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04-two-class-volume-rule.html

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/* ===== Page script ===== */
/* ===== Shared numerics for KP15: seeded RNG, Gamma function, textbook nearest-neighbour distance ===== */
const C1='#63b3ed',C2='#ff7b72';                        /* class omega_1 (blue) / omega_2 (coral) */
function rng(seed){let a=seed>>>0;return()=>{a=(a+0x6D2B79F5)>>>0;let t=a;t=Math.imul(t^(t>>>15),t|1);t^=t+Math.imul(t^(t>>>7),t|61);return((t^(t>>>14))>>>0)/4294967296}}
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const SUP={'0':'⁰','1':'¹','2':'²','3':'³','4':'⁴','5':'⁵','6':'⁶','7':'⁷','8':'⁸','9':'⁹','-':'⁻'};
const sup=n=>String(n).split('').map(c=>SUP[c]||c).join('');
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/* 04 — Two-class decision: the class that needs the smaller volume wins (worked example from the lecture) */
let lg=0,st=3;                                       /* lg = log10 of the prior*loss multiplier m */
const Q=[.7,.6],RHO=.1*Math.SQRT2;
const W2=[[.66,.57],[.73,.66],[.62,.65],[.75,.52],[.8,.7]];
const W1=[[.55,.45],[.85,.55],[.6,.8],[.5,.65],[.9,.8]];
const N1=59,N2=61,M=mapper(110,266,520,520,.3,1.1,.2,1.0);
addFormula({id:'rule',tex:'\\dfrac{V_2}{V_1}\\ \\underset{\\omega_2}{\\overset{\\omega_1}{\\gtrless}}\\ \\dfrac{N_1}{N_2}\\cdot\\dfrac{P(\\omega_2)}{P(\\omega_1)}\\cdot\\dfrac{\\lambda_{21}-\\lambda_{22}}{\\lambda_{12}-\\lambda_{11}}',x:1185,y:280,w:700,size:27});
addFormula({id:'s1',tex:'V_2=\\pi\\rho^2=\\pi(0.1414)^2\\approx0.0628',x:1185,y:420,w:700,size:26,color:C2});
addFormula({id:'s2',tex:'V_1=\\pi(2\\rho)^2=4\\pi\\rho^2\\approx0.2513',x:1185,y:484,w:700,size:26,color:C1});
addFormula({id:'s3',tex:'',x:1185,y:566,w:700,size:27,color:P.ink});
function render(){
  reset();title('兩類分類判斷：誰的圈小，誰就稱霸','Two-class rule: the class needing the smaller volume wins',4);
  const V2=Math.PI*RHO*RHO,V1=4*V2,ratio=V2/V1,m=Math.pow(10,lg),thr=N1/N2*m,toW2=ratio<thr;
  rr(60,170,640,680,24,P.panel,'#ffffff14',2);
  txt('待分類點 x = (0.7, 0.6)，k = 5',90,205,23,P.ink,'left',850);txt('N₁ = 59 (blue ω₁), N₂ = 61 (coral ω₂)',90,234,15,P.cyan,'left',650);
  axes(M,[.4,.6,.8,1],[.2,.4,.6,.8,1],v=>v.toFixed(1),v=>v.toFixed(1));
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  circ(2*RHO,C1);circ(RHO,C2);
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  txt('紅圈：半徑 ρ ≈ 0.1414，包住 5 個 ω₂',M.px+10,M.py+16,15,C2,'left',750);txt('藍圈：半徑 2ρ ≈ 0.2828，包住 5 個 ω₁',M.px+10,M.py+40,15,C1,'left',750);
  rr(730,170,810,680,24,P.panel,'#ffffff14',2);
  txt('判決準則',760,200,20,P.cyan,'left',850);
  const vis=['rule'];
  [['s1'],['s2'],['s3']].forEach((a,i)=>{if(i<st)vis.push(a[0])});
  setTex('s3',`\\dfrac{V_2}{V_1}=\\dfrac{0.0628}{0.2513}=0.25\\ ${toW2?'<':'>'}\\ \\dfrac{N_1}{N_2}\\,m=${thr.toFixed(3)}`);
  slider(770,666,730,-1,1,lg,v=>{lg=Math.round(v*20)/20;render()},'先驗 × 損失倍率 m（m = 1 即先驗與損失相等）',v=>Math.pow(10,v).toFixed(2));
  rr(770,722,730,100,18,P.panel2,(toW2?C2:C1),3);
  txt(toW2?'判決：x 歸屬 ω₂':'判決：x 歸屬 ω₁',796,756,28,toW2?C2:C1,'left',900);
  txt(toW2?'ω₂ 只需極小的圈就抓滿 5 個鄰居，局部密度較高。':'倍率把門檻拉到低於 0.25，結論被翻轉。',796,796,17,P.ink,'left',650);
  btn(st<3?'下一步 Next':'重看 Replay',860,184,190,40,()=>{st=st<3?st+1:1;render()},true,P.cyan,16);
  showF(vis);
}
render();

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