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<!DOCTYPE html>
<html lang="zh-CN">
<head>
<meta charset="UTF-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>方差修正四种公式 · 动态演示</title>
<link rel="preconnect" href="https://fonts.googleapis.com" />
<link rel="preconnect" href="https://fonts.gstatic.com" crossorigin />
<link href="https://fonts.googleapis.com/css2?family=IBM+Plex+Sans:wght@400;500;600&family=IBM+Plex+Mono:wght@400;500&display=swap" rel="stylesheet" />
<style>
:root {
--bg: #0c0f14; --surface: #151a22; --surface2: #1c2330; --border: #2a3344;
--text: #e8ecf4; --muted: #8b95a8; --accent: #3d9cf5; --green: #2dd4a8; --red: #f07178; --orange: #f5a623;
--radius: 10px; --font: "IBM Plex Sans", system-ui, sans-serif; --mono: "IBM Plex Mono", ui-monospace, monospace;
}
* { box-sizing: border-box; margin: 0; padding: 0; }
html, body { height: 100%; overflow: hidden; }
body { font-family: var(--font); background: var(--bg); color: var(--text); line-height: 1.5; }
.app { height: 100vh; display: grid; grid-template-rows: auto auto 1fr; gap: 8px; padding: 10px 14px; max-width: 1400px; margin: 0 auto; }
.topbar { display: flex; align-items: center; justify-content: space-between; gap: 12px; padding-bottom: 6px; border-bottom: 1px solid var(--border); }
.topbar h1 { font-size: 1rem; font-weight: 600; }
.topbar .sub { font-size: 0.72rem; color: var(--muted); margin-top: 2px; }
.pill { font-size: 0.74rem; color: var(--muted); background: var(--surface); border: 1px solid var(--border); border-radius: 999px; padding: 4px 12px; white-space: nowrap; }
.tabs { display: flex; flex-wrap: wrap; gap: 6px; }
.tab { font-family: var(--font); font-size: 0.72rem; padding: 6px 12px; border-radius: 8px; border: 1px solid var(--border); background: var(--surface); color: var(--muted); cursor: pointer; }
.tab:hover { border-color: var(--accent); color: var(--text); }
.tab.active { background: var(--accent); border-color: var(--accent); color: #0a0e12; font-weight: 600; }
.scene { display: none; grid-template-columns: 320px 1fr; gap: 10px; min-height: 0; }
.scene.active { display: grid; }
@media (max-width: 980px) { html, body { overflow: auto; height: auto; } .app { height: auto; min-height: 100vh; } .scene.active { grid-template-columns: 1fr; } }
.card { background: var(--surface); border: 1px solid var(--border); border-radius: var(--radius); overflow: hidden; }
.left { display: grid; grid-template-rows: 1fr auto auto; gap: 8px; min-height: 0; }
.section-title { padding: 7px 12px; font-size: 0.68rem; font-weight: 600; color: var(--muted); text-transform: uppercase; letter-spacing: 0.06em; border-bottom: 1px solid var(--border); }
.formula { padding: 10px 12px; font-family: var(--mono); font-size: 0.72rem; line-height: 1.68; white-space: pre-line; overflow-y: auto; max-height: 220px; }
.formula .hl { color: var(--accent); }
.formula .result { font-size: 0.95rem; font-weight: 500; margin-top: 4px; color: var(--green); }
.stats { display: grid; grid-template-columns: repeat(2, 1fr); gap: 5px; padding: 0 10px 10px; }
.stat { background: var(--surface2); border: 1px solid var(--border); border-radius: 8px; padding: 7px; text-align: center; }
.stat label { display: block; font-size: 0.58rem; color: var(--muted); text-transform: uppercase; margin-bottom: 2px; }
.stat span { font-family: var(--mono); font-size: 0.8rem; font-weight: 500; }
.controls { padding: 9px 12px; }
.btn-row { display: grid; grid-template-columns: repeat(4, 1fr); gap: 5px; margin-bottom: 8px; }
.btn { font-family: var(--font); font-size: 0.72rem; font-weight: 500; padding: 6px 3px; border-radius: 7px; border: 1px solid var(--border); background: var(--surface2); color: var(--text); cursor: pointer; }
.btn:hover { border-color: var(--accent); }
.btn.primary { background: var(--accent); border-color: var(--accent); color: #0a0e12; }
.speed { display: flex; align-items: center; gap: 8px; font-size: 0.72rem; color: var(--muted); }
.speed input { flex: 1; accent-color: var(--accent); }
.progress { height: 3px; background: var(--border); margin-top: 8px; }
.progress-fill { height: 100%; background: var(--accent); width: 0%; transition: width 0.08s linear; }
.legend { padding: 7px 12px; display: flex; flex-wrap: wrap; gap: 8px 12px; font-size: 0.66rem; color: var(--muted); }
.legend-item { display: flex; align-items: center; gap: 4px; }
.sw { width: 9px; height: 9px; border-radius: 50%; }
.sw.line { width: 12px; height: 3px; border-radius: 2px; }
.right { display: grid; grid-template-rows: 1fr 1fr; gap: 8px; min-height: 0; }
.right.single { grid-template-rows: 1fr; }
.chart { display: flex; flex-direction: column; min-height: 0; }
.chart-wrap { flex: 1; min-height: 0; }
canvas { display: block; width: 100%; height: 100%; }
</style>
</head>
<body>
<div class="app">
<header class="topbar">
<div>
<h1 id="sceneTitle">方差修正 · 动态演示</h1>
<div class="sub" id="sceneEq">四种公式合一 · 点击标签切换</div>
</div>
<div class="pill" id="stepPill">—</div>
</header>
<nav class="tabs">
<button class="tab active" data-scene="2-1b">(2-1b) 忽略均值</button>
<button class="tab" data-scene="2-2">(2-2)(2-3) 无偏估计</button>
<button class="tab" data-scene="jensen">詹森不等式</button>
<button class="tab" data-scene="2-4">(2-4) s 的分布</button>
</nav>
<!-- Scene 1: 2-1b -->
<div class="scene active" id="scene-2-1b">
<aside class="left">
<div class="card">
<div class="section-title">逐步代入 (2-1b) · 假设 x̄≈0</div>
<div class="formula" id="f1">▶ 点击「播放」</div>
<div class="stats">
<div class="stat"><label>当前 xⱼ</label><span id="s1xj">—</span></div>
<div class="stat"><label>xⱼ²</label><span id="s1sq">—</span></div>
<div class="stat"><label>Σxⱼ²</label><span id="s1sum">—</span></div>
<div class="stat"><label>s²</label><span id="s1var">—</span></div>
</div>
</div>
<div class="card controls" data-ctrl="step">
<div class="btn-row">
<button class="btn primary play">▶ 播放</button>
<button class="btn pause">⏸ 暂停</button>
<button class="btn step">⏭ 单步</button>
<button class="btn reset">↺ 重置</button>
</div>
<div class="speed"><span>速度</span><input type="range" class="speed-range" min="1" max="10" value="9" /></div>
<div class="progress"><div class="progress-fill prog"></div></div>
</div>
<div class="card legend">
<div class="legend-item"><span class="sw" style="background:#4da3ff"></span> xⱼ</div>
<div class="legend-item"><span class="sw line" style="background:var(--orange)"></span> 零线（μ=0）</div>
</div>
</aside>
<main class="right">
<div class="card chart"><div class="section-title">对数收益率 xⱼ（均值为 0 假设）</div><div class="chart-wrap" id="w1a"><canvas id="c1a"></canvas></div></div>
<div class="card chart"><div class="section-title">xⱼ² 累加 → s² = (1/N)Σxⱼ²</div><div class="chart-wrap" id="w1b"><canvas id="c1b"></canvas></div></div>
</main>
</div>
<!-- Scene 2: 2-2 / 2-3 -->
<div class="scene" id="scene-2-2">
<aside class="left">
<div class="card">
<div class="section-title">无偏修正 (2-2) 与直接公式 (2-3)</div>
<div class="formula" id="f2">▶ 点击「播放」</div>
<div class="stats">
<div class="stat"><label>s²(÷N)</label><span id="s2pop">—</span></div>
<div class="stat"><label>N/(N-1)</label><span id="s2fac">—</span></div>
<div class="stat"><label>σ̂² (2-2)</label><span id="s2hat">—</span></div>
<div class="stat"><label>s²(÷N-1)</label><span id="s2unb">—</span></div>
</div>
</div>
<div class="card controls" data-ctrl="step">
<div class="btn-row">
<button class="btn primary play">▶ 播放</button>
<button class="btn pause">⏸ 暂停</button>
<button class="btn step">⏭ 单步</button>
<button class="btn reset">↺ 重置</button>
</div>
<div class="speed"><span>速度</span><input type="range" class="speed-range" min="1" max="10" value="9" /></div>
<div class="progress"><div class="progress-fill prog"></div></div>
</div>
<div class="card legend">
<div class="legend-item"><span class="sw line" style="background:var(--red)"></span> 有偏 ÷N</div>
<div class="legend-item"><span class="sw line" style="background:var(--green)"></span> 无偏 ÷N-1</div>
</div>
</aside>
<main class="right">
<div class="card chart"><div class="section-title">两种分母下的方差估计对比</div><div class="chart-wrap" id="w2a"><canvas id="c2a"></canvas></div></div>
<div class="card chart"><div class="section-title">(xⱼ-x̄)² 累加过程</div><div class="chart-wrap" id="w2b"><canvas id="c2b"></canvas></div></div>
</main>
</div>
<!-- Scene 3: Jensen -->
<div class="scene" id="scene-jensen">
<aside class="left">
<div class="card">
<div class="section-title">E(s) < σ · 詹森不等式</div>
<div class="formula" id="f3">▶ 点击「播放」：蒙特卡洛重复抽样</div>
<div class="stats">
<div class="stat"><label>真实 σ</label><span id="s3sig">—</span></div>
<div class="stat"><label>样本量 n</label><span id="s3n">—</span></div>
<div class="stat"><label>E(s) 估计</label><span id="s3es">—</span></div>
<div class="stat"><label>低估幅度</label><span id="s3bias">—</span></div>
</div>
</div>
<div class="card controls" data-ctrl="jensen">
<div class="btn-row">
<button class="btn primary play">▶ 播放</button>
<button class="btn pause">⏸ 暂停</button>
<button class="btn step">⏭ +500次</button>
<button class="btn reset">↺ 重置</button>
</div>
<div class="speed"><span>速度</span><input type="range" class="speed-range" min="1" max="10" value="7" /></div>
<div class="progress"><div class="progress-fill prog"></div></div>
</div>
<div class="card legend">
<div class="legend-item"><span class="sw line" style="background:var(--green)"></span> 真实 σ</div>
<div class="legend-item"><span class="sw line" style="background:var(--orange)"></span> E(s)</div>
</div>
</aside>
<main class="right">
<div class="card chart"><div class="section-title">√ 函数的凹性：E(√s²) < √(E(s²))</div><div class="chart-wrap" id="w3a"><canvas id="c3a"></canvas></div></div>
<div class="card chart"><div class="section-title">重复抽样得到的样本标准差 s 的分布</div><div class="chart-wrap" id="w3b"><canvas id="c3b"></canvas></div></div>
</main>
</div>
<!-- Scene 4: 2-4 PDF -->
<div class="scene" id="scene-2-4">
<aside class="left">
<div class="card">
<div class="section-title">正态假设下 f_N(s) · 公式 (2-4)</div>
<div class="formula" id="f4">▶ 点击「播放」：观察 N 增大时 PDF 形态</div>
<div class="stats">
<div class="stat"><label>样本量 N</label><span id="s4N">—</span></div>
<div class="stat"><label>总体 σ</label><span id="s4sig">—</span></div>
<div class="stat"><label>PDF 峰值 s*</label><span id="s4peak">—</span></div>
<div class="stat"><label>峰值→σ</label><span id="s4note">—</span></div>
</div>
</div>
<div class="card controls" data-ctrl="pdf">
<div class="btn-row">
<button class="btn primary play">▶ 播放</button>
<button class="btn pause">⏸ 暂停</button>
<button class="btn step">⏭ N+1</button>
<button class="btn reset">↺ 重置</button>
</div>
<div class="speed"><span>速度</span><input type="range" class="speed-range" min="1" max="10" value="6" /></div>
<div class="progress"><div class="progress-fill prog"></div></div>
<div class="speed" style="margin-top:8px"><span>N</span><input type="range" id="nSlider" min="3" max="120" value="3" /></div>
</div>
<div class="card legend">
<div class="legend-item"><span class="sw line" style="background:var(--accent)"></span> f_N(s)</div>
<div class="legend-item"><span class="sw line" style="background:var(--green)"></span> σ</div>
</div>
</aside>
<main class="right single">
<div class="card chart"><div class="section-title">样本标准差 s 的概率密度 f_N(s)</div><div class="chart-wrap" id="w4"><canvas id="c4"></canvas></div></div>
</main>
</div>
</div>
<script>
const LOG = [
-0.004566, -0.016261, 0.004302, 0.000452, 0.017849, 0.004151, -0.009377, -0.018738,
0.007921, -0.003217, -0.000283, -0.006752, -0.02362, 0.011419, 0.006377, 0.003259,
0.008526, -0.01448, 0.023943, -0.003201, 0.004881, 0.009525, -0.032625, 0.020466,
0.00129, -0.019982, -0.003723, 0.032799, -0.00228, -0.016614, -0.001416, 0.005257,
-0.006902, -0.006664, 0.014693, 0.010532, 0.000557, -0.000223, 0.002337, -0.004624,
0.000502, 0.00223, 0.018154, -0.013487, 0.011791, 0.020439, -0.009707, 0.002814,
0.003614, -0.001347, -0.005135, -0.011446, -0.026326, 0.007513, -0.039478, 0.006718,
0.011477, -0.011824, 0.007306, 0.009129, 0.009159, -0.004739, 0.004795, 0.003034,
-0.00913, -0.005735, 0.008336, -0.00589, 0.006003, 0.001693, -0.002201, 0.001975,
0.004669, 0.001514, 0.002798, 0.008845, -0.001386, 0.003212, -0.00166, 0.005522,
-0.000716, 0.009216, 0.013936, -0.016178, 0.017308, 0.014843, -0.008729, 0.029032,
0.003473, 0.006962, 0.020902, -0.006868, -0.01275, -0.003239, 0.002725, 0.005378,
0.009101, 0.020837, -0.008713, -0.013999, 0.004402, 0.000353, -0.000252, -0.015876,
0.008681, 0.008456, -0.002777, 0.032677, -0.013697, 0.012816, -0.006338, -0.009707,
-0.013529, 0.012931, -0.006695, -0.005227, -0.004242, 0.005753, 0.014686, 0.004897,
0.015958, 0.017141, -0.004066, 0.005116, 0.011851, 0.010174, 0.002795, 0.020079,
-0.001323, -0.001005, 0.005923, -0.000364, -0.01037, 0.014678, 0.00217, 0.013366,
0.016, -0.012664, 0.008666, -0.003749, 0.005201, 0.009537, -0.006421, -0.009555,
0.00221, -0.001546, -0.003987, 0.006151, 0.000309, 0.007907, 0.009754, -0.006085,
-0.004106, 0.005762, 0.004906, 0.004106, 0.019182, 0.019941, -0.017701, 0.00253,
-0.000674, -0.00974, -0.017254, -0.024676, -0.001732, -0.003875, 0.004097, -0.002136,
0.01028, 0.014529, -0.007107, -0.005397, 0.016622, -0.00904, 0.021168, 0.025366,
-0.002963, 0.00238, 0.005904, -0.013816, 0.025275, -0.024602, 0.000504, -0.003238,
0.024669, 0.002093, -0.01633, 0.007309, 0.004747, 0.016338, -0.02001, 0.012773,
0.013779, -0.006906, -0.010184, -0.006902, 0.013676, 0.010558, -0.004881, 0.011379,
0.016141, -0.002876, 0.014881, -0.004051, -0.003519, -0.000862, 0.004264, -0.024297,
-0.012111, -0.027829, 0.008207, 0.015687, 0.017923, -0.023179, 0.001225, -0.013722,
-0.015718, -0.006323, -0.015126, 0.024806, -0.01732, -0.0114, -0.001596, 0.001165,
0.000776, -0.013884, 0.003358, 0.003738, 0.008897, -0.001635, 0.011816, -0.016954,
0.01606, 0.010635, 0.004374, 0.006066, 0.012106, 0.010657, -0.012142, 0.013731,
0.00313, 0.014383, 0.005346, -0.011364, -0.012676, 0.005649, 0.013525, -0.017504,
-0.003587, 0.013103, -0.003836,
];
const DATA_N = LOG.length;
const META = {
'2-1b': { title: '(2-1b) 忽略均值的简化方差', eq: 's² = (1/N) · Σxⱼ²' },
'2-2': { title: '(2-2)(2-3) 总体方差无偏估计', eq: 'σ̂² = N/(N-1)·s² ≡ Σ(xⱼ-x̄)²/(N-1)' },
'jensen': { title: '詹森不等式与波动率偏差', eq: 'E(s) = E(√s²) < √σ² = σ' },
'2-4': { title: '(2-4) 样本标准差分布', eq: 'f_N(s) · 正态假设' },
};
let activeScene = '2-1b';
let frame = 0, playing = false, timer = null;
// --- shared canvas helpers ---
function pad() { return { left: 48, right: 14, top: 16, bottom: 32 }; }
function xPos(i, span, p, w) { return p.left + (i / Math.max(1, span - 1)) * (w - p.left - p.right); }
function setupCanvas(wrapId, canvas) {
const wrap = document.getElementById(wrapId);
const dpr = window.devicePixelRatio || 1;
const w = wrap.clientWidth, h = wrap.clientHeight;
canvas.width = Math.max(1, w * dpr);
canvas.height = Math.max(1, h * dpr);
canvas.style.width = w + 'px';
canvas.style.height = h + 'px';
const ctx = canvas.getContext('2d');
ctx.setTransform(dpr, 0, 0, dpr, 0, 0);
return { w, h, ctx };
}
function drawGrid(ctx, w, h, p) {
ctx.strokeStyle = '#232a38'; ctx.lineWidth = 1;
for (let i = 0; i <= 4; i++) {
const y = p.top + (i / 4) * (h - p.top - p.bottom);
ctx.beginPath(); ctx.moveTo(p.left, y); ctx.lineTo(w - p.right, y); ctx.stroke();
}
}
// --- Scene routing ---
function switchScene(id) {
activeScene = id;
document.querySelectorAll('.tab').forEach(t => t.classList.toggle('active', t.dataset.scene === id));
document.querySelectorAll('.scene').forEach(s => s.classList.toggle('active', s.id === 'scene-' + id));
document.getElementById('sceneTitle').textContent = META[id].title;
document.getElementById('sceneEq').textContent = META[id].eq;
pauseAll();
resetActive();
drawActive();
}
document.querySelectorAll('.tab').forEach(t => t.addEventListener('click', () => switchScene(t.dataset.scene)));
window.addEventListener('hashchange', applyHash);
function applyHash() {
const h = (location.hash || '#2-1b').slice(1);
if (META[h]) switchScene(h);
}
// --- Step-through scenes (1 & 2) ---
function zeroMeanStats(k) {
const slice = LOG.slice(0, k + 1);
const n = slice.length;
const sq = slice.map(x => x * x);
const sumSq = sq.reduce((a, b) => a + b, 0);
return { slice, sq, sumSq, var0: sumSq / n, n };
}
function devStats(k) {
const slice = LOG.slice(0, k + 1);
const n = slice.length;
const mean = slice.reduce((a, b) => a + b, 0) / n;
const sq = slice.map(x => (x - mean) ** 2);
const sumSq = sq.reduce((a, b) => a + b, 0);
const pop = sumSq / n;
const fac = n > 1 ? n / (n - 1) : NaN;
const hat = n > 1 ? pop * fac : NaN;
const unb = n > 1 ? sumSq / (n - 1) : NaN;
return { slice, mean, sq, sumSq, pop, fac, hat, unb, n };
}
const c1a = document.getElementById('c1a'), c1b = document.getElementById('c1b');
const c2a = document.getElementById('c2a'), c2b = document.getElementById('c2b');
const c3a = document.getElementById('c3a'), c3b = document.getElementById('c3b');
const c4 = document.getElementById('c4');
function drawScene1(k) {
const { w: w1, h: h1, ctx } = setupCanvas('w1a', c1a);
const { w: w2, h: h2, ctx: ctx2 } = setupCanvas('w1b', c1b);
const p = pad();
const { slice, sq, sumSq, var0, n } = zeroMeanStats(k);
const xj = LOG[k];
ctx.clearRect(0, 0, w1, h1);
drawGrid(ctx, w1, h1, p);
const minV = Math.min(-0.04, ...slice) * 1.1;
const maxV = Math.max(0.035, ...slice) * 1.1;
const y = v => p.top + (1 - (v - minV) / (maxV - minV)) * (h1 - p.top - p.bottom);
const y0 = y(0);
ctx.strokeStyle = '#f5a623'; ctx.lineWidth = 1.8;
ctx.beginPath(); ctx.moveTo(p.left, y0); ctx.lineTo(w1 - p.right, y0); ctx.stroke();
ctx.fillStyle = '#f5a623'; ctx.font = '10px IBM Plex Sans,sans-serif';
ctx.fillText('μ=0', p.left + 4, y0 - 6);
ctx.strokeStyle = '#4da3ff'; ctx.lineWidth = 1.6; ctx.beginPath();
for (let i = 0; i <= k; i++) {
const x = xPos(i, n, p, w1), yy = y(LOG[i]);
i === 0 ? ctx.moveTo(x, yy) : ctx.lineTo(x, yy);
}
ctx.stroke();
const x = xPos(k, n, p, w1);
ctx.fillStyle = '#4da3ff';
ctx.beginPath(); ctx.arc(x, y(xj), 5, 0, Math.PI * 2); ctx.fill();
ctx2.clearRect(0, 0, w2, h2);
drawGrid(ctx2, w2, h2, p);
const maxSq = Math.max(0.0008, ...sq) * 1.15;
const y2 = v => p.top + (1 - v / maxSq) * (h2 - p.top - p.bottom);
const yb = y2(0);
const barW = Math.max(2, Math.min(10, (w2 - p.left - p.right) / n * 0.72));
for (let i = 0; i <= k; i++) {
const xx = xPos(i, n, p, w2), yy = y2(sq[i]);
ctx2.fillStyle = i === k ? '#4da3ff' : 'rgba(61,156,245,0.55)';
ctx2.fillRect(xx - barW / 2, yy, barW, Math.max(1, yb - yy));
}
document.getElementById('stepPill').textContent = `第 ${n} / ${DATA_N} 个样本`;
document.getElementById('s1xj').textContent = `${(xj * 100).toFixed(2)}%`;
document.getElementById('s1sq').textContent = sq[k].toExponential(3);
document.getElementById('s1sum').textContent = sumSq.toExponential(3);
document.getElementById('s1var').textContent = var0.toExponential(3);
document.getElementById('f1').innerHTML =
`j = <span class="hl">${n}</span>\n` +
`xⱼ² = (${xj.toFixed(4)})² = ${sq[k].toExponential(3)}\n` +
`Σxⱼ² = ${sumSq.toExponential(3)}\n` +
`<span class="result">s² = ${sumSq.toExponential(3)} / ${n} = ${var0.toExponential(3)}</span>`;
setProg(document.querySelector('#scene-2-1b .prog'), n / DATA_N);
}
function drawScene2(k) {
const { w: w1, h: h1, ctx } = setupCanvas('w2a', c2a);
const { w: w2, h: h2, ctx: ctx2 } = setupCanvas('w2b', c2b);
const p = pad();
const st = devStats(k);
const { sq, sumSq, pop, fac, hat, unb, n, mean } = st;
const atEnd = k >= DATA_N - 1;
const RUN_WINDOW = 42;
ctx.clearRect(0, 0, w1, h1);
drawGrid(ctx, w1, h1, p);
if (n >= 2) {
const iEnd = k;
const iStart = atEnd ? 1 : Math.max(1, k - RUN_WINDOW + 1);
const vals = [];
for (let i = iStart; i <= iEnd; i++) vals.push(devStats(i).pop, devStats(i).unb);
let minV, maxV;
if (atEnd) {
for (let i = 1; i <= k; i++) vals.push(devStats(i).pop, devStats(i).unb);
minV = Math.min(...vals) * 0.85;
maxV = Math.max(...vals) * 1.15;
} else {
const curPop = devStats(k).pop;
const curUnb = devStats(k).unb;
const gap = Math.max(curUnb - curPop, curPop * 0.02);
const mid = (curPop + curUnb) / 2;
minV = Math.min(...vals, mid - gap * 3);
maxV = Math.max(...vals, mid + gap * 3);
const margin = (maxV - minV) * 0.18 || gap;
minV -= margin;
maxV += margin;
}
const y = v => p.top + (1 - (v - minV) / (maxV - minV)) * (h1 - p.top - p.bottom);
const xFor = i => atEnd
? xPos(i - 1, DATA_N, p, w1)
: p.left + ((i - iStart) / Math.max(1, iEnd - iStart)) * (w1 - p.left - p.right);
ctx.strokeStyle = '#f07178'; ctx.lineWidth = 2; ctx.beginPath();
for (let i = iStart; i <= iEnd; i++) {
const x = xFor(i), yy = y(devStats(i).pop);
i === iStart ? ctx.moveTo(x, yy) : ctx.lineTo(x, yy);
}
ctx.stroke();
ctx.strokeStyle = '#2dd4a8'; ctx.lineWidth = 2; ctx.beginPath();
for (let i = iStart; i <= iEnd; i++) {
const x = xFor(i), yy = y(devStats(i).unb);
i === iStart ? ctx.moveTo(x, yy) : ctx.lineTo(x, yy);
}
ctx.stroke();
if (!atEnd) {
const xCur = xFor(k);
const yPop = y(pop), yUnb = y(unb);
ctx.fillStyle = '#f07178';
ctx.beginPath(); ctx.arc(xCur, yPop, 5, 0, Math.PI * 2); ctx.fill();
ctx.fillStyle = '#2dd4a8';
ctx.beginPath(); ctx.arc(xCur, yUnb, 5, 0, Math.PI * 2); ctx.fill();
ctx.strokeStyle = 'rgba(139,149,168,0.45)'; ctx.lineWidth = 1;
ctx.beginPath(); ctx.moveTo(xCur, yPop); ctx.lineTo(xCur, yUnb); ctx.stroke();
}
ctx.fillStyle = '#8b95a8'; ctx.font = '10px IBM Plex Sans,sans-serif';
ctx.fillText(
atEnd ? '全貌：N 增大后 ÷N 与 ÷N-1 两线趋同' : `跟随前沿 N=${n} · 红=÷N 绿=÷N-1`,
p.left + 2, p.top + 12
);
}
ctx2.clearRect(0, 0, w2, h2);
drawGrid(ctx2, w2, h2, p);
const maxSq = Math.max(0.0015, ...sq) * 1.1;
const y2 = v => p.top + (1 - v / maxSq) * (h2 - p.top - p.bottom);
const yb = y2(0);
const barW = Math.max(2, Math.min(10, (w2 - p.left - p.right) / n * 0.72));
for (let i = 0; i <= k; i++) {
const xx = xPos(i, n, p, w2), yy = y2(sq[i]);
ctx2.fillStyle = i === k ? '#4da3ff' : 'rgba(61,156,245,0.55)';
ctx2.fillRect(xx - barW / 2, yy, barW, Math.max(1, yb - yy));
}
document.getElementById('stepPill').textContent = `第 ${n} / ${DATA_N} 个样本`;
document.getElementById('s2pop').textContent = n >= 1 ? pop.toExponential(3) : '—';
document.getElementById('s2fac').textContent = n >= 2 ? fac.toFixed(4) : '—';
document.getElementById('s2hat').textContent = n >= 2 ? hat.toExponential(3) : '—';
document.getElementById('s2unb').textContent = n >= 2 ? unb.toExponential(3) : '—';
const eq = n >= 2
? `<span class="result">σ̂² = (${n}/${n-1})×${pop.toExponential(3)} = ${hat.toExponential(3)}</span>\n` +
`<span class="result">s² = ${sumSq.toExponential(3)}/(${n}-1) = ${unb.toExponential(3)}</span>\n` +
`<span class="hl">两式相等 ✓</span>`
: `<span class="hl">至少需要 2 个样本</span>`;
document.getElementById('f2').innerHTML =
`N=${n}, Σ(xⱼ-x̄)²=${sumSq.toExponential(3)}\n` +
`有偏: s²=${sumSq.toExponential(3)}/${n}=${pop.toExponential(3)}\n` + eq;
setProg(document.querySelector('#scene-2-2 .prog'), n / DATA_N);
}
// --- Scene 3: Jensen Monte Carlo ---
const SIG_TRUE = 0.015;
const JENSEN_N = 30;
let jensenSamples = [];
let jensenTarget = 8000;
function randn() {
let u = 0, v = 0;
while (u === 0) u = Math.random();
while (v === 0) v = Math.random();
return Math.sqrt(-2 * Math.log(u)) * Math.cos(2 * Math.PI * v);
}
function sampleStd(arr) {
const n = arr.length;
const m = arr.reduce((a, b) => a + b, 0) / n;
const ss = arr.reduce((a, x) => a + (x - m) ** 2, 0);
return Math.sqrt(ss / (n - 1));
}
function addJensenBatch(batch) {
for (let b = 0; b < batch; b++) {
const draw = Array.from({ length: JENSEN_N }, () => randn() * SIG_TRUE);
jensenSamples.push(sampleStd(draw));
}
}
function drawScene3() {
const { w: w1, h: h1, ctx } = setupCanvas('w3a', c3a);
const { w: w2, h: h2, ctx: ctx2 } = setupCanvas('w3b', c3b);
const p = pad();
ctx.clearRect(0, 0, w1, h1);
drawGrid(ctx, w1, h1, p);
const xMin = 0, xMax = 0.0012;
const yMin = 0, yMax = 0.038;
const tx = v => p.left + (v - xMin) / (xMax - xMin) * (w1 - p.left - p.right);
const ty = v => p.top + (1 - (v - yMin) / (yMax - yMin)) * (h1 - p.top - p.bottom);
ctx.strokeStyle = '#3d9cf5'; ctx.lineWidth = 2; ctx.beginPath();
for (let i = 0; i <= 200; i++) {
const v = xMin + (xMax - xMin) * i / 200;
const yy = Math.sqrt(v);
const x = tx(v), y = ty(yy);
i === 0 ? ctx.moveTo(x, y) : ctx.lineTo(x, y);
}
ctx.stroke();
if (jensenSamples.length > 0) {
const last = jensenSamples[jensenSamples.length - 1];
const v2 = last * last;
ctx.fillStyle = '#4da3ff';
ctx.beginPath(); ctx.arc(tx(v2), ty(last), 4, 0, Math.PI * 2); ctx.fill();
ctx.strokeStyle = 'rgba(245,166,35,0.6)'; ctx.setLineDash([4, 4]);
ctx.beginPath(); ctx.moveTo(tx(v2), ty(Math.sqrt(v2))); ctx.lineTo(tx(v2), ty(Math.sqrt(xMax)));
ctx.stroke(); ctx.setLineDash([]);
}
ctx.fillStyle = '#8b95a8'; ctx.font = '10px IBM Plex Sans,sans-serif';
ctx.fillText('y=√x（凹函数）', p.left + 4, p.top + 14);
ctx2.clearRect(0, 0, w2, h2);
drawGrid(ctx2, w2, h2, p);
const m = jensenSamples.length;
if (m > 0) {
const bins = 40;
const lo = 0.005, hi = 0.025;
const hist = new Array(bins).fill(0);
jensenSamples.forEach(s => {
const b = Math.min(bins - 1, Math.max(0, Math.floor((s - lo) / (hi - lo) * bins)));
hist[b]++;
});
const maxC = Math.max(...hist);
const bw = (w2 - p.left - p.right) / bins;
hist.forEach((c, i) => {
const bh = (c / maxC) * (h2 - p.top - p.bottom) * 0.9;
ctx2.fillStyle = 'rgba(61,156,245,0.65)';
ctx2.fillRect(p.left + i * bw, h2 - p.bottom - bh, bw - 1, bh);
});
const meanS = jensenSamples.reduce((a, b) => a + b, 0) / m;
const xSig = p.left + (SIG_TRUE - lo) / (hi - lo) * (w2 - p.left - p.right);
const xEs = p.left + (meanS - lo) / (hi - lo) * (w2 - p.left - p.right);
ctx2.strokeStyle = '#2dd4a8'; ctx2.lineWidth = 2;
ctx2.beginPath(); ctx2.moveTo(xSig, p.top); ctx2.lineTo(xSig, h2 - p.bottom); ctx2.stroke();
ctx2.strokeStyle = '#f5a623'; ctx2.beginPath(); ctx2.moveTo(xEs, p.top); ctx2.lineTo(xEs, h2 - p.bottom); ctx2.stroke();
document.getElementById('s3es').textContent = `${(meanS * 100).toFixed(3)}%`;
document.getElementById('s3bias').textContent = `${((SIG_TRUE - meanS) * 100).toFixed(3)}%`;
document.getElementById('f3').innerHTML =
`已模拟 <span class="hl">${m}</span> 次（每次 n=${JENSEN_N}）\n` +
`σ = ${SIG_TRUE.toFixed(4)}\n` +
`E(s) ≈ ${(meanS * 100).toFixed(3)}% < σ = ${(SIG_TRUE * 100).toFixed(2)}%\n` +
`<span class="result">E(√s²) < √E(s²) = σ</span>`;
}
document.getElementById('stepPill').textContent = `${jensenSamples.length} / ${jensenTarget} 次模拟`;
document.getElementById('s3sig').textContent = `${(SIG_TRUE * 100).toFixed(2)}%`;
document.getElementById('s3n').textContent = String(JENSEN_N);
setProg(document.querySelector('#scene-jensen .prog'), jensenSamples.length / jensenTarget);
}
// --- Scene 4: PDF ---
let pdfN = 3;
const PDF_SIG = SIG_TRUE;
const PDF_N_MAX = 120;
function lnGamma(z) {
const g = 7;
const c = [0.99999999999980993, 676.5203681218851, -1259.1392167224028,
771.32342877765313, -176.61502916214059, 12.507343278686905,
-0.13857109526572012, 9.984369578019571e-6, 1.5056327351493116e-7];
if (z < 0.5) return Math.log(Math.PI / Math.sin(Math.PI * z)) - lnGamma(1 - z);
z -= 1;
let x = c[0];
for (let i = 1; i < g + 2; i++) x += c[i] / (z + i);
const t = z + g + 0.5;
return 0.5 * Math.log(2 * Math.PI) + (z + 0.5) * Math.log(t) - t + Math.log(x);
}
function pdfStd(s, N, sigma) {
if (s <= 0 || N < 2) return 0;
const a = (N - 1) / 2;
const logF = Math.log(2) - lnGamma(a) + a * Math.log(N / (2 * sigma * sigma))
- (N * s * s) / (2 * sigma * sigma) + (N - 2) * Math.log(s);
return Math.exp(logF);
}
function drawScene4() {
const { w, h, ctx } = setupCanvas('w4', c4);
const p = pad();
ctx.clearRect(0, 0, w, h);
drawGrid(ctx, w, h, p);
const N = pdfN;
const lo = 0.002, hi = PDF_SIG * 2.2;
const pts = [];
let peak = 0, peakS = 0;
for (let i = 0; i <= 300; i++) {
const s = lo + (hi - lo) * i / 300;
const f = pdfStd(s, N, PDF_SIG);
pts.push([s, f]);
if (f > peak) { peak = f; peakS = s; }
}
const maxF = peak * 1.08;
const tx = s => p.left + (s - lo) / (hi - lo) * (w - p.left - p.right);
const ty = f => p.top + (1 - f / maxF) * (h - p.top - p.bottom);
ctx.strokeStyle = '#3d9cf5'; ctx.lineWidth = 2.2; ctx.beginPath();
pts.forEach(([s, f], i) => { i === 0 ? ctx.moveTo(tx(s), ty(f)) : ctx.lineTo(tx(s), ty(f)); });
ctx.stroke();
const xSig = tx(PDF_SIG);
ctx.strokeStyle = '#2dd4a8'; ctx.lineWidth = 2; ctx.setLineDash([5, 4]);
ctx.beginPath(); ctx.moveTo(xSig, p.top); ctx.lineTo(xSig, h - p.bottom); ctx.stroke();
ctx.setLineDash([]);
ctx.fillStyle = '#8b95a8'; ctx.font = '10px IBM Plex Sans,sans-serif';
ctx.fillText(`N=${N}`, p.left + 4, p.top + 14);
ctx.fillText(`σ=${(PDF_SIG * 100).toFixed(1)}%`, xSig + 4, p.top + 14);
document.getElementById('stepPill').textContent = `N = ${N}`;
document.getElementById('s4N').textContent = String(N);
document.getElementById('s4sig').textContent = `${(PDF_SIG * 100).toFixed(2)}%`;
document.getElementById('s4peak').textContent = `${(peakS * 100).toFixed(2)}%`;
document.getElementById('s4note').textContent = N >= 20 ? '趋近 σ' : '左偏';
document.getElementById('nSlider').value = N;
document.getElementById('f4').innerHTML =
`f_${N}(s) = 2/Γ(${((N-1)/2).toFixed(1)}) · (N/2σ²)^(${((N-1)/2).toFixed(1)})\n` +
`· exp(-Ns²/2σ²) · s^${N-2}\n` +
`<span class="result">N 越大，分布越集中在 σ 附近</span>`;
setProg(document.querySelector('#scene-2-4 .prog'), (N - 3) / (PDF_N_MAX - 3));
}
function setProg(el, r) { if (el) el.style.width = `${Math.min(100, r * 100)}%`; }
function drawActive() {
if (activeScene === '2-1b') drawScene1(frame);
else if (activeScene === '2-2') drawScene2(frame);
else if (activeScene === 'jensen') drawScene3();
else if (activeScene === '2-4') drawScene4();
}
function pauseAll() { playing = false; clearTimeout(timer); document.querySelectorAll('.play').forEach(b => b.textContent = '▶ 播放'); }
function pause() { pauseAll(); }
function getSpeedMs(sceneEl) {
const v = Number(sceneEl.querySelector('.speed-range').value);
return 1150 - v * 95;
}
function resetActive() {
if (activeScene === '2-1b' || activeScene === '2-2') frame = 0;
else if (activeScene === 'jensen') { jensenSamples = []; }
else if (activeScene === '2-4') { pdfN = 3; }
}
function stepActive() {
if (activeScene === '2-1b' || activeScene === '2-2') {
if (frame < DATA_N - 1) { frame++; drawActive(); }
else pause();
} else if (activeScene === 'jensen') {
addJensenBatch(500);
drawActive();
if (jensenSamples.length >= jensenTarget) pause();
} else if (activeScene === '2-4') {
if (pdfN < PDF_N_MAX) { pdfN++; drawActive(); }
else pause();
}
}
function schedule() {
clearTimeout(timer);
if (!playing) return;
const sceneEl = document.getElementById('scene-' + activeScene);
const ms = getSpeedMs(sceneEl);
timer = setTimeout(() => {
if (activeScene === '2-1b' || activeScene === '2-2') {
if (frame < DATA_N - 1) { frame++; drawActive(); schedule(); }
else pause();
} else if (activeScene === 'jensen') {
addJensenBatch(200);
drawActive();
if (jensenSamples.length >= jensenTarget) pause();
else schedule();
} else if (activeScene === '2-4') {
if (pdfN < PDF_N_MAX) { pdfN++; drawActive(); schedule(); }
else pause();
}
}, ms);
}
function playActive() {
const sceneEl = document.getElementById('scene-' + activeScene);
if (activeScene === '2-1b' || activeScene === '2-2') {
if (frame >= DATA_N - 1) { frame = 0; drawActive(); }
} else if (activeScene === 'jensen') {
if (jensenSamples.length >= jensenTarget) { jensenSamples = []; drawActive(); }
} else if (activeScene === '2-4') {
if (pdfN >= PDF_N_MAX) { pdfN = 3; drawActive(); }
}
playing = true;
sceneEl.querySelector('.play').textContent = '▶ 播放中';
schedule();
}
document.querySelectorAll('.scene').forEach(scene => {
const ctrl = scene.querySelector('.controls');
ctrl.querySelector('.play').addEventListener('click', () => {
if (document.getElementById('scene-' + activeScene) !== scene) return;
playing ? pause() : playActive();
});
ctrl.querySelector('.pause').addEventListener('click', () => {
if (document.getElementById('scene-' + activeScene) === scene) pause();
});
ctrl.querySelector('.step').addEventListener('click', () => {
if (document.getElementById('scene-' + activeScene) === scene) { pause(); stepActive(); }
});
ctrl.querySelector('.reset').addEventListener('click', () => {
if (document.getElementById('scene-' + activeScene) === scene) { pause(); resetActive(); drawActive(); }
});
ctrl.querySelector('.speed-range').addEventListener('input', () => { if (playing) schedule(); });
});
document.getElementById('nSlider').addEventListener('input', e => {
pdfN = Number(e.target.value);
pause();
drawActive();
});
window.addEventListener('resize', drawActive);
applyHash();
if (!location.hash) drawActive();
</script>
</body>
</html>