{"id":2095,"date":"2019-12-23T23:00:57","date_gmt":"2019-12-23T14:00:57","guid":{"rendered":"http:\/\/141.164.34.82\/?p=2095"},"modified":"2019-12-23T23:06:58","modified_gmt":"2019-12-23T14:06:58","slug":"k-means%ec%99%80-gmm-%eb%b9%84%ea%b5%90","status":"publish","type":"post","link":"http:\/\/ds.sumeun.org\/?p=2095","title":{"rendered":"k-means\uc640 GMM \ube44\uad50"},"content":{"rendered":"

\\(k\\) -means \ubc29\ubc95<\/h2>\n

\\(k\\) – means \ubc29\ubc95\uc740 \ub2e4\ubcc0\ub7c9 \uc790\ub8cc\ub97c \ud074\ub7ec\uc2a4\ud130\ub9c1\uc744 \ud558\ub294 \uac00\uc7a5 \ub300\ud45c\uc801\uc778 \ubc29\ubc95\uc774\ub2e4. \uac1c\ub150\uc740 \uac04\ub2e8\ud558\ub2e4. \\(k\\) \uac1c\uc758 \ud3c9\uade0\uc774 \uc788\uace0, \uc774 \ud3c9\uade0\uc740 \ubaa8\ub450 \uc11c\ub85c \ub2e4\ub978 \uc9d1\ub2e8\uc744 \ub098\ud0c0\ub0b8\ub2e4. \ubaa8\ub4e0 \uc790\ub8cc\ub294 \uac00\uc7a5 \uac00\uae4c\uc6b4 \ud3c9\uade0\uc5d0 \uc18c\uc18d\ub41c\ub2e4.<\/p>\n

\uac00\uc6b0\uc2dc\uc548 \ud63c\ud569 \ubaa8\ud615(GMM<\/strong>: G<\/strong>aussian M<\/strong>ixture M<\/strong>odel)<\/h2>\n

1\ubcc0\uc218 \ub450 \uc9d1\ub2e8\uc758 \uac00\uc6b0\uc2dc\uc548 \ud63c\ud569 \ubaa8\ud615\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4. <\/p>\n

\\[f(X=x|\\mu_1, \\sigma_1, \\mu_2, \\sigma_2) = w_1\\frac{1}{\\sigma_1\\sqrt{2\\pi}}\\exp\\left[-\\frac{1}{2}(\\frac{x-\\mu_1}{\\sigma_1})^2\\right] + w_2\\frac{1}{\\sigma_2\\sqrt{2\\pi}}\\exp\\left[-\\frac{1}{2}(\\frac{x-\\mu_2}{\\sigma_2})^2\\right]\\]<\/p>\n

\uc5ec\uae30\uc11c \\(\\mu_1\\) , \\(\\mu_2\\) \ub294 \ub450 \uc9d1\ub2e8\uc758 \ud3c9\uade0\uc774\uace0, \\(\\sigma_1\\) , \\(\\sigma_2\\) \ub294 \ub450 \uc9d1\ub2e8\uc758 \ud45c\uc900\ud3b8\ucc28\uc774\ub2e4. \uadf8\ub9ac\uace0 \\(w_1\\) \uacfc \\(w_2\\) \ub294 \uac01 \uc9d1\ub2e8\uc758 \ube44\uc728\uc744 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n

\ub9cc\uc57d 1\ubcc0\uc218\uac00 \uc544\ub2c8\ub77c \ub2e4\ubcc0\uc218\uc758 \uc790\ub8cc\ub97c \ub2e4\ub8ec\ub2e4\uba74 \\(x\\) \ub294 \\(\\vec{x}\\) \ub85c, \\(\\mu_1\\) \ub294 \\(\\vec{\\mu_1}\\) , \\(\\sigma_1\\) \uc740 \\(\\Sigma_1\\) (\ubd84\uc0b0-\uacf5\ubd84\uc0b0 \ud589\ub82c)\ub85c \ub098\ud0c0\ub0b4\uc57c \ud560 \uac83\uc774\ub2e4. \ub9cc\uc57d \uc9d1\ub2e8\uc774 \uc5ec\ub7ff\uc774\ub77c\uba74 \\(w_3\\frac{1}{\\sigma_3\\sqrt{2\\pi}}\\exp\\left[-\\frac{1}{2}(\\frac{x-\\mu_3}{\\sigma_3})^2\\right]\\) \uac00 \ucd94\uac00\ub41c\ub2e4.<\/p>\n

\\(k\\) -means\uc640 GMM\uc758 \ucc28\uc774<\/h2>\n

\\(k\\) -\ud3c9\uade0\uacfc GMM\uc740 \ube44\uc2b7\ud558\ub2e4. \uc5ec\ub7ec \uc9d1\ub2e8\uc774 \uc788\uace0, \uc774 \uc9d1\ub2e8\uc740 \ud3c9\uade0\uc73c\ub85c \uad6c\ubd84\ub41c\ub2e4. \ud558\uc9c0\ub9cc \\(k\\) -\ud3c9\uade0\uc5d0\ub294 \ubd84\uc0b0\uc774 \uc5c6\ub2e4. \uadf8\ub9ac\uace0 \ubaa8\ub4e0 \uc0ac\ub840\ub294 \ud558\ub098\uc758 \uc9d1\ub2e8\uc5d0 \uc18c\uc18d\ub41c\ub2e4. (\ubc18\uba74 GMM\uc5d0\uc11c \ud55c \uc0ac\ub840\ub294 \ubcf4\ud1b5 \uc5ec\ub7ec \uc9d1\ub2e8\uc5d0 \ud655\ub960\uc801\uc73c\ub85c \uc18c\uc18d\ub41c\ub2e4.)<\/p>\n

\uc0ac\uc2e4 PRML\uc5d0\uc11c Bishop\uc774 \uc598\uae30\ud588\ub4ef\uc774 GMM\uc5d0\uc11c \ubaa8\ub4e0 \uc9d1\ub2e8\uc758 \ubd84\uc0b0\uc744 \\(\\sigma\\textbf{I}\\) ( \\(\\sigma \\rightarrow 0\\) )\uc73c\ub85c \ubb34\ud55c\ud788 \ucd95\uc18c\uc2dc\ud0a4\uba74 GMM\uc740 \\(k\\) -\ud3c9\uade0\uacfc \uac19\uc544\uc9c4\ub2e4.<\/p>\n

\uc774\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \ud655\uc778\ud560 \uc218 \uc788\ub2e4. \ub2e4\uc74c\uc740 \uac01 \uc0ac\ub840\uc758 \uc18c\uc18d\uc774 \uacb0\uc815\ub418\uc5c8\uc744 \ub54c \ud655\ub960\ubcc0\uc218 \\(X\\) \uc758 \ud655\ub960\ubd84\ud3ec\ub97c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n

\\[f_1(X=x|\\mu_1, \\sigma_1) = \\frac{1}{\\sigma_1\\sqrt{2\\pi}}\\exp\\left[-\\frac{1}{2}(\\frac{x-\\mu_1}{\\sigma_1})^2\\right]\\]<\/p>\n

\\[f_2(X=x|\\mu_2, \\sigma_2) = \\frac{1}{\\sigma_2\\sqrt{2\\pi}}\\exp\\left[-\\frac{1}{2}(\\frac{x-\\mu_2}{\\sigma_2})^2\\right]\\]<\/p>\n

\uc5b4\ub5a4 \uc0ac\ub840 \\(x\\) \uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \\(f_1(x)\\) \uc640 \\(f_2(x)\\) \uc744 \ube44\uad50\ud558\uc5ec \\(x\\) \uc758 \uc18c\uc18d\uc744 \uacb0\uc815\ud55c\ub2e4\uace0 \ud574\ubcf4\uc790. \ubca0\uc774\uc9c0\uc548\uc73c\ub85c \uc0dd\uac01\ud558\uba74 \\(P(g=1 |x) = P(x|g=1)P(g=1)\/P(x)\\) \uc774\uace0 \uc774\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \ud480\uc5b4 \uc4f8 \uc218 \uc788\ub2e4.<\/p>\n

\\[P(g=1|x)= \\frac{P(x|g=1)P(g=1)}{P(x)} = \\frac{f_1(X=x)P(g=1)}{f_1(x)P(g=1)+f_2(x)P(g=2)}\\]<\/p>\n

\uc5ec\uae30\uc11c \ubd84\uc790, \ubd84\ubaa8\uc5d0 \uacf5\ud1b5\uc778 \\(f_1(x)P(g=1)\\) \uc744 \ub098\ub220\uc8fc\uba74,<\/p>\n

\\[P(g=1|x)= \\frac{1}{1+f_2(x)P(g=2)\/f_1(x)P(g=1)}\\]<\/p>\n

\\(\\frac{1}{1+x}\\) \uc758 \uadf8\ub798\ud504\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4. ( \\(f_2(x)P(g=2)\/f_1(x)P(g=1)\\) \ub294 \uc591\uc218\uc784\uc744 \uc8fc\ubaa9\ud558\uc790.)<\/p>\n

curve(1\/(1+x), xlim=c(0,3))\n<\/code><\/pre>\n
\"plot<\/p>\n
\uc774\ub54c \ub9cc\uc57d \\(\\sigma_1\\) \uacfc \\(\\sigma_2\\) \uac00 \ubaa8\ub450 \\(0\\) \uacfc \ub9e4\uc6b0 \uac00\uae4c\uc6cc\uc9c4\ub2e4\uba74 \\(f_2(x)\/f_1(x)\\) \ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n
\\[\\frac{f_2(x)}{f_1(x)} = \\frac{\\frac{1}{\\sigma_2\\sqrt{2\\pi}}\\exp\\left[-\\frac{1}{2}(\\frac{x-\\mu_2}{\\sigma_2})^2\\right]}{\\frac{1}{\\sigma_1\\sqrt{2\\pi}}\\exp\\left[-\\frac{1}{2}(\\frac{x-\\mu_1}{\\sigma_1})^2\\right]}\\]<\/p>\n
\uc5ec\uae30\uc11c \\(\\frac{\\sigma_1}{\\sigma_2} \\approx 1\\) \ub85c \uc0dd\uac01\ud558\uba74 \uc704\uc758 \uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\ub9ac\ub41c\ub2e4. <\/p>\n
\\[\\frac{f_2(x)}{f_1(x)} = \\exp\\left[-\\frac{1}{2}[(\\frac{x-\\mu_2}{\\sigma_2})^2-(\\frac{x-\\mu_1}{\\sigma_1})^2]\\right]\\]<\/p>\n
\\[\\phantom{\\frac{f_2(x)}{f_1(x)}} = \\exp\\left[-\\frac{1}{2}\\frac{-2(\\mu_2-\\mu_1)x+\\mu_2^2-\\mu_1^2}{\\sigma_1^2}\\right]\\]<\/p>\n
\\[\\phantom{\\frac{f_2(x)}{f_1(x)}} = \\exp\\left[-\\frac{1}{2}\\frac{-2(\\mu_2-\\mu_1)(x-(\\mu_1+\\mu_2)\/2)}{\\sigma_1^2}\\right]\\]<\/p>\n
\\(\\mu_1 \\neq \\mu_2\\) \uc774\uace0, \\(\\sigma_1^2 > 0\\) \uc774\ubbc0\ub85c, \\(\\exp\\) \uc548\uc758 \ubd80\ud638\ub294 \\(x-(\\mu_1+\\mu_2)\/2\\) \uc758 \ubd80\ud638\uc5d0 \uc758\ud574 \uacb0\uc815\ub418\ub294\ub370, \ub9cc\uc57d \\(x-(\\mu_1+\\mu_2)\/2 > 0\\) \uc774\ub77c\uba74, \\(\\exp\\) \uc548\uc740 \\(\\infty\\) \uac00 \ub41c\uace0, \\(\\frac{f_2(x)}{f_1(x)}\\) \uc5ed\uc2dc \ubb34\ud55c\ub300\uac00 \ub41c\ub2e4. \ubc18\ub300\ub85c \\(x-(\\mu_1+\\mu_2)\/2 < 0\\) \ub77c\uba74, \\(\\frac{f_2(x)}{f_1(x)}\\) \uc740 \\(0\\) \uc774 \ub41c\ub2e4.<\/p>\n
\ub530\ub77c\uc11c \\(P(g=1)\\) \uacfc \\(P(g=2)\\) \uc5d0 \uc0c1\uad00\uc5c6\uc774(0\ubcf4\ub2e4 \ud06c\uace0, \uc0c1\uc218\ub77c\uba74) \\(P(g=1|x)\\) \uc740 \\(1\\) \ub610\ub294 \\(0\\) \uc774 \ub418\uc5b4, \ud754\ud788 \ub9d0\ud558\ub294 hard assignment\uac00 \ub41c\ub2e4. \uadf8\ub9ac\uace0 \uc18c\uc18d\uc740 \uc870\uae08\uc774\ub77c\ub3c4 \uac00\uae4c\uc6b4 \ud3c9\uade0\uc758 \uc18c\uc18d\uc774 \ub41c\ub2e4.<\/p>\n
\uadf8\ub7f0\ub370 \uc194\uc9c1\ud788 \uc5b4\ub5a4 \uc9d1\ub2e8\uc758 \uc0dd\uc131 \ubd84\ud3ec\uac00 \uac00\uc6b0\uc2dc\uc548\uc77c \ub54c \ubd84\uc0b0\uc774 0\uc778 \uacbd\uc6b0\uac00 \uc5bc\ub9c8\ub098 \ub418\uaca0\ub098? (\uadf8\ub9ac\uace0 \ubd84\uc0b0\uc774 0\uc774\ub77c\uba74 \ub370\uc774\ud130\ub294 \uac70\uc758 \ud655\uc2e4\ud788 \ud3c9\uade0\uacfc \ub3d9\uc77c\ud574\uc57c \ud558\uc9c0 \uc54a\uc740\uac00?)<\/strong> \ub530\ub77c\uc11c \\(k\\) -\ud3c9\uade0 \ubaa8\ud615\uc740 \ud655\ub960 \ubaa8\ud615\uc73c\ub85c \uc0dd\uac01\ud588\uc744 \ub54c \uc870\uae08\uc740 \uc9c0\ub098\uce58\uac8c \uc81c\uc57d\uc801\uc774\uba70, \ube44\ud604\uc2e4\uc801\uc778 \ubaa8\ud615\uc774\ub77c\uace0 \ud560 \uc218 \uc788\ub2e4.<\/p>\n
\uadf8\ub7fc\uc5d0\ub3c4 \\(k\\) -means \uac00 \uc2ec\uc2ec\uce58 \uc54a\uac8c \uc4f0\uc774\ub294 \uac83\uc740 \ubb34\uc2a8 \uc5f0\uc720\uc77c\uae4c\uc694?<\/p>\n
    \n
  1. \uacc4\uc0b0 \uc18d\ub3c4\uac00 \ube68\ub77c\uc11c<\/li>\n
  2. \ub0a8\ub4e4\uc774 \ub9ce\uc774 \uc4f0\ub2c8\uae4c<\/li>\n
  3. GMM\uc744 \uc368 \ubcf8 \uc801\uc774 \uc5c6\uc5b4\uc11c<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"
    \\(k\\) -means \ubc29\ubc95 \\(k\\) – means \ubc29\ubc95\uc740 \ub2e4\ubcc0\ub7c9 \uc790\ub8cc\ub97c \ud074\ub7ec\uc2a4\ud130\ub9c1\uc744 \ud558\ub294 \uac00\uc7a5 \ub300\ud45c\uc801\uc778 \ubc29\ubc95\uc774\ub2e4. \uac1c\ub150\uc740 \uac04\ub2e8\ud558\ub2e4. \\(k\\) \uac1c\uc758 \ud3c9\uade0\uc774 \uc788\uace0, \uc774 \ud3c9\uade0\uc740 \ubaa8\ub450 \uc11c\ub85c \ub2e4\ub978 \uc9d1\ub2e8\uc744 \ub098\ud0c0\ub0b8\ub2e4. \ubaa8\ub4e0 \uc790\ub8cc\ub294 \uac00\uc7a5 \uac00\uae4c\uc6b4 \ud3c9\uade0\uc5d0 \uc18c\uc18d\ub41c\ub2e4. \uac00\uc6b0\uc2dc\uc548 \ud63c\ud569 \ubaa8\ud615(GMM: Gaussian Mixture Model) 1\ubcc0\uc218 \ub450 \uc9d1\ub2e8\uc758 \uac00\uc6b0\uc2dc\uc548 \ud63c\ud569 \ubaa8\ud615\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4. \\[f(X=x|\\mu_1, \\sigma_1, \\mu_2, \\sigma_2) = w_1\\frac{1}{\\sigma_1\\sqrt{2\\pi}}\\exp\\left[-\\frac{1}{2}(\\frac{x-\\mu_1}{\\sigma_1})^2\\right] + w_2\\frac{1}{\\sigma_2\\sqrt{2\\pi}}\\exp\\left[-\\frac{1}{2}(\\frac{x-\\mu_2}{\\sigma_2})^2\\right]\\] […]<\/p>\n","protected":false},"author":1,"featured_media":2098,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[146,319],"tags":[423,422,424,425],"jetpack_featured_media_url":"http:\/\/ds.sumeun.org\/wp-content\/uploads\/2019\/12\/kmeans_vs_GMM.png","_links":{"self":[{"href":"http:\/\/ds.sumeun.org\/index.php?rest_route=\/wp\/v2\/posts\/2095"}],"collection":[{"href":"http:\/\/ds.sumeun.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/ds.sumeun.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/ds.sumeun.org\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/ds.sumeun.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2095"}],"version-history":[{"count":4,"href":"http:\/\/ds.sumeun.org\/index.php?rest_route=\/wp\/v2\/posts\/2095\/revisions"}],"predecessor-version":[{"id":2100,"href":"http:\/\/ds.sumeun.org\/index.php?rest_route=\/wp\/v2\/posts\/2095\/revisions\/2100"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/ds.sumeun.org\/index.php?rest_route=\/wp\/v2\/media\/2098"}],"wp:attachment":[{"href":"http:\/\/ds.sumeun.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2095"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/ds.sumeun.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2095"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/ds.sumeun.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2095"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}