{"id":480,"date":"2019-12-19T18:21:01","date_gmt":"2019-12-19T09:21:01","guid":{"rendered":"https:\/\/hyandmj.asuscomm.com\/hblog\/?p=480"},"modified":"2020-04-16T16:10:14","modified_gmt":"2020-04-16T07:10:14","slug":"normalized-convolution","status":"publish","type":"post","link":"https:\/\/hyandmj.asuscomm.com\/hblog\/?p=480","title":{"rendered":"Normalized Convolution"},"content":{"rendered":"\n

Motive<\/h3>\n\n\n\n

\ub370\uc774\ud130 \ubd84\uc11d\uc744 \ud558\uba74\uc11c \uc591\uc131\uc790\uac00 \uc2e0\ud2f8\ub808\uc774\ud130\uc5d0 \ub0a8\uae34 \uc2e0\ud638\ub97c fitting\ud558\ub294 \uacfc\uc815\uc5d0\uc11c Gaussian \ubd84\ud3ec\uc640 Landau \ubd84\ud3ec\uc758 convolution\uc744 \ubaa8\ub378\ub85c \uc0ac\uc6a9\ud558\ub824\uace0 \ud588\ub294\ub370, CERN ROOT<\/a>\uc5d0\ub294 \uae30\ubcf8\uc801\uc73c\ub85c \ud568\uc218\uac00 \uad6c\ud604\ub418\uc5b4 \uc788\uc9c0 \uc54a\uc544 \uc0c8\ub85c \ub9cc\ub4e4\uc5b4\uc57c \ud588\ub2e4. \ub2e4\ub9cc fitting tutorial \ud30c\uc77c \uc911 langaus.C<\/code>\ub77c\ub294 \ud30c\uc77c\uc774 \ubc14\ub85c Landau\uc640 Gaussian\uc758 convolution \ubaa8\ub378\uc744 \uc774\uc6a9\ud574 fitting\uc744 \ud558\ub294 \uc608\uc81c\ub97c \uc18c\uac1c\ud558\uace0 \uc788\ub2e4. \ud558\uc9c0\ub9cc \uc774 \uc608\uc81c\uc758 “langaus”\ub294 \uc81c\ub300\ub85c normalization\uc774 \uc548\ub418\uc5b4 \uc788\uc5b4\uc11c \ub9e4\uc6b0 \uc0ac\uc6a9\ud558\uae30 \ubd88\ud3b8\ud558\ub2e4. normalized convolution\uc744 \uad6c\ud604\ud558\ub2e4 \ubcf4\ub2c8 \ubb38\ub4dd \ub4dc\ub294 \uc758\ubb38.<\/p>\n\n\n\n

Q. \ub450 PDF\uac00 normalized\uc77c \ub54c \ub450 PDF\uc758 convolution\ub3c4 \uc790\ub3d9\uc73c\ub85c normalize\uac00 \ub418\uc5b4\uc788\ub294\uac00?<\/p>\n\n\n\n

Convolution\uc740 Normalization\uc744 \ubcf4\uc874\ud558\ub294\uac00?<\/h3>\n\n\n\n

\ub9e4\uc6b0 \uac04\ub2e8\ud55c \uc608\uc2dc\ub97c \ub4e4\uc5b4\ubcf8\ub2e4. \ub2e4\uc74c\uacfc \uac19\uc740 \ub450 \ud655\ub960 \ubd84\ud3ec\ub97c \uac00\uc815\ud55c\ub2e4.<\/p>\n\n\n\n

(1) <\/span>   <\/span>\"\begin{align*}f(1)&=x_1, \\ f(2)&=x_2, \\ f(3)&=x_3\end{align*}\"<\/p>

(2) <\/span>   <\/span>\"\begin{align*}g(1)&=y_1, \\ g(2)&=y_2, \\ g(3)&=y_3\end{align*}\"<\/p><\/p>\n\n\n\n

\ubb3c\ub860 f\uc640 g\ub294 \uac01\uac01 \ub2e4 \ud569\ud558\uba74 1\uc774 \ub41c\ub2e4. \uc774\uc81c \ub450 \ud655\ub960 \ubcc0\uc218 x\uc640 y\uc758 \ud569 z=x+y\uc758 \ud655\ub960 \ubd84\ud3ec\ub97c \uad6c\ud574\ubcf4\uc790.<\/p>\n\n\n\n

(3) <\/span>   <\/span>\"\begin{align*} h(1)&=0, \\ h(2)&=f(1)g(1)=x_1 y_1, \\ h(3)&=f(1)g(2)+f(2)g(1)=x_1 y_2+x_2 y_1\end{align*}\"<\/p><\/p>\n\n\n\n

3\uc774 \ub098\uc62c \ud655\ub960\uc740 1+2\uc640 2+1 \ub450 \uac00\uc9c0 \uacbd\uc6b0\uc758 \uc218\uac00 \uc788\ub294 \uc2dd\uc774\ub2e4. 6\uae4c\uc9c0 \ub098\uc62c \uc218 \uc788\uace0, \ub098\uba38\uc9c0 \ud655\ub960\uc744 \ubaa8\ub450 \uad6c\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n\n\n\n

(4) <\/span>   <\/span>\"\begin{align*} h(4)&=x_1 y_3+x_2 y_2+x_3 y_1, \\ h(5)&=x_2 y_3+x_3 y_2, \\ h(6)&=x_3 y_3\end{align*}\"<\/p><\/p>\n\n\n\n

h \uc5ed\uc2dc normalize\uac00 \ub418\uc5b4\uc788\ub294\uc9c0 \ubcf4\uae30 \uc704\ud574 \ubaa8\ub4e0 \ud655\ub960\uc744 \ub2e4 \ub354\ud574 \ubcf8\ub2e4.<\/p>\n\n\n\n

(5) <\/span>   <\/span>\"\begin{align*}\sum_i h(z_i)&=h(2)+h(3)+h(4)+h(5)+h(6) \\&=(x_1 y_1)+( x_1 y_2+x_2 y_1 )+( x_1 y_3+x_2 y_2+x_3 y_1 )+( x_2 y_3+x_3 y_2 )+( x_3 y_3 ) \\&=x_1(y_1+y_2+y_3)+ x_2(y_1+y_2+y_3) + x_3(y_1+y_2+y_3) \\&= (x_1+x_2+x_3)(y_1+y_2+y_3)\end{align*}\"<\/p><\/p>\n\n\n\n

f\uc640 g\uac00 normalization\uc774 \ub418\uc5b4\uc788\ub2e4\ub294 \uc544\ub798\uc640 \uac19\uc740 \uac00\uc815 \ud558\uc5d0 \ub2e4\ud589\ud788\ub3c4 h \uc5ed\uc2dc normalization\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n\n\n\n

(6) <\/span>   <\/span>\"\begin{align*}&x_1+x_2+x_3=y_1+y_2+y_3=1& \\&\therefore \sum_i h(z_i) =1&\end{align*}\"<\/p><\/p>\n\n\n\n

\uac04\ub2e8\ud55c \uc608\ub97c \ub4e4\uc5b4\ubd24\uc744 \ub54c \uc9c8\ubb38\uc5d0 \ub300\ud55c \ub2f5\uc740 yes\uc778\ub370… \uc77c\ubc18\uc801\uc778 \uacbd\uc6b0\uc5d0\ub3c4 \uc131\ub9bd\ud560\uae4c?<\/p>\n\n\n\n

General Case<\/h3>\n\n\n\n

\uc0dd\uac01\ud574\ubcf4\ub2c8 \ubaa8\ub4e0 \ud655\ub960\uc744 \ub2e4 \ub354\ud558\uba74 \ub2f9\uc5f0\ud788 1\uc774 \ub098\uc640\uc57c \ud558\ubbc0\ub85c \ub450 \ud655\ub960 \ubd84\ud3ec\uc758 convolution\ub3c4 \ud655\ub960 \ubd84\ud3ec\uac00 \ub428\uc740 \uc790\uba85\ud558\ub2e4. \uc218\uc2dd\uc73c\ub85c \uc880 \ub354 \uc5c4\ubc00\ud788 \uc0b4\ud3b4\ubcf4\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4. \ub2e4\uc74c\uacfc \uac19\uc740 \ub450 \ud655\ub960 \ubd84\ud3ec\ub97c \uac00\uc815\ud55c\ub2e4.<\/p>\n\n\n\n

(7) <\/span>   <\/span>\"\begin{align*}f(x_i)&, \; g(y_i) \\\sum_i f(x_i) &= \sum_i g(y_i)=1\end{align*}\"<\/p><\/p>\n\n\n\n

\uc5ec\uae30\uc11c \ub450 \ud568\uc218\uac00 \ud655\ub960 \ubd84\ud3ec\ub77c\ub294 \uac83, \uc989 \ud655\ub960 \uc9c8\ub7c9 \ud568\uc218probability mass function\uc758 \uc815\uc758\ub97c \ub9cc\uc871\ud55c\ub2e4\ub294 \uc758\ubbf8\ub294 \ud569\uc774 1\uc778 \uac83 \uc774\uc678\uc5d0\ub3c4 \ub2e4\uc74c \ub450 \uc870\uac74\uc744 \ucda9\uc871\ud55c\ub2e4\ub294 \uac83\uacfc \uac19\ub2e4. (Reference<\/a>)<\/p>\n\n\n\n

(8) <\/span>   <\/span>\"\begin{align*} f(x_i) &> 0 \\f(x_i) &=0 \; \textrm{for all other } i\end{align*}\"<\/p><\/p>\n\n\n\n

\ub450 \ud568\uc218\uc758 convolution\uc740 \uc815\uc758\uc5d0 \uc758\ud574 \ub2e4\uc74c\uacfc \uac19\uc774 \ud45c\ud604\ub41c\ub2e4.<\/p>\n\n\n\n

(9) <\/span>   <\/span>\"\begin{align*}h(z_j) &= \sum_{i} f(x_i) g(z_j-x_i)\end{align*}\"<\/p><\/p>\n\n\n\n

\uc774\uc81c \uc2dd (9)\ub97c \ubaa8\ub4e0 j\uc5d0 \ub300\ud574 sum \ud574\ubcf4\uc790.<\/p>\n\n\n\n

(10) <\/span>   <\/span>\"\begin{align*}\sum_j h(z_j) &= \sum_j \sum_i f(x_i) g(z_j-x_i) \\ &= \sum_i f(x_i) \left[ \sum_j g(z_j-x_i) \right]\end{align*}\"<\/p><\/p>\n\n\n\n

\uc2dd (10)\uc758 \ub9c8\uc9c0\ub9c9 \ud56d\uc740 \ud655\ub960 \ubd84\ud3ec g\ub97c \ubaa8\ub450 \ub354\ud558\ub418, x_i\ub9cc\ud07c \ud3c9\ud589\uc774\ub3d9\ud558\uc5ec \ub354\ud55c \uac83\uc778\ub370 \uc2dd (8)\uc758 \uc870\uac74\uc5d0 \uc758\ud558\uc5ec \uc544\ubb34\ub9ac \ud3c9\ud589\uc774\ub3d9\uc774 \ub41c\ub2e4 \ud55c\ub4e4 \uadf8 \ud569\uc740 \uc5ec\uc804\ud788 1\uc774\ub2e4. \ub530\ub77c\uc11c,<\/p>\n\n\n\n

(11) <\/span>   <\/span>\"\begin{align*}\sum_j h(z_j) &= \sum_i f(x_i) \left[ \sum_j g(z_j-x_i) \right] \\&= \sum_i f(x_i) \times 1 \\&= 1\end{align*}\"<\/p><\/p>\n\n\n\n

\uacb0\ub860: \ud655\ub960 \ubd84\ud3ec \ub07c\ub9ac convolution\ud55c \ud568\uc218\ub3c4 \ud655\ub960 \ubd84\ud3ec\ub2e4. Discrete \ubd84\ud3ec\uc5d0 \ub300\ud574\uc11c\ub9cc \ud574\ubcf4\uc558\uc9c0\ub9cc PDF\ub97c \uac00\uc9c0\uace0 \uc801\ubd84\ud574\ubd10\ub3c4 \uac19\uc740 \uacb0\ub860\uc744 \uc5bb\uc744 \uc218 \uc788\ub2e4.<\/p>\n\n\n\n

\uadf8\ub9ac\uace0 \ub09c \uc544\ubb34 \uac71\uc815 \uc5c6\uc774 \ub9c8\uc74c \ub193\uace0 fitting code\ub97c \uc791\uc131\ud560 \uc218 \uc788\uc5c8\ub2e4.<\/p>\n","protected":false},"excerpt":{"rendered":"

Motive \ub370\uc774\ud130 \ubd84\uc11d\uc744 \ud558\uba74\uc11c \uc591\uc131\uc790\uac00 \uc2e0\ud2f8\ub808\uc774\ud130\uc5d0 \ub0a8\uae34 \uc2e0\ud638\ub97c fitting\ud558\ub294 \uacfc\uc815\uc5d0\uc11c Gaussian \ubd84\ud3ec\uc640 Landau \ubd84\ud3ec\uc758 convolution\uc744 \ubaa8\ub378\ub85c \uc0ac\uc6a9\ud558\ub824\uace0 \ud588\ub294\ub370, CERN ROOT\uc5d0\ub294 \uae30\ubcf8\uc801\uc73c\ub85c \ud568\uc218\uac00 \uad6c\ud604\ub418\uc5b4 \uc788\uc9c0 \uc54a\uc544 \uc0c8\ub85c \ub9cc\ub4e4\uc5b4\uc57c \ud588\ub2e4. \ub2e4\ub9cc fitting tutorial \ud30c\uc77c \uc911 langaus.C\ub77c\ub294 \ud30c\uc77c\uc774 \ubc14\ub85c Landau\uc640 Gaussian\uc758 convolution \ubaa8\ub378\uc744 \uc774\uc6a9\ud574 fitting\uc744 \ud558\ub294 \uc608\uc81c\ub97c \uc18c\uac1c\ud558\uace0 \uc788\ub2e4. \ud558\uc9c0\ub9cc \uc774 \uc608\uc81c\uc758 “langaus”\ub294 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[17],"tags":[38,41,42,43,37,39,40],"class_list":["post-480","post","type-post","status-publish","format-standard","hentry","category-math","tag-convolution","tag-density","tag-function","tag-normalization","tag-normalized","tag-pdf","tag-probability"],"_links":{"self":[{"href":"https:\/\/hyandmj.asuscomm.com\/hblog\/index.php?rest_route=\/wp\/v2\/posts\/480","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/hyandmj.asuscomm.com\/hblog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/hyandmj.asuscomm.com\/hblog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/hyandmj.asuscomm.com\/hblog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/hyandmj.asuscomm.com\/hblog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=480"}],"version-history":[{"count":62,"href":"https:\/\/hyandmj.asuscomm.com\/hblog\/index.php?rest_route=\/wp\/v2\/posts\/480\/revisions"}],"predecessor-version":[{"id":546,"href":"https:\/\/hyandmj.asuscomm.com\/hblog\/index.php?rest_route=\/wp\/v2\/posts\/480\/revisions\/546"}],"wp:attachment":[{"href":"https:\/\/hyandmj.asuscomm.com\/hblog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=480"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/hyandmj.asuscomm.com\/hblog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=480"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/hyandmj.asuscomm.com\/hblog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=480"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}