9076276018
## Rhode Island   1.5485337745 1.8603206882 0.6465916089 1.7324209040 0.7726884642
## South Carolina 0.0291288261 0.5217321656 1.2706656015 0.0177942668 1.7211633716
## South Dakota   0.2970145365 0.1388506123 1.2490997809 0.1148539078 1.3342480484
## Tennessee      0.2041030039 0.4628624009 1.6851841442 0.2161213362 1.6514791316
## Texas          0.9358083387 1.6985755420 1.8148596616 1.3185998185 1.5345413026
## Utah           1.9936607906 1.3916603633 0.5992137279 1.9512088364 0.1923868967
## Vermont        0.4044726325 0.0833334212 1.1898013199 0.1805657123 1.2046037022
## Virginia       0.3445471205 0.8152198710 1.9274904038 0.5026132238 1.7289903339
## Washington     1.9568457805 0.9983021047 0.4482645444 1.7558264095 0.0322727276
## West Virginia  0.0384036580 0.6152466715 1.6034021040 0.1112667168 1.8337534041
## Wisconsin      1.7930675184 1.7768919032 0.9950200526 1.9729672358 0.6165352255
## Wyoming        0.3432852855 1.3277086920 1.0167123371 0.5400904982 1.7004642321
##                    Colorado  Connecticut     Delaware      Florida      Georgia
## Arizona
## Arkansas
## California
## Connecticut    1.0185123544
## Delaware       1.3582485780 0.4608111033
## Florida        1.7129152223 1.5460029888 0.7181523250
## Georgia        1.5295690153 1.7039265274 1.5418713236 0.4170422475
## Hawaii         0.6324566539 0.2432644035 1.1318870533 1.9136686462 1.4965738366
## Idaho          0.5341813172 0.7928419016 0.3939093803 1.0897845135 1.7939763287
## Illinois       1.4383840525 0.1530107077 0.1489267035 1.0203976420 1.5144605369
## Indiana        0.5474764400 0.7794305428 1.7009930816 1.8152555713 1.0081816430
## Iowa           0.6095631090 0.1429784233 0.9223844015 1.8958223096 1.6786397073
## Kansas         0.7054066822 0.2232956426 1.1096940357 1.8797225866 1.4629520300
## Kentucky       1.4687209001 1.6106288136 1.6449136460 0.5858134248 0.0210269564
## Louisiana      1.8213148508 1.5262510539 1.1317428594 0.1874542948 0.0962874713
## Maine          1.4385240677 0.2420976543 0.0636718795 0.8972729389 1.5300602864
## Maryland       1.3200251263 1.6479985362 0.7081470504 0.1161020758 0.6777247923
## Massachusetts  0.9013147940 0.0087811793 0.4591556130 1.5998373785 1.7894552886
## Michigan       0.4386452325 1.8370934575 1.6148352274 0.9230258769 0.7229795857
## Minnesota      0.5421367089 0.1496278955 0.8821567640 1.9037641259 1.7506853022
## Mississippi    1.6768722831 1.7234165763 1.1447218887 0.1030843838 0.1190046803
## Missouri       0.1270352066 1.2511104277 1.7656621709 1.6717365956 1.0986787646
## Montana        1.0371431493 1.8412147980 1.8478283803 0.7301993195 0.1326503438
## Nebraska       0.4372841195 0.2269913177 0.9862057604 1.9508364344 1.7366422587
## Nevada         0.0850658323 1.3936267318 1.6394405167 1.5027837765 1.1492466894
## New Hampshire  0.8364591781 0.0484850390 0.7406686936 1.7557057927 1.6542207500
## New Jersey     1.0179595210 0.0138671533 0.5956434120 1.6088477225 1.6182796640
## New Mexico     0.9022934338 1.8030190487 1.0093153099 0.3718183417 0.7501278679
## New York       1.3346274313 0.0716783704 0.2405499199 1.1954467959 1.5698015171
## North Carolina 1.6376229111 1.6246802788 0.7780225601 0.0058800900 0.4066814869
## North Dakota   0.2664591169 0.5634348110 0.6325884541 1.5593899587 1.9431513897
## Ohio           0.6860637020 0.2505176081 1.1510907482 1.8896044569 1.4419702702
## Oklahoma       0.6440972507 0.0815588033 0.7357293514 1.8269177319 1.7885487360
## Oregon         0.0112610188 0.9909631670 1.2268326777 1.6579431786 1.6105452936
## Pennsylvania   1.1007741760 0.1066979873 0.8275177418 1.6135748630 1.3807940318
## Rhode Island   1.2047706713 0.0484596718 0.2346510620 1.2809684478 1.6800323905
## South Carolina 1.5360150109 1.8199134790 1.1317094997 0.0883713273 0.1943872899
## South Dakota   0.9669701935 1.9992774379 1.5707904560 0.4797572639 0.2883765233
## Tennessee      1.2407565041 1.8616146897 1.7108604023 0.5290963708 0.0535222009
## Texas          1.4771065310 0.6252472586 1.2468021394 1.2025497890 0.6424671557
## Utah           0.3952580801 0.2154653086 0.8155301702 1.8888326485 1.8747589639
## Vermont        0.8261595348 1.9871700490 1.6139688377 0.5886898330 0.3731091961
## Virginia       1.3394222476 1.5720254303 1.7552683347 0.7483007177 0.0674516539
## Washington     0.1443213664 0.5086289742 0.9579044238 1.8504888961 1.8621468964
## West Virginia  1.5421860929 1.7997828180 1.4086435375 0.2550494908 0.0322204217
## Wisconsin      0.8324040739 0.0689103714 0.8034949135 1.7750903478 1.6116405473
## Wyoming        1.9129645619 1.0249404824 0.3839982882 0.1540326117 0.6726365889
##                      Hawaii        Idaho     Illinois      Indiana         Iowa
## Arizona
## Arkansas
## California
## Connecticut
## Delaware
## Florida
## Georgia
## Hawaii
## Idaho          1.1194637192
## Illinois       0.6914531794 0.7465587060
## Indiana        0.2013613293 1.4793914393 1.2840194438
## Iowa           0.0282225069 0.9034570743 0.5480239849 0.3523302067
## Kansas         0.0029975967 1.1649380722 0.6496748393 0.2145303950 0.0348817924
## Kentucky       1.3110295308 1.8924747942 1.5171894830 0.8111626802 1.5188948361
## Louisiana      1.6202379913 1.6533727507 1.1521202197 1.3410960799 1.7319619872
## Maine          0.8513847169 0.6173772594 0.0189375246 1.4536026973 0.6799165640
## Maryland       1.9809006430 0.7020143344 1.1875458869 1.8853274782 1.9074566364
## Massachusetts  0.2378864446 0.6868917486 0.1901128578 0.7809217595 0.1224235643
## Michigan       1.4435858401 0.8870923265 1.9286084543 0.9653039349 1.5065738607
## Minnesota      0.0541245177 0.7979379705 0.5538445586 0.3986556827 0.0061350369
## Mississippi    1.8142598556 1.4687214214 1.3188066041 1.4803110344 1.9025429436
## Missouri       0.6317248457 1.0108902036 1.7176157354 0.2903752822 0.7268623875
## Montana        1.3710765709 1.6608269229 1.8636395079 0.7458418733 1.5718759316
## Nebraska       0.0549854976 0.7948162656 0.6741582764 0.3389140713 0.0195338128
## Nevada         0.8746092973 0.7793340325 1.7638460774 0.5474904019 0.9225815521
## New Hampshire  0.0788727931 0.9310950364 0.3507917748 0.4931429210 0.0321473524
## New Jersey     0.1655594681 0.9379933225 0.2183672402 0.6393231759 0.1021636020
## New Mexico     1.8892976931 0.6209766683 1.5249749381 1.6479179073 1.8415117997
## New York       0.5213352450 0.7908364720 0.0171515854 1.1069439752 0.3947126025
## North Carolina 1.9507527897 1.0621327220 1.1147797397 1.8081850272 1.9366596466
## North Dakota   0.6568052017 0.1242187891 0.7797410414 1.0037593052 0.4857325583
## Ohio           0.0024548531 1.1842842571 0.6891693557 0.1892748734 0.0422265760
## Oklahoma       0.0927185454 0.7476089769 0.4222362362 0.5135359785 0.0195685768
## Oregon         0.6938055995 0.4072638253 1.3630857867 0.6731154411 0.6369366915
## Pennsylvania   0.1218279259 1.2333890666 0.3500496379 0.4609426946 0.1287849470
## Rhode Island   0.4808735307 0.6814258072 0.0362817055 1.0858894086 0.3378788928
## South Carolina 1.9127627993 1.2963552424 1.4116417088 1.5656382261 1.9756124256
## South Dakota   1.7320021389 1.2303197685 1.8638402260 1.1835050738 1.8393697570
## Tennessee      1.5176048483 1.6748264034 1.7623384622 0.9299519728 1.7040434728
## Texas          0.4717715849 1.8289408291 0.7412902397 0.4434643356 0.6167798742
## Utah           0.1541697506 0.5720533433 0.6069461768 0.5184719237 0.0665213145
## Vermont        1.6686135300 1.1621915322 1.9134622503 1.1157721485 1.7714347203
## Virginia       1.1727496872 1.9195043682 1.5755631320 0.6467402868 1.3974859251
## Washington     0.3523820515 0.4041477984 0.9165183156 0.5847754624 0.2598098819
## West Virginia  1.6991330263 1.6192564786 1.5139502191 1.2381033254 1.8409717107
## Wisconsin      0.0579469122 0.9901532458 0.3908860240 0.4391707809 0.0272786223
## Wyoming        1.5829463662 1.0859923304 0.5040705614 1.7725351795 1.5212227682
##                      Kansas     Kentucky    Louisiana        Maine     Maryland
## Arizona
## Arkansas
## California
## Connecticut
## Delaware
## Florida
## Georgia
## Hawaii
## Idaho
## Illinois
## Indiana
## Iowa
## Kansas
## Kentucky       1.2769869476
## Louisiana      1.5671655737 0.1666374285
## Maine          0.8143828375 1.5708956778 1.1347819452
## Maryland       1.9821235619 0.8765575743 0.5059016414 1.0106304327
## Massachusetts  0.2277022005 1.7002032521 1.6299578502 0.2679847940 1.6390192621
## Michigan       1.5005628044 0.7728802563 1.0195154196 1.8489898544 0.6150082648
## Minnesota      0.0673887259 1.6023967656 1.8001473208 0.6700882699 1.8653760357
## Mississippi    1.7787032838 0.2307929344 0.0465917420 1.2520280757 0.2976238695
## Missouri       0.6920571260 0.9944204833 1.5130565286 1.7707556534 1.4186673280
## Montana        1.3743202082 0.1221136254 0.4169708548 1.8817092914 0.8062347963
## Nebraska       0.0766833137 1.5832122171 1.8392595313 0.7896306497 1.8731089707
## Nevada         0.9457530653 1.1043225410 1.5214576814 1.7554477567 1.1456075438
## New Hampshire  0.0685318801 1.5114611931 1.6067550329 0.4788920208 1.8471321619
## New Jersey     0.1428893334 1.4977856935 1.4894347169 0.3349330489 1.7528642471
## New Mexico     1.9220152639 0.9211102866 0.7521270863 1.3514751901 0.1026385722
## New York       0.4832462132 1.5362466083 1.2593194566 0.0658178933 1.3635339873
## North Carolina 1.9239337577 0.5800226666 0.2093686710 0.9829108486 0.0822455043
## North Dakota   0.7111361480 1.9433442437 1.9375995064 0.7318710437 1.1802334153
## Ohio           0.0008935177 1.2528560274 1.5637871555 0.8560339620 1.9887584301
## Oklahoma       0.0990717118 1.6569136357 1.7697115639 0.5275210878 1.8093139913
## Oregon         0.7680844013 1.5730414633 1.8532631357 1.3402045079 1.2334575070
## Pennsylvania   0.0898935464 1.2269824824 1.3208890394 0.5058662614 1.8534931631
## Rhode Island   0.4526942781 1.6448300708 1.3889194427 0.0785015366 1.3871954150
## South Carolina 1.8919058548 0.3318660820 0.1379721449 1.3109064705 0.1798307688
## South Dakota   1.7523766779 0.3746721665 0.4799104973 1.7809199909 0.3813680915
## Tennessee      1.5082578283 0.0748635158 0.2546363442 1.7608737943 0.6565209615
## Texas          0.4157088181 0.4998564380 0.6630374235 0.9120786613 1.6343534163
## Utah           0.1829280349 1.7576986240 1.9223507859 0.6830908113 1.7344374675
## Vermont        1.6989726818 0.4493189605 0.6033963743 1.8320021142 0.4365675289
## Virginia       1.1453515261 0.0154710865 0.2749515567 1.6523181429 1.0215192876
## Washington     0.4072060469 1.7797189100 1.9958533418 0.9441436823 1.5355199270
## West Virginia  1.6706212924 0.1033420753 0.0759578361 1.4803915776 0.4500151060
## Wisconsin      0.0470896154 1.4592503680 1.5872433001 0.5285754991 1.8802894951
## Wyoming        1.5244615169 0.8017745127 0.3003023924 0.4299878625 0.4022482853
##                Massachusetts     Michigan    Minnesota  Mississippi
## Arizona
## Arkansas
## California
## Connecticut
## Delaware
## Florida
## Georgia
## Hawaii
## Idaho
## Illinois
## Indiana
## Iowa
## Kansas
## Kentucky
## Louisiana
## Maine
## Maryland
## Massachusetts
## Michigan        1.7636956971
## Minnesota       0.1157396102 1.4612041342
## Mississippi     1.7974872506 0.7711279050 1.9436498986
## Missouri        1.1735391052 0.3305334603 0.7042272760 1.4206237200
## Montana         1.8730991836 0.3433380776 1.6170350389 0.3584598904
## Nebraska        0.1824258831 1.3411654388 0.0089307360 1.9593131552
## Nevada          1.2928106546 0.1799187457 0.8715526902 1.3428922638
## New Hampshire   0.0509214002 1.7062389241 0.0480827529 1.8150046016
## New Jersey      0.0326997869 1.8310994222 0.1232585125 1.7146406808
## New Mexico      1.7499684754 0.2620238564 1.7788262138 0.4725455920
## New York        0.1050970275 1.9436978329 0.4052902860 1.4511438066
## North Carolina  1.6685156442 0.8164723416 1.9381605386 0.0970832642
## North Dakota    0.4491264156 0.9427670048 0.3940440918 1.8246843513
## Ohio            0.2540963869 1.4684033389 0.0755899327 1.7721853651
## Oklahoma        0.0540700850 1.5697834149 0.0121268718 1.9227417583
## Oregon          0.8662163263 0.4624564817 0.5575111879 1.6884461405
## Pennsylvania    0.1515599083 1.8091395300 0.1791232178 1.5908606017
## Rhode Island    0.0637188788 1.8902319776 0.3354040132 1.5575833130
## South Carolina  1.8677509035 0.6092607189 1.9937310600 0.0252834548
## South Dakota    1.9909615767 0.1567991577 1.8348274254 0.2873621766
## Tennessee       1.9110334834 0.4341594478 1.7540036170 0.2022109158
## Texas           0.7327297983 1.5982444923 0.7210151568 0.9515120290
## Utah            0.1516827453 1.3196166264 0.0326533416 1.9953372574
## Vermont         1.9624947204 0.0899487427 1.7575429666 0.3935102643
## Virginia        1.6536061860 0.7380994391 1.4823152089 0.3492030759
## Washington      0.4054086920 0.9355572949 0.1958827615 1.9519460641
## West Virginia   1.8717236676 0.6424320814 1.8927685307 0.0382369041
## Wisconsin       0.0747904671 1.6894267804 0.0486114956 1.8031850365
## Wyoming         1.1066952366 1.4637686881 1.5469507120 0.3486644824
## Arizona
## Arkansas
## California
## Connecticut
## Delaware
## Florida
## Georgia
## Hawaii
## Idaho
## Illinois
## Indiana
## Iowa
## Kansas
## Kentucky
## Louisiana
## Maine
## Maryland
## Massachusetts
## Michigan
## Minnesota
## Mississippi
## Missouri
## Montana        0.6045316904
## New Hampshire  0.9753270942 1.6762160610 0.0944258198 1.1740787998
## New Jersey     1.1764327210 1.7402593530 0.1938250919 1.3638772833
## New Mexico     0.9786622378 0.6464394712 1.7283779674 0.6941197972
## New York       1.5915268979 1.8682669904 0.5165855203 1.6816349620
## North Carolina 1.5924581459 0.6737159453 1.9718775908 1.4063365826
## North Dakota   0.6830710674 1.6803315621 0.3685595897 0.5721775917
## Ohio           0.6576215264 1.3382688581 0.0799277409 0.9141278109
## Oklahoma       0.8504004103 1.7213697970 0.0400813081 1.0038690131
## Oregon         0.2061823801 1.1389256129 0.4609496709 0.1269683518
## Pennsylvania   1.1087728289 1.5094403567 0.2418010391 1.3642063363
## Rhode Island   1.5058012553 1.9218780747 0.4367512096 1.5797984267
## South Carolina 1.3388614631 0.3642685539 1.9926190787 1.2000626101
## South Dakota   0.7214375288 0.1412606043 1.7548117796 0.5883807964
## Tennessee      0.8194671802 0.0260503959 1.6988835050 0.8305488553
## Texas          1.1609617032 0.8604839970 0.7712377545 1.4587503116
## Utah           0.6463927063 1.6766182529 0.0282609326 0.7422426424
## Vermont        0.6016542016 0.1584814585 1.6638475230 0.4649848933
## Virginia       0.8433111101 0.1011766981 1.4488244017 0.9852602218
## Washington     0.4159543628 1.5139937403 0.1479398762 0.4230425992
## West Virginia  1.1873957397 0.1775088901 1.8808474690 1.1585425398
## Wisconsin      0.9388344464 1.6237643420 0.0907331230 1.1531610665
## Wyoming        1.9606476936 1.1492687297 1.6534450382 1.8782873229
##                New Hampshire   New Jersey   New Mexico     New York
## Arizona
## Arkansas
## California
## Connecticut
## Delaware
## Florida
## Georgia
## Hawaii
## Idaho
## Illinois
## Indiana
## Iowa
## Kansas
## Kentucky
## Louisiana
## Maine
## Maryland
## Massachusetts
## Michigan
## Minnesota
## Mississippi
## Missouri
## Montana
## New Hampshire
## New Jersey      0.0203728912
## New Mexico      1.8957741979 1.8898013836
## New York        0.2222963805 0.1167466764 1.6576106697
## North Carolina  1.8198707129 1.6882031327 0.2961511702 1.2898437488
## North Dakota    0.5767658153 0.6537637867 0.9886387711 0.7317483654
## Ohio            0.0844858459 0.1653658184 1.9141075942 0.5195838485
## Oklahoma        0.0257607594 0.0723185219 1.7843815492 0.2912931360
## Oregon          0.8487673525 1.0152130104 0.8384753777 1.2771795023
## Pennsylvania    0.0555306843 0.0458419220 1.9848712613 0.2293760187
## Rhode Island    0.1891617632 0.1022025050 1.6356190421 0.0100380185
## South Carolina  1.9184134692 1.8343221684 0.2961797316 1.5562559201
## South Dakota    1.9394642587 1.9795876396 0.2170833923 1.9390582991
## Tennessee       1.7575234093 1.7770806331 0.5877749579 1.8030281330
## Texas           0.5300029920 0.4979806053 1.8114475771 0.6573595425
## Utah            0.1338905442 0.2212266249 1.5962259834 0.4740991019
## Vermont         1.9022004740 1.9674825759 0.2155586374 1.9727487501
## Virginia        1.4233295694 1.4410928294 1.0074305682 1.5633590655
## Washington      0.4011662888 0.5321616975 1.2625519956 0.7986747615
## West Virginia   1.8063579661 1.7517474666 0.5358830622 1.6088350316
## Wisconsin       0.0023869041 0.0295769366 1.9179607226 0.2551727018
## Wyoming         1.2901545938 1.0922252594 0.8161265668 0.6545829567
##                North Carolina North Dakota         Ohio     Oklahoma
## Arizona
## Arkansas
## California
## Connecticut
## Delaware
## Florida
## Georgia
## Hawaii
## Idaho
## Illinois
## Indiana
## Iowa
## Kansas
## Kentucky
## Louisiana
## Maine
## Maryland
## Massachusetts
## Michigan
## Minnesota
## Mississippi
## Missouri
## Montana
## New Hampshire
## New Jersey
## New Mexico
## New York
## North Carolina
## North Dakota     1.5334660335
## Ohio             1.9305660879 0.7224273197
## Oklahoma         1.8730569173 0.3894724849 0.1135825137
## Oregon           1.5840246033 0.1874986867 0.7535614934 0.6421908533
## Pennsylvania     1.6948438658 0.8972866596 0.1041933456 0.1528087139
## Rhode Island     1.3670234039 0.5992124162 0.4885377431 0.2280287820
## South Carolina   0.0629087954 1.6882794009 1.8840309592 1.9789921433
## South Dakota     0.4004425720 1.4433502477 1.7239773779 1.9077298547
## Tennessee        0.4849926609 1.7895126804 1.4781317108 1.8294151211
## Texas            1.2753845397 1.5920099336 0.4137677415 0.7259247856
## Utah             1.9051389528 0.2132190165 0.1918326612 0.0445269026
## Vermont          0.4999187795 1.3338138932 1.6682254904 1.8439010025
## Virginia         0.7371528881 1.8870575997 1.1168773891 1.5582034780
## Washington       1.8241715155 0.0918509747 0.4069737096 0.2395915914
## West Virginia    0.2356779973 1.8762239882 1.6534685677 1.9128679762
## Wisconsin        1.8380878909 0.6165079814 0.0606859470 0.0344655601
## Wyoming          0.2151566980 1.4818868846 1.5516658256 1.4171228531
##                      Oregon Pennsylvania Rhode Island South Carolina
## Arizona
## Arkansas
## California
## Connecticut
## Delaware
## Florida
## Georgia
## Hawaii
## Idaho
## Illinois
## Indiana
## Iowa
## Kansas
## Kentucky
## Louisiana
## Maine
## Maryland
## Massachusetts
## Michigan
## Minnesota
## Mississippi
## Missouri
## Montana
## New Hampshire
## New Jersey
## New Mexico
## New York
## North Carolina
## North Dakota
## Ohio
## Oklahoma
## Oregon
## Pennsylvania   1.1402152025
## Rhode Island   1.1406450586 0.2419360239
## South Carolina 1.5299656498 1.7541267394 1.6393268340
## South Dakota   0.9998229194 1.8783760662 1.9621421149   0.1941116530
## Tennessee      1.3261513949 1.5547510403 1.8786284310   0.2195857744
## Texas          1.5792856110 0.2728562511 0.7503357874   1.1713485991
## Utah           0.3846795223 0.3382295113 0.3766769616   1.9892920549
## Vermont        0.8603052484 1.8791156203 1.9772079214   0.2819179152
## Virginia       1.4605553633 1.1540094022 1.6597640598   0.4524906907
## Washington     0.1250356268 0.6790861261 0.6693868115   1.8639160485
## West Virginia  1.5911289550 1.5687910119 1.7113045304   0.0716453706
## Wisconsin      0.8550107222 0.0449914095 0.2253652497   1.9138657713
## Wyoming        1.8441095424 1.1245036904 0.7536875801   0.4132556859
##                South Dakota    Tennessee        Texas         Utah      Vermont
## Arizona
## Arkansas
## California
## Connecticut
## Delaware
## Florida
## Georgia
## Hawaii
## Idaho
## Illinois
## Indiana
## Iowa
## Kansas
## Kentucky
## Louisiana
## Maine
## Maryland
## Massachusetts
## Michigan
## Minnesota
## Mississippi
## Missouri
## Montana
## New Hampshire
## New Jersey
## New Mexico
## New York
## North Carolina
## North Dakota
## Ohio
## Oklahoma
## Oregon
## Pennsylvania
## Rhode Island
## South Carolina
## South Dakota
## Tennessee      0.1273627621
## Texas          1.3532551592 0.8523188788
## Utah           1.7739220612 1.8219725457 0.9734951576
## Vermont        0.0100223889 0.1736851072 1.4144547921 1.6770332905
## Virginia       0.4074246189 0.0924766016 0.4570537882 1.6435425977 0.4631778030
## Washington     1.4820379795 1.6902263278 1.3043144506 0.0818347191 1.3541598701
## West Virginia  0.2010383453 0.0699588182 0.8590835241 1.9658528281 0.2891370076
## Wisconsin      1.9166857823 1.7120146536 0.4867300251 0.1456776132 1.8791013765
## Wyoming        1.0026095155 0.9216589901 0.9040447147 1.5974824356 1.1320832930
##                    Virginia   Washington West Virginia    Wisconsin
## Arizona
## Arkansas
## California
## Connecticut
## Delaware
## Florida
## Georgia
## Hawaii
## Idaho
## Illinois
## Indiana
## Iowa
## Kansas
## Kentucky
## Louisiana
## Maine
## Maryland
## Massachusetts
## Michigan
## Minnesota
## Mississippi
## Missouri
## Montana
## New Hampshire
## New Jersey
## New Mexico
## New York
## North Carolina
## North Dakota
## Ohio
## Oklahoma
## Oregon
## Pennsylvania
## Rhode Island
## South Carolina
## South Dakota
## Tennessee
## Texas
## Utah
## Vermont
## Virginia
## Washington     1.6597348538
## West Virginia  0.1816146555 1.8976998489
## Wisconsin      1.3640483557 0.4147626496  1.7773345725
## Wyoming        0.9669925767 1.7609179290  0.5618676063 1.3197538043