{"id":368,"date":"2025-02-14T23:13:03","date_gmt":"2025-02-14T15:13:03","guid":{"rendered":"https:\/\/www.loyeh.top\/?p=368"},"modified":"2025-02-14T23:13:04","modified_gmt":"2025-02-14T15:13:04","slug":"%e6%88%aa%e6%96%ad%e6%ad%a3%e6%80%81%e5%88%86%e5%b8%83%ef%bc%88truncated-normal-distribution%ef%bc%89","status":"publish","type":"post","link":"https:\/\/loyeh.top\/index.php\/2025\/02\/14\/%e6%88%aa%e6%96%ad%e6%ad%a3%e6%80%81%e5%88%86%e5%b8%83%ef%bc%88truncated-normal-distribution%ef%bc%89\/","title":{"rendered":"\u622a\u65ad\u6b63\u6001\u5206\u5e03\uff08Truncated Normal Distribution\uff09"},"content":{"rendered":"\n<p>\u622a\u65ad\u6b63\u6001\u5206\u5e03\u662f\u6b63\u6001\u5206\u5e03\u7684\u4e00\u79cd\u53d8\u4f53\uff0c\u5176\u5b9a\u4e49\u57df\u88ab\u9650\u5236\u5728\u4e00\u4e2a\u6709\u9650\u7684\u533a\u95f4\u5185\u3002\u4e0e\u6807\u51c6\u6b63\u6001\u5206\u5e03\u4e0d\u540c\uff0c\u622a\u65ad\u6b63\u6001\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u5728\u6307\u5b9a\u533a\u95f4\u4e4b\u5916\u4e3a\u96f6\uff0c\u800c\u5728\u533a\u95f4\u5185\u5219\u4e0e\u6b63\u6001\u5206\u5e03\u6210\u6bd4\u4f8b\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">1. \u5b9a\u4e49<\/h3>\n\n\n\n<p>\u7ed9\u5b9a\u4e00\u4e2a\u6b63\u6001\u5206\u5e03 $X \\sim \\mathcal{N}(\\mu, \\sigma^2)$\uff0c\u622a\u65ad\u6b63\u6001\u5206\u5e03\u662f\u6307\u5c06 $X$ \u9650\u5236\u5728\u533a\u95f4 $[a, b]$ \u5185\u7684\u5206\u5e03\u3002\u5176\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u4e3a\uff1a<\/p>\n\n\n<p>f(x; \\mu, \\sigma, a, b) =<br \/>\n\\begin{cases}<br \/>\n\\frac{\\phi\\left(\\frac{x - \\mu}{\\sigma}\\right)}{\\sigma \\left( \\Phi\\left(\\frac{b - \\mu}{\\sigma}\\right) - \\Phi\\left(\\frac{a - \\mu}{\\sigma}\\right) \\right)}, & \\text{if } a \\leq x \\leq b \\\\<br \/>\n0, & \\text{otherwise}<br \/>\n\\end{cases}<\/p>\n\n\n\n<p>\u5176\u4e2d\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$\\phi(x)$ \u662f\u6807\u51c6\u6b63\u6001\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\uff1a<br>$$<br>\\phi(x) = \\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{x^2}{2}}<br>$$<\/li>\n\n\n\n<li>$\\Phi(x)$ \u662f\u6807\u51c6\u6b63\u6001\u5206\u5e03\u7684\u7d2f\u79ef\u5206\u5e03\u51fd\u6570\uff08CDF\uff09\uff1a<br>$$<br>\\Phi(x) = \\int_{-\\infty}^x \\phi(t) \\, dt<br>$$<\/li>\n\n\n\n<li>$\\mu$ \u548c $\\sigma$ \u662f\u539f\u59cb\u6b63\u6001\u5206\u5e03\u7684\u5747\u503c\u548c\u6807\u51c6\u5dee\u3002<\/li>\n\n\n\n<li>$[a, b]$ \u662f\u622a\u65ad\u533a\u95f4\u3002<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">2. \u6027\u8d28<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u5f52\u4e00\u5316<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u622a\u65ad\u6b63\u6001\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u5728\u533a\u95f4 $[a, b]$ \u5185\u662f\u5f52\u4e00\u5316\u7684\uff0c\u5373\uff1a<br>$$<br>\\int_a^b f(x; \\mu, \\sigma, a, b) \\, dx = 1<br>$$<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u5747\u503c\u4e0e\u65b9\u5dee<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u622a\u65ad\u6b63\u6001\u5206\u5e03\u7684\u5747\u503c $\\mu_T$ \u548c\u65b9\u5dee $\\sigma_T^2$ \u4e0e\u539f\u59cb\u6b63\u6001\u5206\u5e03\u7684\u5747\u503c\u548c\u65b9\u5dee\u4e0d\u540c\u3002<\/li>\n\n\n\n<li>\u5747\u503c\u516c\u5f0f\uff1a<br>$$<br>\\mu_T = \\mu + \\sigma \\cdot \\frac{\\phi\\left(\\frac{a - \\mu}{\\sigma}\\right) - \\phi\\left(\\frac{b - \\mu}{\\sigma}\\right)}{\\Phi\\left(\\frac{b - \\mu}{\\sigma}\\right) - \\Phi\\left(\\frac{a - \\mu}{\\sigma}\\right)}<br>$$<\/li>\n\n\n\n<li>\u65b9\u5dee\u516c\u5f0f\uff1a<br>$$<br>\\sigma_T^2 = \\sigma^2 \\left[ 1 + \\frac{\\left(\\frac{a - \\mu}{\\sigma}\\right) \\phi\\left(\\frac{a - \\mu}{\\sigma}\\right) - \\left(\\frac{b - \\mu}{\\sigma}\\right) \\phi\\left(\\frac{b - \\mu}{\\sigma}\\right)}{\\Phi\\left(\\frac{b - \\mu}{\\sigma}\\right) - \\Phi\\left(\\frac{a - \\mu}{\\sigma}\\right)} - \\left( \\frac{\\phi\\left(\\frac{a - \\mu}{\\sigma}\\right) - \\phi\\left(\\frac{b - \\mu}{\\sigma}\\right)}{\\Phi\\left(\\frac{b - \\mu}{\\sigma}\\right) - \\Phi\\left(\\frac{a - \\mu}{\\sigma}\\right)} \\right)^2 \\right]<br>$$<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<ol start=\"3\" class=\"wp-block-list\">\n<li><strong>\u533a\u95f4\u9009\u62e9\u7684\u5f71\u54cd<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u5982\u679c\u622a\u65ad\u533a\u95f4 $[a, b]$ \u8fdc\u79bb\u5747\u503c $\\mu$\uff0c\u4f1a\u8d8b\u8fd1\u4e8e\u5747\u5300\u5206\u5e03\u3002<\/li>\n\n\n\n<li>\u5982\u679c\u622a\u65ad\u533a\u95f4 $[a, b]$ \u5305\u542b\u5747\u503c $\\mu$\uff0c\u4f1a\u63a5\u8fd1\u539f\u59cb\u6b63\u6001\u5206\u5e03\u3002<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">3. \u622a\u65ad\u6b63\u6001\u5206\u5e03\u7684\u5e94\u7528<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u9650\u5236\u53d8\u91cf\u8303\u56f4<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u5728\u5b9e\u9645\u95ee\u9898\u4e2d\uff0c\u67d0\u4e9b\u53d8\u91cf\u7684\u53d6\u503c\u8303\u56f4\u53ef\u80fd\u53d7\u5230\u9650\u5236\u3002\u4f8b\u5982\uff0c\u8eab\u9ad8\u3001\u4f53\u91cd\u3001\u6e29\u5ea6\u7b49\u7269\u7406\u91cf\u4e0d\u80fd\u4e3a\u8d1f\u503c\uff0c\u56e0\u6b64\u53ef\u4ee5\u4f7f\u7528\u622a\u65ad\u6b63\u6001\u5206\u5e03\u6765\u5efa\u6a21\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u8d1d\u53f6\u65af\u63a8\u65ad<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u5728\u8d1d\u53f6\u65af\u7edf\u8ba1\u4e2d\uff0c\u622a\u65ad\u6b63\u6001\u5206\u5e03\u5e38\u7528\u4e8e\u5148\u9a8c\u5206\u5e03\u6216\u540e\u9a8c\u5206\u5e03\uff0c\u7279\u522b\u662f\u5f53\u53c2\u6570\u6709\u660e\u786e\u7684\u4e0a\u4e0b\u754c\u65f6\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u91d1\u878d\u5efa\u6a21<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u5728\u91d1\u878d\u9886\u57df\uff0c\u8d44\u4ea7\u6536\u76ca\u7387\u6216\u6ce2\u52a8\u7387\u901a\u5e38\u6709\u4e0a\u4e0b\u9650\uff0c\u622a\u65ad\u6b63\u6001\u5206\u5e03\u53ef\u4ee5\u7528\u4e8e\u5efa\u6a21\u8fd9\u4e9b\u53d7\u9650\u53d8\u91cf\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u673a\u5668\u5b66\u4e60<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u5728\u6df1\u5ea6\u5b66\u4e60\u4e2d\uff0c\u622a\u65ad\u6b63\u6001\u5206\u5e03\u5e38\u7528\u4e8e\u521d\u59cb\u5316\u795e\u7ecf\u7f51\u7edc\u6743\u91cd\uff0c\u4ee5\u786e\u4fdd\u6743\u91cd\u503c\u5728\u4e00\u4e2a\u5408\u7406\u7684\u8303\u56f4\u5185\u3002<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">4. \u622a\u65ad\u6b63\u6001\u5206\u5e03\u7684\u91c7\u6837<\/h3>\n\n\n\n<p>\u4ece\u622a\u65ad\u6b63\u6001\u5206\u5e03\u4e2d\u91c7\u6837\u901a\u5e38\u4f7f\u7528\u4ee5\u4e0b\u65b9\u6cd5\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u62d2\u7edd\u91c7\u6837\u6cd5<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u4ece\u539f\u59cb\u6b63\u6001\u5206\u5e03\u4e2d\u91c7\u6837\uff0c\u5982\u679c\u6837\u672c\u843d\u5728\u622a\u65ad\u533a\u95f4 $[a, b]$ \u5185\uff0c\u5219\u63a5\u53d7\uff1b\u5426\u5219\u62d2\u7edd\u5e76\u91cd\u65b0\u91c7\u6837\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u9006\u53d8\u6362\u91c7\u6837\u6cd5<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u5229\u7528\u7d2f\u79ef\u5206\u5e03\u51fd\u6570\uff08CDF\uff09\u7684\u9006\u51fd\u6570\u8fdb\u884c\u91c7\u6837\u3002\u5177\u4f53\u6b65\u9aa4\u5982\u4e0b\uff1a\n<ol class=\"wp-block-list\">\n<li>\u8ba1\u7b97\u622a\u65ad\u533a\u95f4\u7684 CDF \u503c\uff1a<br>$$<br>u_a = \\Phi\\left(\\frac{a - \\mu}{\\sigma}\\right), \\quad u_b = \\Phi\\left(\\frac{b - \\mu}{\\sigma}\\right)<br>$$<\/li>\n\n\n\n<li>\u4ece\u5747\u5300\u5206\u5e03 $U(u_a, u_b)$ \u4e2d\u91c7\u6837\u4e00\u4e2a\u503c $u$\u3002<\/li>\n\n\n\n<li>\u4f7f\u7528\u9006 CDF \u51fd\u6570\u8ba1\u7b97\u6837\u672c\uff1a<br>$$<br>x = \\mu + \\sigma \\cdot \\Phi^{-1}(u)<br>$$<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u5e93\u51fd\u6570<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u8bb8\u591a\u79d1\u5b66\u8ba1\u7b97\u5e93\uff08\u5982 NumPy\u3001SciPy\u3001TensorFlow \u7b49\uff09\u63d0\u4f9b\u4e86\u622a\u65ad\u6b63\u6001\u5206\u5e03\u7684\u91c7\u6837\u51fd\u6570\u3002\u4f8b\u5982\uff0c\u5728 SciPy \u4e2d\uff1a <\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<pre class=\"wp-block-code\"><code>from scipy.stats import truncnorm\n\n# \u5b9a\u4e49\u622a\u65ad\u533a\u95f4 &#91;a, b]\na, b = 0, 1\n# \u5b9a\u4e49\u5747\u503c mu \u548c\u6807\u51c6\u5dee sigma\nmu, sigma = 0.5, 0.1\n# \u521b\u5efa\u622a\u65ad\u6b63\u6001\u5206\u5e03\u5bf9\u8c61\ndist = truncnorm((a - mu) \/ sigma, (b - mu) \/ sigma, loc=mu, scale=sigma)\n# \u91c7\u6837\nsamples = dist.rvs(size=1000)<\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\">5. \u622a\u65ad\u6b63\u6001\u5206\u5e03\u4e0e\u6807\u51c6\u6b63\u6001\u5206\u5e03\u7684\u533a\u522b<\/h3>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>\u7279\u6027<\/th><th>\u6807\u51c6\u6b63\u6001\u5206\u5e03<\/th><th>\u622a\u65ad\u6b63\u6001\u5206\u5e03<\/th><\/tr><\/thead><tbody><tr><td>\u5b9a\u4e49\u57df<\/td><td>$(-\\infty, +\\infty)$<\/td><td>$[a, b]$<\/td><\/tr><tr><td>\u6982\u7387\u5bc6\u5ea6\u51fd\u6570<\/td><td>$\\phi(x)$<\/td><td>\u5f52\u4e00\u5316\u7684 $\\phi(x)$<\/td><\/tr><tr><td>\u5747\u503c\u4e0e\u65b9\u5dee<\/td><td>$\\mu$ \u548c $\\sigma^2$<\/td><td>$\\mu_T$ \u548c $\\sigma_T^2$<\/td><\/tr><tr><td>\u5e94\u7528\u573a\u666f<\/td><td>\u65e0\u9650\u5236\u7684\u8fde\u7eed\u53d8\u91cf<\/td><td>\u6709\u9650\u5236\u7684\u8fde\u7eed\u53d8\u91cf<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">6. \u622a\u65ad\u6b63\u6001\u5206\u5e03\u7684\u5c40\u9650\u6027<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u8ba1\u7b97\u590d\u6742\u5ea6<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u622a\u65ad\u6b63\u6001\u5206\u5e03\u7684\u5747\u503c\u548c\u65b9\u5dee\u8ba1\u7b97\u6d89\u53ca\u590d\u6742\u7684\u79ef\u5206\u548c\u7279\u6b8a\u51fd\u6570\uff0c\u8ba1\u7b97\u6210\u672c\u8f83\u9ad8\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u533a\u95f4\u9009\u62e9\u7684\u5f71\u54cd<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u622a\u65ad\u533a\u95f4\u7684\u9009\u62e9\u5bf9\u5206\u5e03\u7684\u5f62\u72b6\u548c\u53c2\u6570\u6709\u663e\u8457\u5f71\u54cd\uff0c\u5982\u679c\u533a\u95f4\u9009\u62e9\u4e0d\u5f53\uff0c\u53ef\u80fd\u5bfc\u81f4\u6a21\u578b\u504f\u5dee\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u591a\u53d8\u91cf\u6269\u5c55\u56f0\u96be<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u622a\u65ad\u6b63\u6001\u5206\u5e03\u7684\u591a\u53d8\u91cf\u6269\u5c55\uff08\u5373\u622a\u65ad\u591a\u5143\u6b63\u6001\u5206\u5e03\uff09\u8ba1\u7b97\u66f4\u52a0\u590d\u6742\uff0c\u5e94\u7528\u53d7\u9650\u3002<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">\u603b\u7ed3<\/h3>\n\n\n\n<p>\u622a\u65ad\u6b63\u6001\u5206\u5e03\u662f\u4e00\u79cd\u5c06\u6b63\u6001\u5206\u5e03\u9650\u5236\u5728\u6709\u9650\u533a\u95f4\u5185\u7684\u6982\u7387\u5206\u5e03\uff0c\u5e7f\u6cdb\u5e94\u7528\u4e8e\u9700\u8981\u9650\u5236\u53d8\u91cf\u8303\u56f4\u7684\u573a\u666f\u3002\u5b83\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u662f\u539f\u59cb\u6b63\u6001\u5206\u5e03\u5728\u622a\u65ad\u533a\u95f4\u5185\u7684\u5f52\u4e00\u5316\u5f62\u5f0f\uff0c\u5176\u5747\u503c\u548c\u65b9\u5dee\u4e0e\u539f\u59cb\u6b63\u6001\u5206\u5e03\u4e0d\u540c\u3002\u622a\u65ad\u6b63\u6001\u5206\u5e03\u7684\u91c7\u6837\u53ef\u4ee5\u901a\u8fc7\u62d2\u7edd\u91c7\u6837\u6cd5\u3001\u9006\u53d8\u6362\u91c7\u6837\u6cd5\u6216\u5e93\u51fd\u6570\u5b9e\u73b0\u3002\u867d\u7136\u8ba1\u7b97\u8f83\u4e3a\u590d\u6742\uff0c\u4f46\u5b83\u5728\u8d1d\u53f6\u65af\u63a8\u65ad\u3001\u91d1\u878d\u5efa\u6a21\u548c\u673a\u5668\u5b66\u4e60\u7b49\u9886\u57df\u5177\u6709\u91cd\u8981\u4ef7\u503c\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u622a\u65ad\u6b63\u6001\u5206\u5e03\u662f\u6b63\u6001\u5206\u5e03\u7684\u4e00\u79cd\u53d8\u4f53\uff0c\u5176\u5b9a\u4e49\u57df\u88ab\u9650\u5236\u5728\u4e00\u4e2a\u6709\u9650\u7684\u533a\u95f4\u5185\u3002\u4e0e\u6807\u51c6\u6b63\u6001\u5206\u5e03\u4e0d\u540c\uff0c\u622a\u65ad\u6b63\u6001\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u5728\u6307\u5b9a\u533a\u95f4\u4e4b\u5916\u4e3a 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