Neural Network神经网络论文

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关于Neural Network神经网络论文,包括中文和英文的论文。
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内容介绍
<html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta charset="utf-8"> <meta name="generator" content="pdf2htmlEX"> <meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"> <link rel="stylesheet" href="https://static.pudn.com/base/css/base.min.css"> <link rel="stylesheet" href="https://static.pudn.com/base/css/fancy.min.css"> <link rel="stylesheet" href="https://static.pudn.com/prod/directory_preview_static/62795bb5517cd20ea4ca41e7/raw.css"> <script src="https://static.pudn.com/base/js/compatibility.min.js"></script> <script src="https://static.pudn.com/base/js/pdf2htmlEX.min.js"></script> <script> try{ pdf2htmlEX.defaultViewer = new pdf2htmlEX.Viewer({}); }catch(e){} </script> <title></title> </head> <body> <div id="sidebar" style="display: none"> <div id="outline"> </div> </div> <div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/62795bb5517cd20ea4ca41e7/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">Novelty <span class="_ _0"></span>detection <span class="_ _1"></span>and <span class="_ _2"></span><span class="ls1">neural <span class="_ _3"></span><span class="ls0">network <span class="_ _3"></span><span class="ls2">validation </span></span></span></div><div class="t m1 x1 h3 y2 ff2 fs1 fc0 sc0 ls3 ws0">C.M. </div><div class="t m2 x2 h4 y2 ff1 fs2 fc0 sc0 ls4 ws0">Bishop </div><div class="t m3 x1 h5 y3 ff3 fs3 fc0 sc0 ls5 ws0">Indexing </div><div class="t m4 x3 h6 y3 ff4 fs4 fc0 sc0 ls6 ws0">terms: <span class="_ _4"></span>Novelty <span class="_ _2"></span><span class="ls7">assessment, Neural <span class="_ _5"> </span>networks </span></div><div class="t m5 x4 h7 y4 ff3 fs5 fc0 sc0 ls1 ws0">Abstract: </div><div class="t m6 x5 h8 y4 ff2 fs6 fc0 sc0 ls8 ws0">One <span class="_ _2"></span><span class="ls9">of <span class="_ _6"> </span><span class="lsa">the <span class="_ _4"></span><span class="lsb">key <span class="_"> </span><span class="ls1">factors <span class="_ _4"></span><span class="lsc">which <span class="_ _5"></span></span>limits <span class="_ _4"></span>the </span></span></span></span></div><div class="t m6 x4 h8 y5 ff2 fs6 fc0 sc0 lsb ws0">use <span class="_ _5"></span><span class="ls9">of <span class="_ _7"> </span><span class="lsa">neural <span class="_ _2"></span><span class="ls1">networks <span class="_ _4"></span><span class="lsc">in <span class="_ _2"></span></span>many <span class="_ _4"></span></span>industrial <span class="_ _2"></span><span class="ls1">applica- </span></span></span></div><div class="t m6 x4 h8 y6 ff2 fs6 fc0 sc0 lsa ws0">tions <span class="_ _2"></span><span class="ls1">has <span class="_ _4"></span>been <span class="_ _2"></span></span>the <span class="_ _2"></span><span class="lsc">difficulty <span class="_ _2"></span><span class="ls9">of <span class="_ _8"> </span></span></span>demonstrating <span class="_ _4"></span><span class="ls8">that </span></div><div class="t m6 x4 h8 y7 ff2 fs6 fc0 sc0 lsa ws0">a <span class="_ _7"> </span><span class="ls1">trained <span class="_"> </span>network <span class="_"> </span><span class="lsd">will <span class="_ _8"> </span></span></span>continue <span class="_ _7"> </span><span class="lse">to <span class="_ _2"></span><span class="ls1">generate <span class="_"> </span>reli- </span></span></div><div class="t m6 x4 h8 y8 ff2 fs6 fc0 sc0 lsa ws0">able <span class="_ _2"></span><span class="ls8">outputs <span class="ls1">once <span class="_ _4"></span></span></span>it <span class="_ _4"></span><span class="lsc">is <span class="_ _4"></span>in <span class="_ _5"></span></span>routine <span class="_ _5"></span><span class="lsc">use. <span class="_ _4"></span><span class="lsd">An <span class="_"> </span></span></span>import- </div><div class="t m6 x4 h8 y9 ff2 fs6 fc0 sc0 lsa ws0">ant <span class="_ _5"></span><span class="ls1">potential <span class="_ _7"> </span></span>source <span class="lsf">of <span class="_"> </span><span class="ls8">errors <span class="lsc">is <span class="_ _2"></span><span class="ls1">novel <span class="_ _2"></span></span></span></span></span>input <span class="_ _9"> </span><span class="lse">data; </span></div><div class="t m6 x4 h8 ya ff2 fs6 fc0 sc0 ls8 ws0">that <span class="_ _5"></span><span class="lsc">is, <span class="_ _4"></span><span class="lsa">input <span class="_ _9"> </span><span class="lse">data <span class="_ _2"></span></span></span>which <span class="_"> </span>differ <span class="_ _4"></span><span class="ls1">significantly from </span></span></div><div class="t m6 x4 h8 yb ff2 fs6 fc0 sc0 lsa ws0">the <span class="_ _9"> </span><span class="lse">data <span class="_ _9"> </span><span class="lsc">used <span class="_ _8"> </span></span>to <span class="_ _5"></span></span>train <span class="_ _a"> </span>the <span class="_ _9"> </span><span class="ls1">network. <span class="_ _a"> </span></span>The <span class="_ _7"> </span><span class="ls8">author </span></div><div class="t m6 x4 h8 yc ff2 fs6 fc0 sc0 ls1 ws0">investigates <span class="_"> </span><span class="lsa">the <span class="_"> </span></span>relationship <span class="_ _6"> </span>between <span class="_"> </span><span class="lsa">the <span class="_ _a"> </span></span>degree </div><div class="t m6 x4 h8 yd ff2 fs6 fc0 sc0 ls9 ws0">of <span class="_ _b"> </span><span class="ls1">novelty <span class="_ _8"> </span></span>of <span class="_ _b"> </span><span class="lsa">input <span class="_ _a"> </span><span class="lse">data <span class="_ _9"> </span></span>and <span class="_"> </span>the <span class="_"> </span>corresponding </span></div><div class="t m6 x4 h8 ye ff2 fs6 fc0 sc0 ls1 ws0">reliability <span class="_"> </span><span class="lsf">of <span class="_ _c"> </span><span class="lsa">the <span class="_"> </span><span class="ls8">outputs <span class="_ _5"></span></span></span></span>from <span class="_ _a"> </span>the <span class="_ _a"> </span>network. <span class="_ _6"> </span><span class="ls8">He </span></div><div class="t m6 x4 h8 yf ff2 fs6 fc0 sc0 ls1 ws0">describes <span class="_"> </span><span class="lsa">a <span class="_ _a"> </span>quantitative <span class="_ _a"> </span>procedure <span class="_"> </span></span>for <span class="_ _a"> </span>assessing </div><div class="t m6 x4 h8 y10 ff2 fs6 fc0 sc0 ls1 ws0">novelty, <span class="_ _c"> </span><span class="ls8">and <span class="_ _c"> </span><span class="lsa">demonstrates <span class="_ _b"> </span>its <span class="_ _c"> </span></span></span>performance <span class="_ _d"> </span><span class="lsb">by </span></div><div class="t m6 x4 h8 y11 ff2 fs6 fc0 sc0 ls1 ws0">using <span class="_"> </span><span class="lsa">an <span class="_ _a"> </span>application <span class="_"> </span><span class="lsc">which <span class="_ _8"> </span></span></span>involves <span class="_ _9"> </span><span class="lsa">monitoring </span></div><div class="t m6 x4 h8 y12 ff2 fs6 fc0 sc0 ls1 ws0">oil <span class="_ _1"></span><span class="lsc">flow <span class="ls1">in multiphase <span class="_ _2"></span>pipelines. </span></span></div><div class="t m7 x1 h9 y13 ff1 fs7 fc0 sc0 ls1 ws0">1 </div><div class="t m8 x6 h9 y13 ff1 fs7 fc0 sc0 ls10 ws0">Introduction </div><div class="t m6 x1 h8 y14 ff2 fs6 fc0 sc0 lsa ws0">Neural <span class="_ _b"> </span><span class="ls1">networks <span class="_ _c"> </span>have <span class="_ _c"> </span>been <span class="_ _c"> </span>shown <span class="_ _c"> </span><span class="lse">to <span class="_ _6"> </span></span>have <span class="_ _c"> </span></span>a <span class="_ _c"> </span><span class="lsc">useful </span></div><div class="t m6 x1 h8 y15 ff2 fs6 fc0 sc0 ls1 ws0">degree <span class="_ _9"> </span><span class="ls9">of <span class="_ _d"> </span></span>performance <span class="_"> </span>over <span class="_"> </span><span class="lsa">a <span class="_ _7"> </span><span class="lsc">wide <span class="_"> </span></span></span>range <span class="_ _8"> </span><span class="ls9">of <span class="_ _b"> </span><span class="lsa">industrial </span></span></div><div class="t m6 x1 h8 y16 ff2 fs6 fc0 sc0 ls8 ws0">and <span class="_ _5"></span><span class="ls1">medical <span class="_ _9"> </span>applications. <span class="_"> </span>However, <span class="_ _5"></span><span class="lsa">a <span class="_ _7"> </span><span class="lsb">key <span class="_ _8"> </span></span>factor <span class="_ _5"></span></span>which </span></div><div class="t m6 x1 h8 y17 ff2 fs6 fc0 sc0 ls1 ws0">limits <span class="_ _4"></span><span class="lsa">the <span class="_ _5"></span></span>widespread <span class="_ _5"></span>implementation <span class="_"> </span><span class="lsf">of <span class="_ _8"> </span></span>neural-network </div><div class="t m6 x7 h8 y18 ff2 fs6 fc0 sc0 lsa ws0">solutions <span class="lsc">in <span class="_ _2"></span><span class="ls1">many <span class="_ _5"></span></span></span>areas has <span class="_ _2"></span><span class="ls1">been <span class="_ _2"></span></span>the <span class="_ _2"></span><span class="lsc">difficulty <span class="ls9">of <span class="_ _7"> </span></span></span>demon- </div><div class="t m6 x1 h8 y19 ff2 fs6 fc0 sc0 lsa ws0">strating <span class="_ _4"></span><span class="ls8">that <span class="_ _2"></span><span class="ls1">the <span class="_ _2"></span></span>outputs <span class="ls1">generated <span class="_ _4"></span><span class="lsb">by <span class="_ _5"></span></span></span></span>the <span class="_ _2"></span><span class="ls1">network <span class="_ _5"></span><span class="lsc">in <span class="_ _4"></span></span></span>the </div><div class="t m6 x1 h8 y1a ff2 fs6 fc0 sc0 lsb ws0">field <span class="_ _9"> </span><span class="ls8">are <span class="_ _1"></span><span class="ls1">reliable. <span class="ls8">In <span class="_ _2"></span></span>general, <span class="lsa">the <span class="_ _2"></span></span>problem <span class="_ _9"> </span><span class="lsf">of <span class="_"> </span></span>network <span class="_ _9"> </span>val- </span></span></div><div class="t m6 x1 h8 y1b ff2 fs6 fc0 sc0 lsa ws0">idation <span class="_"> </span><span class="ls1">is <span class="_"> </span></span>a <span class="_"> </span><span class="lsc">difficult <span class="_ _9"> </span><span class="ls1">one, <span class="_"> </span><span class="ls8">and <span class="_ _7"> </span></span></span></span>it <span class="_"> </span><span class="ls1">involves <span class="_ _9"> </span>many <span class="_"> </span><span class="lsc">issues, </span></span></div><div class="t m6 x1 h8 y1c ff2 fs6 fc0 sc0 ls1 ws0">some </div><div class="t m9 x8 ha y1d ff2 fs8 fc0 sc0 ls11 ws0">of </div><div class="t m6 x9 h8 y1d ff2 fs6 fc0 sc0 lsc ws0">which <span class="_ _9"> </span><span class="lsa">are <span class="_ _4"></span><span class="ls1">generic <span class="_ _2"></span><span class="lse">to <span class="_ _2"></span></span></span>any <span class="_ _9"> </span><span class="ls1">software <span class="_ _2"></span>system. <span class="_ _2"></span>Here </span></span></div><div class="t m6 x1 h8 y1e ff2 fs6 fc0 sc0 lsf ws0">we <span class="_ _b"> </span><span class="ls1">consider <span class="_ _c"> </span>only <span class="_ _6"> </span>those <span class="_ _c"> </span>aspects <span class="_ _6"> </span><span class="lsc">which <span class="_ _c"> </span><span class="lsa">are <span class="_ _a"> </span></span>specific <span class="_"> </span><span class="lse">to </span></span></span></div><div class="t m6 x7 h8 y1f ff2 fs6 fc0 sc0 ls1 ws0">neural <span class="_ _4"></span>networks <span class="_ _2"></span><span class="ls8">and </span>related methods. </div><div class="t ma xa hb y20 ff2 fs9 fc0 sc0 ls12 ws0">It </div><div class="t m6 xb h8 y20 ff2 fs6 fc0 sc0 lsc ws0">is <span class="_ _4"></span>useful <span class="_ _4"></span><span class="lse">to <span class="ls1">distinguish <span class="_ _4"></span>between <span class="_ _4"></span>two <span class="_ _5"></span><span class="ls8">broad <span class="_ _4"></span></span></span></span>levels <span class="_ _2"></span><span class="ls1">of </span></div><div class="t m6 x1 h8 y21 ff2 fs6 fc0 sc0 ls1 ws0">network <span class="_ _9"> </span>validation. <span class="_ _4"></span><span class="lsf">At <span class="_ _7"> </span><span class="lsa">the <span class="_ _4"></span></span></span>first <span class="lsb">level <span class="_ _4"></span><span class="lsf">we <span class="_ _7"> </span></span></span>provide, <span class="_ _2"></span><span class="lsc">in <span class="_ _4"></span><span class="lsa">addi- </span></span></div><div class="t m6 x7 h8 y22 ff2 fs6 fc0 sc0 ls1 ws0">tion <span class="_ _5"></span><span class="lse">to <span class="_ _2"></span><span class="lsa">the <span class="_ _2"></span></span></span>network <span class="_ _7"> </span><span class="lsa">outputs, <span class="_ _2"></span></span>some <span class="_ _4"></span>associated <span class="_ _4"></span>measure <span class="_ _5"></span><span class="ls9">of </span></div><div class="t m6 x7 h8 y23 ff2 fs6 fc0 sc0 ls1 ws0">confidence </div><div class="t mb xc ha y24 ff2 fs8 fc0 sc0 ls1 ws0">so </div><div class="t m6 xd h8 y24 ff2 fs6 fc0 sc0 ls8 ws0">that <span class="_ _4"></span><span class="ls1">potentially <span class="_"> </span>unreliable <span class="_ _5"></span></span>outputs <span class="_ _4"></span><span class="lsa">can <span class="_ _4"></span><span class="lsc">be </span></span></div><div class="t m6 x1 h8 y25 ff2 fs6 fc0 sc0 ls1 ws0">detected. <span class="_ _a"> </span>In <span class="_ _6"> </span><span class="lsa">many <span class="_ _a"> </span>applications <span class="_ _6"> </span></span>this <span class="_ _8"> </span><span class="lsd">will <span class="_ _b"> </span><span class="lsc">give <span class="_"> </span><span class="lsa">a <span class="_ _6"> </span></span>useful </span></span></div><div class="t m6 x7 h8 y26 ff2 fs6 fc0 sc0 ls1 ws0">degree <span class="_ _2"></span><span class="ls9">of <span class="_ _6"> </span></span>validation, <span class="_ _9"> </span>since <span class="_ _2"></span>it <span class="_ _4"></span>could <span class="_ _5"></span>allow <span class="_ _4"></span>the <span class="_ _9"> </span>bulk <span class="_ _5"></span><span class="ls9">of <span class="_ _6"> </span><span class="lsa">the </span></span></div><div class="t m6 x7 h8 y27 ff2 fs6 fc0 sc0 lsa ws0">input <span class="_"> </span><span class="lse">data <span class="_ _9"> </span>to <span class="_ _9"> </span><span class="lsc">be <span class="_ _8"> </span><span class="ls1">processed <span class="_"> </span>with <span class="_"> </span>high <span class="_"> </span>confidence, <span class="_ _4"></span>with </span></span></span></div><div class="t m6 x7 h8 y28 ff2 fs6 fc0 sc0 ls1 ws0">remaining <span class="_ _8"> </span><span class="lse">data <span class="_ _9"> </span></span>either <span class="_"> </span>being <span class="_ _9"> </span>discarded </div><div class="t mc xe hc y29 ff2 fs2 fc0 sc0 ls13 ws0">or </div><div class="t m6 xf h8 y29 ff2 fs6 fc0 sc0 ls1 ws0">processed <span class="_"> </span><span class="lsb">by </span></div><div class="t m6 x7 h8 y2a ff2 fs6 fc0 sc0 lsa ws0">other <span class="_"> </span><span class="ls1">means. <span class="_ _7"> </span><span class="ls14">An <span class="_ _6"> </span></span>example <span class="_ _9"> </span>would <span class="_"> </span><span class="lsc">be <span class="_"> </span></span></span>the <span class="_"> </span><span class="ls8">automation <span class="_ _4"></span><span class="ls9">of </span></span></div><div class="t m6 x7 h8 y2b ff2 fs6 fc0 sc0 ls1 ws0">medical <span class="_ _a"> </span>image <span class="_ _c"> </span><span class="lsa">interpretation <span class="_ _c"> </span></span>for <span class="_ _6"> </span>mass <span class="_ _c"> </span>screening <span class="_ _a"> </span><span class="lsa">pro- </span></div><div class="t m6 x7 h8 y2c ff2 fs6 fc0 sc0 ls1 ws0">grammes, <span class="_ _7"> </span><span class="lsc">in <span class="_ _8"> </span>which <span class="_ _a"> </span></span>images <span class="_"> </span><span class="ls8">that <span class="_"> </span></span>gave <span class="_ _9"> </span>network <span class="_ _c"> </span><span class="ls8">outputs </span></div><div class="t m6 x7 h8 y2d ff2 fs6 fc0 sc0 ls1 ws0">with <span class="_ _5"></span><span class="lsa">a <span class="_ _9"> </span><span class="lsc">low <span class="_ _4"></span></span></span>confidence <span class="_ _2"></span>could <span class="_ _9"> </span><span class="lsc">be <span class="_ _9"> </span>rejected <span class="_ _5"></span><span class="lsb">by <span class="_ _8"> </span><span class="lsa">the <span class="_ _5"></span></span></span></span>network </div><div class="t m6 x7 h8 y2e ff2 fs6 fc0 sc0 lsa ws0">and <span class="_"> </span><span class="ls1">interpreted <span class="_ _6"> </span>instead <span class="_ _8"> </span><span class="lsf">by <span class="_ _e"> </span></span></span>human <span class="_"> </span><span class="ls1">experts. <span class="_ _9"> </span><span class="ls8">The <span class="_ _5"></span></span>second </span></div><div class="t m6 x7 h8 y2f ff2 fs6 fc0 sc0 lsb ws0">level <span class="_ _5"></span><span class="lsf">of <span class="_ _7"> </span><span class="ls1">validation <span class="_ _7"> </span>requires <span class="_ _4"></span><span class="lsa">a <span class="_ _4"></span>guarantee <span class="_ _4"></span><span class="ls8">that <span class="_ _4"></span></span>the <span class="_ _4"></span></span>network </span></span></div><div class="t m6 x7 h8 y30 ff2 fs6 fc0 sc0 ls8 ws0">outputs are <span class="ls1">reliable for <span class="_ _4"></span>any <span class="_ _4"></span>achievable <span class="lsa">input <span class="_ _5"></span></span>vector. <span class="_ _2"></span>This </span></div><div class="t m6 x7 h8 y31 ff2 fs6 fc0 sc0 ls1 ws0">represents <span class="_"> </span><span class="lsa">a <span class="_"> </span></span>much <span class="_"> </span>more <span class="_ _8"> </span><span class="lsa">rigorous <span class="_"> </span></span>degree <span class="_ _9"> </span><span class="ls9">of <span class="_ _b"> </span></span>validation, </div><div class="t m6 x7 h8 y32 ff2 fs6 fc0 sc0 ls8 ws0">and </div><div class="t md x10 ha y33 ff2 fs8 fc0 sc0 ls15 ws0">it </div><div class="t m6 x11 h8 y33 ff2 fs6 fc0 sc0 ls1 ws0">would <span class="_ _5"></span><span class="lsd">be <span class="_ _9"> </span><span class="ls8">appropriate <span class="_ _0"></span><span class="ls1">for <span class="_ _4"></span><span class="lsa">networks <span class="_ _2"></span><span class="lsc">used <span class="_ _9"> </span>in <span class="_ _4"></span></span></span>safety- </span></span></span></div><div class="t m6 x7 h8 y34 ff2 fs6 fc0 sc0 ls1 ws0">critical <span class="_ _5"></span>applications, <span class="_ _9"> </span>for <span class="_ _5"></span>example. <span class="_ _4"></span><span class="ls8">In <span class="_ _4"></span></span>this <span class="_ _9"> </span><span class="lsa">paper <span class="_ _9"> </span><span class="ls9">we <span class="_ _6"> </span></span></span>shall </div><div class="t m6 x7 h8 y35 ff2 fs6 fc0 sc0 ls1 ws0">consider only <span class="_ _2"></span><span class="ls8">the <span class="_ _3"></span><span class="ls1">first <span class="lsc">level <span class="_ _3"></span><span class="lsf">of <span class="_ _5"></span><span class="ls1">validation. </span></span></span></span></span></div><div class="t me x7 hd y36 ff5 fs6 fc0 sc0 ls1 ws0">0 </div><div class="t m3 x12 h5 y36 ff3 fs3 fc0 sc0 ls5 ws0">IEE, <span class="_ _4"></span>1994 </div><div class="t m3 x13 h5 y37 ff3 fs3 fc0 sc0 ls5 ws0">Paper <span class="_ _4"></span>1330K <span class="_ _2"></span><span class="ls16">(C4, <span class="_ _0"></span>E5), <span class="_ _0"></span><span class="ls17">first <span class="_ _4"></span><span class="ls5">received 29th <span class="_ _2"></span><span class="ls16">November <span class="_ _5"> </span></span>1993 <span class="ls16">and </span>in revised </span></span></span></div><div class="t m3 x7 h5 y38 ff3 fs3 fc0 sc0 ls5 ws0">form <span class="_ _2"></span>19th <span class="_ _2"></span><span class="ls18">May <span class="_ _4"></span></span>1994 </div><div class="t m3 x7 h5 y39 ff3 fs3 fc0 sc0 ls5 ws0">The <span class="_"> </span>author <span class="_"> </span><span class="ls19">is <span class="_"> </span></span>with <span class="_"> </span>the <span class="_"> </span>Neural <span class="_ _7"> </span><span class="ls16">Computing <span class="_"> </span></span>Research <span class="_ _5"> </span><span class="ls16">Group, <span class="_ _5"> </span></span>Depart- </div><div class="t m3 x7 h5 y3a ff3 fs3 fc0 sc0 ls5 ws0">ment </div><div class="t mf x14 he y3b ff3 fsa fc0 sc0 ls1a ws0">of </div><div class="t m3 x15 h5 y3b ff3 fs3 fc0 sc0 ls5 ws0">Computer <span class="_ _a"> </span>Science, <span class="_ _7"> </span>Aston <span class="_ _6"> </span>University, <span class="_ _8"> </span>Birmingham <span class="_ _a"> </span><span class="ls1b">84 <span class="_ _0"></span><span class="ls16">7ET, </span></span></div><div class="t m3 x7 h5 y3c ff3 fs3 fc0 sc0 ls16 ws0">United <span class="_ _2"></span>Kingdom </div><div class="t m3 x7 h5 y3d ff3 fs3 fc0 sc0 ls1c ws0">IEE <span class="_ _3"></span><span class="ls16">Proc.-Vis. </span></div><div class="t m4 x16 h6 y3e ff4 fs4 fc0 sc0 ls7 ws0">Image <span class="_ _2"></span><span class="ls1">Signal <span class="_ _4"></span></span>Process., <span class="_ _5"> </span><span class="ls1d">Vol. <span class="_ _f"></span><span class="ls7">141, <span class="ls1d">No. </span></span></span></div><div class="t m10 xe hf y3e ff4 fsb fc0 sc0 ls1e ws0">4, </div><div class="t m11 x17 h10 y3e ff6 fsc fc0 sc0 ls1 ws0">August </div><div class="t m4 x18 h6 y3e ff4 fs4 fc0 sc0 ls1 ws0">1994 </div><div class="t m6 x19 h8 y3f ff2 fs6 fc0 sc0 ls1 ws0">Intuitively <span class="_ _4"></span><span class="lsf">we <span class="_"> </span></span>expect <span class="ls8">that <span class="_ _4"></span><span class="lsa">a <span class="_ _2"></span></span></span>network <span class="_ _5"></span><span class="lsd">will <span class="_ _5"></span></span>generate <span class="_ _4"></span><span class="lsc">reli- </span></div><div class="t m6 x1a h8 y40 ff2 fs6 fc0 sc0 ls1 ws0">able <span class="_ _5"></span>results <span class="_ _5"></span>when <span class="_ _9"> </span>presented <span class="_ _9"> </span>with <span class="_ _4"></span><span class="ls8">data <span class="_ _4"></span><span class="lsc">which <span class="_ _9"> </span></span>are <span class="_ _4"></span></span>similar </div><div class="t m6 x1a h8 y41 ff2 fs6 fc0 sc0 lse ws0">to <span class="lsa">those <span class="lsc">used <span class="_ _4"></span></span>during <span class="ls1">training, <span class="_ _2"></span></span>but <span class="_ _4"></span><span class="ls8">that <span class="lsc">when <span class="_ _2"></span><span class="ls1">substantially </span></span></span></span></div><div class="t m6 x1a h8 y42 ff2 fs6 fc0 sc0 lsc ws0">novel <span class="_ _a"> </span><span class="lse">data <span class="_ _7"> </span><span class="ls8">are <span class="_ _9"> </span><span class="ls1">presented <span class="_ _6"> </span>the <span class="_ _8"> </span>network <span class="_ _c"> </span></span>outputs <span class="_ _9"> </span><span class="lsd">will <span class="_ _c"> </span></span></span></span>be </div><div class="t m6 x1a h8 y43 ff2 fs6 fc0 sc0 ls8 ws0">prone </div><div class="t m12 x1b hc y44 ff2 fs2 fc0 sc0 ls1f ws0">to </div><div class="t m6 x1c h8 y44 ff2 fs6 fc0 sc0 lsa ws0">serious <span class="_ _5"></span>error. <span class="_ _7"> </span><span class="ls8">In <span class="_ _9"> </span><span class="ls1">Section </span></span></div><div class="t m5 x1d h8 y44 ff2 fs6 fc0 sc0 ls1 ws0">2 </div><div class="t m6 x1e h8 y44 ff2 fs6 fc0 sc0 ls9 ws0">we <span class="_ _e"> </span><span class="ls1">investigate <span class="_ _9"> </span><span class="lsa">the </span></span></div><div class="t m6 x1a h8 y45 ff2 fs6 fc0 sc0 ls1 ws0">relationship <span class="_ _5"></span>between <span class="_ _2"></span><span class="lsa">the <span class="_ _4"></span></span>novelty <span class="ls9">of <span class="_ _8"> </span><span class="lsa">input <span class="_ _5"></span><span class="lse">data <span class="ls8">and </span></span></span></span>valid- </div><div class="t m6 x1a h8 y46 ff2 fs6 fc0 sc0 lsc ws0">ity <span class="_ _5"></span><span class="ls9">of <span class="_ _6"> </span><span class="ls1">network <span class="_ _9"> </span><span class="lsa">outputs, <span class="_ _5"></span><span class="ls8">and <span class="_ _2"></span></span></span></span>we <span class="_ _a"> </span></span>use <span class="_ _5"></span><span class="ls1">this <span class="_ _4"></span><span class="lsa">as <span class="_ _4"></span></span>the <span class="_ _4"></span>basis <span class="_ _4"></span><span class="lsf">of <span class="_ _8"> </span><span class="lsa">a </span></span></span></div><div class="t m6 x1a h8 y47 ff2 fs6 fc0 sc0 ls1 ws0">practical <span class="_ _6"> </span>system <span class="_ _9"> </span>for <span class="_"> </span>network <span class="_ _6"> </span>validation </div><div class="t m13 x1f h11 y48 ff2 fsd fc0 sc0 ls20 ws0">[2]. </div><div class="t m6 x20 h8 y48 ff2 fs6 fc0 sc0 ls8 ws0">The <span class="_ _9"> </span><span class="ls1">tech- </span></div><div class="t m6 x1a h8 y49 ff2 fs6 fc0 sc0 ls1 ws0">nique <span class="_ _9"> </span><span class="lsc">is <span class="_ _9"> </span><span class="lsa">illustrated <span class="_ _5"></span></span>in <span class="_"> </span></span>Section </div><div class="t m14 x21 h12 y4a ff3 fs9 fc0 sc0 ls1 ws0">3 </div><div class="t m6 x22 h8 y4a ff2 fs6 fc0 sc0 ls1 ws0">using <span class="_ _9"> </span><span class="lsa">an <span class="_ _9"> </span></span>example <span class="_ _4"></span>from </div><div class="t m6 x1a h8 y4b ff2 fs6 fc0 sc0 ls1 ws0">the <span class="_ _4"></span><span class="lsa">monitoring <span class="_ _9"> </span><span class="ls9">of <span class="_ _8"> </span></span></span>multiphase <span class="_ _9"> </span><span class="lsc">flows <span class="_ _2"></span></span>in <span class="_ _4"></span>oil <span class="_ _5"></span>pipelines. Pos- </div><div class="t m6 x1a h8 y4c ff2 fs6 fc0 sc0 ls1 ws0">sible extensions <span class="_ _4"></span><span class="ls9">of <span class="_ _7"> </span></span>this <span class="_ _4"></span><span class="lsa">approach <span class="_ _5"></span><span class="ls8">are </span></span>discussed <span class="lsc">in <span class="_ _4"></span></span>Section </div><div class="t m15 x1a h4 y4d ff1 fs2 fc0 sc0 ls21 ws0">4. </div><div class="t m16 x1a h13 y4e ff1 fse fc0 sc0 ls1 ws0">2 </div><div class="t m17 x23 h14 y4e ff1 fsf fc0 sc0 ls22 ws0">Network </div><div class="t m8 x24 h9 y4e ff1 fs7 fc0 sc0 ls10 ws0">validation </div><div class="t m6 x25 h8 y4f ff2 fs6 fc0 sc0 ls1 ws0">Consider <span class="_ _4"></span><span class="lsa">a <span class="_ _4"></span></span>feedforward network <span class="_ _5"></span><span class="lsa">trained <span class="_ _5"></span><span class="lsb">by <span class="_ _5"></span></span></span>minimising <span class="_ _2"></span><span class="lsa">a </span></div><div class="t m6 x25 h8 y50 ff2 fs6 fc0 sc0 ls1 ws0">sum-of-squares <span class="_"> </span><span class="ls8">error <span class="_"> </span></span>function. <span class="_ _a"> </span><span class="ls14">If <span class="_ _d"> </span><span class="lsf">we <span class="_ _d"> </span><span class="lsa">denote <span class="_ _6"> </span>the <span class="_"> </span>joint </span></span></span></div><div class="t m6 x25 h8 y51 ff2 fs6 fc0 sc0 ls1 ws0">probability <span class="_ _e"> </span>density <span class="_ _a"> </span>functions <span class="_ _a"> </span>for <span class="_ _a"> </span><span class="lsa">the <span class="_ _6"> </span></span>training <span class="_ _c"> </span><span class="lse">data <span class="_ _7"> </span><span class="lsb">by </span></span></div><div class="t m18 x25 h15 y52 ff4 fs5 fc0 sc0 ls23 ws0">p(x, </div><div class="t m19 x26 h16 y52 ff7 fs10 fc0 sc0 ls24 ws0">fj), </div><div class="t m6 x27 h8 y52 ff2 fs6 fc0 sc0 ls1 ws0">then <span class="_ _2"></span><span class="lsf">we <span class="_ _5"></span></span>can <span class="_ _2"></span>write <span class="lsa">the </span></div><div class="t m5 x28 h8 y52 ff2 fs6 fc0 sc0 ls8 ws0">error </div><div class="t m6 x29 h8 y52 ff2 fs6 fc0 sc0 ls1 ws0">in <span class="_ _2"></span><span class="lsa">the </span>form </div><div class="t m1a x2a h17 y53 ff4 fs2 fc0 sc0 ls1 ws0">E </div><div class="t m1b x2b h18 y53 ff8 fs4 fc0 sc0 ls1 ws0">= </div><div class="t m1c x2c h19 y53 ff9 fs11 fc0 sc0 ls1 ws0">1 </div><div class="t m1d x2d h1a y53 ff6 fse fc0 sc0 ls25 ws0">[yAx; </div><div class="t m1e x2e h1a y53 ff6 fse fc0 sc0 ls25 ws0">w) </div><div class="t m1f x2f h1b y53 ff8 fs12 fc0 sc0 ls1 ws0">- </div><div class="t m20 x30 h17 y53 ff4 fs2 fc0 sc0 ls26 ws0">tj]&#8217;p(x, </div><div class="t m21 x31 h17 y53 ff4 fs2 fc0 sc0 ls27 ws0">tj) <span class="_ _10"></span><span class="ls28">dx <span class="_ _0"></span><span class="ls29">dtj </span></span></div><div class="t m22 x32 h1c y54 ff8 fs13 fc0 sc0 ls2a ws0">(1) </div><div class="t m23 x27 h1d y55 ffa fs14 fc0 sc0 ls2b ws0">j= </div><div class="t m24 x33 h1e y55 ff8 fs15 fc0 sc0 ls2c ws0">*s </div><div class="t m25 x1c h1f y55 ff1 fs16 fc0 sc0 ls1 ws0">1 </div><div class="t m6 x1a h8 y56 ff2 fs6 fc0 sc0 ls1 ws0">where </div><div class="t m26 x34 h20 y56 ff2 fs13 fc0 sc0 ls1 ws0">j </div><div class="t m27 x2c h21 y56 ff8 fsf fc0 sc0 ls1 ws0">= </div><div class="t m28 x35 h22 y56 ff2 fs5 fc0 sc0 ls2d ws0">1, </div><div class="t m29 x36 h21 y56 ff8 fsf fc0 sc0 ls1 ws0">. <span class="_ _11"></span>. </div><div class="t m5 x37 h23 y56 ff8 fs17 fc0 sc0 ls1 ws0">. </div><div class="t m2a x38 h24 y56 ff8 fs18 fc0 sc0 ls1 ws0">, </div><div class="t m9 x39 ha y56 ff2 fs8 fc0 sc0 ls1 ws0">c </div><div class="t m6 x3a h8 y56 ff2 fs6 fc0 sc0 ls1 ws0">labels <span class="_ _4"></span><span class="lsa">the <span class="_ _4"></span><span class="ls8">output <span class="_ _4"></span></span></span>units, </div><div class="t m2b x3b h13 y56 ff1 fse fc0 sc0 ls1 ws0">x </div><div class="t m6 x3c h8 y56 ff2 fs6 fc0 sc0 lsc ws0">is <span class="_ _4"></span><span class="lsa">the <span class="_ _4"></span>input </span></div><div class="t m6 x1a h8 y57 ff2 fs6 fc0 sc0 ls1 ws0">vector <span class="_ _4"></span><span class="lse">to <span class="_ _2"></span><span class="lsa">the <span class="_ _4"></span>network, </span></span></div><div class="t m1d x3d h1a y58 ff6 fse fc0 sc0 ls2e ws0">yj </div><div class="t m6 x3e h8 y58 ff2 fs6 fc0 sc0 lsa ws0">denotes <span class="_ _2"></span>the <span class="_ _4"></span><span class="ls8">output <span class="_ _2"></span><span class="ls1">from <span class="_ _2"></span>unit </span></span></div><div class="t m2c x3f h25 y58 ffa fs19 fc0 sc0 ls2f ws0">j, </div><div class="t m6 x25 h8 y59 ff2 fs6 fc0 sc0 lsa ws0">and </div><div class="t m4 x2b h17 y59 ff7 fs2 fc0 sc0 ls30 ws0">tj </div><div class="t m6 x40 h8 y59 ff2 fs6 fc0 sc0 lsc ws0">is <span class="_ _8"> </span><span class="lsa">the <span class="_ _a"> </span><span class="ls1">target <span class="_ _c"> </span>value <span class="_"> </span>for <span class="_ _8"> </span><span class="ls8">that <span class="_ _6"> </span></span>unit. <span class="_"> </span><span class="ls8">The <span class="_ _7"> </span></span>network </span></span></div><div class="t m6 x1a h8 y5a ff2 fs6 fc0 sc0 lsa ws0">corresponds <span class="_"> </span><span class="lse">to <span class="_"> </span></span>a <span class="_"> </span><span class="ls1">set <span class="_ _a"> </span><span class="ls9">of <span class="_ _d"> </span></span>functional <span class="_ _6"> </span>mappings </span></div><div class="t m1d x41 h1a y5b ff6 fse fc0 sc0 ls31 ws0">yjx; </div><div class="t m2d x32 h26 y5b ff6 fs7 fc0 sc0 ls32 ws0">w), </div><div class="t m6 x1a h8 y5c ff2 fs6 fc0 sc0 ls1 ws0">parametrised <span class="_ _e"> </span><span class="lsb">by <span class="_ _c"> </span><span class="lsa">a <span class="_"> </span><span class="lsc">set- <span class="_ _1"></span><span class="lsf">of <span class="_ _e"> </span><span class="ls1">weights <span class="_ _9"> </span><span class="ls8">and <span class="_ _7"> </span></span>biases </span></span></span></span></span></div><div class="t m2d x42 h26 y5c ff6 fs7 fc0 sc0 ls1 ws0">w </div><div class="t m6 x43 h8 y5c ff2 fs6 fc0 sc0 ls1 ws0">whose </div><div class="t m6 x25 h8 y5d ff2 fs6 fc0 sc0 lsc ws0">values <span class="ls8">are <span class="_ _0"></span><span class="ls1">found <span class="_ _4"></span><span class="lsb">by <span class="_ _4"></span></span>minimising </span></span></div><div class="t m2e x44 h27 y5e ff7 fs9 fc0 sc0 ls12 ws0">E. </div><div class="t m6 x2a h8 y5f ff2 fs6 fc0 sc0 lsc ws0">We <span class="_ _2"></span><span class="lsa">note <span class="_ _2"></span><span class="ls8">that <span class="_ _2"></span><span class="ls1">the <span class="_ _1"></span><span class="lsa">joint <span class="_ _5"></span><span class="ls1">density </span></span></span></span></span></div><div class="t m2f x22 h28 y5f ff1 fs1a fc0 sc0 ls33 ws0">p(x, </div><div class="t m21 x45 h17 y5f ff4 fs2 fc0 sc0 ls34 ws0">tj) </div><div class="t m6 x46 h8 y5f ff2 fs6 fc0 sc0 lsa ws0">can <span class="_ _2"></span><span class="lsc">be <span class="_ _2"></span><span class="ls1">factorised </span></span></div><div class="t m6 x25 h8 y60 ff2 fs6 fc0 sc0 lsa ws0">into <span class="_ _2"></span>the <span class="_ _2"></span>product <span class="_ _4"></span><span class="ls9">of <span class="_ _7"> </span></span>the <span class="_ _2"></span>unconditional <span class="_ _2"></span><span class="ls1">density </span></div><div class="t mb x47 ha y61 ff2 fs8 fc0 sc0 ls35 ws0">of </div><div class="t m6 x42 h8 y61 ff2 fs6 fc0 sc0 lsa ws0">the <span class="_ _2"></span>input </div><div class="t m6 x25 h8 y62 ff2 fs6 fc0 sc0 lse ws0">data </div><div class="t m2f x23 h28 y63 ff1 fs1a fc0 sc0 ls36 ws0">p(x) </div><div class="t m6 x48 h8 y63 ff2 fs6 fc0 sc0 ls8 ws0">and <span class="_ _4"></span><span class="lsa">the <span class="_ _5"></span>conditional <span class="_ _5"></span><span class="ls1">density <span class="_ _9"> </span><span class="lsf">of <span class="_ _a"> </span></span>the <span class="_ _9"> </span></span>target <span class="_ _9"> </span></span>data </div><div class="t m18 x25 h15 y64 ff4 fs5 fc0 sc0 ls23 ws0">p(tj </div><div class="t m30 x49 h29 y64 ff8 fs1b fc0 sc0 ls1 ws0">I </div><div class="t m31 x4a h9 y64 ff1 fs7 fc0 sc0 ls37 ws0">x). </div><div class="t m6 x2c h8 y64 ff2 fs6 fc0 sc0 lsc ws0">After <span class="_ _9"> </span><span class="ls1">some <span class="_ _5"></span>simple <span class="_ _5"></span>algebra, <span class="_"> </span><span class="lsf">we <span class="_ _8"> </span><span class="lsa">can <span class="_ _7"> </span></span></span>then <span class="_"> </span>rewrite </span></div><div class="t m6 x25 h8 y65 ff2 fs6 fc0 sc0 lsa ws0">the <span class="ls8">error <span class="_ _3"></span><span class="ls9">of <span class="_ _7"> </span><span class="ls1">eqn. <span class="_ _4"></span>1 <span class="_ _0"></span><span class="lsc">in <span class="lsa">the <span class="_ _2"></span><span class="ls1">form </span></span></span></span></span></span></div><div class="t m32 x27 h2a y66 ff7 fs14 fc0 sc0 ls38 ws0">j=-l </div><div class="t m33 x4b h2b y66 ff2 fs1c fc0 sc0 ls1 ws0">J </div><div class="t m6 x25 h8 y67 ff2 fs6 fc0 sc0 ls1 ws0">where <span class="_ _9"> </span><span class="lsf">we <span class="_ _6"> </span></span>have <span class="_"> </span>defined <span class="_ _5"></span>the <span class="_"> </span><span class="lsa">conditional <span class="_ _9"> </span></span>averages <span class="_ _9"> </span><span class="ls9">of <span class="_ _b"> </span></span>the </div><div class="t m6 x25 h8 y68 ff2 fs6 fc0 sc0 lsa ws0">target <span class="_ _2"></span><span class="lse">data <span class="_ _3"></span><span class="lsa">as </span></span></div><div class="t m34 x4c h2c y69 ff6 fsa fc0 sc0 ls30 ws0">(tjlx) </div><div class="t m35 x48 h2d y69 ff8 fs1d fc0 sc0 ls1 ws0">= </div><div class="t m36 x4d h15 y69 ff4 fs5 fc0 sc0 ls39 ws0">rip(tilx) </div><div class="t md x4e h2e y69 ff4 fs8 fc0 sc0 ls3a ws0">dtj </div><div class="t m37 x4f h12 y69 ff3 fs9 fc0 sc0 ls3b ws0">(3) </div><div class="t m24 x50 h1e y6a ff8 fs15 fc0 sc0 ls1 ws0">s </div><div class="t m21 x4c h17 y6b ff4 fs2 fc0 sc0 ls3c ws0">(tj&#8217; </div><div class="t m38 x34 h2f y6b ff8 fs1e fc0 sc0 ls1 ws0">I </div><div class="t m39 x27 h30 y6b ff1 fs14 fc0 sc0 ls3d ws0">X) </div><div class="t m21 x37 h17 y6c ff4 fs2 fc0 sc0 ls3c ws0">tj&#8217;p(tj </div><div class="t m3a x51 h29 y6c ff8 fs1b fc0 sc0 ls1 ws0">1 </div><div class="t m2b x52 h13 y6c ff1 fse fc0 sc0 ls1 ws0">x) </div><div class="t m21 x53 h17 y6c ff4 fs2 fc0 sc0 ls34 ws0">dtj </div><div class="t m3b x36 h31 y6d ff8 fs1f fc0 sc0 ls1 ws0">s </div><div class="t m3c x4f h32 y6e ff1 fs13 fc0 sc0 ls1 ws0">(4) </div><div class="t m6 x25 h8 y6f ff2 fs6 fc0 sc0 ls8 ws0">Note <span class="_ _5"></span>that <span class="_ _9"> </span><span class="ls1">only <span class="_ _9"> </span><span class="lsa">the <span class="_ _5"></span></span>first <span class="_ _9"> </span>term <span class="_"> </span><span class="lsc">in <span class="_ _9"> </span></span>eqn. </span></div><div class="t m13 x54 h11 y6f ff2 fsd fc0 sc0 ls1 ws0">2 </div><div class="t m6 x55 h8 y6f ff2 fs6 fc0 sc0 ls1 ws0">depends <span class="_ _9"> </span><span class="lse">on <span class="_ _5"></span></span>the </div><div class="t m6 x56 h8 y70 ff2 fs6 fc0 sc0 ls1 ws0">network <span class="_ _c"> </span>weights. <span class="_ _9"> </span>Provided <span class="_ _c"> </span><span class="ls8">that <span class="_"> </span><span class="lsa">the <span class="_ _a"> </span></span></span>functions </div><div class="t m3d x57 h28 y71 ff1 fs1a fc0 sc0 ls3e ws0">yjx; </div><div class="t m3e x58 h1a y71 ff6 fse fc0 sc0 ls3f ws0">w) </div><div class="t m6 x56 h8 y72 ff2 fs6 fc0 sc0 ls1 ws0">have <span class="_ _5"></span><span class="lsc">sufficient <span class="_ _4"></span>flexibility, </span>for <span class="_ _4"></span>instance <span class="_ _9"> </span><span class="ls40">if <span class="_ _c"> </span></span>they <span class="_ _4"></span><span class="lsa">correspond </span></div><div class="t m6 x56 h8 y73 ff2 fs6 fc0 sc0 lse ws0">to <span class="_ _4"></span><span class="lsa">a <span class="_"> </span><span class="ls1">network <span class="_"> </span>with <span class="_ _9"> </span></span>a <span class="_ _9"> </span><span class="lsc">sufficient <span class="_ _9"> </span></span>number <span class="_ _9"> </span><span class="lsf">of <span class="_ _a"> </span><span class="ls1">hidden <span class="_ _a"> </span>units, </span></span></span></div><div class="t m6 x56 h8 y74 ff2 fs6 fc0 sc0 lsa ws0">then <span class="_ _2"></span>the <span class="ls1">minimum <span class="lsf">of <span class="_"> </span></span>this <span class="ls8">error <span class="_ _0"></span><span class="ls1">function <span class="_ _2"></span>occurs <span class="_ _2"></span><span class="lsc">when </span></span></span></span></div><div class="t ma x56 hb y75 ff2 fs9 fc0 sc0 ls1 ws0">so </div><div class="t m6 x59 h8 y75 ff2 fs6 fc0 sc0 ls8 ws0">that <span class="_ _9"> </span><span class="ls1">the <span class="_ _9"> </span>network <span class="_"> </span></span>outputs <span class="_ _4"></span><span class="ls1">represent <span class="_"> </span><span class="lsa">the <span class="_ _4"></span></span>regression </span></div><div class="t m3f x3f h33 y75 ff3 fs8 fc0 sc0 ls41 ws0">of </div><div class="t m6 x56 h8 y76 ff2 fs6 fc0 sc0 lsa ws0">the <span class="_ _7"> </span>target <span class="_ _7"> </span><span class="ls8">data, <span class="_ _5"></span><span class="ls1">conditioned <span class="_ _6"> </span></span></span>on <span class="_ _7"> </span>the <span class="_ _7"> </span>input <span class="_"> </span><span class="ls1">vector. <span class="_ _5"></span><span class="ls8">Note </span></span></div><div class="t m40 x5a he y77 ff3 fsa fc0 sc0 ls1 ws0">217 </div><div class="t m5 x5b h34 y78 ffb fs20 fc1 sc0 ls1 ws0">Authorized licensed use limited to: IEEE Xplore. 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