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02-invert-parzen-window.html

<!doctype html><html lang="zh-Hant"><head><meta charset="utf-8"><meta name="viewport" content="width=device-width,initial-scale=1"><title>KP15 02 Inverting the Parzen Window</title>
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</style></head><body><div class="wrap"><div class="stage" id="stage"><canvas id="c" width="1600" height="900"></canvas><div id="math-layer"></div><div class="hint">互動：點擊 / 拖曳 • Click / Drag</div></div></div>
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function footer(n){txt('PR115 · Ch02 · KP-15  維度詛咒與 kNN 密度估計 Curse of Dimensionality & kNN',70,878,15,'#6f8da3','left',600);txt(`${n} / 11`,1530,878,15,'#6f8da3','right',700)}
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/* ---------- plotting ---------- */
function mapper(px,py,pw,ph,x0,x1,y0,y1){return{X:x=>px+(x-x0)/(x1-x0)*pw,Y:y=>py+ph-(y-y0)/(y1-y0)*ph,px,py,pw,ph,x0,x1,y0,y1}}
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/* ===== Page script ===== */
/* ===== Shared numerics for KP15: seeded RNG, Gamma function, textbook nearest-neighbour distance ===== */
const C1='#63b3ed',C2='#ff7b72';                        /* class omega_1 (blue) / omega_2 (coral) */
function rng(seed){let a=seed>>>0;return()=>{a=(a+0x6D2B79F5)>>>0;let t=a;t=Math.imul(t^(t>>>15),t|1);t^=t+Math.imul(t^(t>>>7),t|61);return((t^(t>>>14))>>>0)/4294967296}}
function gaussRand(r){let u=0;while(u===0)u=r();const v=r();return Math.sqrt(-2*Math.log(u))*Math.cos(2*Math.PI*v)}
/* Lanczos approximation of ln Gamma(z), z > 0 */
function lgamma(z){const c=[0.99999999999980993,676.5203681218851,-1259.1392167224028,771.32342877765313,-176.61502916214059,12.507343278686905,-0.13857109526572012,9.9843695780195716e-6,1.5056327351493116e-7];
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/* textbook average nearest-neighbour distance in the unit hypercube [0,1]^l with N samples */
const dForm=(l,N)=>2*Math.exp((Math.log(l)+lgamma(l/2)-Math.log(2)-(l/2)*Math.log(Math.PI)-Math.log(N))/l);
const SUP={'0':'⁰','1':'¹','2':'²','3':'³','4':'⁴','5':'⁵','6':'⁶','7':'⁷','8':'⁸','9':'⁹','-':'⁻'};
const sup=n=>String(n).split('').map(c=>SUP[c]||c).join('');
function sci(v){if(!isFinite(v)||v===0)return String(v);const e=Math.floor(Math.log10(Math.abs(v)));if(e>=-2&&e<=3)return v.toPrecision(3).replace(/\.?0+$/,m=>m.includes('.')?'':m);return (v/Math.pow(10,e)).toFixed(2)+'×10'+sup(e)}
/* 02 — Parzen window (fixed volume, count points) vs kNN (fixed k, adaptive volume) on 1-D data */
let bw=.08,K=5,x0=.3;
const R=rng(21),XS=[];for(let i=0;i<40;i++)XS.push(Math.min(.98,Math.max(.02,.3+.06*gaussRand(R))));for(let i=0;i<20;i++)XS.push(Math.min(.98,Math.max(.02,.7+.12*gaussRand(R))));XS.sort((a,b)=>a-b);
const N=XS.length,YM=8;
const M=mapper(130,282,880,400,0,1,0,YM);
const parz=(x,b)=>XS.filter(v=>Math.abs(v-x)<=b/2).length/(N*b);
const rk=(x,k)=>XS.map(v=>Math.abs(v-x)).sort((a,b)=>a-b)[k-1];
const knn=(x,k)=>k/(N*2*Math.max(rk(x,k),1e-4));
addFormula({id:'fp',tex:'',x:1305,y:600,w:440,size:24,color:P.cyan});
addFormula({id:'fk',tex:'',x:1305,y:722,w:440,size:24,color:P.amber});
function render(){
  reset();title('反轉 Parzen 窗：固定鄰居數，讓體積自己伸縮','Inverting the Parzen window: fixed k, adaptive volume',2);
  rr(60,170,980,680,24,P.panel,'#ffffff14',2);
  txt('同一組 60 筆資料，兩種密度估計',90,205,24,P.ink,'left',850);txt('Parzen: fixed volume, count points  ·  kNN: fixed count, grow the volume',90,236,16,P.cyan,'left',650);
  axes(M,[0,.2,.4,.6,.8,1],[0,2,4,6,8],v=>v.toFixed(1),v=>v.toFixed(0));
  /* Parzen window and kNN radius at the query point */
  ctx.fillStyle=P.cyan+'22';ctx.fillRect(M.X(x0-bw/2),M.py,M.X(x0+bw/2)-M.X(x0-bw/2),M.ph);
  const r=rk(x0,K);
  line(M.X(x0-r),M.py+16,M.X(x0+r),M.py+16,P.amber,4);line(M.X(x0-r),M.py+6,M.X(x0-r),M.py+26,P.amber,4);line(M.X(x0+r),M.py+6,M.X(x0+r),M.py+26,P.amber,4);
  const gx=[];for(let i=0;i<=500;i++)gx.push(i/500);
  const path=(f,col,lw)=>{ctx.strokeStyle=col;ctx.lineWidth=lw;ctx.lineJoin='round';ctx.beginPath();gx.forEach((x,i)=>{const y=Math.min(f(x),YM);i?ctx.lineTo(M.X(x),M.Y(y)):ctx.moveTo(M.X(x),M.Y(y))});ctx.stroke()};
  path(x=>parz(x,bw),P.cyan,3);path(x=>knn(x,K),P.amber,3);
  XS.forEach(v=>dot(M.X(v),M.py+M.ph-8,4,'#cbd5e1aa'));
  line(M.X(x0),M.py,M.X(x0),M.py+M.ph,P.ink,2,[6,6]);
  txt('青：Parzen（窗寬固定）',M.px+M.pw-8,M.py+46,16,P.cyan,'right',750);txt('橘：kNN（鄰居數固定）',M.px+M.pw-8,M.py+72,16,P.amber,'right',750);
  txt('灰點：訓練資料　縱軸截斷於 8',M.px,M.py+M.ph+54,15,P.muted,'left',600);
  rr(1060,170,480,680,24,P.panel,'#ffffff14',2);
  slider(1090,262,420,.02,.3,bw,v=>{bw=Math.round(v*100)/100;render()},'Parzen 窗寬 b',v=>v.toFixed(2));
  slider(1090,342,420,1,15,K,v=>{K=Math.round(v);render()},'鄰居數 k',v=>v.toFixed(0));
  slider(1090,422,420,.02,.98,x0,v=>{x0=Math.round(v*100)/100;render()},'查詢位置 x',v=>v.toFixed(2));
  const cnt=XS.filter(v=>Math.abs(v-x0)<=bw/2).length,pp=cnt/(N*bw),V=2*Math.max(r,1e-4),pk=K/(N*V);
  txt(`Parzen：窗內抓到 ${cnt} 個`+(cnt===0?'（空無一人）':''),1090,498,19,P.cyan,'left',800);
  txt(`kNN：包住 ${K} 個需半徑 ${r.toFixed(3)}`,1090,528,19,P.amber,'left',800);
  setTex('fp',`\\hat p=\\dfrac{${cnt}}{${N}\\times${bw.toFixed(2)}}=${pp.toFixed(3)}`);
  setTex('fk',`\\hat p=\\dfrac{${K}}{${N}\\times${V.toFixed(3)}}=${pk.toFixed(3)}`);
  wrap('Parzen 在稀疏處常抓不到人（k_N = 0）；kNN 自動擴大圈子，永遠湊得到 k 個人。',1090,780,420,24,16,P.muted,600);
  showF(['fp','fk']);
}
render();

</script></body></html>