{"id":423,"date":"2019-09-01T23:50:52","date_gmt":"2019-09-01T14:50:52","guid":{"rendered":"https:\/\/best-biostatistics.com\/toukei-er\/entry\/correspondence-analysis-with-corresp-and-ca-in-r\/"},"modified":"2024-11-17T21:33:59","modified_gmt":"2024-11-17T12:33:59","slug":"correspondence-analysis-with-corresp-and-ca-in-r","status":"publish","type":"post","link":"https:\/\/best-biostatistics.com\/toukei-er\/entry\/correspondence-analysis-with-corresp-and-ca-in-r\/","title":{"rendered":"R \u3067\u30b3\u30ec\u30b9\u30dd\u30f3\u30c7\u30f3\u30b9\u5206\u6790\u306e\u8a08\u7b97\u3092 Step by Step \u3067\u78ba\u8a8d\u3059\u308b"},"content":{"rendered":"\n

\u30b3\u30ec\u30b9\u30dd\u30f3\u30c7\u30f3\u30b9\u5206\u6790\uff08\u5bfe\u5fdc\u5206\u6790\u3068\u3082\u8a00\u3046\uff09 \u306f\u3001\u5927\u304d\u306a\u5206\u5272\u8868\u306b\u96c6\u8a08\u3055\u308c\u305f\u30c7\u30fc\u30bf\u3092\u898b\u3084\u3059\u304f\u3059\u308b\u5206\u6790\u65b9\u6cd5\u3002<\/p>\n\n\n\n

\u4e8c\u6b21\u5143 \u3064\u307e\u308a X\u8ef8\u3068Y\u8ef8\u306b\u5909\u63db\u3057\u3066\u3001\u6563\u5e03\u56f3\u306b\u3057\u3066\u50be\u5411\u3092\u898b\u308b\u3002<\/p>\n\n\n\n\n\n\n\n

\u30b3\u30ec\u30b9\u30dd\u30f3\u30c7\u30f3\u30b9\u5206\u6790\u3068\u306f\uff1f<\/h2>\n\n\n\n

\u30b3\u30ec\u30b9\u30dd\u30f3\u30c7\u30f3\u30b9\u5206\u6790\u3068\u306f\u3001\u5bfe\u5fdc\u5206\u6790\u3068\u3082\u547c\u3070\u308c\u3001\u5206\u5272\u8868\u306b\u96c6\u8a08\u3057\u305f\u30c7\u30fc\u30bf\u3092\u3001\u884c\u3068\u5217\u305d\u308c\u305e\u308c\u7279\u5fb4\u3092\u5206\u6790\u3057\u3001\u4e8c\u6b21\u5143\u6563\u5e03\u56f3\u306b\u3057\u3066\u50be\u5411\u3092\u773a\u3081\u3066\u307f\u308b\u5206\u6790\u65b9\u6cd5\u3060\u3002<\/p>\n\n\n\n

\u4e09\u6b21\u5143\u3082\u53ef\u80fd\u3060\u304c\u3001\u4e8c\u6b21\u5143\u304c\u4e00\u822c\u7684\u3067\u898b\u3084\u3059\u3044\u3068\u601d\u3046\u3002<\/p>\n\n\n\n

\u904e\u53bb\u8a18\u4e8b\u306f\u3053\u3061\u3089\u3002<\/p>\n\n\n

\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\u3042\u308f\u305b\u3066\u8aad\u307f\u305f\u3044<\/span>\n\t\t\t\t\t
\"\"<\/figure><\/div>\t\t\t\t\t
\n\t\t\t\t\t\tR \u3067\u30b3\u30ec\u30b9\u30dd\u30f3\u30c7\u30f3\u30b9\u5206\u6790\u3092\u5b9f\u65bd\u3059\u308b\u65b9\u6cd5<\/a>\n\t\t\t\t\t\t\u30b3\u30ec\u30b9\u30dd\u30f3\u30c7\u30f3\u30b9\u5206\u6790\uff08\u5bfe\u5fdc\u5206\u6790\uff09\u3068\u306f\u3001\u4eba\u306e\u5c5e\u6027\u306e\u985e\u4f3c\u6027(\u3082\u3057\u304f\u306f\u5bfe\u5fdc)\u3092\u56f3\u306b\u8868\u3059\u5206\u6790\u3002 \u7d50\u679c\u3092\u898b\u308b\u3068\u3044\u308d\u3044\u308d\u3068\u89e3\u91c8\u3057\u305f\u304f\u306a\u308b\u3001\u9762\u767d\u3044\u5206\u6790\u3002 R \u3067\u306f\u3001ca\u3068\u3044\u3046\u30d1\u30c3…<\/span>\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\n\n\n\n\n\n

\u30b3\u30ec\u30b9\u30dd\u30f3\u30c7\u30f3\u30b9\u5206\u6790\u306e\u5b9f\u969b\u4f8b<\/h2>\n\n\n\n

\u30b5\u30f3\u30d7\u30eb\u30c7\u30fc\u30bf<\/h3>\n\n\n\n

MASS \u30d1\u30c3\u30b1\u30fc\u30b8\u306e caith\u30c7\u30fc\u30bf\u3092\u4f7f\u7528\u3059\u308b\u3002<\/p>\n\n\n\n

MASS \u30d1\u30c3\u30b1\u30fc\u30b8\u3092\u4f7f\u3046\u6e96\u5099\u3092\u3059\u308b\u3002<\/p>\n\n\n\n

MASS \u306f\u7d71\u8a08\u30bd\u30d5\u30c8R\u3092\u30a4\u30f3\u30b9\u30c8\u30fc\u30eb\u3057\u305f\u6642\u306b\u3001\u540c\u6642\u306b\u30a4\u30f3\u30b9\u30c8\u30fc\u30eb\u3055\u308c\u3066\u3044\u308b\u306e\u3067\u3001\u6539\u3081\u3066\u306e\u30a4\u30f3\u30b9\u30c8\u30fc\u30eb\u306f\u4e0d\u8981\u3002<\/p>\n\n\n\n

library<\/span>(<\/span>MASS)<\/span>\n<\/code><\/pre>\n\n\n\n
caith \u30c7\u30fc\u30bf\u306f\u3001\u30b9\u30b3\u30c3\u30c8\u30e9\u30f3\u30c9\u306e\u30b1\u30a4\u30b9\u30cd\u30b9 \uff08Caithness\uff09\u306b\u4f4f\u3080\u4eba\u3005\u306e\u76ee\uff08\u5149\u5f69\uff09\u306e\u8272\u3068\u6bdb\u9aea\u306e\u8272\u3092\u96c6\u8a08\u3057\u305f\u884c\u5217\u30c7\u30fc\u30bf\u3067\u3042\u308b\u3002<\/p>\n\n\n\n
> caith\n       fair red medium dark black\nblue    326  38    241  110     3\nlight   688 116    584  188     4\nmedium  343  84    909  412    26\ndark     98  48    403  681    85\n<\/code><\/pre>\n\n\n\n
\u5b9f\u969b\u306b\u5206\u6790\u3057\u3066\u307f\u308b<\/h3>\n\n\n\n
\u5206\u6790\u3059\u308b\u305f\u3081\u306e\u95a2\u6570\u306fcorresp()\u3067\u3042\u308b\u3002<\/p>\n\n\n\n
\u884c\u5217\u3092corresp()\u5185\u306b\u5165\u308c\u308b\u3002<\/p>\n\n\n\n
nf=\u3067\u6b21\u5143\u306e\u6570\u3092\u6307\u5b9a\u3059\u308b\u3002<\/p>\n\n\n\n
nf=2\u3068\u6307\u5b9a\u3059\u308b\u3068\u6700\u7d42\u7684\u306b\u4e8c\u6b21\u5143\u30d7\u30ed\u30c3\u30c8\u306b\u306a\u308b\u3002<\/p>\n\n\n\n
First canonical correlation\u306f\u3001\u7279\u7570\u5024 Singular values \u3068\u3082\u8a00\u308f\u308c\u3001\u56fa\u6709\u5024 Eigenvalues\u306e\u5e73\u65b9\u6839\u3067\u3042\u308b\u3002<\/p>\n\n\n\n
\u884c\uff08row\uff09\u3068\u5217\uff08column\uff09\u306e\u30b9\u30b3\u30a2\u304c\u8a08\u7b97\u3055\u308c\u308b\u3002\u6a19\u6e96\u5ea7\u6a19 Standard coordinates\u3068\u3082\u8a00\u308f\u308c\u308b\u3002<\/p>\n\n\n\n
corresp1 <-<\/span> corresp<\/span>(<\/span>caith,<\/span>nf=<\/span>2<\/span>)<\/span>\ncorresp1\ncorresp1$<\/span>cor^<\/span>2<\/span> # eigenvalues\nbiplot<\/span>(<\/span>corresp1)<\/span> # biplot\ncorresp1$<\/span>rscore %*%<\/span> diag<\/span>(<\/span>corresp1$<\/span>cor)<\/span>\ncorresp1$<\/span>cscore %*%<\/span> diag<\/span>(<\/span>corresp1$<\/span>cor)<\/span>\n<\/code><\/pre>\n\n\n\n
> corresp1 <- corresp(caith, nf=2)\n> corresp1\nFirst canonical correlation(s): 0.4463684 0.1734554 \n\n Row scores:\n              [,1]       [,2]\nblue    0.89679252  0.9536227\nlight   0.98731818  0.5100045\nmedium -0.07530627 -1.4124778\ndark   -1.57434710  0.7720361\n\n Column scores:\n              [,1]       [,2]\nfair    1.21871379  1.0022432\nred     0.52257500  0.2783364\nmedium  0.09414671 -1.2009094\ndark   -1.31888486  0.5992920\nblack  -2.45176017  1.6513565\n\n><\/span> corresp1$<\/span>cor^2<\/span> # eigenvalues<\/span>\n[1]<\/span> 0.19924475<\/span> 0.03008677<\/span>\n<\/code><\/pre>\n\n\n\n
\u30d7\u30ed\u30c3\u30c8\u3057\u305f\u7d50\u679c\u306f\u3053\u3061\u3089\u3002<\/p>\n\n\n\n
\u9ed2\u5b57\u304c\u76ee\u306e\u8272\u3001\u8d64\u5b57\u304c\u9aea\u306e\u6bdb\u306e\u8272\u3067\u3042\u308b\u3002<\/p>\n\n\n\n
\u76ee\u306e\u8272\u3068\u9aea\u306e\u8272\u306f\u540c\u3058\u50be\u5411\u304c\u3042\u308a\u3001\u6697\u3044\u8272\u3068\u660e\u308b\u3044\u8272\u304c\u4e00\u7dd2\u306b\u306a\u3063\u3066\u3044\u308b\u3068\u8a00\u3048\u308b\u3002<\/p>\n\n\n\n
X\u8ef8\u306f\u30d7\u30e9\u30b9\u5024\u304c\u6697\u3044\u8272\u3001\u30de\u30a4\u30ca\u30b9\u5024\u304c\u660e\u308b\u3044\u8272\u3068\u8a00\u3048\u308b\u3002<\/p>\n\n\n\n
Y\u8ef8\u306f\u3001\u89e3\u91c8\u304c\u601d\u3044\u6d6e\u304b\u3070\u306a\u3044\u3002<\/p>\n\n\n\n
\"\"<\/figure>\n\n\n\n
<\/span><\/p>\n\n\n\n
\u30d7\u30ed\u30c3\u30c8\u306e\u5024\u306f\u3001\u4ee5\u4e0b\u306e\u8a08\u7b97\u7d50\u679c\u3067\u3042\u308b\u3002<\/p>\n\n\n\n
\u884c\u306e\u30b9\u30b3\u30a2\u3068\u5217\u306e\u30b9\u30b3\u30a2\u305d\u308c\u305e\u308c\u306bCanonical Correlation\u3092\u639b\u3051\u3066\u3044\u308b\u3002<\/p>\n\n\n\n
\u30b9\u30b3\u30a2\u306e\u53f3\u5074\u304b\u3089\u3001Canonical Correlation\u3092\u8981\u7d20\u3068\u3059\u308b\u5bfe\u89d2\u884c\u5217\u3092\u639b\u3051\u308b\u3068\u30d7\u30ed\u30c3\u30c8\u5024\u304c\u8a08\u7b97\u3055\u308c\u308b\u3002<\/p>\n\n\n\n
\u4e3b\u6210\u5206\u5ea7\u6a19\uff08Principal coordinates\uff09\u3068\u547c\u3070\u308c\u308b\u3002<\/p>\n\n\n\n
\u884c\u306e\u30d7\u30ed\u30c3\u30c8\uff08\u9ed2\uff09\u306f\u5de6\u3068\u4e0b\u306e\u76ee\u76db\u308a\u3001\u5217\u306e\u30d7\u30ed\u30c3\u30c8\uff08\u8d64\uff09\u306f\u53f3\u3068\u4e0a\u306e\u76ee\u76db\u308a\u3092\u4f7f\u3063\u3066\u3044\u308b\u3002<\/p>\n\n\n\n
\u300c\u4e8c\u3064\u300d\u306e\u30bb\u30c3\u30c8\uff08\u9ed2\u3068\u8d64\uff09\u3092\u4e00\u7dd2\u306b\u30d7\u30ed\u30c3\u30c8\u3057\u3066\u3044\u308b\u306e\u304c biplot \u306e\u3086\u3048\u3093\u3067\u3042\u308b\u3002<\/p>\n\n\n\n
> corresp1$rscore %*% diag(corresp1$cor)\n              [,1]        [,2]\nblue    0.40029985  0.16541100\nlight   0.44070764  0.08846303\nmedium -0.03361434 -0.24500190\ndark   -0.70273880  0.13391383\n\n> corresp1$cscore %*% diag(corresp1$cor)\n              [,1]        [,2]\nfair    0.54399533  0.17384449\nred     0.23326097  0.04827895\nmedium  0.04202412 -0.20830421\ndark   -0.58870853  0.10395044\nblack  -1.09438828  0.28643670<\/code><\/pre>\n\n\n\n
\u5225\u306e\u30d1\u30c3\u30b1\u30fc\u30b8\u3067\u5206\u6790\u30fb\u30d7\u30ed\u30c3\u30c8\u3057\u3066\u307f\u308b<\/h3>\n\n\n\n
ca \u30d1\u30c3\u30b1\u30fc\u30b8\u3067\u3082\u5206\u6790\u30fb\u30d7\u30ed\u30c3\u30c8\u304c\u3067\u304d\u308b\u3002<\/p>\n\n\n\n
\u30a4\u30f3\u30b9\u30c8\u30fc\u30eb\u3092\u3057\u3066\u304b\u3089\u5229\u7528\u3059\u308b\u3002<\/p>\n\n\n\n
install.packages<\/span>(<\/span>\"ca\"<\/span>)<\/span>\n<\/code><\/pre>\n\n\n\n
ca()\u3067\u5206\u6790\u3059\u308b\u3002<\/p>\n\n\n\n
library<\/span>(<\/span>ca)<\/span>\nca1 <-<\/span> ca<\/span>(<\/span>caith)<\/span>\nplot<\/span>(<\/span>ca1)<\/span>\nplot<\/span>(<\/span>ca1,<\/span>mass=<\/span>TRUE<\/span>)<\/span>\nca1$<\/span>rowcoord %*%<\/span> diag<\/span>(<\/span>ca1$<\/span>sv)<\/span>\nca1$<\/span>colcoord %*%<\/span> diag<\/span>(<\/span>ca1$<\/span>sv)<\/span>\n<\/code><\/pre>\n\n\n\n
\u56fa\u6709\u5024\u3068\u884c\u30fb\u5217\u305d\u308c\u305e\u308c\u306e\u30b9\u30b3\u30a2\u304c\u8a08\u7b97\u3055\u308c\u308b\u3002<\/p>\n\n\n\n
\u30b9\u30b3\u30a2\u306f\u6a19\u6e96\u5ea7\u6a19 Standard coordinates\u3068\u3082\u8a00\u308f\u308c\u308b\u3002<\/p>\n\n\n\n
$rowcoord<\/code>, $colcoord<\/code>\u3067\u3001\u884c\u3068\u5217\u306e\u6a19\u6e96\u5ea7\u6a19\u3092\u6307\u5b9a\u3057\u3066\u3044\u308b\u3002<\/p>\n\n\n\n
$sv<\/code>\u306f\u7279\u7570\u5024 Singular values\uff08\u56fa\u6709\u5024\u306e\u5e73\u65b9\u6839\uff09\u3060\u3002<\/p>\n\n\n\n
> ca1\n\n Principal inertias (eigenvalues):\n           1        2        3       \nValue      0.199245 0.030087 0.000859\nPercentage 86.56%   13.07%   0.37%   \n\n\n Rows:\n            blue    light    medium      dark\nMass    0.133284 0.293299  0.329311  0.244106\nChiDist 0.437855 0.450620  0.247359  0.715398\nInertia 0.025553 0.059557  0.020149  0.124932\nDim. 1  0.896793 0.987318 -0.075306 -1.574347\nDim. 2  0.953623 0.510004 -1.412478  0.772036\n\n\n Columns:\n            fair      red    medium      dark     black\nMass    0.270095 0.053091  0.396696  0.258214  0.021905\nChiDist 0.571235 0.265854  0.212526  0.597901  1.132193\nInertia 0.088134 0.003752  0.017918  0.092308  0.028079\nDim. 1  1.218714 0.522575  0.094147 -1.318885 -2.451760\nDim. 2  1.002243 0.278336 -1.200909  0.599292  1.651357<\/code><\/pre>\n\n\n\n
\u30d7\u30ed\u30c3\u30c8\u306e\u7d50\u679c\u306f\u3053\u3061\u3089\u3002<\/p>\n\n\n\n
corresp()\u306ebiplot()\u3068\u9055\u3063\u3066\u3001\u884c\u3082\u5217\u3082\u540c\u3058\u76ee\u76db\u308a\u3092\u4f7f\u3063\u3066\u3044\u308b\u3002<\/p>\n\n\n\n
\u9752\u5b57\u304c\u76ee\u306e\u8272\u3067\u3001\u8d64\u5b57\u304c\u9aea\u306e\u6bdb\u306e\u8272\u3060\u3002<\/p>\n\n\n\n
\"\"<\/figure>\n\n\n\n
<\/span><\/p>\n\n\n\n
Dimension1\uff08X\u8ef8\uff09\u306f\u3001\u76ee\u306e\u8272\u3068\u9aea\u306e\u8272\u304c\u3001\u30d7\u30e9\u30b9\u5024\u3067\u6697\u3044\u8272\u3001\u30de\u30a4\u30ca\u30b9\u5024\u3067\u660e\u308b\u3044\u8272\u3067\u3042\u308b\u3053\u3068\u306f\u308f\u304b\u308b\u3002<\/p>\n\n\n\n
Dimension2\uff08Y\u8ef8\uff09\u306e\u610f\u5473\u5408\u3044\u306f\u660e\u3089\u304b\u3067\u306a\u3044\u3002<\/p>\n\n\n\n
X\u8ef8\u3068Y\u8ef8\u306e\u30e9\u30d9\u30eb\u306b86.6%\u306813.1%\u3068\u66f8\u3044\u3066\u3042\u308b\u3002<\/p>\n\n\n\n
\u3053\u308c\u306f\u56fa\u6709\u5024\u5168\u4f53\u306b\u304a\u3051\u308b\u5272\u5408\u3092\u793a\u3057\u3066\u3044\u308b\u3002<\/p>\n\n\n\n
Dimension1\u306f86.6%\u3067\u56fa\u6709\u5024\u306e\u5927\u534a\u3092\u5360\u3081\u3066\u3044\u3066\u3001Dimension1\u3060\u3051\u3067\u5927\u534a\u8aac\u660e\u304c\u3064\u304f\u3068\u3044\u3046\u610f\u5473\u306b\u306a\u308b\u3002<\/p>\n\n\n\n
\u3086\u3048\u306bDimension2\u306b\u3046\u307e\u3044\u89e3\u91c8\u304c\u3064\u304b\u306a\u3044\u306e\u3082\u7d0d\u5f97\u3067\u304d\u308b\u3002<\/p>\n\n\n\n
mass=TRUE\u3092\u8ffd\u52a0\u3059\u308b\u3068\u3001n\u306e\u5927\u304d\u3055\u306b\u3088\u3063\u3066\u30dd\u30a4\u30f3\u30c8\u306e\u5927\u304d\u3055\u304c\u5909\u308f\u308b\u3002<\/p>\n\n\n\n
n\u304c\u591a\u3044\u3068\u30dd\u30a4\u30f3\u30c8\u304c\u5927\u304d\u304f\u306a\u308b\u3002<\/p>\n\n\n\n
\"\"<\/figure>\n\n\n\n
<\/span><\/p>\n\n\n\n
\u30d7\u30ed\u30c3\u30c8\u306e\u5ea7\u6a19\u306f\u3001\u6a19\u6e96\u5ea7\u6a19\u306b\u7279\u7570\u5024\uff08Singular values\uff09\u3092\u639b\u3051\u305f\u4e3b\u6210\u5206\u5ea7\u6a19\uff08Principal coordinates\uff09\u3092\u7528\u3044\u3066\u3044\u308b\u3002<\/p>\n\n\n\n
\u30d7\u30ed\u30c3\u30c8\u306b\u306f\u4e00\u5217\u76ee\u3092Dimension1\uff08X\u8ef8\uff09\u3001\u4e8c\u5217\u76ee\u3092Dimension2\uff08Y\u8ef8\uff09\u306b\u7528\u3044\u3066\u3044\u308b\u3002<\/p>\n\n\n\n
> ca1$rowcoord %*% diag(ca1$sv)\n              [,1]        [,2]         [,3]\nblue    0.40029985  0.16541100 -0.064157519\nlight   0.44070764  0.08846303  0.031773257\nmedium -0.03361434 -0.24500190 -0.005552885\ndark   -0.70273880  0.13391383  0.004345377\n\n> ca1$colcoord %*% diag(ca1$sv)\n              [,1]        [,2]         [,3]\nfair    0.54399533  0.17384449 -0.012522082\nred     0.23326097  0.04827895  0.118054940\nmedium  0.04202412 -0.20830421 -0.003236468\ndark   -0.58870853  0.10395044 -0.010116315\nblack  -1.09438828  0.28643670  0.046135954<\/code><\/pre>\n\n\n\n
\uff1e\uff1e\u3082\u3046\u7d71\u8a08\u3067\u60a9\u3080\u306e\u306f\u7d42\u308f\u308a\u306b\u3057\u307e\u305b\u3093\u304b\uff1f\u00a0<\/a><\/strong><\/span><\/p>\r\n\"\"<\/a>\r\n
\u21911\u4e07\u4eba\u4ee5\u4e0a\u306e\u533b\u7642\u5f93\u4e8b\u8005\u304c\u8cfc\u8aad\u4e2d<\/span><\/strong><\/span><\/p><\/div>
\u30b3\u30ec\u30b9\u30dd\u30f3\u30c7\u30f3\u30b9\u5206\u6790\u3092step by step\u3067\u5b9f\u65bd\u3057\u3066\u307f\u308b<\/h2>\n\n\n\n
\u6a19\u6e96\u6b8b\u5dee Standard Residuals \u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u304c\u30b3\u30ec\u30b9\u30dd\u30f3\u30c7\u30f3\u30b9\u5206\u6790\u306e\u30ad\u30e2\u3068\u306a\u308b\u3002<\/p>\n\n\n\n
\u6a19\u6e96\u6b8b\u5dee\u306f\u3001\u4e8c\u4e57\u3057\u3066\u5408\u8a08\u3059\u308b\u3068\u72ec\u7acb\u6027\u306e\u691c\u5b9a\u306b\u4f7f\u3046\u30ab\u30a4\u4e8c\u4e57 $ \\chi^2 $ \u5024\u306b\u306a\u308b\u3002<\/p>\n\n\n\n
\u6a19\u6e96\u6b8b\u5dee\u306f\u3001\u89b3\u5bdf\u5024\u3068\u671f\u5f85\u5024\u306e\u300c\u9694\u305f\u308a\u300d\u3092\u8868\u73fe\u3057\u305f\u3082\u306e\u306b\u306a\u308b\u3002<\/p>\n\n\n\n
n <- sum(caith)\nf.row <- rowSums(caith)\nf.col <- colSums(caith)\nf.mat <- caith\nz.ij <- matrix(rep(0,20),nr=4)\nfor (i in 1:4){\n  for (j in 1:5){\n    z.ij[i,j] <- (f.mat[i,j]\/n-f.row[i]*f.col[j]\/n^2)\/(sqrt(f.row[i]*f.col[j])\/n)\n  }\n}\nz.ij # Standard Residuals \u6a19\u6e96\u6b8b\u5dee\nQ.t <- z.ij %*% t(z.ij)\nQ <- t(z.ij) %*% z.ij\neigen.row <- eigen(Q.t) # \u884c\u306e\u7279\u5fb4\u3092\u56fa\u6709\u5024\u30fb\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u306b\u51dd\u7e2e\u3059\u308b\neigen.col <- eigen(Q) # \u5217\u306e\u7279\u5fb4\u3092\u56fa\u6709\u5024\u30fb\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u306b\u51dd\u7e2e\u3059\u308b\n# \u884c\u3068\u5217\u306e\u5468\u8fba\u5272\u5408\u306e\u5e73\u65b9\u6839\u306e\u9006\u6570\u3067\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u3092\u898f\u6e96\u5316\u3057\u305f\u3082\u306e\u304c\u30b9\u30b3\u30a2\uff08\u6a19\u6e96\u5ea7\u6a19\uff09\u306b\u306a\u308b\nD.r <- diag(1\/sqrt(f.row\/n))\nD.c <- diag(1\/sqrt(f.col\/n))\n# \u6a19\u6e96\u5ea7\u6a19 Standard coordinates\nD.r %*% eigen.row$vectors\nD.c %*% eigen.col$vectors\n# \u4e3b\u6210\u5206\u5ea7\u6a19 Pricipal coordinates\nD.r %*% eigen.row$vectors %*% diag(sqrt(eigen.row$values))\nD.c %*% eigen.col$vectors %*% diag(sqrt(eigen.col$values))\n<\/code><\/pre>\n\n\n\n
\u6a19\u6e96\u6b8b\u5dee\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8a08\u7b97\u3055\u308c\u308b\u3002<\/p>\n\n\n\n
> z.ij # Standard Residuals \u6a19\u6e96\u6b8b\u5dee\n           [,1]          [,2]        [,3]        [,4]        [,5]\n[1,]  0.1292162 -0.0002629922 -0.03538203 -0.07544543 -0.04372599\n[2,]  0.1723046  0.0477768696 -0.02328106 -0.14838434 -0.07088969\n[3,] -0.0847428 -0.0142960921  0.10542144 -0.02932898 -0.02810479\n[4,] -0.1859232 -0.0355710546 -0.07078163  0.25246345  0.14265843\n<\/code><\/pre>\n\n\n\n
\u6a19\u6e96\u6b8b\u5dee\u306e\u884c\u65b9\u5411\u306e\u4e8c\u4e57\u306b\u3042\u305f\u308bQ.t\u3068\u5217\u65b9\u5411\u306e\u4e8c\u4e57\u306b\u5f53\u305f\u308bQ\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3002<\/p>\n\n\n\n
> Q.t <- z.ij %*% t(z.ij)\n> Q <- t(z.ij) %*% z.ij\n\n> Q.t\n            [,1]        [,2]         [,3]         [,4]\n[1,]  0.02555277  0.03737037 -0.011234763 -0.046795636\n[2,]  0.03737037  0.05955678 -0.011394621 -0.079661660\n[3,] -0.01123476 -0.01139462  0.020149468 -0.002611605\n[4,] -0.04679564 -0.07966166 -0.002611605  0.124931987\n> Q\n             [,1]          [,2]          [,3]        [,4]         [,5]\n[1,]  0.088134492  1.602317e-02 -4.357128e-03 -0.07976947 -0.042006558\n[2,]  0.016023167  3.752377e-03 -9.232812e-05 -0.01563060 -0.008048110\n[3,] -0.004357128 -9.232812e-05  1.791761e-02 -0.01483772 -0.009862943\n[4,] -0.079769470 -1.563060e-02 -1.483772e-02  0.09230791  0.050658169\n[5,] -0.042006558 -8.048110e-03 -9.862943e-03  0.05065817  0.028078616\n<\/code><\/pre>\n\n\n\n
Q.t\u3068Q\u305d\u308c\u305e\u308c\u304b\u3089\u56fa\u6709\u5024\u30fb\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u3092\u6c42\u3081\u308b\u3068\u4ee5\u4e0b\u306e\u901a\u308a\u3002\u56fa\u6709\u5024($values<\/code>)\u304c\u884c\u3001\u5217\u3068\u3082\u306b3\u3064\u307e\u3067\u306f\u540c\u3058\u5024\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n\n\n\n
> eigen.row <- eigen(Q.t) # \u884c\u306e\u7279\u5fb4\u3092\u56fa\u6709\u5024\u30fb\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u306b\u51dd\u7e2e\u3059\u308b\n> eigen.col <- eigen(Q) # \u5217\u306e\u7279\u5fb4\u3092\u56fa\u6709\u5024\u30fb\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u306b\u51dd\u7e2e\u3059\u308b\n\n> eigen.row\neigen() decomposition\n$values\n[1]  1.992448e-01  3.008677e-02  8.594814e-04 -9.238255e-18\n\n$vectors\n            [,1]       [,2]        [,3]      [,4]\n[1,] -0.32740153  0.3481491  0.79894718 0.3650806\n[2,] -0.53470247  0.2762034 -0.58694655 0.5415706\n[3,]  0.04321499 -0.8105596  0.10869354 0.5738565\n[4,]  0.77783929  0.3814407 -0.07323162 0.4940710\n\n> eigen.col\neigen() decomposition\n$values\n[1]  1.992448e-01  3.008677e-02  8.594814e-04  7.761727e-18 -5.800482e-18\n\n$vectors\n            [,1]       [,2]        [,3]       [,4]        [,5]\n[1,] -0.63337328  0.5208721  0.22198126 -0.5274987  0.00000000\n[2,] -0.12040878  0.0641327 -0.92784507 -0.1825513 -0.29524104\n[3,] -0.05929716 -0.7563782  0.06953154 -0.6464176  0.04105527\n[4,]  0.67018848  0.3045289  0.17534533 -0.4302105 -0.49222197\n[5,]  0.36286535  0.2444040 -0.23291035 -0.2923755  0.81784150\n<\/code><\/pre>\n\n\n\n
\u884c\u3068\u5217\u306e\u5468\u8fba\u5272\u5408\uff08f.row\/n, f.col\/n\uff09\u306e\u5e73\u65b9\u6839\u306e\u9006\u6570\u3067\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u3092\u898f\u6e96\u5316\u3057\u305f\u3082\u306e\u304c\u30b9\u30b3\u30a2\u306b\u306a\u308b\u3002<\/p>\n\n\n\n
\u300c\u5272\u5408\u306e\u5e73\u65b9\u6839\u306e\u9006\u6570\u300d\u30d9\u30af\u30c8\u30eb\u306e\u5bfe\u89d2\u884c\u5217\uff08D.r, D.c\uff09\u3092\u305d\u308c\u305e\u308c\u5de6\u304b\u3089\u639b\u3051\u308b\u3002<\/p>\n\n\n\n
D.r, D.c\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3002<\/p>\n\n\n\n
> # \u884c\u3068\u5217\u306e\u5468\u8fba\u5272\u5408\u306e\u5e73\u65b9\u6839\u306e\u9006\u6570\u3067\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u3092\u898f\u6e96\u5316\u3057\u305f\u3082\u306e\u304c\u30b9\u30b3\u30a2\uff08\u6a19\u6e96\u5ea7\u6a19\uff09\u306b\u306a\u308b\n> D.r <- diag(1\/sqrt(f.row\/n))\n> D.c <- diag(1\/sqrt(f.col\/n))\n\n> D.r\n         [,1]     [,2]     [,3]  [,4]\n[1,] 2.739121 0.000000 0.000000 0.000\n[2,] 0.000000 1.846481 0.000000 0.000\n[3,] 0.000000 0.000000 1.742596 0.000\n[4,] 0.000000 0.000000 0.000000 2.024\n\n> D.c\n         [,1]     [,2]    [,3]     [,4]     [,5]\n[1,] 1.924164 0.000000 0.00000 0.000000 0.000000\n[2,] 0.000000 4.340007 0.00000 0.000000 0.000000\n[3,] 0.000000 0.000000 1.58771 0.000000 0.000000\n[4,] 0.000000 0.000000 0.00000 1.967931 0.000000\n[5,] 0.000000 0.000000 0.00000 0.000000 6.756667\n<\/code><\/pre>\n\n\n\n
\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u3068\u639b\u3051\u7b97\u3059\u308b\u3068\u30b9\u30b3\u30a2\u306b\u306a\u308b\u3002<\/p>\n\n\n\n
\u6a19\u6e96\u5ea7\u6a19 Standard coordinates \u3068\u3082\u8a00\u308f\u308c\u308b\u3002<\/p>\n\n\n\n
> # \u6a19\u6e96\u5ea7\u6a19 Standard coordinates\n> D.r %*% eigen.row$vectors\n            [,1]       [,2]       [,3] [,4]\n[1,] -0.89679252  0.9536227  2.1884132    1\n[2,] -0.98731818  0.5100045 -1.0837859    1\n[3,]  0.07530627 -1.4124778  0.1894089    1\n[4,]  1.57434710  0.7720361 -0.1482208    1\n\n> D.c %*% eigen.col$vectors\n            [,1]       [,2]       [,3]       [,4]        [,5]\n[1,] -1.21871379  1.0022432  0.4271282 -1.0149937  0.00000000\n[2,] -0.52257500  0.2783364 -4.0268545 -0.7922741 -1.28134829\n[3,] -0.09414671 -1.2009094  0.1103959 -1.0263237  0.06518388\n[4,]  1.31888486  0.5992920  0.3450676 -0.8466247 -0.96865900\n[5,]  2.45176017  1.6513565 -1.5736976 -1.9754840  5.52588230\n<\/code><\/pre>\n\n\n\n
\u4e0a\u8a18\u306e\u7d50\u679c\u306e\u4e8c\u5217\u76ee\u307e\u3067\u304c\u3001corresp()\u3084ca()\u306e\u7d50\u679c\u3068\u4e00\u81f4\u3059\u308b\u3002<\/p>\n\n\n\n
\u7b2c\u4e00\u6210\u5206\u306f\u3001\u7b26\u53f7\u304c\u9006\u3067\u3042\u308b\u304c\u3001\u540c\u3058\u3082\u306e\u3068\u8003\u3048\u3066\u3088\u3044\u3002<\/p>\n\n\n\n
> corresp1\nFirst canonical correlation(s): 0.4463684 0.1734554 \n\n Row scores:\n              [,1]       [,2]\nblue    0.89679252  0.9536227\nlight   0.98731818  0.5100045\nmedium -0.07530627 -1.4124778\ndark   -1.57434710  0.7720361\n\n Column scores:\n              [,1]       [,2]\nfair    1.21871379  1.0022432\nred     0.52257500  0.2783364\nmedium  0.09414671 -1.2009094\ndark   -1.31888486  0.5992920\nblack  -2.45176017  1.6513565<\/code><\/pre>\n\n\n\n
Dim.1, Dim.2 \u306e\u884c\u3092\u898b\u308b\u3068\u3001\u4e0a\u8a18\u306e\u6700\u521d\u306e 2 \u5217\u3068\u540c\u3058\u5024\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n\n\n\n
> ca1\n\n Principal inertias (eigenvalues):\n           1        2        3       \nValue      0.199245 0.030087 0.000859\nPercentage 86.56%   13.07%   0.37%   \n\n\n Rows:\n            blue    light    medium      dark\nMass    0.133284 0.293299  0.329311  0.244106\nChiDist 0.437855 0.450620  0.247359  0.715398\nInertia 0.025553 0.059557  0.020149  0.124932\nDim. 1  0.896793 0.987318 -0.075306 -1.574347\nDim. 2  0.953623 0.510004 -1.412478  0.772036\n\n\n Columns:\n            fair      red    medium      dark     black\nMass    0.270095 0.053091  0.396696  0.258214  0.021905\nChiDist 0.571235 0.265854  0.212526  0.597901  1.132193\nInertia 0.088134 0.003752  0.017918  0.092308  0.028079\nDim. 1  1.218714 0.522575  0.094147 -1.318885 -2.451760\nDim. 2  1.002243 0.278336 -1.200909  0.599292  1.651357<\/code><\/pre>\n\n\n\n
\u6a19\u6e96\u5ea7\u6a19 Standard coordinates\u306b\u3001\u53f3\u5074\u304b\u3089\u56fa\u6709\u5024\u306e\u5e73\u65b9\u6839\uff08\u7279\u7570\u5024 Singular values\uff09\u3092\u8981\u7d20\u3068\u3059\u308b\u5bfe\u89d2\u884c\u5217\u3092\u639b\u3051\u308b\u3068\u3001\u4e3b\u6210\u5206\u5ea7\u6a19 Principal coordinates\u306b\u306a\u308b\u3002<\/p>\n\n\n\n
\u3053\u308c\u304ccorresp()\u3084ca()\u306e\u30d7\u30ed\u30c3\u30c8\u5ea7\u6a19\u306b\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n\n\n\n
\u5de6\u304b\u3089\u4e00\u5217\u76ee\u3092Dimension1\uff08X\u8ef8\uff09\u306b\u3001\u4e8c\u5217\u76ee\u3092Dimension2\uff08Y\u8ef8\uff09\u306b\u3057\u3066\u3044\u308b\u3002<\/p>\n\n\n\n
> # \u4e3b\u6210\u5206\u5ea7\u6a19 Pricipal coordinates\n> D.r %*% eigen.row$vectors %*% diag(sqrt(eigen.row$values))\n            [,1]        [,2]         [,3] [,4]\n[1,] -0.40029985  0.16541100  0.064157519  NaN\n[2,] -0.44070764  0.08846303 -0.031773257  NaN\n[3,]  0.03361434 -0.24500190  0.005552885  NaN\n[4,]  0.70273880  0.13391383 -0.004345377  NaN\n\u8b66\u544a\u30e1\u30c3\u30bb\u30fc\u30b8:\nsqrt(eigen.row$values) \u3067: \u8a08\u7b97\u7d50\u679c\u304c NaN \u306b\u306a\u308a\u307e\u3057\u305f\n\n> D.c %*% eigen.col$vectors %*% diag(sqrt(eigen.col$values))\n            [,1]        [,2]         [,3]          [,4] [,5]\n[1,] -0.54399533  0.17384449  0.012522082 -2.827760e-09  NaN\n[2,] -0.23326097  0.04827895 -0.118054940 -2.207266e-09  NaN\n[3,] -0.04202412 -0.20830421  0.003236468 -2.859325e-09  NaN\n[4,]  0.58870853  0.10395044  0.010116315 -2.358686e-09  NaN\n[5,]  1.09438828  0.28643670 -0.046135954 -5.503674e-09  NaN\n\u8b66\u544a\u30e1\u30c3\u30bb\u30fc\u30b8:\nsqrt(eigen.col$values) \u3067: \u8a08\u7b97\u7d50\u679c\u304c NaN \u306b\u306a\u308a\u307e\u3057\u305f\n<\/code><\/pre>\n\n\n\n
\u56fa\u6709\u5024\u30fb\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u3068\u306f\uff1f<\/h3>\n\n\n\n
\u56fa\u6709\u5024\u30fb\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u3068\u306f\u3001\u30c7\u30fc\u30bf\u304b\u3089\u4f5c\u3063\u305f\u30c7\u30fc\u30bf\u306e\u7279\u5fb4\u3092\u8868\u3059\u884c\u5217\u3092\u3001\u5909\u63db\u3057\u3066\u8a08\u7b97\u3057\u305f\u5024\u30fb\u30d9\u30af\u30c8\u30eb\u3068\u8a00\u3048\u308b\u3002<\/p>\n\n\n\n
\u305f\u3068\u3048\u3070\u3001\u30c7\u30fc\u30bf\u304b\u3089\u4f5c\u3063\u305f\u7279\u5fb4\u306e\u884c\u5217\u3092 A \u3068\u3059\u308b\u3002<\/p>\n\n\n\n
\u4e0b\u8a18\u3092\u6e80\u305f\u3059 \u03bb \u304c\u56fa\u6709\u5024\u3001x \u304c\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u3068\u8a00\u308f\u308c\u308b\u3002<\/p>\n\n\n\n
$$ \\boldsymbol{Ax} = \\lambda \\boldsymbol{x} $$<\/p>\n\n\n\n\n\n\n\n
\u56fa\u6709\u5024\u3001\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u306f\u3044\u304f\u3064\u3082\u5b58\u5728\u3059\u308b\u3002<\/p>\n\n\n\n
\u4e0a\u8a18Q.t\u306b\u95a2\u3057\u3066\u306f\u3001eigen()\u3067\u6c42\u3081\u308b\u3068\uff14\u3064\u8a08\u7b97\u3055\u308c\u308b\u3002<\/p>\n\n\n\n
\u7279\u5fb4\u306e\u884c\u5217\uff08\u4eca\u56de\u306fQ.t\uff09\u306e\u884c\uff08\u307e\u305f\u306f\u5217\uff09\u3068\u540c\u3058\u6570\u3060\u3051\u8a08\u7b97\u3055\u308c\u308b\u3002<\/p>\n\n\n\n
\u30c7\u30fc\u30bf\u304b\u3089\u4f5c\u3063\u305f\u884c\u5217\u3092\u3001\u30ae\u30e5\u30c3\u3068\u51dd\u7e2e\u3057\u305f\u3082\u306e\u304c\u3001\u56fa\u6709\u5024\u30fb\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u3060\u3068\u8a00\u3048\u308b\u3002<\/p>\n\n\n\n
> Q.t\n            [,1]        [,2]         [,3]         [,4]\n[1,]  0.02555277  0.03737037 -0.011234763 -0.046795636\n[2,]  0.03737037  0.05955678 -0.011394621 -0.079661660\n[3,] -0.01123476 -0.01139462  0.020149468 -0.002611605\n[4,] -0.04679564 -0.07966166 -0.002611605  0.124931987\n\n> eigen.row <- eigen(Q.t) # \u884c\u306e\u7279\u5fb4\u3092\u56fa\u6709\u5024\u30fb\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u306b\u51dd\u7e2e\u3059\u308b\n> eigen.row\neigen() decomposition\n$values\n[1]  1.992448e-01  3.008677e-02  8.594814e-04 -9.238255e-18\n\n$vectors\n            [,1]       [,2]        [,3]      [,4]\n[1,] -0.32740153  0.3481491  0.79894718 0.3650806\n[2,] -0.53470247  0.2762034 -0.58694655 0.5415706\n[3,]  0.04321499 -0.8105596  0.10869354 0.5738565\n[4,]  0.77783929  0.3814407 -0.07323162 0.4940710\n<\/code><\/pre>\n\n\n\n
\u691c\u7b97\u3057\u3066\u307f\u308b\u3068 $ \\boldsymbol{Ax} = \\lambda \\boldsymbol{x} $ \u3067\u3042\u308b\u3053\u3068\u304c\u78ba\u8a8d\u3067\u304d\u308b\u3002<\/p>\n\n\n\n
## \u5de6\u8fba<\/span>\n> Q.t %*% eigen.row$vector[,1]\n            [,1]\n[1,] -0.06523304\n[2,] -0.10653666\n[3,]  0.00861036\n[4,]  0.15498040\n\n## \u53f3\u8fba<\/span>\n> matrix(eigen.row$value[1]*eigen.row$vector[,1])\n            [,1]\n[1,] -0.06523304\n[2,] -0.10653666\n[3,]  0.00861036\n[4,]  0.15498040\n<\/code><\/pre>\n\n\n\n
Singular Value Decomposition\u3067Step by step<\/h2>\n\n\n\n
\u3055\u304d\u307b\u3069\u306feigen()\u3092\u4f7f\u3063\u3066\u3001\u56fa\u6709\u5024\u30fb\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u3092\u6c42\u3081\u305f\u304c\u3001svd()\u3092\u4f7f\u3046\u65b9\u6cd5\u3082\u3042\u308b\u3002<\/p>\n\n\n\n
SVD \u306f\u3001Singular Value Decomposition \u306e\u982d\u6587\u5b57\u8a9e\u3067\u3001\u7279\u7570\u5024\uff08Singular values\uff09\u3068\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u306b\u5206\u89e3\u3059\u308b\u65b9\u6cd5\u3060\u3002<\/p>\n\n\n\n
\u7279\u7570\u5024\u306f\u3001\u56fa\u6709\u5024\u306e\u5e73\u65b9\u6839\u3060\u3002<\/p>\n\n\n\n
canonical correlation\u3068\u3082\u547c\u3070\u308c\u308b\u3002<\/p>\n\n\n\n
\u6a19\u6e96\u6b8b\u5dee\u884c\u5217\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5206\u89e3\u3067\u304d\u308b\u3002<\/p>\n\n\n\n
$$ \\boldsymbol{z_{ij}} = \\boldsymbol{U D_{\\alpha} V^T } $$<\/p>\n\n\n\n\n\n\n\n
\u3053\u3053\u3067 $ D_{\\alpha} $ \u306f\u3001\u7279\u7570\u5024\u3092\u8981\u7d20\u3068\u3059\u308b\u5bfe\u89d2\u884c\u5217\u3001U\u3001V \u306f\u305d\u308c\u305e\u308c\u3001\u884c\u3068\u5217\u306e\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b\u3002<\/p>\n\n\n\n
S.svd <-<\/span> svd(<\/span>z.ij)<\/span> # Singular Value Decompsition<\/span>\nS.svd$<\/span>d # Singular values<\/span>\nS.svd$<\/span>d^2<\/span> # Eigenvalues or principal inertia<\/span>\nS.svd$<\/span>u # Eigenvector of row<\/span>\nS.svd$<\/span>v # Eigenvector of column<\/span>\nS.svd$<\/span>u %*%<\/span> diag(<\/span>S.svd$<\/span>d)<\/span> %*%<\/span> t(<\/span>S.svd$<\/span>v)<\/span> # \u6a19\u6e96\u6b8b\u5dee\u884c\u5217\nz.ij # for confirmation<\/span>\nD.r %*% S.svd$u # \u6a19\u6e96\u5ea7\u6a19 Standard coordinates\nD.c %*% S.svd$v # \u6a19\u6e96\u5ea7\u6a19 Standard coordinates\nD.r %*% S.svd$u %*% diag(S.svd$d) # \u4e3b\u6210\u5206\u5ea7\u6a19 Principal coordinates\nD.c %*% S.svd$v %*% diag(S.svd$d) # \u4e3b\u6210\u5206\u5ea7\u6a19 Principal coordinates<\/code><\/pre>\n\n\n\n
\u7279\u7570\u5024 Singular values \u306e\u4e8c\u4e57\u304c\u56fa\u6709\u5024\u306b\u306a\u308b\u3002<\/p>\n\n\n\n
><\/span> S.svd$<\/span>d # Singular values<\/span>\n[<\/span>1<\/span>]<\/span> 4.463684e-01<\/span> 1.734554e-01<\/span> 2.931691e-02<\/span> 1.829859e-17<\/span>\n><\/span> S.svd$<\/span>d^<\/span>2<\/span> # Eigenvalues<\/span>\n[<\/span>1<\/span>]<\/span> 1.992448e-01<\/span> 3.008677e-02<\/span> 8.594814e-04<\/span> 3.348386e-34<\/span>\n<\/code><\/pre>\n\n\n\n
\u884c\u306e\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u3068\u5217\u306e\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3002<\/p>\n\n\n\n
> S.svd$u # Eigenvector of row\n            [,1]       [,2]        [,3]      [,4]\n[1,]  0.32740153  0.3481491 -0.79894718 0.3650806\n[2,]  0.53470247  0.2762034  0.58694655 0.5415706\n[3,] -0.04321499 -0.8105596 -0.10869354 0.5738565\n[4,] -0.77783929  0.3814407  0.07323162 0.4940710\n\n> S.svd$v # Eigenvector of column\n            [,1]       [,2]        [,3]        [,4]\n[1,]  0.63337328  0.5208721 -0.22198126 -0.06089658\n[2,]  0.12040878  0.0641327  0.92784507 -0.31434151\n[3,]  0.05929716 -0.7563782 -0.06953154 -0.03384428\n[4,] -0.67018848  0.3045289 -0.17534533 -0.53859621\n[5,] -0.36286535  0.2444040  0.23291035  0.77862039\n<\/code><\/pre>\n\n\n\n
\u7279\u7570\u5024\u3092\u6210\u5206\u3068\u3057\u305f\u5bfe\u89d2\u884c\u5217\u306e\u5de6\u53f3\u304b\u3089\u3001\u884c\u306e\u56fa\u6709\u30d9\u30af\u30c8\u30eb U \u3068\u5217\u306e\u56fa\u6709\u30d9\u30af\u30c8\u30eb V \uff08\u306e\u8ee2\u7f6e\u884c\u5217\uff09\u3092\u639b\u3051\u308b\u3068\u3001\u3082\u3068\u306e\u6a19\u6e96\u6b8b\u5dee\u884c\u5217\u306b\u306a\u308b\u3002<\/p>\n\n\n\n
\u4ee5\u4e0b\u306e\u5f0f\u306e\u901a\u308a\u3060\u3002<\/p>\n\n\n\n
$$ \\boldsymbol{z_{ij}} = \\boldsymbol{U D_{\\alpha} V^T} $$<\/p>\n\n\n\n\n\n\n\n
> S.svd$u %*% diag(S.svd$d) %*% t(S.svd$v) # \u6a19\u6e96\u6b8b\u5dee\u884c\u5217\n           [,1]          [,2]        [,3]        [,4]        [,5]\n[1,]  0.1292162 -0.0002629922 -0.03538203 -0.07544543 -0.04372599\n[2,]  0.1723046  0.0477768696 -0.02328106 -0.14838434 -0.07088969\n[3,] -0.0847428 -0.0142960921  0.10542144 -0.02932898 -0.02810479\n[4,] -0.1859232 -0.0355710546 -0.07078163  0.25246345  0.14265843\n\n> z.ij # for confirmation\n           [,1]          [,2]        [,3]        [,4]        [,5]\n[1,]  0.1292162 -0.0002629922 -0.03538203 -0.07544543 -0.04372599\n[2,]  0.1723046  0.0477768696 -0.02328106 -0.14838434 -0.07088969\n[3,] -0.0847428 -0.0142960921  0.10542144 -0.02932898 -0.02810479\n[4,] -0.1859232 -0.0355710546 -0.07078163  0.25246345  0.14265843\n<\/code><\/pre>\n\n\n\n
\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u306b\u5de6\u5074\u304b\u3089\u300c\u5272\u5408\u306e\u5e73\u65b9\u6839\u306e\u9006\u6570\u300d\u30d9\u30af\u30c8\u30eb\u306e\u5bfe\u89d2\u884c\u5217\uff08D.r, D.c\uff09\u3092\u305d\u308c\u305e\u308c\u5de6\u304b\u3089\u639b\u3051\u308b\u3068\u3001\u30b9\u30b3\u30a2\u306b\u306a\u308b\u306e\u306f\u4e0a\u8ff0\u3068\u540c\u69d8\u3002<\/p>\n\n\n\n
corresp()\u3084ca()\u304c\u8fd4\u3059\u30b9\u30b3\u30a2\u3060\u3002<\/p>\n\n\n\n
\u6a19\u6e96\u5ea7\u6a19 Standard coordinates\u3068\u3082\u8a00\u3046\u3002<\/p>\n\n\n\n
> D.r %*% S.svd$u # \u6a19\u6e96\u5ea7\u6a19 Standard coordinates\n            [,1]       [,2]       [,3] [,4]\n[1,]  0.89679252  0.9536227 -2.1884132    1\n[2,]  0.98731818  0.5100045  1.0837859    1\n[3,] -0.07530627 -1.4124778 -0.1894089    1\n[4,] -1.57434710  0.7720361  0.1482208    1\n\n> D.c %*% S.svd$v # \u6a19\u6e96\u5ea7\u6a19 Standard coordinates\n            [,1]       [,2]       [,3]        [,4]\n[1,]  1.21871379  1.0022432 -0.4271282 -0.11717498\n[2,]  0.52257500  0.2783364  4.0268545 -1.36424450\n[3,]  0.09414671 -1.2009094 -0.1103959 -0.05373491\n[4,] -1.31888486  0.5992920 -0.3450676 -1.05992031\n[5,] -2.45176017  1.6513565  1.5736976  5.26087829\n<\/code><\/pre>\n\n\n\n
\u3055\u3089\u306b\u53f3\u5074\u304b\u3089\u7279\u7570\u5024\u3092\u8981\u7d20\u3068\u3059\u308b\u5bfe\u89d2\u884c\u5217\u3092\u639b\u3051\u308b\u3068\u3001\u4e3b\u6210\u5206\u5ea7\u6a19 Principal coordinates \u306b\u306a\u308b\u3002<\/p>\n\n\n\n
\u3053\u308c\u304ccorresp()\u3084ca()\u306e\u30d7\u30ed\u30c3\u30c8\u5ea7\u6a19\u306b\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n\n\n\n
\u5de6\u304b\u3089\u4e00\u5217\u76ee\u3092Dimension1\uff08X\u8ef8\uff09\u306b\u3001\u4e8c\u5217\u76ee\u3092Dimension2\uff08Y\u8ef8\uff09\u306b\u3057\u3066\u3044\u308b\u3002<\/p>\n\n\n\n
> D.r %*% S.svd$u %*% diag(S.svd$d) # \u4e3b\u6210\u5206\u5ea7\u6a19 Principal coordinates\n            [,1]        [,2]         [,3]         [,4]\n[1,]  0.40029985  0.16541100 -0.064157519 1.829859e-17\n[2,]  0.44070764  0.08846303  0.031773257 1.829859e-17\n[3,] -0.03361434 -0.24500190 -0.005552885 1.829859e-17\n[4,] -0.70273880  0.13391383  0.004345377 1.829859e-17\n\n> D.c %*% S.svd$v %*% diag(S.svd$d) # \u4e3b\u6210\u5206\u5ea7\u6a19 Principal coordinates\n            [,1]        [,2]         [,3]          [,4]\n[1,]  0.54399533  0.17384449 -0.012522082 -2.144138e-18\n[2,]  0.23326097  0.04827895  0.118054940 -2.496376e-17\n[3,]  0.04202412 -0.20830421 -0.003236468 -9.832734e-19\n[4,] -0.58870853  0.10395044 -0.010116315 -1.939505e-17\n[5,] -1.09438828  0.28643670  0.046135954  9.626668e-17<\/code><\/pre>\n\n\n\n
\u307e\u3068\u3081<\/h2>\n\n\n\n
\u30b3\u30ec\u30b9\u30dd\u30f3\u30c7\u30f3\u30b9\u5206\u6790\u3092corresp()\u3068ca()\u3092\u4f7f\u3063\u3066\u5b9f\u65bd\u3057\u3066\u307f\u305f\u3002<\/p>\n\n\n\n
\u3055\u3089\u306beigen()\u3084svd()\u3092\u4f7f\u3063\u3066\u3001step by step\u3067\u5b9f\u65bd\u3057\u3066\u307f\u305f\u3002<\/p>\n\n\n\n
\u5206\u5272\u8868\u306b\u307e\u3068\u3081\u3089\u308c\u305f\u7d50\u679c\u304c\u3001\u884c\u306e\u8981\u7d20\u3068\u5217\u306e\u8981\u7d20\u305d\u308c\u305e\u308c\u3001\u3069\u306e\u3088\u3046\u306a\u95a2\u4fc2\u306b\u3042\u308b\u304b\u3092\u56f3\u306b\u3088\u3063\u3066\u773a\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u8208\u5473\u6df1\u3044\u65b9\u6cd5\u3060\u3002<\/p>\n\n\n\n
\u5927\u304d\u306a\u5206\u5272\u8868\u3067\u3001\u884c\u3068\u5217\u306e\u5bfe\u5fdc\u304c\u628a\u63e1\u3057\u304d\u308c\u306a\u3044\u5834\u5408\u3001\u6709\u52b9\u306a\u5206\u6790\u65b9\u6cd5\u3068\u8a00\u3048\u308b\u3002<\/p>\n\n\n\n
\u53c2\u8003\u306b\u306a\u308c\u3070\u3002<\/p>\n\n\n\n
\u53c2\u8003\u6587\u732e<\/h2>\n\n\n\n
\u5bfe\u5fdc\u5206\u6790\u306b\u3088\u308b\u30c7\u30fc\u30bf\u89e3\u6790<\/p>\n\n\n\n
https:\/\/www.kwansei.ac.jp\/s_sociology\/attached\/6779_56340_ref.pdf<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"
\u30b3\u30ec\u30b9\u30dd\u30f3\u30c7\u30f3\u30b9\u5206\u6790\uff08\u5bfe\u5fdc\u5206\u6790\u3068\u3082\u8a00\u3046\uff09 \u306f\u3001\u5927\u304d\u306a\u5206\u5272\u8868\u306b\u96c6\u8a08\u3055\u308c\u305f\u30c7\u30fc\u30bf\u3092\u898b\u3084\u3059\u304f\u3059\u308b\u5206\u6790\u65b9\u6cd5\u3002 \u4e8c\u6b21\u5143 \u3064\u307e\u308a X\u8ef8\u3068Y\u8ef8\u306b\u5909\u63db\u3057\u3066\u3001\u6563\u5e03\u56f3\u306b\u3057\u3066\u50be\u5411\u3092\u898b\u308b\u3002<\/p>\n","protected":false},"author":2,"featured_media":3277,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"swell_btn_cv_data":"","_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[5,123],"tags":[],"class_list":["post-423","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-r","category-123"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"https:\/\/best-biostatistics.com\/toukei-er\/wp-content\/uploads\/2019\/09\/\u30b9\u30af\u30ea\u30fc\u30f3\u30b7\u30e7\u30c3\u30c8-2024-11-17-213012.png","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/posts\/423","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/comments?post=423"}],"version-history":[{"count":7,"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/posts\/423\/revisions"}],"predecessor-version":[{"id":3279,"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/posts\/423\/revisions\/3279"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/media\/3277"}],"wp:attachment":[{"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/media?parent=423"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/categories?post=423"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/tags?post=423"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}