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07-uniform-entropy-calculation.html
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/* ---------- maximum-entropy candidate shapes (all normalised on [0,W]) ---------- */
const gs=(u,m,s)=>Math.exp(-.5*((u-m)/s)**2)/(s*Math.sqrt(2*Math.PI));
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uniform :{zh:'均勻',en:'Uniform',color:P.lime,q:u=>1},
unimodal:{zh:'單峰',en:'Unimodal',color:P.cyan,q:u=>6*u*(1-u)},
bimodal :{zh:'雙峰',en:'Bimodal',color:P.violet,q:u=>gs(u,.25,.11)+gs(u,.75,.11)},
sharp :{zh:'高尖峰',en:'Sharp peak',color:P.coral,q:u=>gs(u,.5,.08)}};
const SHAPE_KEYS=['uniform','unimodal','bimodal','sharp'];
(function(){const N=4000;for(const k of SHAPE_KEYS){const s=SHAPES[k];let Z=0;
for(let i=0;i<N;i++)Z+=s.q((i+.5)/N)/N;
s.Z=Z;s.qn=u=>(u<0||u>1)?0:s.q(u)/Z;
let h=0;for(let i=0;i<N;i++){const p=s.qn((i+.5)/N);if(p>1e-300)h-=p*Math.log(p)/N}
s.HQ=h}})(); /* entropy on the unit interval; H(W)=HQ+ln W */
const pdf=(k,x,W)=>SHAPES[k].qn(x/W)/W;
const entropy=(k,W)=>SHAPES[k].HQ+Math.log(W);
function areaOf(k,W){const N=2000;let a=0;for(let i=0;i<N;i++)a+=pdf(k,(i+.5)*W/N,W)*W/N;return a}
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/* ===== Page script ===== */
/* 07 — Hand-calculating the uniform entropy on [0, W] (W is adjustable) */
let Wv=4,st=3;
const nf=v=>Number.isInteger(v)?String(v):v.toFixed(1);
addFormula({id:'s1',tex:'',x:1165,y:268,w:690,size:27});
addFormula({id:'s2',tex:'',x:1165,y:402,w:690,size:28});
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const MR=mapper(130,282,560,310,0,8,0,1.1);
function setTexts(){
const W=nf(Wv),h=(1/Wv).toFixed(3).replace(/0+$/,'').replace(/\.$/,''),Hn=Math.log(Wv).toFixed(3);
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}
function render(){
reset();title('數值演示：親手計算均勻分佈的熵','Numerical demo: hand-calculating the uniform entropy',7);
rr(60,170,700,680,24,P.panel,'#ffffff14',2);
txt('長方形面積驗證 · Rectangle check',90,205,25,P.ink,'left',850);
txt('Height 1/W, width W: the area is always exactly 1',90,236,17,P.cyan,'left',650);
axes(MR,[0,2,4,6,8],[0,.25,.5,.75,1],v=>v,v=>v.toFixed(2),'x');
const h=1/Wv;
plotCurve(x=>x<=Wv?h:0,MR,0,8,P.lime,5,'44');
txt('p = 1/'+nf(Wv)+' = '+h.toFixed(3),MR.X(Wv/2),MR.Y(h)-22,19,P.lime,'center',850);
txt('W = '+nf(Wv),MR.X(Wv/2),MR.Y(0)-22,19,P.ink,'center',850);
slider(130,712,560,1,8,Wv,v=>{Wv=Math.round(v*2)/2;render()},'區間寬度 W = x₂ − x₁',v=>nf(v));
txt('H = ln W = '+Math.log(Wv).toFixed(3)+' nats',130,780,26,P.amber,'left',900);
txt('W=4 時 H = ln 4 ≈ 1.386；W 越寬，最大熵越大。',130,818,17,P.muted,'left',600);
rr(790,170,750,680,24,P.panel,'#ffffff14',2);
const rows=[{top:188,h:120,zh:'第一步：計算密度高度',en:'Step 1: density height',col:P.cyan},{top:320,h:112,zh:'第二步：驗證合法機率面積',en:'Step 2: verify unit area',col:P.violet},{top:444,h:282,zh:'第三步：代入熵積分並化簡',en:'Step 3: substitute into the entropy integral',col:P.amber}];
const vis=[];
rows.forEach((r,i)=>{const on=i<st;rr(810,r.top,710,r.h,20,on?P.panel2:'#ffffff06',on?r.col:'#ffffff12',on?3:2);
txt(r.zh,836,r.top+26,22,on?r.col:'#5f7a8d','left',850);txt(r.en,1496,r.top+26,15,on?P.muted:'#4a6272','right',600);
if(on)vis.push('s'+(i+1));else txt('按「下一步」揭露 · Press Next',1165,r.top+r.h/2+16,19,'#4a6272','center',700)});
setTexts();
btn('上一步',810,752,110,52,()=>{if(st>1){st--;render()}},false,P.muted,17);
btn('下一步 Next',934,752,250,52,()=>{st=st%3+1;render()},true,P.cyan,19);
showF(vis);
}
render();
</script></body></html>