qnorm(p=c(0.025, 0.975), mean=0, sd=1)\r\n<\/code><\/pre>\n## [1] -1.959964 1.959964\r\n<\/pre>\nquantile regression\uc740 \uc870\uac74\ubd80 \ubd84\uc704\uc218\ub97c \ucd94\uc815\ud55c\ub2e4\uace0 \uc0dd\uac01\ud558\uba74 \ub41c\ub2e4.<\/p>\n
\\[P(Y< y_{(q)}(x) | X = x) = q\\]<\/p>\n
\uc704\uc758 \uc2dd\uc740 \uc608\uce21 \ubcc0\uc218\uac00 \\(x\\) \ub85c \uc8fc\uc5b4\uc84c\uc744 \ub54c, \ud655\ub960\ubcc0\uc218 \\(Y\\) \uc758 \\(q\\) -\ubd84\uc704\uc218 \\(y(q)\\) \uc5d0 \ub300\ud55c \uc815\uc758\ub97c \ub098\ud0c0\ub0b8\ub2e4. \ud655\ub960\ubcc0\uc218 \\(Y\\) \uc758 \\(q\\) -\ubd84\uc704\uc218\ub294 \uc608\uce21 \ubcc0\uc218 \\(x\\) \uc758 \uac12\uc5d0 \ub530\ub77c \ub2ec\ub77c\uc9c8 \uc218 \uc788\uc73c\ubbc0\ub85c \\(x\\) \uc758 \ud568\uc218 \\(y_{(q)}(x)\\) \ub85c \ub098\ud0c0\ub0b4\uc5c8\ub2e4.<\/p>\n
\ubd84\uc704\uc218 \uc190\uc2e4(Quantile loss)<\/h2>\n
\uadf8\ub7f0\ub370 q-\ubd84\uc704\uc218\ub97c \uc5b4\ub5bb\uac8c \ucd94\uc815\ud574\uc57c \ud560\uae4c?<\/p>\n
\uc7a0\uc2dc <\uae30\ucd08 \ud1b5\uacc4\ud559\uc758 \uc228\uc740 \uc6d0\ub9ac><\/a>\ub780 \ucc45\uc744 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n\ub9c1\ud06c : <\uae30\ucd08 \ud1b5\uacc4\ud559\uc758 \uc228\uc740 \uc6d0\ub9ac \ubc1c\ucdcc: \uc65c \ud45c\ubcf8\uc911\uc559\uac12\uc774 \uc544\ub2c8\ub77c \ud45c\ubcf8\ud3c9\uade0\uc778\uac00?><\/a><\/p>\n\uacb0\ub860\uc801\uc73c\ub85c 0.5-\ubd84\uc704\uc218(\uc911\uc559\uac12)\uc744 \ucd94\uc815\ud558\uae30 \uc704\ud574\uc11c MAE(M<\/strong>ean A<\/strong>bsolute E<\/strong>rror)\ub97c \ub9ce\uc774 \uc4f4\ub2e4. \uc65c\ub0d0\ud558\uba74 MAE\ub97c \ucd5c\uc18c\ub85c \ud558\ub294 \ucd94\uc815\ub7c9\uc740 \ud45c\ubcf8 \uc911\uc559\uac12(\ud45c\ubcf8\uc5d0\uc11c 0.5-\ubd84\uc704\uc218)\uc774\uae30 \ub54c\ubb38\uc5d0 \uc77c\uce58(Consistent)\ucd94\uc815\ub7c9(\ud45c\ubcf8 \ud06c\uae30\uac00 \ub9e4\uc6b0 \ud074 \ub54c\uc5d0 \ubaa8\uc218\uc5d0 \uac70\uc758 \ud655\uc2e4\ud788 \uc218\ub834\ud558\ub294 \ucd94\uc815\ub7c9)\uc774\ub77c\uace0 \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4.<\/p>\n\ubb3c\ub860 \ud2b9\uc815\ud55c \ubd84\ud3ec\ub97c \uac00\uc815\ud55c\ub2e4\uba74 \ud45c\ubcf8 \uc911\uc559\uac12\ubcf4\ub2e4 \ub354 \ud6a8\uc728\uc801\uc778 \ucd94\uc815\ub7c9\uc744 \uad6c\ud560 \uc218 \uc788\uaca0\uc9c0\ub9cc, \uc6b0\ub9ac\ub294 \ubd84\ud3ec\ub97c \ubaa8\ub978\ub2e4!<\/p>\n
\ubc18\uba74 q-\ubd84\uc704\uc218\ub97c \ucd94\uc815\ud560 \ub54c\uc5d0\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 Loss\ub97c \ucd5c\uc18c\ud654\ud558\ub294 \uac12\uc744 \ucd94\uc815\uac12\uc73c\ub85c \uc0ac\uc6a9\ud55c\ub2e4.<\/p>\n
\\[L(y_{(q)},q, \\vec{y}) = \\sum (q-\\mathbb{I}(y_i-y_{(q)} > 0))(y_i-y_{(q)})\\]<\/p>\n
\ucc98\uc74c \uc774 \uc2dd\uc744 \ubd24\uc744 \ub54c, \uc65c \uc774\ub798\uc57c \ud558\ub294\uc9c0 \uad49\uc7a5\ud788 \ud5f7\uac08\ub838\ub2e4. \uc628\ud1b5 \uc218\uc2dd\uc73c\ub85c \ub3c4\ubc30\ub418\uc5b4 \uc788\ub294 \ub17c\ubb38\uc744 \ubd24\uc744 \ub54c\uc5d0\ub3c4 \ud63c\ub780\uc2a4\ub7fd\uae30 \uadf8\uc9c0 \uc5c6\uc5c8\ub2e4. \u201c\ud558\uc9c0\ub9cc \uc218\uc2dd\uc758 \uc7a5\ubcbd\uc744, \ucc44 100\uac1c\ub3c4 \ub418\uc9c0 \uc54a\ub294 \uc218\ud559 \uae30\ud638\ub97c \ub118\uc5b4, \uadf8 \ub4a4\uc5d0 \uc228\uaca8 \uc788\ub294 \uc774\uc720\uc640 \uc758\ubbf8\ub97c \uc774\ud574\ud558\uba74, \uc5ec\ub7ec\ubd84\uc740 \ub354 \ub9ce\uc740 \uc790\uc720\uc640 \uc5ec\uc720\ub97c \ub204\ub9b4 \uc218 \uc788\uc2b5\ub2c8\ub2e4.\u201d<\/strong><\/p>\n\uc790, \uc704\uc5d0\uc11c \ud45c\uc900 \uc911\uc559\uac12\uc744 \uad6c\ud560 \uc218 \uc788\ub294 \uc190\uc2e4 \ud568\uc218\ub97c \ud655\uc778\ud588\ub2e4. \uc190\uc2e4 \ud568\uc218\ub97c MAE\ub85c \ud558\uba70, MAE\ub97c \ucd5c\uc18c\ud654\ud558\ub294 \uac12\uc740 \ud45c\ubcf8 \uc911\uc559\uac12\uc774 \ub41c\ub2e4.<\/p>\n
\uc5ec\uae30\uc11c \ud45c\ubcf8 0.9-\ubd84\uc704\uc218\ub97c \uad6c\ud558\ub294 \ubc29\ubc95, 0.9-\ubd84\uc704\uc218\ub97c \uad6c\ud560 \uc218 \uc788\ub294 \uc190\uc2e4\ud568\uc218\ub97c \uc0dd\uac01\ud574\ubcf4\uc790.<\/p>\n
\uc911\uc559\uac12\uc740 \uadf8 \uc218(\uc911\uc559\uac12)\ubcf4\ub2e4 \uc791\uc740 \uc218\uc758 \uac2f\uc218\uc640 \uadf8 \uc218\ubcf4\ub2e4 \ud070 \uc218\uc758 \uac2f\uc218\uac00 \uc815\ud655\ud788 1:1\uc774 \ub418\ub294 \uc218\uc774\ub2e4.
\n\ubc18\uba74 0.9-\ubd84\uc704\uc218\ub294 \uadf8 \uc218(0.9-\ubd84\uc704\uc218)\ubcf4\ub2e4 \uc791\uc740 \uc218\uc758 \uac2f\uc218\uac00 \uadf8 \uc218\ubcf4\ub2e4 \ud070 \uc218\uc758 \uac2f\uc218\ubcf4\ub2e4 9\ubc30\uc778 \uc218\uc774\ub2e4.
\n\uadf8\ub807\ub2e4\uba74 \uadf8 \uc218\ubcf4\ub2e4 \ud070 \uc218\ub4e4\uc740 9\ubc30\ub97c \ud574\uc8fc\uba74, \uc591\ucabd\uc758 \uac2f\uc218\uac00 \ub3d9\ub4f1\ud574\uc9c0\uc9c0 \uc54a\uc740\uac00?<\/p>\n
\uc544\uc8fc \uac04\ub2e8\ud55c \uc608\ub97c \ub4e4\uc5b4 \ubcf4\uc790. \ub2e4\uc74c\uc5d0 \ud45c\ubcf8\uc774 \uc788\ub2e4.<\/p>\n
\\(1, 2, 3, 4, 5, 6, 7, 8, 9, 10\\)<\/p>\n
\uc5ec\uae30\uc11c \\(5.5\\) (\ub610\ub294 \uadf8 \uadfc\ubc29\uc758 \uc218)\ub294 \uc704\uc758 \uc218\ub97c \uc815\ud655\ud788 1:1\ub85c \ub098\ub208\ub2e4.
\n\\(9.5\\) \ub294 \uc704\uc758 \uc218\ub97c \uc815\ud655\ud788 9:1\ub85c \ub098\ub208\ub2e4. \uc774\ub54c 10\ub97c 9\uac1c \ub298\ub824\uc918\ubcf4\uc790.<\/p>\n
\\(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10\\)<\/p>\n
\uc704\uc758 \uc0c8\ub85c\uc6b4 \ud45c\ubcf8\uc5d0 \ub300\ud574 \uc911\uc559\uac12\uc744 \uad6c\ud558\uba74 \\(9.5\\) \uac00 \ub41c\ub2e4.<\/p>\n
\ub2e4\uc2dc \ub9d0\ud574 \uc5b4\ub5a4 \ud45c\ubcf8\uc5d0\uc11c \ud45c\ubcf8 q-\ubd84\uc704\uc218\uc758 \ud2b9\uc9d5\uc744 \uc54c \uc218 \uc788\ub2e4.
\nq-\ubd84\uc704\uc218\ubcf4\ub2e4 \uc791\uc740 \uc218\uc640 \ud070 \uc218\uc758 \ube44\uac00 \\(q:(1-q)\\) \uac00 \ub418\uc5b4\uc57c \ud55c\ub2e4.
\n\uadf8\ub9ac\uace0 q-\ubd84\uc704\uc218\ub294 q-\ubd84\uc704\uc218\ubcf4\ub2e4 \uc791\uc740 \uc218\uc640 \ud070 \uc218\uc5d0 \ub300\ud574 \\((1-q)\\) \uc640 \\(q\\) \uc758 \ube44\uc911(weight)\ub97c \uacf1\ud574 \uc92c\uc744 \ub54c, \uc911\uc559\uac12\uc5d0 \ud574\ub2f9\ud55c\ub2e4!<\/p>\n
\uc704\uc758 \uc2dd\uc5d0\uc11c \\(\\mathbb{I}(y_i-y_{(q)} > 0)\\) \ub294 I<\/strong>ndicator \ud568\uc218\ub85c \\(y_i-y_{(q)}>0\\) \uac00 \ub9cc\uc871\ud558\uba74 1, \uc544\ub2c8\uba74 0\uc774 \ub41c\ub2e4. \ub530\ub77c\uc11c \\((q-\\mathbb{I}(y_i-y_{(q)} > 0))\\) \ub294 \\(y_i\\) \uc640 \\(y_{(q)}\\) \uc758 \uc0c1\ub300 \ud06c\uae30\uc5d0 \ub530\ub77c \\(q\\) \ub610\ub294 \\((q-1)\\) \uc774 \ub41c\ub2e4. \\((q-1)\\) \uc5d0 \\(-1\\) \uc744 \uacf1\ud558\uba74 \\((1-q)\\) \uac00 \ub41c\ub2e4. \\(-1\\) \uc740 \\((y_i-y_{(q)})\\) = \\(-|(y_i-y_{(q)})|\\) (if \\((y_i-y_{(q)} < 0\\) ) \uc5d0\uc11c \uacf5\uae09\ub41c\ub2e4!<\/p>\n\uc704\uc758 \uc190\uc2e4\ud568\uc218\ub97c \uac04\ub2e8\ud788 \\(y_i-y{(q)}> 0\\) \uc77c \ub54c, \\(q|(y_i-y_{(q)})|\\) , \\(y_i-y{(q)} \\leq 0\\) \uc77c \ub54c, \\((1-q)|(y_i-y_{(q)})|\\) \ub85c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n
\uadf8\ub0e5 \uc815\ub82c\ud574\uc11c \uad6c\ud558\uc9c0?<\/h2>\n
\uadf8\ub7f0\ub370 \uc65c \uc774\ub807\uac8c \ubcf5\uc7a1\ud558\uac8c \uad6c\ud558\uc9c0? \uadf8\ub0e5 \uc815\ub82c\ud574\uc11c \uc790\ub974\uba74 \ub418\uc9c0 \uc54a\uc740\uac00?<\/p>\n
\ud558\uc9c0\ub9cc \uc815\ub82c\ud558\ub294 \ubc29\ubc95\uc740 quantile regression\uc5d0 \uc801\uc6a9\ud558\uae30 \ud798\ub4e4\uc9c0 \uc2f6\ub2e4. \ub2e4\uc2dc \ub9d0\ud574 \\(Y_{(q)}\\) \ub97c \uad6c\ud560 \uc21c \uc788\uc5b4\ub3c4, \\(Y_{(q)}|X\\) \ub97c \uad6c\ud558\uae34 \ud798\ub4e4\uc5b4 \ubcf4\uc778\ub2e4.<\/p>\n
\uacb0\ub860<\/h2>\n
quantile loss\ub294 q-\ubd84\uc704\uc218\ub97c \ucc3e\uae30 \uc704\ud574 \ud2b9\ubcc4\ud788 \uace0\uc548\ub41c weighted MAE\ub77c\uace0 \uc774\ud574\ud560 \uc218 \uc788\ub2e4. \ubcf4\ud1b5 weighted loss\ub294 \uc5b4\ub5a4 \uc790\ub8cc\uc5d0 \ub300\ud574 weight\uc774 \uc815\ud574\uc9c0\uc9c0\ub9cc quantile loss\uc758 weight\uc740 q-\ubd84\uc704\uc218 \ucd94\uc815\uac12\uc5d0 \ub530\ub77c \\(q\\) \ub610\ub294 \\(1-q\\) \ub85c \ub2ec\ub77c\uc9c0\ub294 weight\uc774\ub77c\uace0 \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4!!!<\/p>\n
PS<\/h2>\n
\ub9c8\uc9c0\ub9c9\uc73c\ub85c quantile-gam\uc758 \uc608\ub85c \uae00\uc744 \ub9c8\uce58\uace0\uc790 \ud55c\ub2e4. \ub2e4\uc74c\uc740 \ub370\uc774\ud130\uc640 qgam<\/code>\ud328\ud0a4\uc9c0\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc608\uce21\ud55c conditional 0.2, 0.4, 0.6, 0.8-quantile\uc744 \ubcf4\uc5ec\uc900\ub2e4. (\ucc38\uace0 : https:\/\/mfasiolo.github.io\/qgam\/articles\/qgam.html<\/a> )<\/p>\n\ub370\uc774\ud130\ub294 R \ud50c\ub86f \ud301<\/a>\uc5d0\uc11c \uac00\uc838\uc654\ub2e4.<\/p>\nlibrary(qgam); library(MASS)\r\nlibrary(data.table)\r\nif( suppressWarnings(require(RhpcBLASctl)) ){ blas_set_num_threads(1) } # Optional\r\n\r\nN <- 7000\r\nx1 <- rnorm(N\/2)\r\ny1 <- 2*sin(x1) + rnorm(N\/2)\r\nx2 <- rnorm(N\/2)\r\ny2 <- 2*cos(x2) + rt(N\/2, df=30)\r\ndat <- data.frame(x=c(x1,x2),\r\n y=c(y1,y2))\r\n\r\nquSeq <- c(0.2, 0.4, 0.6, 0.8)\r\nset.seed(6436)\r\nfit <- mqgam(list(y~ s(x, k=20, bs=\"ad\"), ~ s(x)),\r\n data = dat, \r\n qu = quSeq)\r\n\r\n# Plot the data\r\nxSeq <- data.frame(x = seq(-4, 4, length.out = 1e3))\r\n\r\n#plot(dat$x, dat$y, xlab = \"x\", ylab = \"y\", xlim=c(-3,3))\r\n\r\ndatPred <- data.frame(q = rep(quSeq, each=nrow(xSeq)),\r\n x = xSeq$x, \r\n y = NA)\r\nfor(iq in quSeq) {\r\n datPred[datPred$q == iq, 'y'] = qdo(fit, iq, predict, newdata = xSeq)\r\n #lines(xSeq$x, pred, col = 2)\r\n}\r\n\r\ndatPred$q = factor(datPred$q, levels = c(0.8, 0.6, 0.4, 0.2))\r\n\r\nlibrary(ggplot2)\r\n#png('qloss-qgam.png', width=480, height=360)\r\nggplot() + \r\n geom_point(data=dat, mapping=aes(x=x, y=y), alpha=0.3) + \r\n geom_line(data=datPred, mapping=aes(x=x, y=y, color=q), \r\n size=1.1)\r\n#dev.off()\r\n<\/code><\/pre>\n![\"\"](\"http:\/\/141.164.34.82\/wp-content\/uploads\/2020\/04\/quantileLoss.png\")
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![\"\"](\"http:\/\/141.164.34.82\/wp-content\/uploads\/2020\/04\/qLoss_MDN_gam.png\")
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