{"id":63433,"date":"2025-08-11T18:47:15","date_gmt":"2025-08-11T10:47:15","guid":{"rendered":"https:\/\/swarma.org\/?p=63433"},"modified":"2025-08-11T18:47:15","modified_gmt":"2025-08-11T10:47:15","slug":"%e9%a9%ac%e5%b0%94%e7%a7%91%e5%a4%ab%e9%93%be%ef%bd%9c%e9%9b%86%e6%99%ba%e7%99%be%e7%a7%91","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=63433","title":{"rendered":"\u9a6c\u5c14\u79d1\u592b\u94fe\uff5c\u96c6\u667a\u767e\u79d1"},"content":{"rendered":"
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\u9a6c\u5c14\u79d1\u592b\u94fe\u662f\u6307\u4e00\u7c7b\u5b9a\u4e49\u5728\u53ef\u5217\u96c6\u5408\u4e0a\u3001\u5177\u6709\u9a6c\u5c14\u79d1\u592b\u6027\u7684\u968f\u673a\u8fc7\u7a0b\u3002\u5176\u4e2d\uff0c\u8fd9\u4e2a\u96c6\u5408\u88ab\u79f0\u4e3a\u662f\u72b6\u6001\u7a7a\u95f4\uff0c\u800c\u9a6c\u5c14\u79d1\u592b\u6027\u662f\u6307\u5728\u7ed9\u5b9a\u7cfb\u7edf\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u5bf9\u4e8e\u672a\u6765\u72b6\u6001\u7684\u9884\u6d4b\u4e0d\u518d\u4f9d\u8d56\u4e8e\u8fc7\u53bb\u72b6\u6001\u3002\u5b83\u63cf\u8ff0\u4e86\u968f\u7740\u65f6\u95f4\u7684\u63a8\u79fb\uff0c\u7cfb\u7edf\u72b6\u6001\u7684\u53d8\u5316\u8fc7\u7a0b\u3002\u8fdb\u4e00\u6b65\uff0c\u6839\u636e\u65f6\u95f4\u53c2\u6570\u7684\u79bb\u6563\u6216\u8fde\u7eed\u7279\u6027\uff0c\u5b83\u53ef\u4ee5\u5177\u4f53\u5212\u5206\u4e3a\u79bb\u6563\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\u548c\u8fde\u7eed\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\u3002<\/span><\/span><\/span><\/strong><\/p>\n


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\u672c\u8bcd\u6761\u4e3b\u8981\u53c2\u8003\u5317\u4eac\u5927\u5b66\u674e\u4e1c\u98ce\u8001\u5e08\u7684\u5e94\u7528\u968f\u673a\u8fc7\u7a0b\u8bb2\u4e49[1]\u4ee5\u53ca\u6e05\u534e\u5927\u5b66\u5f20\u98a2\u8001\u5e08\u7684\u4e66\u7c4d\u300a\u968f\u673a\u8fc7\u7a0b\u53ca\u5176\u5e94\u7528\u300b[2]\uff0c\u4f5c\u8005\u4eec\u5bf9\u76f8\u5173\u57fa\u7840\u6982\u5ff5\u8fdb\u884c\u4e86\u68b3\u7406\uff0c\u5e0c\u671b\u4e3a\u540e\u7eed\u76f8\u5173\u5e94\u7528\u505a\u51fa\u94fa\u57ab\u3002<\/span><\/span>
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\u4e3a\u4e86\u7cfb\u7edf\u68b3\u7406\u56e0\u679c\u6d8c\u73b0\u6700\u65b0\u8fdb\u5c55\uff0c\u5317\u4eac\u5e08\u8303\u5927\u5b66\u7cfb\u7edf\u79d1\u5b66\u5b66\u9662\u6559\u6388\u3001\u96c6\u667a\u4ff1\u4e50\u90e8\u521b\u59cb\u4eba\u5f20\u6c5f\u8001\u5e08\u9886\u8854\u56e0\u679c\u6d8c\u73b0\u7cfb\u5217\u8bfb\u4e66\u4f1a\uff0c\u76ee\u524d\u5df2\u7ecf\u6301\u7eed\u5230\u300c\u56e0\u679c\u6d8c\u73b0\u7b2c\u516d\u5b63\u300d\u8bfb\u4e66\u4f1a<\/a>\uff0c\u5982\u679c\u4f60\u5bf9\u8fd9\u4e00\u8bdd\u9898\u611f\u5174\u8da3\uff0c\u975e\u5e38\u63a8\u8350\u4f60\u52a0\u5165\u793e\u533a\uff01<\/span><\/span><\/strong><\/span><\/strong><\/p>\n


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\u201c\u96c6\u667a\u767e\u79d1\u7cbe\u9009\u201d\u662f\u4e00\u4e2a\u957f\u671f\u4e13\u680f\uff0c\u6301\u7eed\u4e3a\u5927\u5bb6\u63a8\u9001\u590d\u6742\u6027\u79d1\u5b66\u76f8\u5173\u7684\u57fa\u672c\u6982\u5ff5\u548c\u8d44\u6e90\u4fe1\u606f\u3002\u4f5c\u4e3a\u96c6\u667a\u4ff1\u4e50\u90e8\u7684\u5f00\u6e90\u79d1\u5b66\u9879\u76ee\uff0c\u96c6\u667a\u767e\u79d1\u5e0c\u671b\u6253\u9020\u590d\u6742\u6027\u79d1\u5b66\u9886\u57df\u6700\u5168\u9762\u7684\u767e\u79d1\u5168\u4e66\uff0c\u6b22\u8fce\u5bf9\u590d\u6742\u6027\u79d1\u5b66\u611f\u5174\u8da3\u3001\u70ed\u7231\u77e5\u8bc6\u6574\u7406\u548c\u5206\u4eab\u7684\u670b\u53cb\u52a0\u5165\uff0c\u6587\u672b\u53ef\u4ee5\u626b\u7801\u62a5\u540d\u52a0\u5165\u767e\u79d1\u5fd7\u613f\u8005\uff01<\/span><\/strong><\/span><\/strong><\/p>\n


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\u2191\u2191\u2191\u626b\u7801\u76f4\u8fbe\u767e\u79d1\u8bcd\u6761<\/span><\/p>\n

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\u5173\u952e\u8bcd\uff1a\u9a6c\u5c14\u79d1\u592b\u94fe\uff0c\u7c97\u7c92\u5316<\/span><\/strong><\/strong><\/strong><\/strong><\/span><\/section>\n
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\u5f20\u6c5f\u3001<\/span>NULL\u3001\u5f20\u4ee3\u5065<\/span> <\/span><\/span><\/span>| \u7f16\u8bd1<\/span><\/span><\/p>\n

\u5f20\u6c5f\u3001\u738b\u5fd7\u9e4f <\/span><\/span><\/span>| \u6574\u7406\u3001\u5ba1\u6821<\/span><\/span><\/section>\n


<\/span><\/p>\n<\/section>\n

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\u76ee\u5f55<\/span><\/strong><\/p>\n

1. \u5386\u53f2\u53d1\u5c55<\/span><\/span><\/span><\/span><\/section>\n

2. \u79bb\u6563\uff08\u6216\u8fde\u7eed\uff09\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u5b9a\u4e49<\/span><\/span><\/p>\n

2.1 \u4f8b\u5b50<\/span><\/span><\/p>\n

2.1.1 \u6d41\u7f51\u7edc<\/span><\/span><\/p>\n

2.1.2 \u9ad8\u5c14\u987f\u9489\u677f<\/span><\/span><\/p>\n

2.1.3 Yahtzee\u7684\u9ab0\u5b50\u6e38\u620f<\/span><\/span><\/p>\n

2.2 \u5b9a\u4e49<\/span><\/span><\/p>\n

3. \u79bb\u6563\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe<\/span><\/span><\/p>\n

3.1 \u57fa\u672c\u6982\u5ff5\u548c\u6027\u8d28<\/span><\/span><\/p>\n

3.1.1 \u591a\u6b65\u8f6c\u79fb\u6982\u7387\u7684\u9057\u5fd8\u6027<\/span><\/p>\n

3.1.2 \u591a\u6b65\u8f6c\u79fb\u6982\u7387\u7684\u8ba1\u7b97\u2014\u2014Chapman-Kolmogorov\u65b9\u7a0b<\/span><\/p>\n

3.1.3 \u65f6\u9f50\u6027\u4e0e\u975e\u65f6\u9f50\u6027<\/span><\/p>\n

3.1.3.1 \u65f6\u9f50\u6027\u4f8b\u5b50\uff1a\u9ad8\u5c14\u987f\u9489\u677f<\/span><\/p>\n

3.1.3.2 \u975e\u65f6\u9f50\u6027\u4f8b\u5b50\uff1a\u6ce2\u5229\u4e9a\u575b\u5b50\u6a21\u578b<\/span><\/p>\n

3.1.3.3 \u5b9a\u4e49<\/span><\/p>\n

3.2 \u65f6\u9f50\u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u8f6c\u79fb\u8fc7\u7a0b\u2014\u2014\u521d\u59cb\u72b6\u6001\u786e\u5b9a<\/span><\/span><\/p>\n

3.2.1 \u591a\u6b65\u8f6c\u79fb\u6982\u7387\u7684\u8ba1\u7b97\u4e0e\u4e00\u6b65\u8f6c\u79fb\u6982\u7387\u77e9\u9635<\/span><\/span><\/p>\n

3.2.2 \u57fa\u4e8e\u8f6c\u79fb\u884c\u4e3a\u7684\u72b6\u6001\u5206\u7c7b<\/span><\/span><\/p>\n

3.2.2.1 \u53ef\u8fbe\u3001\u4e92\u901a\u4e0e\u53ef\u7ea6\u6027<\/span><\/span><\/p>\n

3.2.2.2 \u5468\u671f\u6027<\/span><\/span><\/p>\n

3.3 \u65f6\u9f50\u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u6f14\u5316\u2014\u2014\u521d\u59cb\u72b6\u6001\u968f\u673a<\/span><\/span><\/p>\n

3.3.1 \u6f14\u5316\u8fc7\u7a0b<\/span><\/span><\/p>\n

3.3.2 \u5e73\u7a33\u5206\u5e03\u4e0e\u8f6c\u79fb\u6982\u7387\u77e9\u9635\u7684\u8c31<\/span><\/span><\/p>\n

3.4 \u65f6\u95f4\u53ef\u9006\u7684\u9a6c\u5c14\u79d1\u592b\u94fe<\/span><\/span><\/p>\n

3.4.1 \u65f6\u95f4\u53ef\u9006\u6027\u7684\u5b9a\u4e49(\u7ec6\u81f4\u5e73\u8861\u6761\u4ef6)\u53ca\u53ef\u9006\u5206\u5e03<\/span><\/span><\/p>\n

3.4.2 \u53ef\u9006\u5206\u5e03\u4e0e\u5e73\u7a33\u5206\u5e03<\/span><\/span><\/p>\n

3.4.3 \u65f6\u95f4\u53ef\u9006\u6027\u4e0e\u52a8\u529b\u5b66\u53ef\u9006\u6027<\/span><\/span><\/p>\n

3.5 \u79bb\u6563\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u7c97\u7c92\u5316\u548c\u5ea6\u91cf<\/span><\/span><\/p>\n

3.5.1 \u79bb\u6563\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u7c97\u7c92\u5316<\/span><\/span><\/p>\n

3.5.1.1 \u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u6210\u5757\u6027<\/span><\/span><\/p>\n

3.5.1.2 \u9a6c\u5c14\u79d1\u592b\u94fe\u7684SVD\u7c97\u7c92\u5316<\/span><\/span><\/p>\n

3.5.2 \u79bb\u6563\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u5ea6\u91cf<\/span><\/span><\/p>\n

3.5.2.1 \u8f6c\u79fb\u6982\u7387\u77e9\u9635\u7684\u6709\u6548\u4fe1\u606f<\/span><\/span><\/p>\n

3.5.2.2 \u8f6c\u79fb\u6982\u7387\u77e9\u9635\u7684\u52a8\u529b\u5b66\u53ef\u9006\u6027<\/span><\/span><\/p>\n

4. \u8fde\u7eed\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe<\/span><\/span><\/p>\n

4.1 \u57fa\u672c\u6982\u5ff5<\/span><\/span><\/p>\n

4.1.1 \u4e0e\u79bb\u6563\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\u5bf9\u5e94\u7684\u57fa\u672c\u6982\u5ff5<\/span><\/span><\/p>\n

4.1.1.1 \u8f6c\u79fb\u901f\u7387\u77e9\u9635<\/span><\/span><\/p>\n

4.1.1.2 \u8f6c\u79fb\u901f\u7387\u77e9\u9635\u4e0e\u8f6c\u79fb\u6982\u7387\u77e9\u9635\u5173\u7cfb<\/span><\/span><\/p>\n

4.1.2 \u8fde\u7eed\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\u4e2d\u72ec\u6709\u7684\u6982\u5ff5<\/span><\/span><\/p>\n

4.1.3 \u5177\u4f53\u4f8b\u5b501: \u6392\u961f\u95ee\u9898<\/span><\/span><\/p>\n

4.1.4 \u5177\u4f53\u4f8b\u5b502: \u57fa\u56e0\u8c03\u63a7\u7f51\u7edc<\/span><\/span><\/p>\n

4.2 \u8f6c\u79fb\u8fc7\u7a0b\u548c\u6f14\u5316\u6027\u8d28<\/span><\/span><\/p>\n

4.2.1 \u72b6\u6001\u7684\u5212\u5206\u2014\u2014\u53ef\u8fbe\u3001\u4e92\u901a\u548c\u53ef\u7ea6\u6027<\/span><\/span><\/p>\n

4.2.2 \u72b6\u6001\u5206\u5e03\u968f\u7740\u65f6\u95f4\u53d8\u5316\u4fdd\u6301\u7a33\u5b9a\u2014\u2014\u5e73\u7a33\u5206\u5e03<\/span><\/span><\/p>\n

4.2.3 \u9006\u5411\u8fc7\u7a0b\u548c\u6b63\u5411\u8fc7\u7a0b\u540c\u5206\u5e03\u2014\u2014\u5e73\u7a33\u53ef\u9006\u5206\u5e03<\/span><\/span><\/p>\n

4.2.4 \u5177\u4f53\u4f8b\u5b50<\/span><\/span><\/p>\n

4.3 \u8fde\u7eed\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u7c97\u7c92\u5316<\/span><\/span><\/p>\n

4.3.1 \u5f3a\u6210\u5757\u6027<\/span><\/span><\/p>\n

4.3.2 \u65f6\u95f4\u5c3a\u5ea6\u5206\u89e3<\/span><\/span><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n


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1. \u5386\u53f2\u53d1\u5c55<\/span><\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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<\/span><\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


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1906\u5e74\uff0c\u4fc4\u7f57\u65af\u6570\u5b66\u5bb6Andrey Andreyevich Markov\u5728\u7814\u7a76\u5927\u6570\u5b9a\u5f8b\u65f6\uff0c\u9996\u6b21\u5b9a\u4e49\u4e86\u79bb\u6563\u968f\u673a\u5e8f\u5217\u5728\u7ed9\u5b9a\u73b0\u5728\u7684\u6761\u4ef6\u4e0b\uff0c\u8fc7\u53bb\u4e0e\u672a\u6765\u7684\u6761\u4ef6\u72ec\u7acb\u6027 <\/span><\/span>[3]<\/span><\/span>\u3002\u8fd9\u4e2a\u6027\u8d28\u540e\u6765\u88ab\u79f0\u4e3a\u9a6c\u5c14\u79d1\u592b\u6027\uff0c\u6ee1\u8db3\u8fd9\u4e2a\u6027\u8d28\u7684\u79bb\u6563\u968f\u673a\u8fc7\u7a0b\u88ab\u79f0\u4e3a\u9a6c\u5c14\u79d1\u592b\u94fe\u3002\u968f\u540e\uff0cAndrey Markov\u57fa\u4e8e\u666e\u5e0c\u91d1\u7684\u957f\u8bd7\u300a\u53f6\u752b\u76d6\u5c3c\u30fb\u5965\u6d85\u91d1\u300b\u7814\u7a76\u4e86\u4fc4\u8bed\u5b57\u6bcd\u5e8f\u5217\u4e0a\u7684\u9a6c\u5c14\u79d1\u592b\u94fe\u53ca\u5176\u6781\u9650\u5206\u5e03<\/span><\/span>[4]<\/span><\/span>\u30021931\u5e74\uff0cDavid George Kendall\u7b49\u4eba\u63a8\u5e7f\u4e3a\u8fde\u7eed\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\u5e76\u5efa\u7acb\u4e86\u76f8\u5e94\u7406\u8bba\uff0c\u5176\u4e2d\u5305\u62ec\u8457\u540d\u7684Kolmogorov\u65b9\u7a0b<\/span><\/span>[5]<\/span><\/span>\u30021953\u5e74\uff0cNicholas Metropolis\u901a\u8fc7\u5f15\u5165\u9a6c\u5c14\u79d1\u592b\u94fe\u53d1\u5c55\u51fa\u4e86\u8457\u540d\u7684\u9a6c\u5c14\u79d1\u592b\u94fe\u8499\u7279\u5361\u6d1b\u968f\u673a\u6a21\u62df\u7b97\u6cd5<\/span><\/span>[6]<\/span><\/span>\u3002<\/span><\/span><\/span><\/p>\n

\u8fdb\u4e00\u6b65\uff0c\u9a6c\u5c14\u79d1\u592b\u94fe\u88ab\u53d1\u5c55\u4e3a\u66f4\u590d\u6742\u7684\u6a21\u578b\u30021966\u5e74\uff0cLeonard Esau Baum\u548cTed Petrie\u63d0\u51fa\u4e86\u72b6\u6001\u9690\u85cf\u7684\u53cc\u91cd\u9a6c\u5c14\u79d1\u592b\u8fc7\u7a0b\u2014\u2014\u9690\u9a6c\u5c14<\/span><\/span>\u79d1<\/span><\/span><\/span>\u592b\u6a21\u578b <\/span><\/span>[7]<\/span><\/span>\u3002 1980 \u5e74\uff0cKindermann Richard\u548cJohn Leslie Snell\u5728\u4e13\u8457\u4e2d\u8be6\u7ec6\u4ecb\u7ecd\u4e86\u9488\u5bf9\u591a\u7ef4\u3001\u65e0\u5411\u7684\u9a6c\u5c14<\/span><\/span>\u79d1<\/span><\/span><\/span>\u592b\u968f\u673a\u573a <\/span><\/span>[8]<\/span><\/span>\u3002<\/span><\/span><\/span><\/p>\n


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<\/section>\n<\/section>\n<\/section>\n
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2. \u79bb\u6563\uff08\u6216\u8fde\u7eed\uff09\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u5b9a\u4e49<\/span><\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

\n

<\/span><\/section>\n

<\/span><\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


<\/span><\/span><\/h2>\n

2.1 \u4f8b\u5b50<\/span><\/span><\/span><\/h3>\n

2.1.1 \u6d41\u7f51\u7edc<\/span><\/span><\/span><\/h4>\n

\u5047\u8bbe\u4e00\u4e2a\u6d41\u7f51\u7edc\u6a21\u578b\uff0c\u5982\u56fe\u6240\u793a\uff0c\u5176\u4e2d\u8282\u70b9\u4ee3\u8868\u4e0d\u540c\u7684\u7ba1\u9053\u4ea4\u6c47\u5904\uff0c\u8fde\u8fb9\u4ee3\u8868\u4e0d\u540c\u7684\u6d41\u7ba1\u9053\uff0c\u8fde\u8fb9\u4e0a\u7684\u6743\u91cd\u4ee3\u8868\u6d41\u91cf\uff0c\u65b9\u5411\u4ee3\u8868\u6c34\u6d41\u7684\u65b9\u5411\u3002<\/span><\/span><\/p>\n

<\/section>\n

\u8003\u8651\u4e00\u4e2a\u7c92\u5b50\u5728\u6b64\u6d41\u7f51\u7edc\u4e2d\u6d41\u52a8\uff0c\u5728\u6bcf\u4e00\u4e2a\u8282\u70b9\u5904\uff0c\u5b83\u4f1a\u4ee5\u4e00\u5b9a\u7684\u6982\u7387\u987a\u4e0d\u540c\u7684\u8fde\u8fb9\u6d41\u5411\u4e0b\u4e00\u4e2a\u8282\u70b9\u3002\u5982\u679c\u6211\u4eec\u5c06\u7c92\u5b50\u53ef\u80fd\u5904\u4e8e\u4e0d\u540c\u7684\u8282\u70b9\u770b\u4f5c\u662f\u8be5\u7c92\u5b50\u7684\u4e0d\u540c\u72b6\u6001\uff0c\u5373\u5b9a\u4e49\u72b6\u6001\u7a7a\u95f4<\/span><\/span>S<\/span><\/span>=<\/span><\/span>{<\/span><\/span>1<\/span><\/span>,<\/span><\/span>2<\/span><\/span>,<\/span><\/span>\u22ef<\/span><\/span>,<\/span><\/span>5<\/span><\/span>}<\/span><\/span>\uff0c\u540c\u65f6\u5047\u5b9a\u6bcf\u4e00\u65f6\u523b\u7c92\u5b50\u5fc5\u7136\u5904\u4e8e\u67d0\u4e00\u4e2a\u8282\u70b9\uff0c\u4e14\u4e0d\u4f1a\u5728\u8fde\u8fb9\u5904\u505c\u7559\uff0c\u5219\u6211\u4eec\u53ef\u4ee5\u7528\u4ece\u4efb\u610f\u8282\u70b9<\/span><\/span>i<\/span><\/span>\u51fa\u53d1\uff0c\u5230\u4efb\u610f\u53e6\u5916\u7684\u8282\u70b9<\/span><\/span>j<\/span><\/span>\u7684\u8df3\u8f6c\u6982\u7387<\/span><\/span>\"image.png\"<\/span><\/sub>\u63cf\u8ff0\u8be5\u7c92\u5b50\u7684\u884c\u4e3a\uff0c\u5176\u4e2d<\/span><\/span>f<\/span><\/span>i<\/span><\/sub><\/span>j<\/span><\/sub><\/span>\u4e3a\u8fde\u8fb9<\/span><\/span>i<\/span><\/span>\u5230<\/span><\/span>j<\/span><\/span>\u4e0a\u9762\u7684\u6d41\u91cf\u3002\u663e\u7136\uff0c\u8fd9\u4e2a\u8df3\u8f6c\u6982\u7387\u4e0e\u7c92\u5b50\u5728<\/span><\/span><\/span><<\/span><\/span>t<\/span><\/span>\u7684\u65f6\u95f4\u6b65\u4f4d\u4e8e\u54ea\u4e00\u4e2a\u8282\u70b9\u65e0\u5173\uff0c\u56e0\u6b64\u8fd9\u4e00\u8fc7\u7a0b\u6ee1\u8db3\u9a6c\u5c14\u79d1\u592b\u6027\uff0c\u8be5\u8fc7\u7a0b\u5b9a\u4e49\u4e86\u4e00\u4e2a\u9a6c\u5c14<\/span><\/span>\u79d1<\/span><\/span><\/span>\u592b\u94fe\u3002<\/span><\/span><\/p>\n

2.1.2 \u9ad8\u5c14\u987f\u9489\u677f<\/span><\/span><\/span><\/h4>\n

\u9ad8\u5c14\u987f\u9489\u677f\u662f\u82f1\u56fd\u79d1\u5b66\u5bb6\u9ad8\u5c14\u987f\uff08Francis Galton\uff09\u53d1\u660e\u7684\u4e00\u79cd\u7269\u7406\u88c5\u7f6e\u3002\u5982\u4e0b\u56fe\uff08\u5de6\uff09\u6240\u793a\uff0c\u8fd9\u4e2a\u6709\u8da3\u7684\u88c5\u7f6e\u7531\u4e00\u5927\u5806\u9489\u5b50\u6309\u7167\u4e09\u89d2\u5f62\u5f62\u72b6\u89c4\u5219\u6392\u5217\u800c\u5f62\u6210\u7684\u4e00\u4e2a\u9489\u677f\u3002\u9489\u677f\u7684\u4e0a\u4fa7\u6709\u4e00\u4e2a\u5f00\u53e3\uff0c\u4f60\u53ef\u4ee5\u4ece\u8fd9\u4e2a\u5f00\u53e3\u6295\u653e\u5c0f\u94a2\u7403\u8fdb\u6765\uff0c\u4e0b\u9762\u5219\u662f\u7531\u82e5\u5e72\u4e2a\u67f1\u72b6\u7684\u51f9\u69fd\u6784\u6210\uff0c\u6bcf\u4e2a\u51f9\u69fd\u53ef\u4ee5\u628a\u94a2\u7403\u63a5\u4f4f\uff0c\u5e76\u76db\u653e\u8d77\u6765\u3002\u5b9e\u9a8c\u7684\u65f6\u5019\uff0c\u4f60\u53ef\u4ee5\u4ece\u4e0a\u9762\u7684\u5f00\u53e3\u6295\u653e\u5c0f\u7403\uff0c\u5c0f\u7403\u5c31\u4f1a\u4e0b\u843d\uff0c\u4ece\u800c\u78b0\u649e\u5230\u4e00\u4e9b\u9489\u5b50\uff0c\u7136\u540e\u5760\u5165\u5e95\u90e8\u7684\u51f9\u69fd\u4e4b\u5185\u3002\u7531\u4e8e\u5c0f\u7403\u6bcf\u78b0\u649e\u4e00\u6b21\u9489\u5b50\uff0c\u5c31\u53ef\u80fd\u4ece\u8be5\u9489\u5b50\u7684\u5de6\u4fa7\u4e0b\u843d\uff0c\u6216\u4ece\u53f3\u4fa7\u4e0b\u843d\uff0c\u8fdb\u800c\u518d\u78b0\u649e\u53e6\u4e00\u4e2a\u9489\u5b50\uff0c\u5982\u6b64\u5faa\u73af\u3002\u4e8e\u662f\uff0c\u6700\u7ec8\u843d\u5230\u69fd\u4e2d\u7684\u5c0f\u7403\u5c31\u53ef\u80fd\u5806\u79ef\u51fa\u4e00\u4e2a\u5c0f\u7403\u7684\u5206\u5e03\uff0c\u8fd9\u4e2a\u5206\u5e03\u4f1a\u5f62\u6210\u4e00\u79cd\u4e2d\u95f4\u51f8\u8d77\uff0c\u4e24\u8fb9\u51f9\u9677\u7684\u949f\u5f62\u66f2\u7ebf\u3002\u8fd9\u4e2a\u88c5\u7f6e\u662f\u7531\u82f1\u56fd\u79d1\u5b66\u5bb6\u9ad8\u5c14\u987f\u53d1\u660e\u7684\uff0c\u7528\u4ee5\u8bf4\u660e\u6982\u7387\u8bba\u4e2d\u7684\u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\uff0c\u56e0\u800c\u4eba\u4eec\u5c31\u53eb\u5b83\u201c\u9ad8\u5c14\u987f\u9489\u677f\u201d\u3002<\/span><\/span><\/p>\n

\"Galton<\/section>\n

\u5c0f\u7403\u7684\u5760\u843d\u8fc7\u7a0b\u5b9e\u9645\u4e0a\u53ef\u4ee5\u88ab\u770b\u4f5c\u662f\u4e00\u4e2a\u9a6c\u5c14\u79d1\u592b\u94fe\u3002\u4e3a\u4e86\u65b9\u4fbf\u8d77\u89c1\uff0c\u6211\u4eec\u4ec5\u4ec5\u8003\u8651\u75316\u9897\u9489\u5b50\uff0c4\u4e2a\u5e95\u90e8\u51f9\u69fd\u6784\u6210\u7684\u9ad8\u5c14\u987f\u9489\u677f\u6765\u4f5c\u4e3a\u793a\u4f8b\uff08\u5982\u4e0a\u56fe\u53f3\u6240\u793a\uff09\u3002\u5176\u4e2d\u6bcf\u4e00\u4e2a\u8282\u70b9\u5bf9\u5e94\u4e86\u4e00\u9897\u9489\u5b50\uff0c\u4e5f\u5bf9\u5e94\u4e86\u5c0f\u7403\u5728\u67d0\u4e00\u65f6\u523b\u7684\u53d6\u503c\u72b6\u6001\u3002\u4e00\u4e2a\u5c0f\u7403\u5728\u4e0d\u540c\u7684\u9489\u5b50\u4e4b\u95f4\u8df3\u8dc3\u5c31\u53ef\u4ee5\u7531\u53f3\u56fe\u6240\u793a\u7684\u72b6\u6001\u8f6c\u79fb\u56fe\u8868\u793a\uff0c\u5176\u4e2d\u8fde\u8fb9\u4e0a\u7684\u6570\u503c\u4ee3\u8868\u4ece\u4e0a\u4e00\u4e2a\u8282\u70b9\u8df3\u8f6c\u5230\u4e0b\u4e00\u4e2a\u8282\u70b9\u7684\u6982\u7387\u30027~10\u53f7\u8282\u70b9\u5bf9\u5e94\u5e95\u90e8\u76844\u4e2a\u51f9\u69fd\u3002\u4e00\u65e6\u8fdb\u5165\u8fd94\u4e2a\u8282\u70b9\u4e2d\u7684\u4efb\u610f\u4e00\u4e2a\uff0c\u5c0f\u7403\u5c31\u4e0d\u52a8\u4e86\uff0c\u56e0\u6b64\u5c0f\u7403\u5c31\u505c\u7559\u5728\u5f53\u524d\u8fd9\u4e2a\u8282\u70b9\u3002\u5728\u4e0b\u843d\u8fc7\u7a0b\u4e2d\uff0c\u5c0f\u7403\u5728\u6bcf\u4e00\u5c42\u7684\u4f4d\u7f6e\u53ea\u4f9d\u8d56\u4e8e\u5b83\u5728\u4e0a\u4e00\u5c42\u7684\u4e0b\u843d\u4f4d\u7f6e\uff0c\u56e0\u6b64\uff0c\u8fd9\u4e5f\u662f\u4e00\u4e2a\u9a6c\u5c14<\/span><\/span>\u79d1<\/span><\/span><\/span>\u592b\u8fc7\u7a0b\u3002<\/span><\/span><\/p>\n

2.1.3 Yahtzee\u7684\u9ab0\u5b50\u6e38\u620f<\/span><\/span><\/span><\/h4>\n

\u63a5\u4e0b\u6765\u7684\u8fd9\u4e2a\u4f8b\u5b50\u6765\u6e90\u4e8e\u4e00\u4e2a\u7ecf\u5178\u7684\u9ab0\u5b50\u6e38\u620f\u2014\u2014Yahtzee\u6e38\u620f\u3002\u5728\u6e38\u620f\u4e2d\u5171\u67095\u4e2a\u516d\u9762\u5747\u5300\u7684\u9ab0\u5b50\uff0c\u6e38\u620f\u53c2\u4e0e\u8005\u81f3\u591a\u6709\u4e09\u6b21\u6295\u63b7\u9ab0\u5b50\u7684\u673a\u4f1a\uff0c\u4e00\u6b21\u53ef\u4ee5\u540c\u65f6\u6295\u63b75\u9897\u9ab0\u5b50\u3002\u4e3a\u4e86\u7b80\u5316\uff0c\u6211\u4eec\u5728\u6b64\u5904\u5047\u8bbe\u6e38\u620f\u7684\u76ee\u6807\u662f\u4f7f\u5f97\u6240\u6709\u9ab0\u5b50\u7684\u7ed3\u679c\u76f8\u540c\uff08\u79f0\u4e3a\u83b7\u5f97Yahtzee\uff09\u3002\u540c\u65f6\uff0c\u6e38\u620f\u4e2d\u6709\u4e00\u4e2a\u89c4\u5219\u662f\u5982\u679c\u6e38\u620f\u8005\u5728\u524d\u4e24\u6b21\u7684\u6295\u63b7\u540e\uff0c\u51fa\u73b0\u4e86\u91cd\u590d\u7684\u6570\u5b57\uff0c\u5219\u53ef\u4ee5\u4ec5\u9488\u5bf9\u5269\u4e0b\u7684\u4e0e\u91cd\u590d\u6570\u5b57\u4e0d\u540c\u7684\u9ab0\u5b50\u518d\u8fdb\u884c\u4e00\u6b21\u6295\u63b7\u3002<\/span><\/span><\/p>\n

\u4f8b\u5982\uff0c\u7b2c\u4e00\u6b21\u6295\u63b7\u7ed3\u679c\u4e3a11234\uff0c\u7531\u4e8e\u6709\u4e24\u4e2a1\uff0c\u6240\u4ee5\u4f60\u53ef\u4ee5\u4fdd\u6301\u5bf9\u5e94\u7684\u4e24\u4e2a\u9ab0\u5b50\u4e0d\u52a8\uff0c\u63a5\u4e0b\u6765\uff0c\u4f60\u53ea\u9700\u8981\u91cd\u65b0\u6295\u63b7234\u5bf9\u5e94\u7684\u4e09\u4e2a\u9ab0\u5b50\u5c31\u53ef\u4ee5\u3002<\/span><\/span>[9]<\/span><\/span><\/span><\/p>\n

\u5728\u8fd9\u4e2a\u968f\u673a\u8fc7\u7a0b\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u53d8\u91cf<\/span><\/span>X<\/span><\/span>t<\/span><\/span><\/sub>\u6765\u5efa\u6a21\u5982\u4e0b\u53d8\u91cf\uff1a\u7b2c<\/span><\/span>t<\/span><\/span>\u6b21\u6295\u63b7\u4e2d\uff0c\u7ed3\u679c\u76f8\u540c\u7684\u9ab0\u5b50\u6570\u91cf\uff0c\u8fd9\u6837\uff0c\u8be5\u6e38\u620f\u7684\u72b6\u6001\u7a7a\u95f4\u4e3a<\/span><\/span>S<\/span><\/span>=<\/span><\/span>{<\/span><\/span>0<\/span><\/span>,<\/span><\/span>1<\/span><\/span>,<\/span><\/span>\u22ef<\/span><\/span>,<\/span><\/span>5<\/span><\/span>}<\/span><\/span>\u3002\u5728\u7b2c<\/span><\/span>t<\/span><\/span>+<\/span><\/span>1<\/span><\/span>\u6b65\u4e2d\uff0c\u7cfb\u7edf\u7684\u72b6\u6001<\/span><\/span><\/span>X<\/span><\/span>t<\/span><\/sub><\/span>+<\/span><\/span><\/sub>1<\/span><\/sub><\/span>\u663e\u7136\u53ea\u4f9d\u8d56\u4e8e<\/span><\/span>t<\/span><\/span>\u6b65\u7684\u72b6\u6001\u96c6\u5408<\/span><\/span><\/span>X<\/span><\/span>t<\/span><\/span><\/sub>\uff0c\u800c\u4e0e\u66f4\u65e9\u7684\u72b6\u6001\u65e0\u5173\uff0c\u56e0\u6b64\u8fd9\u4e5f\u662f\u4e00\u4e2a\u9a6c\u5c14<\/span><\/span>\u79d1<\/span><\/span><\/span>\u592b\u8fc7\u7a0b\u3002<\/span><\/span><\/p>\n

2.2 \u5b9a\u4e49<\/span><\/span><\/span><\/h3>\n

\u4e0a\u9762\u7684\u51e0\u4e2a\u4f8b\u5b50\u90fd\u662f\u65f6\u95f4\u548c\u72b6\u6001\u7a7a\u95f4\u5747\u4e3a\u79bb\u6563\u7684\u6570\u5b57\uff0c\u4e14\u7cfb\u7edf\u90fd\u6ee1\u8db3\u9a6c\u5c14\u79d1\u592b\u6027\u8d28\uff0c\u4e5f\u5c31\u662f<\/span><\/span>t<\/span><\/span>+<\/span><\/span>1<\/span><\/span>\u65f6\u95f4\u6b65\u7684\u72b6\u6001\u53ea\u4f9d\u8d56\u4e8e\u7b2c<\/span><\/span>t<\/span><\/span>\u6b65\u7684\u7cfb\u7edf\u72b6\u6001\uff0c\u5219\u79f0\u8fd9\u7c7b\u968f\u673a\u8fc7\u7a0b\u4e3a\u79bb\u6563\u65f6\u95f4\u7684\u9a6c\u5c14\u79d1\u592b\u94fe\uff08\u7b80\u79f0\u9a6c\u5c14\u79d1\u592b\u94fe\uff0cMarkov chain\uff09\uff0c\u4e25\u683c\u7684\u6570\u5b66\u5b9a\u4e49\u5982\u4e0b\uff1a<\/span><\/span><\/span><\/p>\n

\u5b9a\u4e49\uff08\u9a6c\u5c14\u79d1\u592b\u94fe\uff09<\/span><\/span><\/b>\uff1a\u79f0\u72b6\u6001\u7a7a\u95f4<\/span><\/span>S<\/span><\/span>=<\/span><\/span>{<\/span><\/span>1<\/span><\/span>,<\/span><\/span>2<\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>}<\/span><\/span>\u4e0a\u7684\u968f\u673a\u8fc7\u7a0b<\/span><\/span>{<\/span><\/span>X<\/span><\/span>n<\/span><\/span><\/sub>,<\/span><\/span> <\/span><\/span>n<\/span><\/span>=<\/span><\/span>0<\/span><\/span>,<\/span><\/span>1<\/span><\/span>,<\/span><\/span>2<\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>}<\/span><\/span>\u662f<\/span><\/span>\u9a6c\u5c14\u79d1\u592b\u94fe<\/span><\/span><\/b>\u6216\u5177\u6709\u9a6c\u5c14\u79d1\u592b\u6027\u8d28\uff0c\u82e5\u6ee1\u8db3\uff1a<\/span><\/span>\"image.png\"<\/span><\/sub>\uff0c<\/span><\/span>\u5176\u4e2d\uff0c<\/span><\/span>X<\/span><\/span>n<\/span><\/span><\/sub>=<\/span><\/span>j<\/span><\/span>\u2208<\/span><\/span>S<\/span><\/span>\u8868\u793a\u8fc7\u7a0b\u5728\u65f6\u523b<\/span><\/span>n<\/span><\/span>\u5904\u4e8e\u72b6\u6001<\/span><\/span>j<\/span><\/span>\uff0c\u79f0<\/span><\/span>\"image.png\"<\/span><\/sub>\u4e3a\u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u4e00\u6b65\u8f6c\u79fb\u6982\u7387\uff0c\u5e76\u5f15\u5165\u8bb0\u53f7<\/span><\/span>\"image.png\"<\/span><\/sub>\u3002<\/span><\/span><\/span>
<\/span><\/p>\n

\u4e0a\u8ff0\u5b9a\u4e49\u4e2d\uff0c\u6211\u4eec\u5bf9\u65f6\u95f4\u6b65\u6709\u975e\u5e38\u660e\u786e\u7684\u8981\u6c42\uff0c\u5373\u53ea\u80fd\u53d6\u4e00\u7cfb\u5217\u8fde\u7eed\u7684\u6574\u6570\u3002\u7136\u800c\uff0c\u5728\u73b0\u5b9e\u4e16\u754c\u4e2d\uff0c\u65f6\u95f4\u6b65\u5b9e\u9645\u4e0a\u662f\u53d6\u503c\u4e3a\u8fde\u7eed\u7684\u5b9e\u6570\u7684\uff0c\u5b83\u5f88\u96be\u8f6c\u5316\u6210\u4e00\u7cfb\u5217\u8fde\u7eed\u7684\u6574\u6570\u3002\u5728\u8fd9\u6837\u7684\u8fde\u7eed\u7684\u65f6\u95f4\u57df\u4e0a\uff0c\u6211\u4eec\u540c\u6837\u53ef\u4ee5\u5b9a\u4e49\u8fde\u7eed\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u968f\u673a\u8fc7\u7a0b\u3002<\/span><\/span><\/p>\n

\u5177\u4f53\u5730\uff1a<\/span><\/span><\/p>\n

\u5b9a\u4e49\uff08\u8fde\u7eed\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\uff09<\/span><\/span><\/b>\uff1a\u79f0\u72b6\u6001\u7a7a\u95f4<\/span><\/span>S<\/span><\/span>=<\/span><\/span>{<\/span><\/span>1<\/span><\/span>,<\/span><\/span>2<\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>}<\/span><\/span>\u4e0a\u7684\u968f\u673a\u8fc7\u7a0b<\/span><\/span>{<\/span><\/span>X<\/span><\/span>t<\/span><\/span><\/sub>,<\/span><\/span> <\/span><\/span>t<\/span><\/span>\u2265<\/span><\/span>0<\/span><\/span>}<\/span><\/span>\u662f<\/span><\/span>\u8fde\u7eed\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe<\/span><\/span><\/b>\uff0c\u82e5\u6ee1\u8db3<\/span><\/span><\/span><\/p>\n

<\/section>\n

\u65e5\u5e38\u751f\u6d3b\u4e2d\u7684\u8bb8\u591a\u73b0\u8c61\u90fd\u53ef\u4ee5\u5229\u7528\u9a6c\u5c14\u79d1\u592b\u94fe\u6765\u63cf\u8ff0\uff0c\u4f8b\u5982\u5929\u6c14\u7684\u53d8\u5316\u3002\u7b80\u5355\u8003\u8651\u4e24\u79cd\u5929\u6c14\u72b6\u6001\uff0c\u4e0b\u96e8\u548c\u4e0d\u4e0b\u96e8\u3002\u90a3\u4e48\uff0c\u5bf9\u5e94\u7684\u9a6c\u5c14\u79d1\u592b\u5047\u8bbe\u5c31\u662f\u660e\u5929\u662f\u5426\u4e0b\u96e8\u53ea\u53d6\u51b3\u4e8e\u4eca\u5929\u7684\u5929\u6c14\uff0c\u800c\u4e0e\u4e4b\u524d\u7684\u5929\u6c14\u65e0\u5173\u3002<\/span><\/span><\/p>\n


<\/span><\/span><\/h2>\n

\n
\n
\n
\n
\n

<\/span><\/section>\n

<\/span><\/section>\n
\n
<\/section>\n<\/section>\n<\/section>\n
\n

3. \u79bb\u6563\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe<\/span><\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

\n

<\/span><\/section>\n

<\/span><\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


<\/span><\/p>\n

3.1 \u57fa\u672c\u6982\u5ff5\u548c\u6027\u8d28<\/span><\/span><\/span><\/h3>\n

\u4e0a\u8ff0\u5b9a\u4e49\u4e2d\u5f3a\u8c03\u4e86\u4e00\u6b65\u8f6c\u79fb\u6982\u7387\u5bf9\u4e8e\u5386\u53f2\u7684\u9057\u5fd8\u6027\uff0c\u4e5f\u5c31\u662f\u5f53\u524d\u72b6\u6001\u53ea\u4e0e\u4e0a\u4e00\u65f6\u523b\u7684\u72b6\u6001\u6709\u5173\uff0c\u800c\u4e0e\u66f4\u65e9\u4ee5\u524d\u7684\u72b6\u6001\u65e0\u5173\u3002\u90a3\u4e48\uff0c\u6211\u4eec\u5982\u4f55\u5229\u7528\u8fd9\u4e00\u6027\u8d28\u8ba1\u7b97\u4efb\u610f\u6b65\u8f6c\u79fb\u6982\u7387\u5462\uff1f\u5982\u679c\u5bf9\u4e8e\u591a\u6b65\u8f6c\u79fb\u6982\u7387\uff0c\u5386\u53f2\u9057\u5fd8\u6027\u4e5f\u88ab\u6ee1\u8db3\uff0c\u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u5c06\u8def\u5f84\u8fdb\u884c\u5206\u89e3\u6765\u8ba1\u7b97\u4efb\u610f\u6b65\u7684\u8f6c\u79fb\u6982\u7387\u3002<\/span><\/span><\/p>\n

3.1.1 \u591a\u6b65\u8f6c\u79fb\u6982\u7387\u7684\u9057\u5fd8\u6027<\/span><\/span><\/span><\/h4>\n

\u8003\u8651\u79bb\u6563\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\uff0c\u5982\u679c\u5bf9\u4efb\u610f\u7684<\/span><\/span>n<\/span><\/span>\u2265<\/span><\/span>0<\/span><\/span>\u548c<\/span><\/span>m<\/span><\/span>\u2265<\/span><\/span>1<\/span><\/span>\uff0c\u6709<\/span><\/span><\/span><\/h4>\n


<\/span><\/span><\/h4>\n

<\/span><\/span><\/h4>\n


<\/span><\/span><\/h4>\n

\u5373\u5f53<\/span><\/span>X<\/span><\/span>n<\/span><\/span><\/sub>\u7ed9\u5b9a\u540e\uff0c<\/span><\/span>X<\/span><\/span>n<\/span><\/sub><\/span>+<\/span><\/span><\/sub>m<\/span><\/sub><\/span>\u548c<\/span><\/span>X<\/span><\/span>n<\/span><\/sub><\/span>\u2212<\/span><\/span><\/sub>1<\/span><\/sub><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>X<\/span><\/span>0<\/span><\/span><\/sub>\u6761\u4ef6\u72ec\u7acb\u3002\u5219\u6211\u4eec\u79f0\u8fd9\u4e00\u9a6c\u5c14\u79d1\u592b\u94fe\u5177\u6709\u591a\u6b65\u8f6c\u79fb\u7684\u9057\u5fd8\u6027\u3002<\/span><\/span><\/span><\/h4>\n


<\/span><\/span><\/h4>\n

\u6ce8\uff1a\u4e8b\u5b9e\u4e0a\uff0c\u4f60\u53ef\u4ee5\u5bf9<\/span><\/span>m<\/span><\/span>\u8fdb\u884c\u6570\u5b66\u5f52\u7eb3\u6cd5\u8bc1\u660e\uff0c\u4ece\u800c\u5229\u7528\u9a6c\u5c14\u79d1\u592b\u968f\u673a\u8fc7\u7a0b\u7684\u5b9a\u4e49\uff08\u79bb\u6563\u6216\u8fde\u7eed\u65f6\u95f4\u90fd\u53ef\u4ee5\uff09\uff0c\u4ece\u800c\u8bc1\u660e\u4e0a\u8ff0\u6027\u8d28\u3002<\/span><\/span><\/span><\/p>\n

3.1.2 \u591a\u6b65\u8f6c\u79fb\u6982\u7387\u7684\u8ba1\u7b97\u2014\u2014Chapman-Kolmogorov\u65b9\u7a0b<\/span><\/span><\/span><\/h4>\n


<\/span><\/p>\n

\u7531\u4e8e\u4e0a\u8ff0\u6027\u8d28\uff0c\u5373\u7ed9\u5b9a\u73b0\u5728\uff0c\u7cfb\u7edf\u7684\u8fc7\u53bb\u4e0e\u672a\u6765\u90fd\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u56e0\u6b64\u9057\u5fd8\u6027\u5bf9\u4e8e\u591a\u6b65\u8f6c\u79fb\u6982\u7387\u4e5f\u662f\u6210\u7acb\u7684\u3002\u90a3\u4e48\uff0c\u5bf9\u4e8e\u4efb\u610f\u6b65\u8f6c\u79fb\u6982\u7387\u7684\u8ba1\u7b97\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u6839\u636e\u8fd9\u4e2a\u6027\u8d28\u8fdb\u884c\u5206\u89e3\u8ba1\u7b97\u3002<\/span><\/span><\/p>\n

\u5177\u4f53\u6765\u8bf4\uff0c\u5bf9\u4e8e<\/span><\/span>n<\/span><\/span>\u2265<\/span><\/span>2<\/span><\/span>\u6b65\u8f6c\u79fb\u6982\u7387<\/span><\/span>\"image.png\"<\/span><\/sub>\uff0c\u6211\u4eec\u53ef\u4ee5\u5148\u7ed9\u5b9a\u67d0\u4e00\u4e2d\u95f4\u65f6\u523b<\/span><\/span>0<\/span><\/span><<\/span><\/span>m<\/span><\/span><<\/span><\/span>n<\/span><\/span>\u7684\u72b6\u6001<\/span><\/span>k<\/span><\/span>\uff0c\u6b64\u65f6\u7531\u6761\u4ef6\u72ec\u7acb\u6027\uff0c\u4ece<\/span><\/span>i<\/span><\/span>\u5148\u7ecf<\/span><\/span>m<\/span><\/span>\u6b65\u81f3<\/span><\/span>k<\/span><\/span>\uff0c\u518d\u7ecf<\/span><\/span>n<\/span><\/span>\u2212<\/span><\/span>m<\/span><\/span>\u6b65\u81f3<\/span><\/span>j<\/span><\/span>\u7684\u6982\u7387\u5c31\u53ef\u4ee5\u88ab\u5206\u89e3\u4e3a\u4e24\u4e2a\u8f6c\u79fb\u6982\u7387\u7684\u4e58\u79ef\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u53ea\u9700\u8981\u904d\u5386\u72b6\u6001<\/span><\/span>k<\/span><\/span>\u7684\u6240\u6709\u53ef\u80fd\uff0c\u5373\u53ef\u4ee5\u5f97\u5230\u5982\u4e0b\u8868\u8fbe\u5f0f\uff1a<\/span><\/span><\/span><\/p>\n

<\/section>\n

\u8fd9\u4e2a\u5f0f\u5b50\u88ab\u79f0\u4e3a<\/span><\/span>Chapman-Kolmogorov\u65b9\u7a0b<\/span><\/span><\/b>\u3002\u4e0a\u8ff0\u8ba1\u7b97\u8fc7\u7a0b\u53ef\u4ee5\u7ed8\u5236\u4e3a\u793a\u610f\u56fe\uff1a<\/span><\/span><\/span><\/p>\n

\"CK.png\"<\/section>\n

3.1.3 \u65f6\u9f50\u6027\u4e0e\u975e\u65f6\u9f50\u6027<\/span><\/span><\/span><\/h4>\n


<\/span><\/span><\/h4>\n

\u7531\u5f0f<\/span><\/span>1<\/span><\/span><\/b>\u53ef\u77e5\uff0c\u591a\u6b65\u8f6c\u79fb\u6982\u7387\u5176\u5b9e\u662f\u4e00\u6b65\u8f6c\u79fb\u6982\u7387\u7684\u4e58\u79ef\u4e0e\u6c42\u548c\u3002\u90a3\u4e48\uff0c\u6211\u4eec\u81ea\u7136\u5730\u4f1a\u60f3\u5230\uff0c\u5728\u8ba1\u7b97\u4e58\u79ef\u7684\u8fc7\u7a0b\u4e2d\uff0c\u4e00\u6b65\u8f6c\u79fb\u6982\u7387\u662f\u5426\u4f1a\u968f\u65f6\u95f4\u53d8\u5316\uff1f<\/span><\/span><\/span><\/p>\n

3.1.3.1 \u65f6\u9f50\u6027\u4f8b\u5b50\uff1a\u9ad8\u5c14\u987f\u9489\u677f<\/span><\/span><\/span><\/h5>\n

<\/span><\/span><\/h5>\n

\u56de\u987e\u524d\u9762\u7684\u4e09\u5c42\u9ad8\u5c14\u987f\u9489\u677f\u7684\u5b9e\u9a8c\u4f8b\u5b50\uff0c\u5982\u679c\u6211\u4eec\u5c06\u9489\u5b50\u6309\u7167\u5b83\u6240\u5c5e\u5c42\u7684\u987a\u5e8f\u4f9d\u6b21\u7f16\u53f7\u4e3a1-6\uff0c\u51f9\u69fd\u4ece\u5de6\u81f3\u53f3\u7f16\u53f7\u4e3a7-10\uff0c\u90a3\u4e48\u5c0f\u7403\u4e0b\u843d\u8fc7\u7a0b\u6784\u6210\u4e86\u4e00\u4e2a\u72b6\u6001\u7a7a\u95f4<\/span><\/span>S<\/span><\/span>=<\/span><\/span>{<\/span><\/span>1<\/span><\/span>,<\/span><\/span>2<\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>10<\/span><\/span>}<\/span><\/span>\u7684\u9a6c\u5c14\u79d1\u592b\u94fe\uff0c\u5176\u8f6c\u79fb\u884c\u4e3a\u5982\u4e0b\u56fe\u6240\u63cf\u8ff0\uff1a<\/span><\/span><\/span><\/p>\n

\"Galton.png\"<\/section>\n

\u5176\u4e2d\uff0c\u4efb\u610f\u4e00\u6761\u8fde\u8fb9\u4e0a\u7684\u6570\u503c\u4ee3\u8868\u4ece\u8fde\u8fb9\u7684\u8d77\u70b9\u72b6\u6001\u4f5c\u4e3a\u6761\u4ef6\uff0c\u5230\u8fde\u8fb9\u7ec8\u70b9\u72b6\u6001\u7684\u6761\u4ef6\u8f6c\u79fb\u6982\u7387\u3002<\/span><\/span><\/p>\n

\u5f88\u663e\u7136\uff0c\u8fd9\u662f\u4e00\u4e2a\u4e0e\u65f6\u95f4\u65e0\u5173\u7684\u6982\u7387\u8f6c\u79fb\u77e9\u9635\u7684\u4f8b\u5b50\uff0c\u8fd9\u79cd\u60c5\u51b5\u4e5f\u53eb\u65f6\u9f50\u7684\u9a6c\u5c14\u79d1\u592b\u94fe\u3002<\/span><\/span><\/p>\n

3.1.3.2 \u975e\u65f6\u9f50\u6027\u4f8b\u5b50\uff1a\u6ce2\u5229\u4e9a\u575b\u5b50\u6a21\u578b<\/span><\/span><\/span><\/h5>\n

<\/span><\/span><\/h5>\n

\u7136\u800c\uff0c\u5728\u53e6\u4e00\u4e2a\u7ecf\u5178\u6982\u7387\u6a21\u578b\u2014\u2014\u6ce2\u5229\u4e9a\u575b\u5b50\u6a21\u578b\uff08Polya’s urn scheme)\u4e2d\uff0c\u4e00\u6b65\u8f6c\u79fb\u6982\u7387\u5c31\u4e0e\u65f6\u95f4\u6709\u5173\u3002<\/span><\/span><\/p>\n

\u6a21\u578b\u5047\u8bbe\u575b\u5b50\u4e2d\u88c5\u6709<\/span><\/span>b<\/span><\/span>><\/span><\/span>0<\/span><\/span>\u4e2a\u9ed1\u7403\u548c<\/span><\/span>w<\/span><\/span>><\/span><\/span>0<\/span><\/span>\u4e2a\u767d\u7403\u3002\u6bcf\u6b21\u968f\u673a\u5730\u4ece\u575b\u5b50\u4e2d\u6709\u653e\u56de\u5730\u62bd\u51fa\u4e00\u4e2a\u7403\uff0c\u5e76\u4e14\u518d\u653e\u5165\u4e00\u4e2a\u540c\u989c\u8272\u7684\u7403\u3002\u8bb0<\/span><\/span>X<\/span><\/span>n<\/span><\/span><\/sub>\u4e3a\u7b2c<\/span><\/span>n<\/span><\/span>\u6b21\u62bd\u53d6\u540e\u575b\u5b50\u4e2d\u9ed1\u7403\u7684\u4e2a\u6570\uff0c\u5219\u4e00\u6b65\u8f6c\u79fb\u6982\u7387<\/span><\/span><\/span><\/p>\n

<\/section>\n

\u663e\u7136\uff0c\u4e0a\u8ff0\u6982\u7387\u662f\u968f\u7740<\/span><\/span>n<\/span><\/span>\u7684\u53d8\u5316\u800c\u53d8\u5316\u7684\u3002<\/span><\/span><\/span><\/p>\n

\u5b9a\u4e49<\/span><\/span><\/span><\/h5>\n

\u4e00\u822c\u7684\uff0c\u5f53\u4e00\u6b65\u8f6c\u79fb\u6982\u7387<\/span><\/span>\"image.png\"<\/span><\/sub>\u53ea\u4e0e\u72b6\u6001<\/span><\/span>i<\/span><\/span>,<\/span><\/span>j<\/span><\/span>\u6709\u5173\uff0c\u800c\u4e0e\u65f6\u523b<\/span><\/span>n<\/span><\/span>\u65e0\u5173\uff08\u5e73\u7a33\u6027\u5047\u8bbe\uff09\uff0c\u79f0\u6b64\u9a6c\u5c14\u79d1\u592b\u94fe\u4e3a<\/span><\/span>\u65f6\u9f50\u9a6c\u5c14\u79d1\u592b\u94fe\uff08homogeneous\uff09<\/span><\/span><\/b>\uff1b\u82e5<\/span><\/span>\"image.png\"<\/span><\/sub>\u4e0e\u72b6\u6001<\/span><\/span>i<\/span><\/span>,<\/span><\/span>j<\/span><\/span>\u548c\u65f6\u523b<\/span><\/span>n<\/span><\/span>\u5747\u6709\u5173\uff0c\u5219\u79f0\u6b64\u9a6c\u5c14\u79d1\u592b\u94fe\u4e3a<\/span><\/span>\u975e\u65f6\u9f50\u7684<\/span><\/span><\/b>\u3002<\/span><\/span><\/span>
<\/span><\/p>\n

\u4e0b\u9762\u672c\u8bcd\u6761\u7684\u8ba8\u8bba\u4e3b\u8981\u56f4\u7ed5\u7740\u79bb\u6563\u65f6\u95f4\u65f6\u9f50\u9a6c\u5c14\u79d1\u592b\u94fe\u800c\u5c55\u5f00\uff0c\u8fd9\u662f\u6700\u5e38\u89c1\u7684\u4e00\u79cd\u573a\u666f\u3002<\/span><\/span><\/p>\n

3.2 \u65f6\u9f50\u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u8f6c\u79fb\u8fc7\u7a0b\u2014\u2014\u521d\u59cb\u72b6\u6001\u786e\u5b9a<\/span><\/span><\/span><\/h3>\n


<\/span><\/span><\/h4>\n

3.2.1 \u591a\u6b65\u8f6c\u79fb\u6982\u7387\u7684\u8ba1\u7b97\u4e0e\u4e00\u6b65\u8f6c\u79fb\u6982\u7387\u77e9\u9635<\/span><\/span><\/span><\/h4>\n

\u7531\u65f6\u9f50\u6027\u5047\u8bbe\uff0c\u8bb0\u4e00\u6b65\u8f6c\u79fb\u6982\u7387<\/span><\/span>\"image.png\"<\/span><\/sub>\u3002<\/span><\/span><\/span><\/p>\n

\u6b64\u65f6\uff0c\u8f6c\u79fb\u8fc7\u7a0b\u5982\u4e0b<\/span><\/span><\/p>\n

\"CK2.png\"<\/section>\n

\u5bf9\u5e94\u7684Chapman-kolmogorov\u65b9\u7a0b\u7684\u5177\u4f53\u8868\u8fbe\u5f0f\u5982\u4e0b\uff0c<\/span><\/span>
<\/span><\/span><\/p>\n

<\/section>\n

\u4e0d\u96be\u770b\u51fa\uff0c\u4e0a\u8ff0\u65b9\u7a0b\u4e0e\u8d77\u59cb\u65f6\u523b\u65e0\u5173\uff0c\u4ec5\u4e0e\u65f6\u95f4\u95f4\u9694\u6709\u5173\uff0c\u5373\u5982\u679c\u5c06\u65f6\u95f4\u7a97\u4efb\u610f\u5e73\u79fb<\/span><\/span>l<\/span><\/span>\u4e2a\u65f6\u523b\uff0c\u6709<\/span><\/span><\/span><\/p>\n

<\/section>\n

\u6240\u4ee5\uff0c\u4e00\u822c\u7684\uff0c\u8bb0<\/span><\/span>n<\/span><\/span>\u2212<\/span><\/span>m<\/span><\/span>\u6b65\u8f6c\u79fb\u6982\u7387 <\/span><\/span>P<\/span><\/span>i<\/span><\/sub><\/span>j<\/span><\/sub><\/span>(<\/span><\/span>m<\/span><\/span>,<\/span><\/span>n<\/span><\/span>)<\/span><\/span>=<\/span><\/span>P<\/span><\/span>i<\/span><\/sub><\/span>j<\/span><\/sub><\/span>(<\/span><\/span>n<\/span><\/span>\u2212<\/span><\/span>m<\/span><\/span>)<\/span><\/span>\u3002<\/span><\/span><\/span><\/p>\n

\u89c2\u5bdf\u4e0a\u8ff0\u65b9\u7a0b\u5bb9\u6613\u53d1\u73b0\uff0c\u6c42\u548c\u9879\u662f\u5bf9\u5904\u5728\u4e2d\u95f4\u7684\u4e0b\u89d2\u6807\u8fdb\u884c\u904d\u5386\uff0c\u8fd9\u6070\u6070\u5bf9\u5e94\u7740\u77e9\u9635\u4e58\u6cd5\u3002 \u5c06\u4e0a\u8ff0\u4f8b\u5b50\u4e2d\u4e00\u6b65\u8f6c\u79fb\u6982\u7387<\/span><\/span>p<\/span><\/span>i<\/span><\/sub><\/span>j<\/span><\/sub><\/span>(<\/span><\/span>i<\/span><\/span>,<\/span><\/span>j<\/span><\/span>\u2208<\/span><\/span>S<\/span><\/span>)<\/span><\/span>\u6392\u5217\u4e3a\u4e00\u4e2a\u77e9\u9635<\/span><\/span><\/span><\/p>\n

<\/section>\n

\u90a3\u4e48\uff0cChapman-Kolmogorov\u65b9\u7a0b\u53d8\u5f62\u4e3a\uff0c<\/span><\/span><\/span><\/p>\n

P<\/span><\/span>(<\/span><\/span>n<\/span><\/span>)<\/span><\/span>=<\/span><\/span>P<\/span><\/span>\u00d7<\/span><\/span>P<\/span><\/span>(<\/span><\/span>n<\/span><\/span>\u2212<\/span><\/span>1<\/span><\/span>)<\/span><\/span>=<\/span><\/span>\u2026<\/span><\/span>=<\/span><\/span>P<\/span><\/span>n<\/span><\/span><\/sup>,<\/span><\/span><\/span><\/p>\n

\u5176\u4e2d\uff0c<\/span><\/span>P<\/span><\/span>(<\/span><\/span>n<\/span><\/span>)<\/span><\/span>\u79f0\u4e3a<\/span><\/span>n<\/span><\/span>\u6b65\u8f6c\u79fb\u6982\u7387\u77e9\u9635\uff0c<\/span><\/span>P<\/span><\/span>\u79f0\u4e3a<\/span><\/span>\u4e00\u6b65\u8f6c\u79fb\u6982\u7387\u77e9\u9635\uff08transition probability matrix, TPM\uff09<\/span><\/span><\/b>\uff0c\u4e0b\u9762\u7b80\u79f0\u4e3a\u8f6c\u79fb\u6982\u7387\u77e9\u9635\u3002<\/span><\/span><\/span>
<\/span><\/span><\/p>\n

\u901a\u8fc7\u89c2\u5bdf\u53ef\u77e5\uff0c\u77e9\u9635<\/span><\/span>P<\/span><\/span>\u6ee1\u8db3\u5982\u4e0b\u6027\u8d28<\/span><\/span><\/span><\/p>\n

<\/section>\n

\u4e00\u822c\u7684\uff0c\u79f0\u6ee1\u8db3\u4e0a\u8ff0\u6027\u8d28\u7684\u77e9\u9635\u4e3a<\/span><\/span>\u968f\u673a\u77e9\u9635\uff08stochastic matrix\uff09<\/span><\/span><\/b>\u3002<\/span><\/span><\/span><\/p>\n

3.2.2 \u57fa\u4e8e\u8f6c\u79fb\u884c\u4e3a\u7684\u72b6\u6001\u5206\u7c7b<\/span><\/span><\/span><\/h4>\n

<\/span><\/span><\/h5>\n
3.2.2.1 \u53ef\u8fbe\u3001\u4e92\u901a\u4e0e\u53ef\u7ea6\u6027<\/span><\/span><\/span><\/h5>\n

\u4e0b\u9762\u4e3e\u4e00\u4e2a\u7b80\u5355\u7684\u4f8b\u5b50\u5177\u4f53\u89c2\u5bdf\u72b6\u6001\u95f4\u7684\u8f6c\u79fb\u884c\u4e3a\uff0c\u4e0b\u56fe\u4e2d(a)\u548c(b)\u5206\u522b\u4e3a\u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u8f6c\u79fb\u6982\u7387\u77e9\u9635\u548c\u56fe\u8868\u793a\u3002<\/span><\/span><\/p>\n

\u4f8b1.<\/span><\/span><\/span> <\/span>\"\u53ef\u7ea6.png\"<\/span><\/p>\n

\u7531Chapman-Kolmogorov\u65b9\u7a0b\uff0c\u53ef\u5f97100\u6b65\u8f6c\u79fb\u6982\u7387\u77e9\u9635\u548c200\u6b65\u8f6c\u79fb\u6982\u7387\u77e9\u9635\u4e3a<\/span><\/span><\/p>\n

<\/section>\n

\u901a\u8fc7\u89c2\u5bdf\u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u56fe\u8868\u793a\u53ef\u4ee5\u53d1\u73b0\uff0c\u72b6\u60011\u548c\u72b6\u60012\u4e4b\u95f4\u5b58\u5728\u4e92\u76f8\u5230\u8fbe\u7684\u8def\u5f84\uff0c\u4f46\u72b6\u60013\u4e0e\u72b6\u60011\u548c\u72b6\u60012\u4e4b\u95f4\u5747\u4e0d\u80fd\u4e92\u76f8\u5230\u8fbe\u3002\u6b64\u5916\uff0c\u5bb9\u6613\u770b\u51fa\uff0c\u968f\u7740\u8f6c\u79fb\u6b65\u6570\u7684\u589e\u52a0\uff0c\u5bf9\u5e94\u7684\u8f6c\u79fb\u6982\u7387\u77e9\u9635\u7684\u7b2c\u4e00\u5217\u548c\u7b2c\u4e8c\u5217\u5747\u8d8b\u8fd10\uff0c\u5373\u72b6\u60011\u548c\u72b6\u60012\u7684\u8f6c\u79fb\u884c\u4e3a\u9010\u6e10\u4e00\u81f4\uff0c\u4f46\u4e0e\u72b6\u60013\u7684\u8f6c\u79fb\u884c\u4e3a\u5b58\u5728\u660e\u663e\u5dee\u5f02\u3002\u90a3\u4e48\uff0c\u53ef\u4ee5\u5229\u7528\u8fd9\u79cd\u4e92\u76f8\u53ef\u8fbe\u7684\u7279\u6027\u6765\u5c06\u76f8\u540c\u8f6c\u79fb\u884c\u4e3a\u7684\u72b6\u6001\u5212\u5206\u4e3a\u4e00\u7c7b\uff0c\u72b6\u60011\u548c\u72b6\u60012\u662f\u4e92\u901a\u7684\uff0c\u53ef\u88ab\u5f52\u4e3a\u4e00\u7c7b\uff0c\u800c\u72b6\u60013\u53ea\u53ef\u8fbe\u81ea\u8eab\uff0c\u6240\u4ee5\u72b6\u60013\u4e3a\u53e6\u4e00\u4e2a\u7c7b\u3002\u4e0b\u9762\u7ed9\u51fa\u53ef\u8fbe\u548c\u4e92\u901a\u7684\u4e25\u683c\u5b9a\u4e49\u3002<\/span><\/span>
<\/span><\/p>\n

\u9996\u5148\uff0c\u9700\u8981\u660e\u786e\u53ef\u8fbe\u7684\u5b9a\u4e49\u3002\u56fe\u4e2d\u7684\u8def\u5f84\u7684\u6743\u91cd\u662f\u5927\u4e8e0\u7684\u6982\u7387\u503c\uff0c\u6240\u4ee5\u5b9a\u4e49\u5982\u4e0b\uff1a<\/span><\/span><\/p>\n

\u5b9a\u4e49\uff08\u53ef\u8fbe\u5173\u7cfb\uff09<\/span><\/span><\/b>\uff1a <\/span><\/span>\u2203<\/span><\/span>n<\/span><\/span><\/sub>\u2265<\/span><\/span>0<\/span><\/span>,<\/span><\/span> <\/span><\/span> <\/span><\/span>P<\/span><\/span>i<\/span><\/sub><\/span>j<\/span><\/sub><\/span>(<\/span><\/span>n<\/span><\/span>)<\/span><\/span>><\/span><\/span>0<\/span><\/span>\uff0c\u5219\u79f0\u72b6\u6001<\/span><\/span>i<\/span><\/span>\u53ef\u8fbe\u72b6\u6001<\/span><\/span>j<\/span><\/span>\uff0c\u8bb0\u4e3a<\/span><\/span>i<\/span><\/span>\u2192<\/span><\/span>j<\/span><\/span>\u3002<\/span><\/span><\/span><\/p>\n

\u6613\u77e5\uff0c<\/span><\/span>\u53ef\u8fbe\u662f\u4f20\u9012\u5173\u7cfb<\/span><\/span><\/b>\uff0c\u5373 <\/span><\/span>i<\/span><\/span>\u2192<\/span><\/span>j<\/span><\/span>,<\/span><\/span> <\/span><\/span>j<\/span><\/span>\u2192<\/span><\/span>k<\/span><\/span>\u27f9<\/span><\/span>i<\/span><\/span>\u2192<\/span><\/span>k<\/span><\/span>\u3002<\/span><\/span><\/span><\/p>\n

\u5b9a\u4e49\uff08\u4e92\u901a\u5173\u7cfb\uff09<\/span><\/span><\/b>\uff1a \u8fdb\u4e00\u6b65\uff0c\u82e5<\/span><\/span>i<\/span><\/span>\u2192<\/span><\/span>j<\/span><\/span>\u4e14<\/span><\/span>j<\/span><\/span>\u2192<\/span><\/span>i<\/span><\/span>\uff0c\u5219\u79f0\u72b6\u6001<\/span><\/span>i<\/span><\/span>\u548c<\/span><\/span>j<\/span><\/span>\u4e92\u901a\uff0c\u8bb0\u4e3a<\/span><\/span>i<\/span><\/span>\u2194<\/span><\/span>j<\/span><\/span>\u3002<\/span><\/span><\/span><\/p>\n

\u6613\u77e5\uff0c<\/span><\/span>\u4e92\u901a\u662f\u4e00\u79cd\u7b49\u4ef7\u5173\u7cfb<\/span><\/span><\/b>\u3002\u90a3\u4e48\uff0c\u53ef\u4ee5\u5c06\u4e92\u901a\u7684\u72b6\u6001\u5f52\u4e3a\u4e00\u7c7b\uff0c\u4e14\u4efb\u4e00\u72b6\u6001\u4ec5\u5c5e\u4e8e\u4e00\u4e2a\u7c7b\u3002<\/span><\/span><\/span><\/p>\n

\u5b9a\u4e49\uff08\u4e0d\u53ef\u7ea6\uff09<\/span>\uff1a<\/span><\/span><\/b>\u7279\u522b\u7684\uff0c\u82e5\u9a6c\u5c14\u79d1\u592b\u94fe\u4e2d\u53ea\u5b58\u5728\u4e00\u4e2a\u4e92\u901a\u7b49\u4ef7\u7c7b\uff0c\u5219\u79f0\u5b83\u662f<\/span><\/span>\u4e0d\u53ef\u7ea6\u7684\uff08irreducible\uff09<\/span><\/span><\/b>\uff1b\u5426\u5219\uff0c\u79f0\u4e3a\u662f<\/span><\/span>\u53ef\u7ea6\u7684\uff08reducible\uff09<\/span><\/span><\/b>\u3002<\/span><\/span><\/span><\/p>\n

\u82e5\u5bf9\u4f8b1\u4e2d\u7684\u72b6\u60013\u7684\u8f6c\u79fb\u5206\u5e03\u7a0d\u4f5c\u4fee\u6539\uff0c<\/span><\/span><\/span><\/p>\n

<\/section>\n

\u90a3\u4e48\uff0c3\u4e2a\u72b6\u6001\u95f4\u5f7c\u6b64\u4e92\u901a\uff0c\u5373\u5747\u5c5e\u4e8e\u540c\u4e00\u4e2a\u7c7b\uff0c\u5219\u5176\u5bf9\u5e94\u7684\u9a6c\u5c14\u79d1\u592b\u94fe\u662f\u4e0d\u53ef\u7ea6\u7684\u3002<\/span><\/span><\/p>\n

3.2.2.2 \u5468\u671f\u6027<\/span><\/span><\/span><\/h5>\n

\u4f8b2.<\/span><\/span><\/b> \u4e0b\u9762\u8003\u5bdf\u4e00\u4e2a\u7b80\u5355\u7684\u4e24\u72b6\u6001\u9a6c\u5c14\u79d1\u592b\u94fe\uff0c\u5176\u8f6c\u79fb\u6982\u7387\u77e9\u9635<\/span><\/span>\"image.png\"<\/span><\/sub>\u3002<\/span><\/span><\/span><\/p>\n

\"Period.png\"<\/section>\n

\u8ba1\u7b97<\/span><\/span>n<\/span><\/span>\u6b65\u8f6c\u79fb\u6982\u7387\u77e9\u9635\uff0c\u5f97<\/span><\/span><\/span><\/p>\n

<\/section>\n

\u4e0d\u96be\u53d1\u73b0\u8fd9\u4e2a\u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u72b6\u6001\u8f6c\u79fb\u5b58\u5728\u7740\u5468\u671f\u6027\u2014\u2014\u521d\u59cb\u72b6\u6001\u4e0d\u8bba\u662f\u72b6\u60011\u8fd8\u662f\u72b6\u60012\uff0c\u6bcf\u8f6c\u79fb2\u6b65\uff0c\u90fd\u80fd\u591f\u4ee5\u6b63\u6982\u7387\u8fd4\u56de\u81ea\u8eab\u72b6\u6001\uff0c\u5373\u5468\u671f\u4e3a2\u3002\u8fdb\u4e00\u6b65\uff0c\u5982\u679c\u6709\u4efb\u4e00\u72b6\u6001\u80fd\u591f\u4ee5\u4e00\u5b9a\u5468\u671f\u3001\u6b63\u6982\u7387\u8fd4\u56de\u81ea\u8eab\uff0c\u8f6c\u79fb\u77e9\u9635\u5219\u4e0d\u5b58\u5728\u6781\u9650\uff0c\u5373\u72b6\u6001\u7684\u5468\u671f\u6027\u6210\u4e3a\u4e86\u4e00\u4e2a\u8f6c\u79fb\u77e9\u9635\u6536\u655b\u7684\u5224\u65ad\u6761\u4ef6\u3002\u4e0b\u9762\u4ecb\u7ecd\u72b6\u6001\u7684\u5468\u671f\u6027\u7684\u4e25\u683c\u5b9a\u4e49\uff1a<\/span><\/span><\/p>\n

\u5b9a\u4e49\uff08\u5468\u671f\uff09<\/span><\/span><\/b>\uff1a\u8bb0\u4f7f\u5f97\u8f6c\u79fb\u6982\u7387<\/span><\/span>P<\/span><\/span>i<\/span><\/sub><\/span>i<\/span><\/sub><\/span>(<\/span><\/span>n<\/span><\/span>)<\/span><\/span>><\/span><\/span>0<\/span><\/span>\u7684\u8f6c\u79fb\u6b65\u6570<\/span><\/span>n<\/span><\/span>\u6240\u7ec4\u6210\u7684\u96c6\u5408\u4e3a<\/span><\/span>A<\/span><\/span>i<\/span><\/span><\/sub>=<\/span><\/span>{<\/span><\/span>n<\/span><\/span>:<\/span><\/span>n<\/span><\/span>\u2265<\/span><\/span>1<\/span><\/span>,<\/span><\/span>P<\/span><\/span>i<\/span><\/sub><\/span>i<\/span><\/sub><\/span>(<\/span><\/span>n<\/span><\/span>)<\/span><\/span>><\/span><\/span>0<\/span><\/span>}<\/span><\/span>\u3002\u82e5<\/span><\/span>A<\/span><\/span>i<\/span><\/span>\u975e\u7a7a\uff0c\u5219\u79f0\u5176\u6700\u5927\u516c\u7ea6\u6570\u4e3a\u72b6\u6001<\/span><\/span>i<\/span><\/span>\u7684\u5468\u671f\uff0c\u8bb0\u4e3a<\/span><\/span>d<\/span><\/span>i<\/span><\/span><\/sub>\u3002\u82e5<\/span><\/span>d<\/span><\/span>i<\/span><\/span><\/sub>><\/span><\/span>1<\/span><\/span>\uff0c\u79f0<\/span><\/span>i<\/span><\/span>\u662f<\/span><\/span>\u5468\u671f\u7684<\/span><\/span><\/b>\uff1b\u82e5<\/span><\/span>d<\/span><\/span>i<\/span><\/span><\/sub>=<\/span><\/span>1<\/span><\/span>\uff0c\u79f0<\/span><\/span>i<\/span><\/span>\u662f<\/span><\/span>\u975e\u5468\u671f\u7684<\/span><\/span><\/b>\u3002<\/span><\/span><\/span>
<\/span><\/span><\/p>\n

\u6839\u636e<\/span><\/span>n<\/span><\/span>\u6b65\u8f6c\u79fb\u6982\u7387\u77e9\u9635\u53ef\u77e5\uff0c<\/span><\/span>P<\/span><\/span>i<\/span><\/sub><\/span>i<\/span><\/sub><\/span>(<\/span><\/span>n<\/span><\/span>)<\/span><\/span>\u5e8f\u5217\u4e3a<\/span><\/span>0<\/span><\/span>,<\/span><\/span>1<\/span><\/span>,<\/span><\/span>0<\/span><\/span>,<\/span><\/span>1<\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>\uff0c<\/span><\/span>P<\/span><\/span>i<\/span><\/sub><\/span>j<\/span><\/sub><\/span>(<\/span><\/span>n<\/span><\/span>)<\/span><\/span>\u5e8f\u5217\u4e3a<\/span><\/span>1<\/span><\/span>,<\/span><\/span>0<\/span><\/span>,<\/span><\/span>1<\/span><\/span>,<\/span><\/span>0<\/span><\/span>\u2026<\/span><\/span>\uff0c\u56db\u4e2a\u8f6c\u79fb\u6982\u7387\u5747\u4e0d\u5b58\u5728\u6781\u9650\uff0c\u5373\u8f6c\u79fb\u77e9\u9635\u6ca1\u6709\u6781\u9650\u3002\u7531\u6b64\u53ef\u89c1\uff0c\u5468\u671f\u6027\u5bf9\u4e8e\u8f6c\u79fb\u77e9\u9635\u662f\u5426\u5177\u6709\u6781\u9650\u7684\u5224\u65ad\u8d77\u7740\u91cd\u8981\u7684\u4f5c\u7528\u3002<\/span><\/span><\/span>
<\/span><\/span><\/p>\n

\u56de\u987e\u4e0a\u8282\u4f8b1\u4e2d\u7684\u4e09\u72b6\u6001\u9a6c\u5c14\u79d1\u592b\u94fe\uff0c\u7531\u4e8e<\/span><\/span>P<\/span><\/span>33<\/span><\/span><\/sub>=<\/span><\/span>1<\/span><\/span>><\/span><\/span>P<\/span><\/span>22<\/span><\/span><\/sub>=<\/span><\/span>0.2<\/span><\/span>><\/span><\/span>0<\/span><\/span>\uff0c\u90a3\u4e48<\/span><\/span>A<\/span><\/span>2<\/span><\/span><\/sub>=<\/span><\/span>A<\/span><\/span>3<\/span><\/span><\/sub>=<\/span><\/span>{<\/span><\/span>1<\/span><\/span>,<\/span><\/span>2<\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>}<\/span><\/span>\uff0c\u6240\u4ee5\u5b83\u4eec\u90fd\u662f\u975e\u5468\u671f\u7684\u3002\u5bf9\u4e8e\u72b6\u60011\uff0c\u5176\u4e0e\u72b6\u60012\u4e92\u901a\u4e14\u72b6\u60012\u4e00\u6b65\u53ef\u8fbe\u81ea\u8eab\uff0c\u90a3\u4e48<\/span><\/span>A<\/span><\/span>1<\/span><\/span><\/sub>=<\/span><\/span>{<\/span><\/span>2<\/span><\/span>,<\/span><\/span>3<\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>}<\/span><\/span>\uff0c\u6240\u4ee5\u72b6\u60011\u4e5f\u662f\u975e\u5468\u671f\u7684\u3002\u53ef\u4ee5\u770b\u51fa\uff0c\u4e92\u901a\u7684\u72b6\u60011\u548c\u72b6\u60012\u7684\u5468\u671f\u6027\u662f\u76f8\u540c\u7684\u3002\u5b9e\u9645\u4e0a\uff0c\u53ef\u4ee5\u901a\u8fc7\u4e25\u683c\u8bc1\u660e\u5f97\u51fa\uff0c<\/span><\/span>\u4e92\u901a\u72b6\u6001\u7684\u5468\u671f\u662f\u76f8\u540c\u7684<\/span><\/span><\/b>\uff0c\u5373\u5468\u671f\u6027\u4e5f\u662f\u4e00\u4e2a\u72b6\u6001\u7b49\u4ef7\u7c7b\u7684\u6027\u8d28\u3002\u90a3\u4e48\uff0c\u4e92\u901a\u7684\u72b6\u6001\u53ea\u9700\u8981\u5224\u65ad\u5176\u4e2d\u4efb\u4e00\u72b6\u6001\u7684\u5468\u671f\u6027\u5373\u53ef\uff0c\u8fd9\u5927\u5927\u51cf\u5c11\u4e86\u5224\u65ad\u7684\u6b21\u6570\u3002<\/span><\/span><\/span>
<\/span><\/p>\n

\u4e0a\u8ff0\u5206\u7c7b\u65b9\u6cd5\u662f\u57fa\u4e8e\u5c40\u90e8\u6027\u8d28\u8fdb\u884c\u4e24\u4e24\u5224\u65ad\u5206\u7c7b\u3002\u6b64\u5916\uff0c\u8fd8\u53ef\u4ee5\u4ece\u6574\u4f53\u8f6c\u79fb\u673a\u5236\u7684\u89d2\u5ea6\u5bf9\u72b6\u6001\u8fdb\u884c\u5212\u5206\uff0c\u8be6\u7ec6\u4ecb\u7ecd\u53c2\u89c1\u6210\u5757\u6027\uff08Lumpability\uff09\u3002<\/span><\/span><\/p>\n

3.2 \u65f6\u9f50\u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u6f14\u5316\u2014\u2014\u521d\u59cb\u72b6\u6001\u968f\u673a<\/span><\/span><\/span><\/h3>\n


<\/span><\/span><\/h4>\n

3.2.1 \u6f14\u5316\u8fc7\u7a0b<\/span><\/span><\/span><\/h4>\n

\u4e0a\u8282\u4e2d\u5173\u4e8e\u8f6c\u79fb\u8fc7\u7a0b\u7684\u5185\u5bb9\uff0c\u5b9e\u9645\u4e0a\u662f\u56fa\u5b9a\u4e86\u521d\u59cb\u65f6\u523b\u7684\u72b6\u6001\u3002\u4e00\u822c\u7684\uff0c\u521d\u59cb\u65f6\u523b\u7684\u72b6\u6001\u4e5f\u5b58\u5728\u7740\u4e00\u5b9a\u7684\u968f\u673a\u6027\uff0c\u5373\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a\u5206\u5e03\u5217\u6765\u523b\u753b\u3002<\/span><\/span>
<\/span><\/span><\/p>\n

\u8bb0<\/span><\/span>X<\/span><\/span>n<\/span><\/span><\/sub>(<\/span><\/span>n<\/span><\/span>\u2265<\/span><\/span>0<\/span><\/span>)<\/span><\/span>\u7684\u5206\u5e03\u5217 <\/span><\/span>\u03c0<\/span><\/span>n<\/span><\/span><\/sub>=<\/span><\/span>(<\/span><\/span>\u03c0<\/span><\/span>n<\/span><\/span><\/sub>(<\/span><\/span>1<\/span><\/span>)<\/span><\/span>,<\/span><\/span>\u03c0<\/span><\/span>n<\/span><\/span><\/sub>(<\/span><\/span>2<\/span><\/span>)<\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>)<\/span><\/span>\uff0c \u5176\u4e2d\uff0c<\/span><\/span>\"image.png\"<\/span><\/sub>\u3002<\/span><\/span><\/span><\/p>\n

\u6839\u636e\u5168\u6982\u7387\u516c\u5f0f\uff0c<\/span><\/span><\/span>X<\/span><\/span>n<\/span><\/sub><\/span>+<\/span><\/span><\/sub>1<\/span><\/sub><\/span>\u7684\u5206\u5e03\u5217<\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/section>\n

\u8fdb\u4e00\u6b65\uff0c\u9012\u63a8\u53ef\u5f97\uff0c<\/span><\/span><\/p>\n

<\/section>\n

\u6ce8\uff1a<\/span><\/span>\u03c0<\/span><\/span>\u662f\u4e00\u4e2a\u884c\u5411\u91cf\uff0c\u4e0a\u8ff0\u65b9\u7a0b\u4e3a\u5de6\u624b\u65b9\u7a0b\uff08left hand equation\uff09\u3002<\/span><\/span><\/span>
<\/span><\/p>\n

\u4f8b1\u4e2d\u7684\u9a6c\u5c14\u79d1\u592b\u94fe\u5982\u679c\u57280\u65f6\u523b\u4e09\u4e2a\u72b6\u6001\u7b49\u6982\u7387\u5730\u51fa\u73b0\uff0c\u90a3\u4e48\u57281\u65f6\u523b\u548c2\u65f6\u523b\u7684\u72b6\u6001\u5206\u5e03\u5206\u522b\u4e3a<\/span><\/span><\/p>\n

<\/section>\n

\u82e5\u5c06\u5de6\u624b\u65b9\u7a0b\u7684\u5de6\u53f3\u4e24\u8fb9\u540c\u65f6\u8fdb\u884c\u8f6c\u7f6e\uff0c\u5f97<\/span><\/span><\/p>\n

<\/section>\n

\u8fd9\u662f\u4e00\u4e2a\u4e00\u9636\u7ebf\u6027\u9f50\u6b21\u5dee\u5206\u65b9\u7a0b\u3002\u7531\u9012\u63a8\u5173\u7cfb\uff0c\u81ea\u7136\u5730\u4f1a\u8054\u60f3\u5230\uff0c\u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u957f\u671f\u884c\u4e3a\u672c\u8d28\u4e0a\u662f\u4e00\u4e2a\u7279\u5f81\u503c\u95ee\u9898\u3002<\/span><\/span><\/p>\n

3.2.2 \u5e73\u7a33\u5206\u5e03\u4e0e\u8f6c\u79fb\u6982\u7387\u77e9\u9635\u7684\u8c31<\/span><\/span><\/span><\/h4>\n


<\/span><\/span><\/p>\n

\u4e0b\u9762\uff0c\u6211\u4eec\u6765\u8003\u5bdf\u4e00\u4e0b<\/span><\/span>P<\/span><\/span>\u7684\u7279\u5f81\u503c\u548c\u7279\u5f81\u5411\u91cf\u5206\u522b\u4ee3\u8868\u4ec0\u4e48\u542b\u4e49\u3002\u8ba9\u6211\u4eec\u56de\u987e\u4e4b\u524d\u4f8b1\u548c\u4f8b2\u4e2d\u7684\u8f6c\u79fb\u77e9\u9635<\/span><\/span><\/span><\/p>\n

<\/section>\n

\u5b83\u4eec\u7684\u7279\u5f81\u503c\u5206\u522b\u4e3a<\/span><\/span>\u2212<\/span><\/span>0.7544<\/span><\/span>,<\/span><\/span>0.9544<\/span><\/span>,<\/span><\/span>1<\/span><\/span>\u4ee5\u53ca<\/span><\/span>\u2212<\/span><\/span>1<\/span><\/span>,<\/span><\/span>1<\/span><\/span>\u3002<\/span><\/span><\/span>
<\/span><\/p>\n

\u901a\u8fc7\u89c2\u5bdf\u4e0d\u96be\u53d1\u73b0\u4e24\u4e2a\u5171\u540c\u7684\u6027\u8d28\uff1a<\/span><\/span><\/p>\n

    \n
  1. \n
    1\u90fd\u662f\u4e24\u4e2a\u8f6c\u79fb\u77e9\u9635\u7684\u7279\u5f81\u503c\uff1b<\/span><\/span><\/section>\n<\/li>\n
  2. \n
    \u7279\u5f81\u503c\u7684\u6a21\u957f\u5747\u4e0d\u8d85\u8fc71\u3002<\/span><\/span><\/section>\n<\/li>\n<\/ol>\n


    <\/span><\/p>\n

    \u8fd9\u4e24\u6761\u6027\u8d28\u5bf9\u4e8e\u4e00\u822c\u7684\u968f\u673a\u77e9\u9635\u4ecd\u662f\u6210\u7acb\u7684\u3002<\/span><\/span><\/p>\n

    \u4e0b\u9762\uff0c\u6211\u4eec\u7b80\u8981\u5206\u6790\u7279\u5f81\u503c1\u4ee5\u53ca\u5bf9\u5e94\u7684\u7279\u5f81\u5411\u91cf\u3002<\/span><\/span><\/p>\n

    \u7531\u4e8e<\/span><\/span>P<\/span><\/span>\u7684\u884c\u548c\u4e3a<\/span><\/span>1<\/span><\/span>\uff0c\u90a3\u4e48\uff0c<\/span><\/span>P<\/span><\/span>1<\/span><\/span>=<\/span><\/span>1<\/span><\/span>\uff0c\u5373<\/span><\/span>1<\/span><\/span>\u662f<\/span><\/span>P<\/span><\/span>\u7684\u7279\u5f81\u503c\uff0c\u4e14\u5bf9\u5e94\u7684\u7279\u5f81\u5411\u91cf\u4e3a<\/span><\/span>1<\/span><\/span>\u3002<\/span><\/span><\/span><\/p>\n

    \u53e6\u4e00\u65b9\u9762\uff0c\u7531\u4e8e<\/span><\/span><\/p>\n

    <\/section>\n

    \u6240\u4ee5\uff0c<\/span><\/span>1<\/span><\/span>\u4e5f\u662f<\/span><\/span>P<\/span><\/span>T<\/span><\/span><\/sup>\u7684\u7279\u5f81\u503c\u3002<\/span><\/span><\/span><\/p>\n

    \u6ce8\uff1a\u5373\u65b9\u7a0b<\/span><\/span>\u03c0<\/span><\/span>=<\/span><\/span>\u03c0<\/span><\/span>P<\/span><\/span>\u603b\u6709\u975e\u96f6\u89e3\uff0c\u8fd9\u4e0e\u9a6c\u5c14\u79d1\u592b\u94fe\u662f\u5426\u6709\u6781\u9650\u6ca1\u6709\u5173\u7cfb\u3002<\/span><\/span><\/span><\/p>\n

    \u82e5<\/span><\/span>\u03c0<\/span><\/span>0<\/span><\/span><\/sub>\u662f\u65b9\u7a0b<\/span><\/span>\u03c0<\/span><\/span>=<\/span><\/span>\u03c0<\/span><\/span>P<\/span><\/span>\u7684\u89e3\uff0c\u90a3\u4e48\u7531\u9012\u63a8\u5173\u7cfb\uff0c\u6709<\/span><\/span><\/span><\/p>\n

    \u03c0<\/span><\/span>n<\/span><\/span><\/sub>=<\/span><\/span>\u03c0<\/span><\/span>0<\/span><\/span><\/sub>P<\/span><\/span>n<\/span><\/span><\/sup>=<\/span><\/span>\u03c0<\/span><\/span>0<\/span><\/span><\/sub>\u3002\u5373\u82e5<\/span><\/span>\"image.png\"<\/span><\/sub>\uff0c\u6709\uff1a<\/span><\/span><\/span><\/p>\n

    \"image.png\"<\/span><\/sub>\uff0c\u5373\u5982\u679c\u521d\u59cb\u65f6\u523b\u7cfb\u7edf\u72b6\u6001\u670d\u4ece\u8be5\u5206\u5e03\uff0c\u90a3\u4e48\u4efb\u610f\u65f6\u523b\u4e0b\u7cfb\u7edf\u72b6\u6001\u5206\u5e03\u4fdd\u6301\u4e0d\u53d8\u3002<\/span><\/span><\/span>
    <\/span><\/b><\/span><\/p>\n

    \u5b9a\u4e49(\u5e73\u7a33\u5206\u5e03)<\/span><\/span><\/b>\uff1a\u6211\u4eec\u79f0<\/span><\/span>\u03c0<\/span><\/span>0<\/span><\/span><\/sub>\u4e3a<\/span><\/span>\u5e73\u7a33\u5206\u5e03<\/span><\/span><\/b>\uff0c\u5982\u679c<\/span><\/span>\u03c0<\/span><\/span>0<\/span><\/span><\/sub>\u662f\u65b9\u7a0b<\/span><\/span>\u03c0<\/span><\/span>=<\/span><\/span>\u03c0<\/span><\/span>P<\/span><\/span>\u7684\u89e3\u3002<\/span><\/span><\/span>
    <\/span><\/span><\/p>\n

    \u8003\u8651\u6982\u7387\u8f6c\u79fb\u77e9\u9635<\/span><\/span>\"image.png\"<\/span><\/sub>\u3002\u6613\u77e5\uff0c\u5176\u5b58\u5728\u5e73\u7a33\u5206\u5e03<\/span><\/span>\"image.png\"<\/span><\/sub>\u3002<\/span><\/span><\/span>
    <\/span><\/span><\/p>\n

    \u5982\u679c\u4e00\u4e2a\u9a6c\u5c14\u79d1\u592b\u94fe\u5b58\u5728\u5e73\u7a33\u5206\u5e03<\/span><\/span>\u03c0<\/span><\/span>\u4e14<\/span><\/span>X<\/span><\/span>0<\/span><\/span><\/sub>\u223c<\/span><\/span>\u03c0<\/span><\/span>\uff0c\u90a3\u4e48\u6613\u5f97\u968f\u673a\u8fc7\u7a0b<\/span><\/span>{<\/span><\/span>X<\/span><\/span>n<\/span><\/span><\/sub>}<\/span><\/span>\u662f\u4e25\u5e73\u7a33\u7684\uff08\u4efb\u4f55\u6709\u9650\u7ef4\u5206\u5e03\u51fd\u6570\u4e0e\u65f6\u95f4\u8d77\u70b9\u65e0\u5173\uff09\u3002<\/span><\/span><\/span><\/p>\n

    3.4 \u65f6\u95f4\u53ef\u9006\u7684\u9a6c\u5c14\u79d1\u592b\u94fe<\/span><\/span><\/span><\/h3>\n

    \u524d\u9762\u7684\u8ba8\u8bba\u90fd\u6cbf\u7740\u65f6\u95f4\u6b63\u5411\u5728\u8ba8\u8bba\uff0c\u4e0b\u9762\u8003\u8651\u65f6\u95f4\u53cd\u5411\u7684\u8fc7\u7a0b\u3002<\/span><\/span><\/p>\n

    <\/section>\n

    \u5373\u53cd\u5411\u8fc7\u7a0b\u4ecd\u6ee1\u8db3\u9a6c\u5c14\u79d1\u592b\u6027\u8d28\u3002\u8fd9\u5176\u5b9e\u662f\u9a6c\u5c14\u79d1\u592b\u6027\u8d28\u7684\u7b49\u4ef7\u5b9a\u4e49\u2014\u2014\u7ed9\u5b9a\u73b0\u5728\u7684\u6761\u4ef6\u4e0b\uff0c\u8fc7\u53bb\u4e0e\u672a\u6765\u6761\u4ef6\u72ec\u7acb\u3002\u5e76\u4e14\uff0c\u4e00\u822c\u6765\u8bb2\uff0c\u4e0a\u8ff0\u4e00\u6b65\u8f6c\u79fb\u6982\u7387\u662f\u4e0e\u65f6\u95f4<\/span><\/span>n<\/span><\/span>\u6709\u5173\u7684\uff0c\u5373\u53cd\u5411\u8fc7\u7a0b\u662f\u975e\u9f50\u6b21\u7684\u3002<\/span><\/span><\/span>
    <\/span><\/span><\/p>\n

    \u4e0b\u9762\u8003\u5bdf\u6709\u9650\u72b6\u6001\u7a7a\u95f4\u4e0a\u4e0d\u53ef\u7ea6\u3001\u975e\u5468\u671f\u7684\u9a6c\u5c14\u79d1\u592b\u94fe\uff0c\u5176\u552f\u4e00\u7684\u5e73\u7a33\u5206\u5e03\u4e3a<\/span><\/span>\u03c0<\/span><\/span>\u3002\u82e5\u4ece\u5e73\u7a33\u5206\u5e03<\/span><\/span>\u03c0<\/span><\/span>\u51fa\u53d1\uff0c\u6709<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u6b64\u65f6\uff0c\u53cd\u5411\u8fc7\u7a0b\u662f\u9f50\u6b21\u7684\u3002<\/span><\/span><\/p>\n

    3.4.1 \u65f6\u95f4\u53ef\u9006\u6027\u7684\u5b9a\u4e49(\u7ec6\u81f4\u5e73\u8861\u6761\u4ef6)\u53ca\u53ef\u9006\u5206\u5e03<\/span><\/span><\/span><\/h4>\n

    \u8fdb\u4e00\u6b65\uff0c\u8003\u8651\u4e00\u79cd\u7279\u6b8a\u60c5\u5f62\uff0c<\/span><\/span><\/p>\n

    \u5b9a\u4e49\uff08\u65f6\u95f4\u53ef\u9006\u9a6c\u5c14\u79d1\u592b\u94fe\uff09\uff1a<\/span><\/span><\/b>\u82e5<\/span><\/span>q<\/span><\/span>i<\/span><\/sub><\/span>j<\/span><\/sub><\/span>=<\/span><\/span>p<\/span><\/span>i<\/span><\/sub><\/span>j<\/span><\/sub><\/span>\uff0c\u5373\u5bf9\u4e8e\u4efb\u610f\u7684\u72b6\u6001<\/span><\/span>i<\/span><\/span>,<\/span><\/span>j<\/span><\/span>\uff0c\u82e5\u5b58\u5728<\/span><\/span>S<\/span><\/span>\u4e0a\u7684\u6982\u7387\u5206\u5e03<\/span><\/span>\u03c0<\/span><\/span>\u6ee1\u8db3<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u6b64\u65f6\uff0c\u79f0\u9a6c\u5c14\u79d1\u592b\u94fe\u6ee1\u8db3\u7ec6\u81f4\u5e73\u8861\u6761\u4ef6\uff0c\u4e5f\u79f0\u8be5\u9a6c\u5c14\u79d1\u592b\u94fe\u662f<\/span><\/span>\u65f6\u95f4\u53ef\u9006\u7684<\/span><\/span><\/b>\uff0c\u79f0\u6982\u7387\u5206\u5e03<\/span><\/span>\u03c0<\/span><\/span>\u4e3a<\/span><\/span>P<\/span><\/span>\u7684\u4e00\u4e2a<\/span><\/span>\u53ef\u9006\u5206\u5e03<\/span><\/span><\/b>\u3002<\/span><\/span><\/span><\/p>\n

    \u6ce8\uff1a<\/span><\/span>\u03c0<\/span><\/span>i<\/span><\/span><\/sub>p<\/span><\/span>i<\/span><\/sub><\/span>j<\/span><\/sub><\/span>\u53ef\u4ee5\u7406\u89e3\u4e3a\u72b6\u6001<\/span><\/span>i<\/span><\/span>\u5411\u72b6\u6001<\/span><\/span>j<\/span><\/span>\u7684\u8f93\u9001\uff0c\u540c\u7406\uff0c<\/span><\/span>\u03c0<\/span><\/span>j<\/span><\/span><\/sub>p<\/span><\/span>j<\/span><\/sub><\/span>i<\/span><\/sub><\/span>\u53ef\u4ee5\u7406\u89e3\u4e3a\u72b6\u6001<\/span><\/span>j<\/span><\/span>\u5411\u72b6\u6001<\/span><\/span>i<\/span><\/span>\u7684\u8f93\u9001\u3002\u7ec6\u81f4\u5e73\u8861\u6761\u4ef6\u53ef\u4ee5\u7406\u89e3\u4e3a\u4efb\u610f\u7684\u72b6\u6001<\/span><\/span>i<\/span><\/span>\u4e0e\u72b6\u6001<\/span><\/span>j<\/span><\/span>\u4e4b\u95f4\u7684\u76f8\u4e92\u8f93\u9001\u662f\u76f8\u540c\u7684\u3002<\/span><\/span><\/span><\/p>\n

    3.4.2 \u53ef\u9006\u5206\u5e03\u4e0e\u5e73\u7a33\u5206\u5e03<\/span><\/span><\/span><\/h4>\n

    \u8fdb\u4e00\u6b65\uff0c\u7531\u7ec6\u81f4\u5e73\u8861\u6761\u4ef6\uff0c\u5219\u53ef\u9006\u5206\u5e03\u662f\u5e73\u7a33\u5206\u5e03\uff0c\u5373<\/span><\/span><\/p>\n

    <\/section>\n

    \u8bc1\uff1a<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u8fd9\u4e00\u6027\u8d28\u4e3a\u6c42\u89e3\u5e73\u7a33\u5206\u5e03\u63d0\u4f9b\u4e86\u4e00\u79cd\u65b0\u7684\u65b9\u6cd5\u3002\u4f8b\u5982\uff0c\u8003\u8651\u5982\u4e0b\u53ef\u9006\u7684\u9a6c\u5c14\u79d1\u592b\u94fe\uff0c\u5176\u6982\u7387\u8f6c\u79fb\u77e9\u9635\u4e3a\uff1a<\/span><\/span><\/p>\n

    <\/section>\n

    \u5229\u7528\u7ec6\u81f4\u5e73\u8861\u6761\u4ef6\uff0c\u53ef\u4ee5\u5f97\u5230\u5173\u4e8e<\/span><\/span>\u03c0<\/span><\/span>\u7684\u65b9\u7a0b\uff1a<\/span><\/span><\/p>\n

    <\/section>\n

    \u8fdb\u4e00\u6b65\uff0c\u7531\u6c42\u548c\u4e3a1\uff0c\u5f97\u53ef\u9006\u5206\u5e03\uff08\u4e5f\u662f\u5e73\u7a33\u5206\u5e03\uff09<\/span><\/span>\u03c0<\/span><\/span>\uff1a<\/span><\/span><\/p>\n

    <\/section>\n

    \u4f46\u4e0a\u8ff0\u7ed3\u8bba\u53cd\u4e4b\u5219\u4e0d\u7136\uff0c\u8003\u8651\u5982\u4e0b\u6982\u7387\u8f6c\u79fb\u77e9\u9635<\/span><\/span><\/p>\n

    <\/span><\/p>\n

    \u6613\u77e5\uff0c\u5176\u5b58\u5728\u5e73\u7a33\u5206\u5e03<\/span><\/span>\"image.png\"<\/span><\/sub>\uff0c\u4f46<\/span><\/span><\/p>\n

    <\/section>\n

    \u5373\u8be5\u94fe\u4e0d\u662f\u53ef\u9006\u7684\u3002<\/span><\/span><\/p>\n

    3.4.3 \u65f6\u95f4\u53ef\u9006\u6027\u4e0e\u52a8\u529b\u5b66\u53ef\u9006\u6027<\/span><\/span><\/span><\/h4>\n

    \u65f6\u95f4\u53ef\u9006\u6027\u63cf\u8ff0\u7684\u662f\u72b6\u6001\u6982\u7387\u5206\u5e03\u4e0a\u7684\u65f6\u95f4\u53cd\u6f14\u5bf9\u79f0\u6027\uff0c\u5373\u5982\u679c\u628a\u65f6\u95f4\u70b9\u4ee5\u8fbe\u5230\u5e73\u7a33\u5206\u5e03\u7684\u65f6\u95f4\u4e3a\u5bf9\u79f0\u8f74\uff0c\u8fdb\u884c\u53cd\u8f6c\uff0c\u5219\u53ef\u4ee5\u5f97\u5230\u4e00\u7cfb\u5217\u548c\u539f\u9a6c\u5c14<\/span><\/span>\u79d1<\/span><\/span><\/span>\u592b\u94fe\u6b63\u5411\u65f6\u95f4\u6f14\u5316\u8fc7\u7a0b\u5b8c\u5168\u76f8\u540c\u7684\u72b6\u6001\u6982\u7387\u5206\u5e03\u3002<\/span><\/span><\/p>\n

    \u4e0e\u65f6\u95f4\u53ef\u9006\u6027\u4e0d\u540c\uff0c\u52a8\u529b\u5b66\u53ef\u9006\u6027\u5219\u63cf\u8ff0\u7684\u662f\u72b6\u6001\u4e0a\u7684\u53ef\u9006\u6027\u3002\u5982\u679c\u4e00\u4e2a\u5e73\u7a33\u9a6c\u5c14\u79d1\u592b\u8fc7\u7a0b\u7684\u6982\u7387\u8f6c\u79fb\u77e9\u9635\u662f\u53ef\u9006\u7684\uff0c\u5219\u8be5\u8fc7\u7a0b\u7684\u72b6\u6001\u6f14\u5316\u5c31\u5b8c\u5168\u662f\u65f6\u95f4\u53ef\u9006\u7684\u3002\u8fdb\u4e00\u6b65\uff0c\u8fd1\u4f3c\u52a8\u529b\u5b66\u53ef\u9006\u6027\u53ef\u4ee5\u4f5c\u4e3a\u4e00\u79cd\u56e0\u679c\u5ea6\u91cf\uff0c\u8be6\u7ec6\u5185\u5bb9\u53c2\u89c1\u540e\u9762\u6709\u5173\u8f6c\u79fb\u6982\u7387\u77e9\u9635\u7684\u52a8\u529b\u5b66\u53ef\u9006\u6027\u7684\u8ba8\u8bba\uff0c\u4e5f\u53ef\u53c2\u8003\u8bcd\u6761<\/span>\u57fa\u4e8e\u53ef\u9006\u6027\u7684\u56e0\u679c\u6d8c\u73b0\u7406\u8bba\uff5c\u96c6\u667a\u767e\u79d1<\/span><\/a>\u3002<\/span><\/span><\/p>\n

    3.5 \u79bb\u6563\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u7c97\u7c92\u5316\u548c\u5ea6\u91cf<\/span><\/span><\/span><\/h3>\n

    3.5.1 \u79bb\u6563\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u7c97\u7c92\u5316<\/span><\/span><\/span><\/h4>\n

    \u4e00\u822c\u7684\uff0c\u5728\u72b6\u6001\u5206\u7c7b\u7684\u57fa\u7840\u4e0a\uff0c\u5bf9\u5206\u7c7b\u540e\u72b6\u6001\u7a7a\u95f4\uff08\u5b8f\u89c2\u5c3a\u5ea6\u4e0a\uff09\u91cd\u65b0\u5b9a\u4e49\u9a6c\u5c14<\/span><\/span>\u79d1<\/span><\/span><\/span>\u592b\u8f6c\u79fb\u77e9\u9635\u7684\u8fc7\u7a0b\u88ab\u79f0\u4e3a\u9a6c\u5c14\u79d1\u592b\u6982\u7387\u8f6c\u79fb\u77e9\u9635\u7684\u7c97\u7c92\u5316\uff08coarse-graining\uff09\uff0c\u8be6\u7ec6\u4ecb\u7ecd\u53c2\u89c1\u8bcd\u6761\u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u7c97\u7c92\u5316\u3002\u4e0b\u9762\uff0c\u7b80\u8981\u4ecb\u7ecd\u4e24\u79cd\u7c97\u7c92\u5316\u65b9\u6cd5\u2014\u2014\u6210\u5757\u6027\u548c\u57fa\u4e8eSVD\u5206\u89e3\u7684\u7c97\u7c92\u5316\u65b9\u6cd5\u3002<\/span><\/span><\/p>\n

    3.5.1.1 \u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u6210\u5757\u6027<\/span><\/span><\/span><\/h5>\n

    \u53ef\u7ea6\u662f\u57fa\u4e8e\u5c40\u90e8\u8fde\u901a\u6027\u8fdb\u884c\u4e24\u4e24\u5224\u65ad\u4ece\u800c\u5bf9\u72b6\u6001\u7a7a\u95f4\u8fdb\u884c\u5206\u7c7b\uff0c\u4f46\u6709\u65f6\u8fd9\u4e2a\u5212\u5206\u7c92\u5ea6\u4ecd\u5f88\u7c97\u7cd9\uff0c\u5373\u8fde\u901a\u7684\u72b6\u6001\u95f4\u5b58\u5728\u5176\u4ed6\u6027\u8d28\u6216\u7ed3\u6784\u53ef\u4ee5\u7ee7\u7eed\u5212\u5206\u3002<\/span><\/span>
    <\/span><\/p>\n

    \u4f8b\u5982\uff0c\u8003\u8651\u5982\u4e0b\u9a6c\u5c14\u79d1\u592b\u94fe\uff0c<\/span><\/span><\/p>\n

    <\/section>\n

    \u6613\u77e5\uff0c\u8fd9\u4e2a\u9a6c\u5c14\u79d1\u592b\u94fe\u662f\u4e0d\u53ef\u7ea6\u7684\u3002\u6b64\u5916\uff0c\u65e0\u8bba\u4ece\u72b6\u60011\u8fd8\u662f2\u51fa\u53d1\uff0c\u4e0b\u4e00\u65f6\u523b\u8f6c\u79fb\u81f3\u72b6\u60011\u6216\u72b6\u60012\u7684\u6982\u7387\u5747\u4e3a0.6\uff0c\u8f6c\u79fb\u81f3\u5176\u4ed6\u4e24\u4e2a\u72b6\u60013\u548c4\u7684\u6982\u7387\u5747\u4e3a0.4\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u72b6\u60013\u548c\u72b6\u60014\u4e5f\u662f\u5982\u6b64\u3002\u90a3\u4e48\uff0c\u57fa\u4e8e\u8fd9\u6837\u7684\u8f6c\u79fb\u673a\u5236\uff0c\u53ef\u4ee5\u5c06\u72b6\u60011\u548c2\u5212\u5206\u4e3a1\u7ec4\uff0c\u72b6\u60013\u548c\u72b6\u60014\u5212\u5206\u4e3a\u4e00\u7ec4\uff0c\u5f62\u6210\u4e24\u4e2a\u5b8f\u89c2\u72b6\u6001\uff0c\u5f97\u5230\u4e00\u4e2a\u65b0\u7684\u4e24\u72b6\u6001\u7684\u9a6c\u5c14\u79d1\u592b\u8f6c\u79fb\u77e9\u9635<\/span><\/span><\/p>\n

    <\/section>\n

    \u5373\u540c\u5c5e\u4e8e\u4e00\u4e2a\u5b8f\u89c2\u72b6\u6001\u7684\u539f\u59cb\u5fae\u89c2\u72b6\u6001\u8f6c\u79fb\u81f3\u5b8f\u89c2\u72b6\u6001\u7684\u673a\u5236\u662f\u76f8\u540c\u7684\uff0c\u8fd9\u79cd\u6027\u8d28\u88ab\u79f0\u4e3a\u6210\u5757\u6027\uff08Lumpability\uff09\u3002\u5b9a\u4e49<\/span><\/span>A<\/span><\/span>=<\/span><\/span>{<\/span><\/span>A<\/span><\/span>i<\/span><\/span><\/sub>}<\/span><\/span>\u4e3a\u5b8f\u89c2\u6001\uff0c\u6210\u5757\u6027\u7684\u6570\u5b66\u8868\u8fbe\u5f0f\u4e3a\uff1a<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u8be6\u7ec6\u4ecb\u7ecd\u53c2\u89c1\u8bcd\u6761\u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u6210\u5757\u6027\u3002<\/span><\/span><\/span><\/p>\n

    3.5.1.2 \u9a6c\u5c14\u79d1\u592b\u94fe\u7684SVD\u7c97\u7c92\u5316<\/span><\/span><\/span><\/h5>\n

    \u4ece\u8f6c\u79fb\u77e9\u9635\u8c31\u7684\u89d2\u5ea6\u51fa\u53d1\uff0c\u5f20\u6c5f\u56e2\u961f\u63d0\u51fa\u4e86\u57fa\u4e8e\u5947\u5f02\u503c\u5206\u89e3\uff08SVD\uff09\u7684\u7c97\u7c92\u5316\u65b9\u6cd5<\/span><\/span>[10]<\/span><\/span>\u3002<\/span><\/span><\/span>
    <\/span><\/span><\/p>\n

    \u7b2c\u4e00\u6b65\uff0c\u5bf9\u72b6\u6001\u7a7a\u95f4\u8fdb\u884c\u5212\u5206\u3002\u9996\u5148\uff0c\u8003\u8651<\/span><\/span>N<\/span><\/span>\u9636\u8f6c\u79fb\u77e9\u9635<\/span><\/span>P<\/span><\/span>\u7684\u622a\u65ad\u5947\u5f02\u503c\u5206\u89e3 <\/span><\/span>U<\/span><\/span>\u03a3<\/span><\/span>r<\/span><\/sub><\/span>\u00d7<\/span><\/span><\/sub>r<\/span><\/sub><\/span>V<\/span><\/span>T<\/span><\/span><\/sup>\uff0c \u5176\u4e2d\uff0c<\/span><\/span>r<\/span><\/span>\u4e3a\u7c97\u7c92\u5316\u540e\u7684\u72b6\u6001\u4e2a\u6570\u3002\u4e4b\u540e\uff0c\u5229\u7528\u77e9\u9635<\/span><\/span>V<\/span><\/span>\u5bf9\u8f6c\u79fb\u77e9\u9635<\/span><\/span>P<\/span><\/span>\u505a\u6295\u5f71\u64cd\u4f5c\uff0c\u5f97<\/span><\/span>\"image.png\"<\/span><\/sub>\u3002\u63a5\u4e0b\u6765\uff0c\u5229\u7528Kmeans\u7b49\u805a\u7c7b\u7b97\u6cd5\u5c06<\/span><\/span>\"image.png\"<\/span><\/sub>\u7684\u884c\u5411\u91cf\u805a\u4e3a<\/span><\/span>r<\/span><\/span>\u7c7b\uff0c\u5f97\u5230\u6307\u793a\u77e9\u9635<\/span><\/span>\u03a6<\/span><\/span>\uff0c\u5176\u5143\u7d20\u8868\u793a\u5982\u4e0b<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u5176\u4e2d\uff0c<\/span><\/span>a<\/span><\/span>(<\/span><\/span>i<\/span><\/span>)<\/span><\/span>\u2208<\/span><\/span>{<\/span><\/span>1<\/span><\/span>,<\/span><\/span>2<\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>r<\/span><\/span>}<\/span><\/span>\u4e3a<\/span><\/span>\"image.png\"<\/span><\/sub>\u7684\u7b2c<\/span><\/span>i<\/span><\/span>\u884c\u884c\u5411\u91cf\u6240\u5c5e\u7c7b\u522b\u3002<\/span><\/span><\/span><\/p>\n

    \u7b2c\u4e8c\u6b65\uff0c\u5b9a\u4e49\u65b0\u7684\u8f6c\u79fb\u6982\u7387\u77e9\u9635\u3002\u5047\u8bbe\u6709\u9650\u7ef4\u8f6c\u79fb\u77e9\u9635<\/span><\/span><\/span>P<\/span><\/span>\u5bf9\u5e94\u7684\u9a6c\u5c14\u79d1\u592b\u94fe\u662f\u4e0d\u53ef\u7ea6\u4e14\u975e<\/span><\/span><\/span>\u5468\u671f\u7684\uff0c\u5219\u5b58\u5728\u552f\u4e00\u5e73\u7a33\u5206\u5e03<\/span><\/span><\/span>\u03c0<\/span><\/span>\u3002\u8003\u8651\u8fbe\u5230\u5e73\u7a33\u540e\u7684\u4e24\u6b65\u8054\u5408\u6982\u7387<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u79f0\u5176\u7ec4\u6210\u7684\u77e9\u9635<\/span><\/span>F<\/span><\/span>\u4e3a\u5e73\u7a33\u6d41\u77e9\u9635\u3002\u63a5\u4e0b\u6765\uff0c\u57fa\u4e8e\u5e73\u7a33\u6d41\u77e9\u9635\u548c\u6307\u793a\u77e9\u9635\uff0c\u5b9a\u4e49\u7b80\u5316\u5e73\u7a33\u6d41\u77e9\u9635<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u4e3a\u4e86\u6ee1\u8db3\u884c\u548c\u4e3a1\u7684\u7ea6\u675f\uff0c\u5bf9\u77e9\u9635<\/span><\/span>F<\/span><\/span>\u2032<\/span><\/span>\u8fdb\u884c\u884c\u5f52\u4e00\u5316\uff0c\u5f97\u5230\u7c97\u7c92\u5316\u540e\u8f6c\u79fb\u77e9\u9635<\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span>\uff0c\u5176\u5143\u7d20\u4e3a<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u8be6\u7ec6\u4ecb\u7ecd\u53c2\u89c1\u8bcd\u6761\u57fa\u4e8e\u53ef\u9006\u6027\u7684\u56e0\u679c\u6d8c\u73b0\u7406\u8bba\u3002<\/span><\/span><\/p>\n

    3.5.2 \u79bb\u6563\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u5ea6\u91cf<\/span><\/span><\/span><\/h4>\n
    3.5.2.1 \u8f6c\u79fb\u6982\u7387\u77e9\u9635\u7684\u6709\u6548\u4fe1\u606f<\/span><\/span><\/span><\/h5>\n

    \u9a6c\u5c14\u79d1\u592b\u94fe\u53ef\u4ee5\u89c6\u4e3a\u662f\u901a\u4fe1\u9886\u57df\u4e2d\u4e00\u79cd\u7279\u6b8a\u7684\u4fe1\u9053\u2014\u2014\u79bb\u6563\u65e0\u8bb0\u5fc6\u4fe1\u9053\u3002\u4fe1\u9053\u7684\u8f93\u5165\u548c\u8f93\u51fa\u5206\u522b\u8bb0\u4e3a<\/span><\/span>X<\/span><\/span>\u548c<\/span><\/span>Y<\/span><\/span>\uff0c\u5f53\u524d\u8f93\u5165<\/span><\/span>X<\/span><\/span>\u7ed9\u5b9a\u7684\u6761\u4ef6\u4e0b\uff0c\u5bf9\u5e94\u7684\u8f93\u51fa<\/span><\/span>Y<\/span><\/span>\u4e0e\u4e4b\u524d\u7684\u8f93\u5165\u548c\u8f93\u51fa\u72ec\u7acb\u3002<\/span><\/span><\/span><\/p>\n

    \u4e3a\u4e86\u63cf\u8ff0\u4fe1\u9053\u80fd\u591f\u4f20\u8f93\u7684\u6700\u5927\u4fe1\u606f\u91cf\uff0c\u4fe1\u606f\u8bba\u9886\u57df\u63d0\u51fa\u4e86\u4fe1\u9053\u5bb9\u91cf\u7684\u6982\u5ff5\uff0c\u5177\u4f53\u5b9a\u4e49\u4e3a<\/span><\/span><\/p>\n

    <\/section>\n

    \u5176\u4e2d\uff0c<\/span><\/span>\u03c0<\/span><\/span>X<\/span><\/span><\/sub>\u4e3a<\/span><\/span>X<\/span><\/span>\u7684\u53d6\u503c\u5206\u5e03\u3002<\/span><\/span><\/span><\/p>\n

    \u57fa\u4e8e\u4e0a\u8ff0\u4fe1\u9053\u5bb9\u91cf\u7684\u6982\u5ff5\uff0cErik Hoel<\/span><\/span>[11]<\/span><\/span>\u63d0\u51fa\u4e86\u79bb\u6563\u9a6c\u5c14\u79d1\u592b\u52a8\u529b\u5b66\u7684\u6709\u6548\u4fe1\u606f\u5ea6\u91cf<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u5176\u4e2d\uff0c<\/span><\/span>U<\/span><\/span>X<\/span><\/span><\/sub>\u4e3a\u5747\u5300\u5206\u5e03\uff0c<\/span><\/span>\"image.png\"<\/span><\/sub>\u548c<\/span><\/span>\"image.png\"<\/span><\/sub>\u4e3a<\/span><\/span>X<\/span><\/span>\u5e72\u9884\u4e3a\u5747\u5300\u5206\u5e03\u540e<\/span><\/span>Y<\/span><\/span>\u548c<\/span><\/span>X<\/span><\/span>\u5bf9\u5e94\u7684\u72b6\u6001\u3002<\/span><\/span><\/span>
    <\/span><\/span><\/p>\n

    \u901a\u8fc7\u7c97\u7c92\u5316\u64cd\u4f5c\uff0c\u53ef\u4ee5\u5f97\u5230\u5b8f\u89c2\u5c3a\u5ea6\u4e0b\u7684\u8f6c\u79fb\u77e9\u9635<\/span><\/span>P<\/span><\/span>M<\/span><\/span><\/sub>\uff0c\u5e76\u8ba1\u7b97\u6709\u6548\u4fe1\u606f<\/span><\/span>E<\/span><\/span>I<\/span><\/span>(<\/span><\/span>P<\/span><\/span>M<\/span><\/span><\/sub>)<\/span><\/span>\u3002\u8fdb\u4e00\u6b65\uff0c\u53ef\u4ee5\u901a\u8fc7\u5b8f\u89c2\u4e0e\u5fae\u89c2\u5c3a\u5ea6\u4e0b\u7684\u52a8\u529b\u5b66\u6709\u6548\u4fe1\u606f\u5dee\u503c\u6765\u5ea6\u91cf\u9a6c\u5c14\u79d1\u592b\u52a8\u529b\u5b66\u7684\u56e0\u679c\u6548\u5e94\uff0c\u5e76\u79f0\u8fd9\u4e2a\u5dee\u503c<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u4e3a\u6709\u6548\u4fe1\u606f\u589e\u76ca\u3002\u5982\u679c\u5b58\u5728\u67d0\u4e2a\u5b8f\u89c2\u5c3a\u5ea6\uff0c\u4f7f\u5f97\u6709\u6548\u4fe1\u606f\u589e\u76ca\u5927\u4e8e0\uff0c\u5219\u8ba4\u4e3a\u8be5\u7cfb\u7edf\u53d1\u751f\u4e86\u56e0\u679c\u6d8c\u73b0\u3002\u8be6\u7ec6\u5185\u5bb9\u53c2\u89c1\u8bcd\u6761\u6709\u6548\u4fe1\u606f\u3002<\/span><\/span><\/span><\/p>\n

    3.5.2.2 \u8f6c\u79fb\u6982\u7387\u77e9\u9635\u7684\u52a8\u529b\u5b66\u53ef\u9006\u6027<\/span><\/span><\/span><\/h5>\n

    \u6613\u77e5\uff0c\u5f53\u4e14\u4ec5\u5f53<\/span><\/span>P<\/span><\/span>\u4e3a\u7f6e\u6362\u77e9\u9635\uff0c\u6709\u6548\u4fe1\u606f\u7684\u503c\u6700\u5927\u3002\u5e76\u4e14\uff0c\u7f6e\u6362\u77e9\u9635\u7684\u9006\u77e9\u9635\u4e5f\u662f\u4e00\u4e2a\u7f6e\u6362\u77e9\u9635\uff0c\u540c\u65f6\u4e5f\u662f\u4e00\u4e2a\u8f6c\u79fb\u6982\u7387\u77e9\u9635\uff0c\u5373<\/span><\/span>P<\/span><\/span>\u662f\u53ef\u9006\u77e9\u9635\u4e14<\/span><\/span>P<\/span><\/span>\u2212<\/span><\/sup><\/span>1<\/span><\/sup><\/span>\u4e5f\u662f\u4e00\u4e2a\u8f6c\u79fb\u6982\u7387\u77e9\u9635\uff0c\u5219\u79f0\u8fd9\u7c7b\u8f6c\u79fb\u6982\u7387\u77e9\u9635\u662f\u4e25\u683c\u52a8\u529b\u5b66\u53ef\u9006\u7684\u3002\u4f46\u662f\uff0c\u53ea\u6709\u7f6e\u6362\u77e9\u9635\u6ee1\u8db3\u4e25\u683c\u52a8\u529b\u5b66\u53ef\u9006\u3002\u6240\u4ee5\uff0c\u9700\u8981\u5b9a\u4e49\u4e00\u4e2a\u53ef\u4ee5\u5ea6\u91cf\u63a5\u8fd1\u52a8\u529b\u5b66\u53ef\u9006\u7a0b\u5ea6\u7684\u6307\u6807\u3002<\/span><\/span><\/span>
    <\/span><\/span><\/p>\n

    \u901a\u8fc7\u89c2\u5bdf\u53ef\u4ee5\u53d1\u73b0\u7f6e\u6362\u77e9\u9635\u5177\u6709\u4e24\u65b9\u9762\u7279\u6027\uff1a\u4e00\u65b9\u9762\uff0c\u5b83\u662f\u6ee1\u79e9\u77e9\u9635\uff1b\u53e6\u4e00\u65b9\u9762\uff0c\u5b83\u6bcf\u884c\u6216\u5217\u5747\u4e3a\u72ec\u70ed\u5411\u91cf\uff0c\u5177\u6709\u7a00\u758f\u6027\u3002\u8fd9\u4e24\u65b9\u9762\u90fd\u53ef\u4ee5\u901a\u8fc7\u8f6c\u79fb\u6982\u7387\u77e9\u9635<\/span><\/span>P<\/span><\/span>\u7684\u5947\u5f02\u503c\u8fdb\u884c\u5ea6\u91cf\u3002\u8003\u8651\u4e00\u4e2a<\/span><\/span>N<\/span><\/span>\u9636\u8f6c\u79fb\u6982\u7387\u77e9\u9635<\/span><\/span>P<\/span><\/span>\uff0c\u8bbe\u5176\u5947\u5f02\u503c\u4e3a<\/span><\/span>\u03c3<\/span><\/span>1<\/span><\/span><\/sub>\u2265<\/span><\/span>\u03c3<\/span><\/span>2<\/span><\/span><\/sub>\u2265<\/span><\/span>\u2026<\/span><\/span>\u2265<\/span><\/span>\u03c3<\/span><\/span>N<\/span><\/span><\/sub>\u2265<\/span><\/span>0<\/span><\/span>\uff0c\u5219<\/span><\/span>P<\/span><\/span>\u7684\u79e9\u6709\u5982\u4e0b\u8868\u793a<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u5e76\u4e14<\/span><\/span>P<\/span><\/span>\u7684\u7a00\u758f\u6027\u53ef\u4ee5\u901a\u8fc7\u77e9\u9635\u7684Frobenius\u8303\u6570\u5ea6\u91cf\uff0c<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u57fa\u4e8e\u4e0a\u8ff0\u7279\u6027\uff0c\u5f20\u6c5f\u56e2\u961f\u5b9a\u4e49\u8f6c\u79fb\u6982\u7387\u77e9\u9635<\/span><\/span><\/span>P<\/span><\/span>\u7684<\/span><\/span>\u03b1<\/span><\/span>\u2208<\/span><\/span>(<\/span><\/span>0<\/span><\/span>,<\/span><\/span>2<\/span><\/span>)<\/span><\/span>\u9636\u8fd1\u4f3c\u52a8\u529b\u5b66\u53ef\u9006\u6027<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u4f5c\u4e3a\u52a8\u529b\u5b66\u53ef\u9006\u6027\u7684\u8fd1\u4f3c<\/span><\/span>[10]<\/span><\/span>\u3002\u5728\u5e94\u7528\u4e2d\uff0c\u5e38\u91c7\u7528<\/span><\/span>\u0393<\/span><\/span>\u03b1<\/span><\/sub><\/span>=<\/span><\/span><\/sub>1<\/span><\/sub><\/span>\u5373\u8f6c\u79fb\u6982\u7387\u77e9\u9635\u7684\u6838\u8303\u6570\u6765\u5ea6\u91cf\u63a5\u8fd1\u52a8\u529b\u5b66\u53ef\u9006\u7684\u7a0b\u5ea6\u3002<\/span><\/span><\/span>
    <\/span><\/span><\/p>\n

    \u57fa\u4e8e<\/span><\/span>\u0393<\/span><\/span>\u03b1<\/span><\/span><\/sub>\uff0c\u6587\u4e2d\u8fd8\u7ed9\u51fa\u4e86\u6e05\u6670\u56e0\u679c\u6d8c\u73b0\u548c\u6a21\u7cca\u56e0\u679c\u6d8c\u73b0\u7a0b\u5ea6\u7684\u8ba1\u7b97\u65b9\u6cd5\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u5b83\u4eec\u5747\u4e0e\u7c97\u7c92\u5316\u65b9\u6cd5\u65e0\u5173\u3002\u8be6\u7ec6\u5185\u5bb9\u53c2\u89c1\u8bcd\u6761\u57fa\u4e8e\u53ef\u9006\u6027\u7684\u56e0\u679c\u6d8c\u73b0\u7406\u8bba\u3002<\/span><\/span><\/span><\/p>\n


    <\/span><\/span><\/h2>\n

    \n
    \n
    \n
    \n
    \n

    <\/span><\/section>\n

    <\/span><\/section>\n
    \n
    <\/section>\n<\/section>\n<\/section>\n
    \n

    4. \u8fde\u7eed\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe<\/span><\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

    \n

    <\/span><\/section>\n

    <\/span><\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


    <\/span><\/p>\n

    \u6211\u4eec\u5148\u4ee5\u4e00\u4e2a\u7b80\u5355\u7684\u4f8b\u5b50\u8bf4\u660e\u4e3a\u4f55\u8981\u5f15\u5165\u8fde\u7eed\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe(Continuous-Time Markov Chain\uff0cCTMC)\u3002\u5047\u8bbe\u6211\u4eec\u89c2\u5bdf\u4e00\u4e2a\u706f\u6ce1\u7684\u8fd0\u884c\u72b6\u6001\uff0c\u5176\u72b6\u6001\u7a7a\u95f4\u53ea\u6709\u4e24\u79cd\uff1a\u6b63\u5e38\uff08\u5de5\u4f5c\uff09\u548c\u6545\u969c\uff08\u635f\u574f\uff09\u3002\u521d\u59cb\u65f6\u706f\u6ce1\u6b63\u5e38\u5de5\u4f5c\uff0c\u800c\u4ece\u6b63\u5e38\u5230\u6545\u969c\u7684\u8f6c\u79fb\u65f6\u95f4\u670d\u4ece\u6307\u6570\u5206\u5e03\uff0c\u6545\u969c\u7387\u4e3a\u03bb\u3002\u5982\u679c\u6211\u4eec\u8bd5\u56fe\u7528\u79bb\u6563\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\uff08Discrete-Time Markov Chain\uff0cDTMC\uff09\u6765\u63cf\u8ff0\u8fd9\u4e00\u8fc7\u7a0b\uff0c\u5c31\u4f1a\u9047\u5230\u660e\u663e\u7684\u5c40\u9650\u6027\u3002<\/span><\/span><\/p>\n

    \u5728DTMC\u4e2d\uff0c\u72b6\u6001\u8f6c\u79fb\u53ea\u80fd\u53d1\u751f\u5728\u56fa\u5b9a\u7684\u65f6\u95f4\u6b65\u957f\u4e0a\uff08\u5982\u6bcf\u5c0f\u65f6\u3001\u6bcf\u5929\u7b49\uff09\uff0c\u800c\u706f\u6ce1\u7684\u6545\u969c\u5374\u53ef\u80fd\u5728\u4efb\u610f\u8fde\u7eed\u65f6\u95f4\u70b9\u53d1\u751f\u3002\u5982\u679c\u6211\u4eec\u5f3a\u884c\u91c7\u7528\u79bb\u6563\u65f6\u95f4\u5efa\u6a21\uff0c\u5c31\u5fc5\u987b\u4eba\u4e3a\u8bbe\u5b9a\u4e00\u4e2a\u65f6\u95f4\u95f4\u9694<\/span><\/span>\u0394<\/span><\/span>t<\/span><\/span>\uff0c\u5e76\u5047\u8bbe\u706f\u6ce1\u5728\u6bcf\u4e2a\u0394t\u5185\u4ee5\u6982\u7387<\/span><\/span>p<\/span><\/span>=<\/span><\/span>1<\/span><\/span>\u2212<\/span><\/span>e<\/span><\/span>\u2212<\/span><\/sup><\/span>\u03bb<\/span><\/span><\/sup>\u0394<\/span><\/span><\/sup>t<\/span><\/sup><\/span>\uff08\u8be6\u89c1\u8f6c\u79fb\u6982\u7387\u77e9\u9635\u4e0e\u8f6c\u79fb\u901f\u7387\u77e9\u9635\u5173\u7cfb\u4e00\u8282\uff09\u53d1\u751f\u6545\u969c\u3002\u7136\u800c\uff0c\u8fd9\u6837\u7684\u8fd1\u4f3c\u4e0d\u4ec5\u4f9d\u8d56\u4e8e<\/span><\/span>\u0394<\/span><\/span>t<\/span><\/span>\u7684\u9009\u62e9\uff0c\u8fd8\u5ffd\u7565\u4e86\u6545\u969c\u7684\u771f\u5b9e\u968f\u673a\u6027\u2014\u2014\u706f\u6ce1\u7684\u5bff\u547d\u4e0d\u53d7\u56fa\u5b9a\u65f6\u95f4\u6b65\u957f\u7684\u7ea6\u675f\uff0c\u6545\u969c\u53ef\u80fd\u53d1\u751f\u5728\u4efb\u4f55\u65f6\u523b\u3002<\/span><\/span><\/span><\/p>\n

    \u76f8\u6bd4\u4e4b\u4e0b\uff0cCTMC\u66f4\u51c6\u786e\u5730\u523b\u753b\u8fd9\u4e00\u8fc7\u7a0b\u3002\u706f\u6ce1\u7684\u6545\u969c\u88ab\u5efa\u6a21\u4e3a\u77ac\u65f6\u8f6c\u79fb\uff0c\u5176\u8f6c\u79fb\u901f\u7387\u77e9\u9635Q\uff08\u8be6\u89c1\u8f6c\u79fb\u901f\u7387\u77e9\u9635\u4e00\u8282\uff09\u76f4\u63a5\u7531\u6545\u969c\u7387\u03bb\u51b3\u5b9a\uff1a<\/span><\/span><\/p>\n

    <\/section>\n

    \u5176\u4e2d\uff0c\u7b2c\u4e00\u4e2a\u72b6\u6001\u4e3a\u5de5\u4f5c\uff0c\u7b2c\u4e8c\u4e2a\u72b6\u6001\u4e3a\u6545\u969c\u3002\u77e9\u9635\u4e2d\u7684\u7b2c\u4e00\u884c\u7b2c\u4e8c\u5217\u7684\u6570\u5b57<\/span><\/span>\u03bb<\/span><\/span>\u4ee3\u8868\u7684\u662f\u4efb\u610f\u77ac\u65f6\u706f\u6ce1\u7531\u6b63\u5e38\u53d8\u4e3a\u6545\u969c\u7684\u6982\u7387\uff0c\u800c\u7b2c\u4e00\u884c\u7b2c\u4e00\u5217\u7684\u6570\u5b57<\/span><\/span>\u2212<\/span><\/span>\u03bb<\/span><\/span>\u662f\u4e3a\u4e86\u4e0e\u6539\u884c\u53e6\u4e00\u4e2a\u6570\u5b57\u7684\u52a0\u548c\u4e3a0\u800c\u5b58\u5728\u7684\u3002\u5b83\u53ef\u8fd1\u4f3c\u4ee3\u8868\u706f\u6ce1\u72b6\u6001\u4e0d\u53d1\u751f\u8f6c\u79fb\u7684\u901f\u7387\u3002\u7b2c\u4e8c\u884c\u5168\u90e8\u4e3a0\uff0c\u8fd9\u8868\u793a\u8be5\u72b6\u6001\u4e3a\u5438\u6536\u6001\u3002\u56e0\u6b64\uff0c\u6574\u4e2a\u8fc7\u7a0b\u7684\u5b9a\u4e49\u65e0\u9700\u4eba\u4e3a\u5212\u5206\u65f6\u95f4\u6b65\u957f\uff0c\u81ea\u7136\u53cd\u6620\u4e86\u72b6\u6001\u7684\u8fde\u7eed\u65f6\u95f4\u968f\u673a\u8f6c\u79fb\u3002\u8fd9\u4e2a\u4f8b\u5b50\u8bf4\u660e\uff0c\u5f53\u7cfb\u7edf\u72b6\u6001\u53d8\u5316\u53d1\u751f\u5728\u4efb\u610f\u65f6\u95f4\u70b9\u4e14\u8f6c\u79fb\u65f6\u95f4\u670d\u4ece\u6307\u6570\u5206\u5e03\u65f6\uff0cDTMC\u4f1a\u56e0\u79bb\u6563\u5316\u800c\u5931\u771f\uff0c\u800cCTMC\u624d\u662f\u66f4\u672c\u8d28\u7684\u5efa\u6a21\u5de5\u5177\u3002\u706f\u6ce1\u6545\u969c\u7684\u8fde\u7eed\u65f6\u95f4\u7279\u6027\uff0c\u6b63\u662f\u5f15\u5165CTMC\u7684\u5fc5\u8981\u6027\u6240\u5728\u3002<\/span><\/span>
    <\/span><\/span><\/p>\n

    \u56de\u987e\u4e00\u4e0b\u4e4b\u524d\u7684\u5b9a\u4e49\uff0c\u6211\u4eec\u8bbe<\/span><\/span>S<\/span><\/span>\u662f\u72b6\u6001\u7a7a\u95f4\uff0c<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>}<\/span><\/span>=<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>|<\/span><\/span>t<\/span><\/span>\u2265<\/span><\/span>0<\/span><\/span>}<\/span><\/span>\u662f\u4ee5<\/span><\/span>S<\/span><\/span>\u4e3a\u72b6\u6001\u7a7a\u95f4\u7684\u8fde\u7eed\u65f6\u95f4\u968f\u673a\u8fc7\u7a0b\u3002<\/span><\/span><\/span><\/p>\n

    \u5b9a\u4e49\uff08\u8fde\u7eed\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\uff09<\/span><\/span><\/b>\uff1a\u5982\u679c\u5bf9\u4e8e\u4efb\u610f\u6b63\u6574\u6570<\/span><\/span>n<\/span><\/span>,<\/span><\/span>t<\/span><\/span>0<\/span><\/span><\/sub><<\/span><\/span>t<\/span><\/span>1<\/span><\/span><\/sub><<\/span><\/span>\u22ef<\/span><\/span><<\/span><\/span>t<\/span><\/span>n<\/span><\/sub><\/span>+<\/span><\/span><\/sub>1<\/span><\/sub><\/span>\u548c<\/span><\/span>i<\/span><\/span>,<\/span><\/span>j<\/span><\/span>,<\/span><\/span>i<\/span><\/span>0<\/span><\/span><\/sub>,<\/span><\/span>i<\/span><\/span>1<\/span><\/span><\/sub>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>i<\/span><\/span>n<\/span><\/sub><\/span>\u2212<\/span><\/span><\/sub>1<\/span><\/sub><\/span>\u2208<\/span><\/span>S<\/span><\/span>\uff0c\u6709<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u5219\u79f0<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>}<\/span><\/span>\u662f\u8fde\u7eed\u65f6\u95f4\u79bb\u6563\u72b6\u6001\u7684\u9a6c\u5c14\u79d1\u592b\u94fe\uff0c\u7b80\u79f0\u4e3a<\/span><\/span>\u8fde\u7eed\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\uff08CTMC\uff09<\/span><\/span><\/b>\u3002\u5176\u4e2d\u7684\u201c\u94fe\u201d\u6307\u72b6\u6001\u7a7a\u95f4<\/span><\/span>S<\/span><\/span>\u662f\u79bb\u6563\u7684\u3002<\/span><\/span><\/span><\/p>\n

    4.1 \u57fa\u672c\u6982\u5ff5<\/span><\/span><\/span><\/h3>\n

    4.1.1 \u4e0e\u79bb\u6563\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\u5bf9\u5e94\u7684\u57fa\u672c\u6982\u5ff5<\/span><\/span><\/span><\/h4>\n

    \u4e0e\u79bb\u6563\u60c5\u5f62\u4e2d\u4e00\u6837\uff0cCTMC\u5bf9\u4e8e\u4efb\u610f\u65f6\u95f4\u95f4\u9694\u90fd\u5177\u6709<\/span><\/span>\u5386\u53f2\u9057\u5fd8\u6027<\/span><\/span><\/b>\u3002\u5373\u5bf9\u4e8e<\/span><\/span>t<\/span><\/span>><\/span><\/span>0<\/span><\/span>\uff0c\u5df2\u77e5<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>=<\/span><\/span>i<\/span><\/span>\u7684\u6761\u4ef6\u4e0b\uff0c\u5c06\u6765\u968f\u673a\u8fc7\u7a0b<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>u<\/span><\/span>)<\/span><\/span>|<\/span><\/span>u<\/span><\/span>><\/span><\/span>t<\/span><\/span>}<\/span><\/span>\u4e0e\u8fc7\u53bb\u968f\u673a\u8fc7\u7a0b<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>v<\/span><\/span>)<\/span><\/span>|<\/span><\/span>0<\/span><\/span>\u2264<\/span><\/span>v<\/span><\/span><<\/span><\/span>t<\/span><\/span>}<\/span><\/span>\u72ec\u7acb\u3002<\/span><\/span><\/span>
    <\/span><\/span><\/p>\n

    \u548c\u79bb\u6563\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\u76f8\u540c\uff0c\u6211\u4eec\u5b9a\u4e49\u6ee1\u8db3\uff1a<\/span><\/span><\/span><\/sub>\uff0c<\/span><\/span><\/span>s<\/span><\/span>,<\/span><\/span>t<\/span><\/span>\u2265<\/span><\/span>0<\/span><\/span> \u7684CTMC\u5177\u6709<\/span><\/span>\u65f6\u9f50\u6027<\/span><\/span><\/b>\u3002\u6ce8\u610f\uff0c\u5982\u679c\u6ca1\u6709\u7279\u522b\u8bf4\u660e\uff0c\u90fd\u9ed8\u8ba4\u9a6c\u5c14\u79d1\u592b\u94fe\u5177\u6709\u65f6\u9f50\u6027\uff1b\u53ea\u6709\u5f53\u9a6c\u5c14\u79d1\u592b\u94fe\u4e0d\u6ee1\u8db3\u65f6\u9f50\u6027\u65f6\uff0c\u6211\u4eec\u624d\u4f1a\u5f3a\u8c03\u975e\u65f6\u9f50\u7684\u3002\u8fd9\u662f\u56e0\u4e3a\u6211\u4eec\u6781\u5c11\u4f1a\u9047\u5230\u975e\u65f6\u9f50\u7684\u9a6c\u5c14\u79d1\u592b\u94fe\u3002<\/span><\/span><\/span>
    <\/span><\/span><\/p>\n

    \u7531\u65f6\u9f50\u6027\uff0c\u53ef\u77e5<\/span><\/span>\u8f6c\u79fb\u6982\u7387<\/span><\/span><\/b>\u4e0e\u8d77\u59cb\u65f6\u95f4<\/span><\/span>s<\/span><\/span>\u65e0\u5173\uff1a<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u5e76\u4e14\u5f53\u5bf9\u4e8e\u4efb\u610f\u4e00\u4e2aCTMC\uff0c\u5728<\/span><\/span>t<\/span><\/span>=<\/span><\/span>0<\/span><\/span>\u65f6\uff0c\u8f6c\u79fb\u6982\u7387<\/span><\/span>p<\/span><\/span>i<\/span><\/sub><\/span>j<\/span><\/sub><\/span>(<\/span><\/span>0<\/span><\/span>)<\/span><\/span>\u662f\u56fa\u5b9a\u7684\uff0c\u90fd\u662f<\/span><\/span>\u03b4<\/span><\/span>\u51fd\u6570\uff1a<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u6709\u4e86\u4efb\u610f\u4e24\u4e2a\u72b6\u6001\u95f4\u7684\u8f6c\u79fb\u6982\u7387\uff0c\u6211\u4eec\u5c31\u80fd\u5b9a\u4e49<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>}<\/span><\/span>\u7684<\/span><\/span>\u8f6c\u79fb\u6982\u7387\u77e9\u9635<\/span><\/span><\/b>\u6765\u96c6\u4e2d\u63cf\u8ff0CTMC\u7684\u72b6\u6001\u662f\u5982\u4f55\u968f\u7740\u65f6\u95f4\u6539\u53d8\u7684\uff1a<\/span><\/span>P<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>=<\/span><\/span>(<\/span><\/span>p<\/span><\/span>i<\/span><\/sub><\/span>j<\/span><\/sub><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>)<\/span><\/span>i<\/span><\/sub><\/span>,<\/span><\/span><\/sub>j<\/span><\/span><\/sub>\u2208<\/span><\/span><\/sub>I<\/span><\/sub><\/span><\/span><\/p>\n

    \u6211\u4eec\u53ef\u4ee5\u5c06\u4e00\u4e2a\u65f6\u95f4\u95f4\u9694\u6bd4\u8f83\u957f\u7684\u8f6c\u79fb\u6982\u7387\u62c6\u5206\u4e3a\u4e24\u4e2a\u65f6\u95f4\u95f4\u9694\u6bd4\u8f83\u77ed\u7684\u8f6c\u79fb\u6982\u7387\uff0cChapman-Kolmogorov\u65b9\u7a0b\u7ed9\u51fa\u4e86\u4e24\u8005\u7684\u5173\u7cfb<\/span>\"image.png\"<\/span><\/sub><\/span>\u3002\u6216\u8005\u5199\u6210\u77e9\u9635\u5f62\u5f0f\u6709<\/span><\/span>
    <\/span><\/span><\/p>\n

    P<\/span><\/span>(<\/span><\/span>t<\/span><\/span>+<\/span><\/span>s<\/span><\/span>)<\/span><\/span>=<\/span><\/span>P<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>P<\/span><\/span>(<\/span><\/span>s<\/span><\/span>)<\/span><\/span>=<\/span><\/span>P<\/span><\/span>(<\/span><\/span>s<\/span><\/span>)<\/span><\/span>P<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span><\/span><\/span><\/p>\n

    \u53ea\u9700\u8981\u7ed9\u5b9a\u521d\u59cb\u72b6\u6001\u548c\u72b6\u6001\u6539\u53d8\u7684\u89c4\u5219\uff0c\u6211\u4eec\u5c31\u80fd\u552f\u4e00\u786e\u5b9a\u4e00\u4e2a\u8fde\u7eed\u65f6\u95f4\u9a6c\u5c14<\/span><\/span>\u79d1<\/span><\/span><\/span>\u592b\u94fe\u3002\u7528\u66f4\u4e25\u683c\u7684\u8bed\u8a00\u6765\u8868\u8ff0\uff0c<\/span><\/span><\/span><\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>\u7684\u6982\u7387\u5206\u5e03<\/span><\/span>p<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>\u7531\u8f6c\u79fb\u6982\u7387\u77e9\u9635<\/span><\/span>P<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>\u548c<\/span><\/span>X<\/span><\/span>(<\/span><\/span>0<\/span><\/span>)<\/span><\/span>\u7684\u6982\u7387\u5206\u5e03\uff0c\u5373\u521d\u59cb\u5206\u5e03<\/span><\/span>p<\/span><\/span>(<\/span><\/span>0<\/span><\/span>)<\/span><\/span>=<\/span><\/span>(<\/span><\/span>p<\/span><\/span>1<\/span><\/span><\/sub>(<\/span><\/span>0<\/span><\/span>)<\/span><\/span>,<\/span><\/span>p<\/span><\/span>2<\/span><\/span><\/sub>(<\/span><\/span>0<\/span><\/span>)<\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>)<\/span><\/span>\u552f\u4e00\u51b3\u5b9a\uff1a<\/span><\/span><\/span><\/p>\n

    p<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>=<\/span><\/span>p<\/span><\/span>(<\/span><\/span>0<\/span><\/span>)<\/span><\/span>P<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>\uff0c<\/span><\/span><\/span><\/p>\n

    \u5176\u4e2d<\/span><\/span>\"image.png\"<\/span><\/sub>\u3002<\/span><\/span><\/span><\/span><\/p>\n

    4.1.1 \u8fde\u7eed\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\u4e2d\u72ec\u6709\u7684\u6982\u5ff5<\/span><\/span><\/span><\/h4>\n
    4.1.2.1 \u8f6c\u79fb\u901f\u7387\u77e9\u9635<\/span><\/span><\/span><\/h5>\n

    \u5728CTMC\u4e2d\uff0c\u6211\u4eec\u51e0\u4e4e\u90fd\u662f\u7528<\/span><\/span>\u8f6c\u79fb\u901f\u7387\u77e9\u9635<\/span><\/span><\/b>\u6216\u8005<\/span><\/span>\u8f6c\u79fb\u5f3a\u5ea6\u77e9\u9635<\/span><\/span><\/b>\u6765\u63cf\u8ff0\u72b6\u6001\u8f6c\u79fb\u7684\u89c4\u5219\uff0c\u8bb0\u4f5c<\/span><\/span>Q<\/span><\/span>=<\/span><\/span>(<\/span><\/span>q<\/span><\/span>i<\/span><\/sub><\/span>j<\/span><\/sub><\/span>)<\/span><\/span>i<\/span><\/span>,<\/span><\/span>j<\/span><\/span>\u2208<\/span><\/span>S<\/span><\/span>\uff0c\u800c\u51e0\u4e4e\u4e0d\u7528\u8f6c\u79fb\u6982\u7387\u77e9\u9635\uff0c\u5176\u4e2d<\/span><\/span>q<\/span><\/span>i<\/span><\/sub><\/span>j<\/span><\/sub><\/span>\u8868\u793a\u4ecei\u72b6\u6001\u8f6c\u79fb\u5230j\u72b6\u6001\u7684\u901f\u7387\u3002\u56e0\u4e3aCTMC\u7684\u8f6c\u79fb\u6982\u7387\u77e9\u9635\u4e2d\u6709\u4e00\u4e2a\u65f6\u95f4\u53c2\u6570t\uff0c\u5bf9\u4e8e\u4e0d\u540c\u7684\u65f6\u95f4\u53c2\u6570\u8f6c\u79fb\u6982\u7387\u77e9\u9635\u7684\u5927\u5c0f\u4e0d\u540c\uff0c\u4f7f\u7528\u8d77\u6765\u5f88\u9ebb\u70e6\uff1b\u4f46\u5728\u65f6\u9f50\u6027\u7684\u524d\u63d0\u4e0b\uff0c\u8f6c\u79fb\u901f\u7387\u77e9\u9635\u662f\u4fdd\u6301\u4e0d\u53d8\u7684\uff0c\u4f7f\u7528\u8d77\u6765\u66f4\u52a0\u65b9\u4fbf\u3002\u8f6c\u79fb\u901f\u7387\u77e9\u9635\u5177\u6709\u4e0b\u9762\u4e09\u4e2a\u91cd\u8981\u7684\u6027\u8d28\uff1a<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u5e76\u4e14\u5bf9\u4e8e\u6709\u9650\u72b6\u6001\u7684CTMC\uff0c\u8f6c\u79fb\u6982\u7387\u77e9\u9635<\/span><\/span>P<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>\u662f\u8f6c\u79fb\u901f\u7387\u77e9\u9635<\/span><\/span>Q<\/span><\/span>\u7684\u77e9\u9635\u6307\u6570\u51fd\u6570\uff1a<\/span><\/span><\/span><\/p>\n

    <\/section>\n
    4.1.2.2 \u8f6c\u79fb\u901f\u7387\u77e9\u9635\u4e0e\u8f6c\u79fb\u6982\u7387\u77e9\u9635\u5173\u7cfb<\/span><\/span><\/span><\/h5>\n

    \u5728\u8fd9\u4e00\u8282\u6211\u4eec\u7ed9\u51fa\u8f6c\u79fb\u901f\u7387\u77e9\u9635\u548c\u8f6c\u79fb\u6982\u7387\u77e9\u9635\u4e4b\u95f4\u5173\u7cfb\u63a8\u5bfc\u601d\u8def\uff0c\u4e5f\u5c31\u662f\u4e3a\u4ec0\u4e48\u516c\u5f0f(<\/span><\/span>2<\/span><\/span><\/b>)\u6210\u7acb\u7684\u201c\u8bc1\u660e\u201d\uff0c\u4ec5\u4f9b\u611f\u5174\u8da3\u7684\u8bfb\u8005\u53c2\u8003\u3002<\/span><\/span><\/span><\/p>\n

    \u5148\u7ed9\u5b9a\u8f6c\u79fb\u6982\u7387\u77e9\u9635<\/span><\/span>P<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>\uff0c\u770b\u770b\u600e\u4e48\u80fd\u5f97\u5230\u8f6c\u79fb\u901f\u7387\u77e9\u9635<\/span><\/span>Q<\/span><\/span>\u3002\u5bf9\u4e8e\u79bb\u6563\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u4e00\u6b65\u8f6c\u79fb\u6982\u7387\u77e9\u9635<\/span><\/span>P<\/span><\/span>\u53ef\u4ee5\u552f\u4e00\u51b3\u5b9a<\/span><\/span>n<\/span><\/span>\u6b65\u8f6c\u79fb\u6982\u7387\u77e9\u9635\uff1a<\/span><\/span>P<\/span><\/span>(<\/span><\/span>n<\/span><\/span>)<\/span><\/span>=<\/span><\/span>P<\/span><\/span>n<\/span><\/span><\/sup>\u3002\u4f46\u662f\u5bf9\u4e8eCTMC\uff0c\u5e76\u4e0d\u5b58\u5728\u4e00\u4e2a\u6700\u5c0f\u7684\u6b63\u6570<\/span><\/span>\u03b4<\/span><\/span>\uff0c\u4f7f\u5f97\u5bf9\u4e8e\u4efb\u610f\u6b63\u6570<\/span><\/span>t<\/span><\/span>\uff0c\u5b58\u5728\u6b63\u6574\u6570<\/span><\/span>n<\/span><\/span>\uff0c\u6709<\/span><\/span>P<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>=<\/span><\/span>P<\/span><\/span>(<\/span><\/span>\u03b4<\/span><\/span>)<\/span><\/span>n<\/span><\/span><\/sup>\u6210\u7acb\u3002\u5bf9\u4e8e\u4efb\u610f<\/span><\/span>\u03f5<\/span><\/span>><\/span><\/span>0<\/span><\/span>\uff0c\u53ef\u4ee5\u8bc1\u660e<\/span><\/span>{<\/span><\/span>P<\/span><\/span>(<\/span><\/span>s<\/span><\/span>)<\/span><\/span>|<\/span><\/span>0<\/span><\/span><<\/span><\/span>s<\/span><\/span>\u2264<\/span><\/span>\u03f5<\/span><\/span>}<\/span><\/span>\u53ef\u4ee5\u51b3\u5b9a\u6240\u6709\u7684<\/span><\/span>P<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>\u3002\u8fd9\u5c31\u63d0\u793a\u6211\u4eec\uff0c<\/span><\/span>P<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>\u53ef\u80fd\u88ab<\/span><\/span>\"image.png\"<\/span><\/sub>\u552f\u4e00\u51b3\u5b9a\u3002\u8fd9\u4e2a\u6781\u9650\u662f\u5426\u5b58\u5728\u5462\uff1f<\/span><\/span><\/span><\/p>\n

    \u5728\u4efb\u4f55\u6709\u9650\u4e4b\u95f4\u5185\u53ea\u6709\u6709\u9650\u6b21\u8f6c\u79fb\u7684CTMC\uff0c\u79f0\u4f5c<\/span><\/span>\u89c4\u5219\u7684CTMC<\/span><\/span><\/b>\u3002\u4e3a\u4e86\u7b80\u5316\u8ba8\u8bba\uff0c\u4ee5\u540e\u65e0\u7279\u6b8a\u58f0\u660e\u65f6\uff0c\u8ba8\u8bba\u7684\u90fd\u662f\u89c4\u5219\u7684CTMC\u3002CTMC\u7684\u8f68\u8ff9\u662f\u9636\u68af\u51fd\u6570\uff0c\u5e76\u4e14\u5728\u8df3\u8dc3\u70b9\u662f\u53f3\u8fde\u7eed\u7684\uff0c\u66f4\u91cd\u8981\u7684\u662f\uff0c\u5b83\u7684\u6982\u7387\u8f6c\u79fb\u77e9\u9635<\/span><\/span>P<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>\u5728<\/span><\/span>t<\/span><\/span>=<\/span><\/span>0<\/span><\/span>\u65f6\uff0c\u5b58\u5728\u53f3\u5bfc\u6570\uff1a<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u7531\u4e8e<\/span><\/span>p<\/span><\/span>i<\/span><\/sub><\/span>j<\/span><\/sub><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>\u662fCTMC\u4ece<\/span><\/span>i<\/span><\/span>\u51fa\u53d1\uff0c\u7ecf\u8fc7<\/span><\/span>t<\/span><\/span>\u65f6\u95f4\u540e\u5904\u4e8e<\/span><\/span>j<\/span><\/span>\u7684\u6982\u7387\uff0c\u6240\u4ee5\u76f8\u5e94\u7684\uff0c\u6211\u4eec\u79f0<\/span><\/span>q<\/span><\/span>i<\/span><\/sub><\/span>j<\/span><\/sub><\/span>\u662f\u67d0\u4e00\u65f6\u523b\u4ece<\/span><\/span>i<\/span><\/span>\u51fa\u53d1\uff0c\u4e0b\u4e00\u6b65\u5411<\/span><\/span>j<\/span><\/span>\u8f6c\u79fb\u7684\u901f\u7387\u6216\u5f3a\u5ea6\uff0c\u7531\u6b64\u6211\u4eec\u5c31\u5f97\u5230\u4e86\u8f6c\u79fb\u901f\u7387\u77e9\u9635<\/span><\/span>Q<\/span><\/span>=<\/span><\/span>(<\/span><\/span>q<\/span><\/span>i<\/span><\/sub><\/span>j<\/span><\/sub><\/span>)<\/span><\/span>i<\/span><\/span>,<\/span><\/span>j<\/span><\/span>\u2208<\/span><\/span>S<\/span><\/span>\u3002<\/span><\/span><\/span><\/p>\n

    \u63a5\u7740\u8ba8\u8bba\u7ed9\u5b9a\u8f6c\u79fb\u901f\u7387\u77e9\u9635<\/span><\/span>Q<\/span><\/span>\uff0c\u5982\u4f55\u786e\u5b9a\u8f6c\u79fb\u6982\u7387\u77e9\u9635<\/span><\/span>P<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>\uff1f\u4e0b\u9762\u5c31\u901a\u8fc7Kolmogorov\u65b9\u7a0b\u56de\u7b54\u8fd9\u4e2a\u95ee\u9898\u3002<\/span><\/span><\/span><\/p>\n

    \u5148\u56de\u5fc6Chapman-Kolmogorov\u65b9\u7a0b\uff1a<\/span><\/span><\/p>\n

    <\/section>\n

    \u6211\u4eec\u5c06\u53f3\u8fb9\u79f0\u4e3a\u524d\uff0c\u628a\u5de6\u8fb9\u79f0\u4e3a\u540e\uff0c\u5728\u4e0a\u5f0f\u4e24\u8fb9\u5bf9\u540e\u9762\u7684<\/span><\/span>t<\/span><\/span>=<\/span><\/span>0<\/span><\/span>\u6c42\u5bfc\u6570\uff0c\u5f62\u5f0f\u4e0a\u5f97\u5230<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u5199\u6210\u77e9\u9635\u5f62\u5f0f\u5f97\u5230\uff1a<\/span><\/span><\/p>\n

    <\/section>\n

    \u6211\u4eec\u4e0a\u9762\u8fd9\u4e24\u4e2a\u5f0f\u5b50\u79f0\u4f5cCTMC\u7684<\/span><\/span>\u5411\u540e\u65b9\u7a0b<\/span><\/span><\/b>\u3002<\/span><\/span><\/span><\/p>\n

    \u540c\u6837\u53ef\u4ee5\u5bf9Chapman-Kolmogorov\u65b9\u7a0b\u524d\u9762\u7684<\/span><\/span>s<\/span><\/span>=<\/span><\/span>0<\/span><\/span>\u6c42\u5bfc\u6570\uff0c\u5f62\u5f0f\u4e0a\u5f97\u5230\uff1a<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u5199\u6210\u77e9\u9635\u5f62\u5f0f\u5f97\u5230<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u8fd9\u4e24\u4e2a\u5f0f\u5b50\u79f0\u4f5cCTMC\u7684<\/span><\/span>\u5411\u524d\u65b9\u7a0b<\/span><\/span><\/b>\u3002<\/span><\/span><\/span><\/p>\n

    \u5411\u524d\u65b9\u7a0b\u548c\u5411\u540e\u65b9\u7a0b\u7edf\u79f0\u4e3a<\/span><\/span>Kolmogorov\u65b9\u7a0b<\/span><\/span><\/b>\u3002<\/span><\/span><\/span>
    <\/span><\/span><\/p>\n

    \u5982\u679cCTMC<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>}<\/span><\/span>\u7684\u72b6\u6001\u7a7a\u95f4<\/span><\/span>S<\/span><\/span>\u662f\u6709\u9650\u96c6\u5408\uff0c\u5219\u79f0<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>}<\/span><\/span>\u662f<\/span><\/span>\u6709\u9650\u72b6\u6001CTMC<\/span><\/span><\/b>\u3002\u8bbe\u72b6\u6001\u7a7a\u95f4<\/span><\/span>S<\/span><\/span>\u6709<\/span><\/span>n<\/span><\/span>\u4e2a\u72b6\u6001\uff0c\u53d6\uff1a<\/span><\/span>\"image.png\"<\/span><\/sub>\u3002\u5219\u77e9\u9635<\/span><\/span>Q<\/span><\/span>t<\/span><\/span>\u5143\u7d20\u7684\u7edd\u5bf9\u503c\u90fd\u5c0f\u4e8e<\/span><\/span>q<\/span><\/span>t<\/span><\/span>,<\/span><\/span>(<\/span><\/span>Q<\/span><\/span>t<\/span><\/span>)<\/span><\/span>2<\/span><\/span><\/sup>\u5143\u7d20\u7684\u7edd\u5bf9\u503c\u90fd\u5c0f\u4e8e<\/span><\/span>n<\/span><\/span>(<\/span><\/span>q<\/span><\/span>t<\/span><\/span>)<\/span><\/span>2<\/span><\/span><\/sup>\uff0c\u2026\uff0c<\/span><\/span>(<\/span><\/span>Q<\/span><\/span>t<\/span><\/span>)<\/span><\/span>k<\/span><\/span><\/sup>\u5143\u7d20\u7684\u7edd\u5bf9\u503c\u90fd\u5c0f\u4e8e<\/span><\/span>n<\/span><\/span>k<\/span><\/sup><\/span>\u2212<\/span><\/span><\/sup>1<\/span><\/sup><\/span>(<\/span><\/span>q<\/span><\/span>t<\/span><\/span>)<\/span><\/span>k<\/span><\/span><\/sup>\uff0c\u4e8e\u662f\u4e0b\u9762\u7ea7\u6570\u4e2d\u6bcf\u4e2a\u5143\u7d20\u90fd\u6536\u655b<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u518d\u5173\u4e8et\u9010\u9879\u6c42\u5bfc\u5f97\u5230\uff1a<\/span><\/span><\/span>
    <\/span><\/span><\/p>\n

    <\/section>\n

    \u5229\u7528\u8fd9\u4e2a\u7b49\u5f0f\u4ee5\u53ca\u5411\u524d\u65b9\u7a0b\uff08\u4e5f\u53ef\u4ee5\u7528\u5411\u540e\u65b9\u7a0b\u8bc1\u660e\uff09\uff0c\u6211\u4eec\u53ef\u4ee5\u8bc1\u660e\u6709\u9650\u72b6\u6001CTMC\u7684\u8f6c\u79fb\u6982\u7387\u77e9\u9635<\/span><\/span>P<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>\u7531\u8f6c\u79fb\u901f\u7387\u77e9\u9635<\/span><\/span>Q<\/span><\/span>\u552f\u4e00\u51b3\u5b9a\uff0c\u5e76\u4e14\u6709\uff1a<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    4.1.3 \u5177\u4f53\u4f8b\u5b501: \u6392\u961f\u95ee\u9898<\/span><\/span><\/span><\/p>\n

    \u6211\u4eec\u53ef\u4ee5\u7528CTMC\u6765\u63cf\u8ff0\u8d85\u5e02\u6536\u94f6\u53f0\u7684\u987e\u5ba2\u6570\u3002\u5047\u8bbe\u8d85\u5e02\u4e2d\u53ea\u6709\u4e00\u4e2a\u6536\u94f6\u53f0\uff0c\u987e\u5ba2\u968f\u673a\u5230\u8fbe\u6536\u94f6\u53f0\u5e76\u6392\u961f\u7ed3\u8d26\u3002\u987e\u5ba2\u5230\u8fbe\u4eba\u6570\u7a0b\u670d\u4ece\u6cca\u677e\u8fc7\u7a0b\uff0c\u5e73\u5747\u5230\u8fbe\u901f\u7387\u4e3a<\/span><\/span>\u03bb<\/span><\/span>\uff1b\u800c\u670d\u52a1\u65f6\u95f4\u670d\u4ece\u6307\u6570\u5206\u5e03\uff0c\u5e73\u5747\u670d\u52a1\u901f\u7387\u4e3a<\/span><\/span>\u03bc<\/span><\/span>\uff0c\u72b6\u6001\u7a7a\u95f4\u4e3a\u6536\u94f6\u53f0\u524d\u7684\u987e\u5ba2\u6570\u76ee\uff0c\u6b64\u65f6\u6211\u4eec\u6709\u8f6c\u79fb\u901f\u7387\u77e9\u9635<\/span><\/span>Q\uff1a<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u5f88\u5bb9\u6613\u5b83\u6ee1\u8db3\u8f6c\u79fb\u901f\u7387\u77e9\u9635\u7684\u4e09\u6761\u6027\u8d28\u3002\u5176\u4e2d\u5bf9\u89d2\u7ebf\u5143\u7d20\u8868\u793a\u79bb\u5f00\u72b6\u6001n\u7684\u603b\u901f\u7387\u4e3a<\/span><\/span>\u2212<\/span><\/span>(<\/span><\/span>\u03bb<\/span><\/span>+<\/span><\/span>\u03bc<\/span><\/span>)<\/span><\/span>\uff0c\u8d85\u5bf9\u89d2\u7ebf\u8868\u793a\u987e\u5ba2\u5230\u8fbe\u7684\u901f\u7387\uff0c\u6b21\u5bf9\u89d2\u7ebf\u8868\u793a\u670d\u52a1\u5b8c\u6210\u7684\u901f\u7387\u3002\u77e9\u9635\u4e2d\u6bcf\u884c\u4ec5\u67093\u4e2a\u975e\u96f6\u5143\u7d20\uff0c\u4f53\u73b0\u201c\u5c40\u90e8\u72b6\u6001\u8f6c\u79fb\u201d\u7279\u6027\u3002\u5728\u4e0b\u4e00\u8282\u4e2d\uff0c\u6211\u4eec\u8fd8\u4f1a\u4ecb\u7ecd\u5982\u4f55\u6c42\u51fa\u5b83\u7684\u5e73\u7a33\u5206\u5e03\u3002<\/span><\/span><\/span>
    <\/span><\/p>\n

    \u4e0b\u9762\u6211\u4eec\u7ed9\u4e00\u4e2a\u7ef4\u5ea6\u66f4\u5c11\u7684\u4f8b\u5b50\u6765\u8bf4\u660e\u8f6c\u79fb\u901f\u7387\u77e9\u9635\u5728CTMC\u4e2d\u76f8\u5bf9\u4e8e\u8f6c\u79fb\u6982\u7387\u77e9\u9635\u7684\u4f18\u52bf\uff1a<\/span><\/span><\/p>\n

    <\/section>\n

    \u56fa\u5b9a\u65f6\u95f4\u95f4\u9694\u4e3a1\uff0c\u53ef\u4ee5\u901a\u8fc7\u6570\u503c\u8ba1\u7b97\u7684\u65b9\u5f0f\u8ba1\u7b97\u77e9\u9635\u6307\u6570\uff0c\u4ece<\/span><\/span>Q<\/span><\/span>\u5f97\u5230<\/span><\/span>P<\/span><\/span>(<\/span><\/span>1<\/span><\/span>)\uff1a<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u53ef\u4ee5\u770b\u5230\u7528<\/span><\/span>Q<\/span><\/span>\u6765\u63cf\u8ff0\u8fde\u7eed\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\u5f88\u76f4\u63a5\uff0c\u800c\u8bd5\u56fe\u7528<\/span><\/span>P<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>\u63cf\u8ff0\u8fde\u7eed\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u8bdd\uff0c\u5bf9\u6bcf\u4e00\u4e2at\u90fd\u8981\u8ba1\u7b97\u4e00\u6b21\u77e9\u9635\u6307\u6570\u3002<\/span><\/span>P<\/span><\/span>(<\/span><\/span>1<\/span><\/span>)<\/span><\/span>\u7b2c3\u884c<\/span><\/span>[<\/span><\/span>0<\/span><\/span>,<\/span><\/span>0<\/span><\/span>,<\/span><\/span>1<\/span><\/span>]<\/span><\/span>\uff0c\u8868\u793a\u4e00\u65e6\u8fdb\u5165\u7b2c\u4e09\u4e2a\u72b6\u6001\uff0c\u5c06\u6c38\u8fdc\u505c\u7559\uff0c\u8fd9\u4e5f\u53ef\u4ee5\u76f4\u63a5\u4ece<\/span><\/span>Q<\/span><\/span>\u4e2d\u76f4\u63a5\u8bfb\u51fa\u6765\u3002<\/span><\/span>Q<\/span><\/span>\u8fd8\u80fd\u5f88\u65b9\u4fbf\u5730\u63cf\u8ff0<\/span><\/span>P<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>\u4e2d\u65e0\u6cd5\u76f4\u63a5\u63cf\u8ff0\u7684\u6bd4\u4f8b\u6210\u5757\u6027\uff0c\u8be6\u89c1\u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u6210\u5757\u6027\u3002<\/span><\/span><\/span><\/p>\n

    4.1.4 \u5177\u4f53\u4f8b\u5b502: \u57fa\u56e0\u8c03\u63a7\u7f51\u7edc<\/span><\/span><\/span><\/h4>\n


    <\/span><\/p>\n

    \u5728\u7cfb\u7edf\u751f\u7269\u5b66\u4e2d\uff0c\u57fa\u56e0\u8c03\u63a7\u7f51\u7edc\uff08Gene Regulatory Network, GRN\uff09\u63cf\u8ff0\u4e86\u57fa\u56e0\u3001\u86cb\u767d\u8d28\u548c\u5176\u4ed6\u5206\u5b50\u4e4b\u95f4\u590d\u6742\u7684\u76f8\u4e92\u4f5c\u7528\uff0c\u5171\u540c\u63a7\u5236\u57fa\u56e0\u7684\u8868\u8fbe\u6c34\u5e73\u3002\u4e3a\u4e86\u5b9a\u91cf\u5206\u6790\u8fd9\u7c7b\u7f51\u7edc\u7684\u52a8\u6001\u884c\u4e3a\uff0c\u7279\u522b\u662f\u6d89\u53ca\u968f\u673a\u6027\u7684\u60c5\u51b5\uff08\u5982\u57fa\u56e0\u8868\u8fbe\u7684\u566a\u58f0\uff09\uff0c\u4eba\u4eec\u5e38\u7528CTMC\u4f5c\u4e3a\u6570\u5b66\u5efa\u6a21\u7684\u6846\u67b6\u3002\u8003\u8651\u4e00\u4e2a\u7b80\u5316\u7684\u4e24\u8282\u70b9\u57fa\u56e0\u8c03\u63a7\u7f51\u7edc\u6a21\u578b\uff0c\u5305\u542b\u4ee5\u4e0b\u4e24\u4e2a\u7ec4\u4ef6\u3002\u57fa\u56e0A\uff1a\u5177\u6709\u81ea\u53d1\u6fc0\u6d3b\uff08Spontaneous Activation\uff09\u548c\u57fa\u7840\u5931\u6d3b\uff08Basal Inactivation\uff09\u7684\u80fd\u529b\u3002\u5f53\u57fa\u56e0A\u5904\u4e8e\u6fc0\u6d3b\u72b6\u6001\u65f6\uff0c\u5b83\u80fd\u4fc3\u8fdb\u8f6c\u5f55\u8c03\u8282\u5b50R\uff08Transcriptional Regulator R\uff09\u7684\u8868\u8fbe\u6216\u6fc0\u6d3b\u3002\u8c03\u8282\u5b50R \uff1a\u5177\u6709\u57fa\u7840\u5931\u6d3b\uff08Basal Inactivation\uff09\u7684\u80fd\u529b\u3002\u5f53\u8c03\u8282\u5b50R\u5904\u4e8e\u6fc0\u6d3b\u72b6\u6001\u65f6\uff0c\u5b83\u4f1a\u6291\u5236\u57fa\u56e0A\u7684\u8868\u8fbe\u3002\u8fd9\u4e2a\u7f51\u7edc\u5f62\u6210\u4e86\u4e00\u4e2a\u8d1f\u53cd\u9988\u56de\u8def\uff1aA\u6fc0\u6d3bR\uff0c\u800cR\u6291\u5236A\u3002<\/span><\/span><\/p>\n

    <\/section>\n

    \u4e24\u8282\u70b9\u57fa\u56e0\u8c03\u63a7\u7f51\u7edc<\/span><\/span><\/p>\n

    \u6839\u636e\u57fa\u56e0A\u548c\u8c03\u8282\u5b50R\u7684\u72b6\u6001\u7ec4\u5408\uff0c\u6574\u4e2a\u7cfb\u7edf\u53ef\u4ee5\u5904\u5728\u4e00\u4e0b\u56db\u79cd\u79bb\u6563\u72b6\u6001\u4e4b\u4e00\uff1a<\/span><\/span>S<\/span><\/span>=<\/span><\/span>{<\/span><\/span>A<\/span><\/span>\u2212<\/span><\/span><\/sup>R<\/span><\/span>\u2212<\/span><\/span><\/sup>,<\/span><\/span>A<\/span><\/span>\u2212<\/span><\/span><\/sup>R<\/span><\/span>+<\/span><\/span><\/sup>,<\/span><\/span>A<\/span><\/span>+<\/span><\/span><\/sup>R<\/span><\/span>\u2212<\/span><\/span><\/sup>,<\/span><\/span>A<\/span><\/span>+<\/span><\/span><\/sup>R<\/span><\/span>+<\/span><\/span><\/sup>}<\/span><\/span>\u3002\u5e76\u5bf9\u5404\u72b6\u6001\u8f6c\u79fb\u53c2\u6570\u8fdb\u884c\u5efa\u6a21\uff0c<\/span><\/span>k<\/span><\/span>1<\/span><\/span><\/sub>\u662fA\u7684\u81ea\u53d1\u6fc0\u6d3b\u901f\u7387\uff0c<\/span><\/span>\u03b3<\/span><\/span>1<\/span><\/span><\/sub>\u662fA\u7684\u57fa\u7840\u5931\u6d3b\u901f\u7387\uff0c<\/span><\/span>\u03b3<\/span><\/span>2<\/span><\/span><\/sub>\u662fR\u4ecb\u5bfc\u7684A\u6291\u5236\u901f\u7387\uff0c<\/span><\/span>\u03bb<\/span><\/span>\u662fA\u6fc0\u6d3bR\u7684\u901f\u7387\uff0c<\/span><\/span>\u03bc<\/span><\/span>\u662fR\u7684\u57fa\u7840\u5931\u6d3b\u901f\u7387\u3002\u4e0b\u56fe\u5c55\u793a\u4e86\u8be5\u57fa\u56e0\u8c03\u63a7\u7f51\u7edc\u56db\u79cd\u72b6\u6001\u4e4b\u95f4\u53ef\u80fd\u7684\u8f6c\u79fb\u8def\u5f84\u53ca\u5176\u5bf9\u5e94\u7684\u901f\u7387\u53c2\u6570\uff1a<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u57fa\u56e0\u8c03\u63a7\u7f51\u7edc\u72b6\u6001\u8f6c\u79fb\u793a\u610f\u56fe<\/span><\/span><\/p>\n

    \u57fa\u4e8e\u4e0a\u8ff0\u72b6\u6001\u5b9a\u4e49\u548c\u8f6c\u79fb\u901f\u7387\uff0c\u53ef\u4ee5\u6784\u5efa\u63cf\u8ff0\u8be5CTMC\u6a21\u578b\u7684\u8f6c\u79fb\u901f\u7387\u77e9\u9635Q\u3002\u77e9\u9635\u7684\u884c\u548c\u5217\u6309\u987a\u5e8f\u5bf9\u5e94\u72b6\u6001<\/span><\/span>{<\/span><\/span>A<\/span><\/span>\u2212<\/span><\/span><\/sup>R<\/span><\/span>\u2212<\/span><\/span><\/sup>,<\/span><\/span>A<\/span><\/span>\u2212<\/span><\/span><\/sup>R<\/span><\/span>+<\/span><\/span><\/sup>,<\/span><\/span>A<\/span><\/span>+<\/span><\/span><\/sup>R<\/span><\/span>\u2212<\/span><\/span><\/sup>,<\/span><\/span>A<\/span><\/span>+<\/span><\/span><\/sup>R<\/span><\/span>+<\/span><\/span><\/sup>}<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u8fd9\u4e2a\u7b80\u5355\u7684\u4e24\u8282\u70b9\u6a21\u578b\u867d\u7136\u7ed3\u6784\u7b80\u6d01\uff0c\u901a\u8fc7\u8c03\u6574\u53c2\u6570\u7684\u76f8\u5bf9\u5927\u5c0f\uff0c\u80fd\u591f\u5c55\u73b0\u51fa\u53cc\u7a33\u6001\u6216\u8005\u9707\u8361\u7684\u884c\u4e3a\u3002\u540c\u6837\u7684\u5efa\u6a21\u601d\u8def\u53ef\u4ee5\u63a8\u5e7f\u5230\u66f4\u590d\u6742\u7684\u8c03\u63a7\u7f51\u7edc\u4e2d\uff0c\u4e3a\u7814\u7a76\u57fa\u56e0\u8868\u8fbe\u566a\u58f0\u3001\u7ec6\u80de\u547d\u8fd0\u51b3\u7b56\uff08\u5982\u53cc\u7a33\u6001\uff09\u3001\u751f\u7269\u8282\u5f8b\uff08\u5982\u9707\u8361\uff09\u7b49\u751f\u7269\u5b66\u73b0\u8c61\u63d0\u4f9b\u4e86\u5f3a\u5927\u7684\u5b9a\u91cf\u5206\u6790\u5de5\u5177\u3002<\/span><\/span><\/span><\/p>\n

    4.2 \u8f6c\u79fb\u8fc7\u7a0b\u548c\u6f14\u5316\u6027\u8d28<\/span><\/span><\/span><\/h3>\n

    4.2.1 \u72b6\u6001\u7684\u5212\u5206\u2014\u2014\u53ef\u8fbe\u3001\u4e92\u901a\u548c\u53ef\u7ea6\u6027<\/span><\/span><\/span><\/h4>\n

    \u548c\u79bb\u6563\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u4e00\u6837\uff0c\u4e3a\u4e86\u7814\u7a76CTMC\u7684\u6781\u9650\u5206\u5e03\u4ee5\u53ca\u5e73\u7a33\u5206\u5e03\u5b58\u5728\u7684\u6761\u4ef6\uff0c\u9700\u8981\u5f15\u5165\u53ef\u8fbe\u6982\u5ff5\u3002\u8bbe<\/span><\/span>S<\/span><\/span>\u662f<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>}<\/span><\/span>\u7684\u72b6\u6001\u7a7a\u95f4\uff0c\u5982\u679c\u5b58\u5728<\/span><\/span>t<\/span><\/span>><\/span><\/span>0<\/span><\/span>\u4f7f\u5f97<\/span><\/span>p<\/span><\/span>i<\/span><\/sub><\/span>j<\/span><\/sub><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>><\/span><\/span>0<\/span><\/span>\uff0c\u5219\u79f0<\/span><\/span>i<\/span><\/span>\u53ef\u8fbe<\/span><\/span><\/b>j<\/span><\/span>\uff0c\u8bb0\u4f5c<\/span><\/span>i<\/span><\/span>\u2192<\/span><\/span>j<\/span><\/span>\u3002\u5982\u679c<\/span><\/span>i<\/span><\/span>\u2192<\/span><\/span>j<\/span><\/span>\u4e14<\/span><\/span>j<\/span><\/span>\u2192<\/span><\/span>i<\/span><\/span>\uff0c\u5219\u79f0<\/span><\/span>i<\/span><\/span>,<\/span><\/span>j<\/span><\/span>\u4e92\u901a<\/span><\/span><\/b>\uff0c\u8bb0\u4f5c<\/span><\/span>i<\/span><\/span>\u2194<\/span><\/span>j<\/span><\/span>\u3002\u5982\u679c<\/span><\/span>S<\/span><\/span>\u7684\u6240\u6709\u72b6\u6001\u4e0d\u4e92\u901a\uff0c\u5219\u79f0<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>}<\/span><\/span>\u53ef\u7ea6<\/span><\/span><\/b>\u3002\u4e0a\u9762\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\uff0c\u6240\u6709\u72b6\u6001\u4e4b\u95f4\u5747\u5b58\u5728\u8f6c\u79fb\u8def\u5f84\uff0c\u56e0\u6b64\u6240\u6709\u72b6\u6001\u4e92\u901a\uff0c\u8be5CTMC\u662f\u4e0d\u53ef\u7ea6\u7684\u3002<\/span><\/span><\/span>
    <\/span><\/span><\/p>\n

    \u5b9e\u9645\u4e0aCTMC\u7684\u72b6\u6001\u5212\u5206\u95ee\u9898\u4e0eDTMC\u7684\u72b6\u6001\u5212\u5206\u95ee\u9898\u5b8c\u5168\u7b49\u4ef7\u3002\u5bf9\u4e8e\u4efb\u4f55<\/span><\/span>h<\/span><\/span>><\/span><\/span>0<\/span><\/span>\uff0c\u79f0<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>n<\/span><\/span>h<\/span><\/span>)<\/span><\/span>}<\/span><\/span>=<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>n<\/span><\/span>h<\/span><\/span>)<\/span><\/span>|<\/span><\/span>n<\/span><\/span>=<\/span><\/span>0<\/span><\/span>,<\/span><\/span>1<\/span><\/span>,<\/span><\/span>2<\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>}<\/span><\/span>\u4e3a<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>}<\/span><\/span>\u7684\u4e00\u4e2a<\/span><\/span>\u79bb\u6563\u9aa8\u67b6<\/span><\/span><\/b>\u6216<\/span><\/span>h\u9aa8\u67b6<\/span><\/span><\/b>\u3002\u5bb9\u6613\u770b\u51fa\uff0c<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>}<\/span><\/span>\u7684h\u9aa8\u67b6\u662f\u67091\u6b65\u8f6c\u79fb\u6982\u7387<\/span><\/span>\"image.png\"<\/span><\/sub>\u7684\u79bb\u6563\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\uff0c\u5176\u72b6\u6001\u7a7a\u95f4\u548c<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>}<\/span><\/span>\u7684\u72b6\u6001\u7a7a\u95f4\u76f8\u540c\u3002\u5e76\u4e14CTMC\u7684\u72b6\u6001\u5212\u5206\u5b8c\u5168\u7531h\u9aa8\u67b6\u7684\u72b6\u6001\u5212\u5206\u51b3\u5b9a\uff1a<\/span><\/span>i<\/span><\/span>\u2192<\/span><\/span>j<\/span><\/span>\u7684\u5145\u5206\u5fc5\u8981\u6761\u4ef6\u662f\u5bf9\u5b58\u5728\u4e00\u4e2ah\u9aa8\u67b6<\/span><\/span>i<\/span><\/span>\u2192<\/span><\/span>j<\/span><\/span>\u3002<\/span><\/span><\/span>
    <\/span><\/span><\/p>\n

    \u56e0\u4e3a\u603b\u6709<\/span><\/span>p<\/span><\/span>i<\/span><\/sub><\/span>i<\/span><\/sub><\/span>(<\/span><\/span>n<\/span><\/span>h<\/span><\/span>)<\/span><\/span>><\/span><\/span>0<\/span><\/span>\uff0c\u6240\u4ee5\u5bf9\u4e8eh\u9aa8\u67b6\u6765\u8bb2\uff0c\u4ece<\/span><\/span>i<\/span><\/span>\u51fa\u53d1\u53ef\u4ee5\u5728\u4efb\u610f\u65f6\u523b\u56de\u5230<\/span><\/span>i<\/span><\/span>\uff0c\u8bf4\u660e\u5b83\u7684\u6240\u6709\u72b6\u6001\u90fd\u662f<\/span><\/span>\u975e\u5468\u671f\u7684<\/span><\/span><\/b>\uff0c\u8be5\u6027\u8d28\u4e3a\u4ee5\u540e\u7684\u8ba8\u8bba\u5e26\u6765\u8bb8\u591a\u65b9\u4fbf\u3002<\/span><\/span>i<\/span><\/span>\u2192<\/span><\/span>j<\/span><\/span>\u7684\u5145\u5206\u5fc5\u8981\u6761\u4ef6\u662f\u5bf9\u5b58\u5728\u4e00\u4e2ah\u9aa8\u67b6<\/span><\/span>i<\/span><\/span>\u2192<\/span><\/span>j<\/span><\/span>\u3002<\/span><\/span><\/span><\/p>\n

    4.2.2 \u72b6\u6001\u5206\u5e03\u968f\u7740\u65f6\u95f4\u53d8\u5316\u4fdd\u6301\u7a33\u5b9a\u2014\u2014\u5e73\u7a33\u5206\u5e03<\/span><\/span><\/span><\/h4>\n

    \u7c7b\u4f3c\u79bb\u6563\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\uff0c\u53ef\u4ee5\u5f15\u5165CTMC\u7684\u5e73\u7a33\u5206\u5e03\u7684\u5b9a\u4e49\u3002\u8bbe<\/span><\/span>P<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>\u662f<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>}<\/span><\/span>\u7684\u8f6c\u79fb\u6982\u7387\u77e9\u9635\uff0c\u5982\u679c\u6982\u7387\u5206\u5e03\u5217<\/span><\/span>p<\/span><\/span>=<\/span><\/span>(<\/span><\/span>p<\/span><\/span>0<\/span><\/span><\/sub>,<\/span><\/span>p<\/span><\/span>1<\/span><\/span><\/sub>,<\/span><\/span>\u2026<\/span><\/span>)<\/span><\/span>\u6ee1\u8db3<\/span><\/span>p<\/span><\/span>=<\/span><\/span>p<\/span><\/span>P<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>\uff0c\u5219\u79f0<\/span><\/span>(<\/span><\/span>p<\/span><\/span>)<\/span><\/span>\u662f<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>}<\/span><\/span>\u7684<\/span><\/span>\u5e73\u7a33\u5206\u5e03<\/span><\/span><\/b>\uff0c\u6216\u5e73\u7a33\u4e0d\u53d8\u5206\u5e03\u3002<\/span><\/span><\/span>
    <\/span><\/p>\n

    \u66f4\u8fdb\u4e00\u6b65\uff0c\u6211\u4eec\u53ef\u4ee5\u5b9a\u4e49\u5e73\u7a33\u8fc7\u7a0b\u3002<\/span><\/span><\/p>\n

    \u5b9a\u4e49\uff08CTMC\u7684\u5e73\u7a33\u8fc7\u7a0b\uff09<\/span><\/span><\/b>\uff1a\u5982\u679c\u5bf9\u4e8e\u4efb\u4f55<\/span><\/span>n<\/span><\/span>\u2265<\/span><\/span>1<\/span><\/span>,<\/span><\/span>s<\/span><\/span>><\/span><\/span>0<\/span><\/span>,<\/span><\/span>0<\/span><\/span>\u2264<\/span><\/span>t<\/span><\/span>0<\/span><\/span><\/sub><<\/span><\/span>t<\/span><\/span>1<\/span><\/span><\/sub><<\/span><\/span>\u22ef<\/span><\/span><<\/span><\/span>t<\/span><\/span>n<\/span><\/span><\/sub>\u968f\u673a\u5411\u91cf<\/span><\/span>(<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>0<\/span><\/span><\/sub>+<\/span><\/span>s<\/span><\/span>)<\/span><\/span>,<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>1<\/span><\/span><\/sub>+<\/span><\/span>s<\/span><\/span>)<\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>n<\/span><\/span><\/sub>+<\/span><\/span>s<\/span><\/span>)<\/span><\/span>)<\/span><\/span>\u548c<\/span><\/span>(<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>0<\/span><\/span><\/sub>)<\/span><\/span>,<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>1<\/span><\/span><\/sub>)<\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>n<\/span><\/span><\/sub>)<\/span><\/span>)<\/span><\/span>\u540c\u5206\u5e03\uff0c\u5219\u79f0<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>}<\/span><\/span>\u662f<\/span><\/span>\u5e73\u7a33\u8fc7\u7a0b<\/span><\/span><\/b>\u3002\u663e\u7136\uff0c\u5f53\u521d\u59cb\u5206\u5e03\u4e3a\u5e73\u7a33\u5206\u5e03<\/span><\/span>p<\/span><\/span>\u65f6\uff0c<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>}<\/span><\/span>\u4e3a\u5e73\u7a33\u8fc7\u7a0b\u3002<\/span><\/span><\/span>
    <\/span><\/span><\/p>\n

    \u524d\u9762\u6211\u4eec\u63d0\u5230\u4e86\uff0c\u5728\u5b9e\u9645\u95ee\u9898\u4e2d\u6211\u4eec\u5e38\u7528\u8f6c\u79fb\u901f\u7387\u77e9\u9635<\/span><\/span>Q<\/span><\/span>\u63cf\u8ff0CTMC\uff0c\u800c\u4e0d\u4f1a\u7528\u8f6c\u79fb\u6982\u7387\u77e9\u9635<\/span><\/span>P<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>\u3002\u6240\u4ee5\u6700\u597d\u80fd\u4ece\u8f6c\u79fb\u901f\u7387\u77e9\u9635<\/span><\/span>Q<\/span><\/span>\u5165\u624b\u7684\u5230\u5e73\u7a33\u5206\u5e03\u3002\u5e73\u7a33\u5206\u5e03\u7684\u5b9a\u4e49<\/span><\/span>p<\/span><\/span>P<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>=<\/span><\/span>p<\/span><\/span>\u63d0\u793a\u6211\u4eec\uff0c\u5f62\u5f0f\u4e0a\u5173\u4e8et\u6c42\u5bfc\u5e94\u5f53\u6709<\/span><\/span>p<\/span><\/span>Q<\/span><\/span>=<\/span><\/span>p<\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>|<\/span><\/span>t<\/span><\/span>=<\/span><\/span>0<\/span><\/span>=<\/span><\/span>p<\/span><\/span>\u2032<\/span><\/span>=<\/span><\/span>0<\/span><\/span>\u3002\u4e8b\u5b9e\u4e0a\uff0c\u5982\u679c\u4e92\u901a\u7684CTMC<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>}<\/span><\/span>\u6709\u8f6c\u79fb\u901f\u7387\u77e9\u9635<\/span><\/span>Q<\/span><\/span>\uff0c\u5219<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>}<\/span><\/span>\u6709\u552f\u4e00\u7684\u5e73\u7a33\u5206\u5e03\u7684\u5145\u5206\u5fc5\u8981\u6761\u4ef6\u662f\u4e0b\u9762\u65b9\u7a0b\u7ec4\u6709\u552f\u4e00\u975e\u8d1f\u89e3\uff0c\u8fd9\u65f6<\/span><\/span>p<\/span><\/span>\u662f<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>}<\/span><\/span>\u7684\u552f\u4e00\u5e73\u7a33\u5206\u5e03\uff1a<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    4.2.3 \u9006\u5411\u8fc7\u7a0b\u548c\u6b63\u5411\u8fc7\u7a0b\u540c\u5206\u5e03\u2014\u2014\u5e73\u7a33\u53ef\u9006\u5206\u5e03<\/span><\/span><\/span><\/h4>\n

    \u5728\u79bb\u6563\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe<\/span><\/span>{<\/span><\/span>X<\/span><\/span>n<\/span><\/span><\/sub>}<\/span><\/span>\u4e2d\uff0c\u5982\u679c\u6982\u7387\u5206\u5e03<\/span><\/span>\u03c0<\/span><\/span>=<\/span><\/span>(<\/span><\/span>\u03c0<\/span><\/span>0<\/span><\/span><\/sub>,<\/span><\/span>\u03c0<\/span><\/span>1<\/span><\/span><\/sub>,<\/span><\/span>\u2026<\/span><\/span>)<\/span><\/span>\u6ee1\u8db3\u7ec6\u81f4\u5e73\u8861\u6761\u4ef6<\/span><\/span>\u03c0<\/span><\/span>i<\/span><\/span><\/sub><\/span>p<\/span><\/span>i<\/span><\/sub><\/span>j<\/span><\/sub><\/span>=<\/span><\/span>\u03c0<\/span><\/span>j<\/span><\/span><\/sub>p<\/span><\/span>j<\/span><\/sub><\/span>i<\/span><\/sub><\/span>\uff0c\u5219\u79f0<\/span><\/span>\u03c0<\/span><\/span>\u662f<\/span><\/span>{<\/span><\/span>X<\/span><\/span>n<\/span><\/span><\/sub>}<\/span><\/span>\u7684\u5e73\u7a33\u53ef\u9006\u5206\u5e03\uff0c\u4ee5\u5e73\u7a33\u53ef\u9006\u5206\u5e03<\/span><\/span>\u03c0<\/span><\/span>\u4e3a\u521d\u59cb\u5206\u5e03\u7684<\/span><\/span>{<\/span><\/span>X<\/span><\/span>n<\/span><\/span><\/sub>}<\/span><\/span>\u79f0\u4e3a\u5e73\u7a33\u53ef\u9006\u7684\u8fc7\u7a0b\u3002CTCM\u4e2d\u4e5f\u6709\u7c7b\u4f3c\u7ed3\u8bba\u3002<\/span><\/span><\/span>
    <\/span><\/b><\/span><\/p>\n

    \u5b9a\u4e49\uff08\u7ec6\u81f4\u5e73\u8861\u6761\u4ef6\uff09<\/span><\/span><\/b>\uff1a\u8bbe<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>}<\/span><\/span>\u7684\u8f6c\u79fb\u6982\u7387\u77e9\u9635\u662f<\/span><\/span>P<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>\uff0c<\/span><\/span>p<\/span><\/span>=<\/span><\/span>(<\/span><\/span>p<\/span><\/span>0<\/span><\/span><\/sub>,<\/span><\/span>p<\/span><\/span>1<\/span><\/span><\/sub>,<\/span><\/span>\u2026<\/span><\/span>)<\/span><\/span>\u662f\u6982\u7387\u5206\u5e03\uff0c\u79f0<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u4e3a<\/span><\/span>\u7ec6\u81f4\u5e73\u8861\u6761\u4ef6<\/span><\/span><\/b>\uff0c\u82e5<\/span><\/span>p<\/span><\/span>\u6ee1\u8db3\u7ec6\u81f4\u5e73\u8861\u6761\u4ef6\uff0c\u5219\u79f0<\/span><\/span>p<\/span><\/span>\u4e3a<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>}<\/span><\/span>\u7684<\/span><\/span>\u5e73\u7a33\u53ef\u9006\u5206\u5e03<\/span><\/span><\/b>\u6216<\/span><\/span>\u53ef\u9006\u5206\u5e03<\/span><\/span><\/b>\u3002\u5f53<\/span><\/span>p<\/span><\/span>\u4e3a<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>}<\/span><\/span>\u7684\u5e73\u7a33\u53ef\u9006\u5206\u5e03\u65f6\uff0c<\/span><\/span>p<\/span><\/span>\u4e5f\u662f<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>}<\/span><\/span>\u7684\u5e73\u7a33\u5206\u5e03\u3002<\/span><\/span><\/span><\/p>\n


    <\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>}<\/span><\/span>\u4ee5<\/span><\/span>p<\/span><\/span>\u4e3a\u521d\u59cb\u5206\u5e03\u65f6\uff0c<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>}<\/span><\/span>\u662f<\/span><\/span>\u5e73\u7a33\u53ef\u9006\u8fc7\u7a0b<\/span><\/span><\/b>\u3002\u6b64\u65f6\u5bf9\u4e8e\u4efb\u4f55<\/span><\/span>0<\/span><\/span>\u2264<\/span><\/span>t<\/span><\/span>1<\/span><\/span><\/sub><<\/span><\/span>t<\/span><\/span>2<\/span><\/span><\/sub><<\/span><\/span>\u22ef<\/span><\/span><<\/span><\/span>t<\/span><\/span>n<\/span><\/span><\/sub><<\/span><\/span>T<\/span><\/span>\uff0c\u968f\u673a\u5411\u91cf<\/span><\/span>(<\/span><\/span>X<\/span><\/span>(<\/span><\/span>T<\/span><\/span>\u2212<\/span><\/span>t<\/span><\/span>1<\/span><\/span><\/sub>)<\/span><\/span>,<\/span><\/span>X<\/span><\/span>(<\/span><\/span>T<\/span><\/span>\u2212<\/span><\/span>t<\/span><\/span>2<\/span><\/span><\/sub>)<\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>X<\/span><\/span>(<\/span><\/span>T<\/span><\/span>\u2212<\/span><\/span>t<\/span><\/span>n<\/span><\/span><\/sub>)<\/span><\/span>)<\/span><\/span>\u548c<\/span><\/span>(<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>1<\/span><\/span><\/sub>)<\/span><\/span>,<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>2<\/span><\/span><\/sub>)<\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>n<\/span><\/span><\/sub>)<\/span><\/span>)<\/span><\/span>\u540c\u5206\u5e03\u3002\u5982\u679c<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>}<\/span><\/span>\u662f\u5e73\u7a33\u53ef\u9006\u8fc7\u7a0b\uff0c\u5219\u53ef\u4ee5\u628a\u65f6\u95f4\u65b9\u5411\u5012\u8fc7\u6765\u89c2\u6d4b\u8fd9\u4e2a\u8fc7\u7a0b\uff0c\u5e76\u4e14\u5012\u8fc7\u6765\u89c2\u6d4b\u5230\u7684\u8fd9\u4e2a\u8fc7\u7a0b\u4e5f\u662fCTMC\uff0c\u8f6c\u79fb\u6982\u7387\u548c\u539f\u6765\u7684\u8fc7\u7a0b\u4e00\u6837\u3002<\/span><\/span><\/span>
    <\/span><\/span><\/p>\n

    \u540c\u6837\u7684\uff0c\u7531\u4e8e\u5b9e\u9645\u95ee\u9898\u4e2d\u5e38\u7528\u8f6c\u79fb\u901f\u7387\u77e9\u9635<\/span><\/span>Q<\/span><\/span>\u63cf\u8ff0CTMC\uff0c\u800c\u4e0d\u662f<\/span><\/span>P<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>\uff0c\u6240\u4ee5\u6211\u4eec\u4e5f\u60f3\u4ece\u8f6c\u79fb\u901f\u7387\u77e9\u9635<\/span><\/span>Q<\/span><\/span>\u5165\u624b\u5f97\u5230\u5e73\u7a33\u53ef\u9006\u5206\u5e03\u3002\u5bf9\u4e0a\u9762\u7ed9\u51fa\u7684\u7ec6\u81f4\u5e73\u8861\u6761\u4ef6\u7684<\/span><\/span>t<\/span><\/span>\u6c42\u5bfc\u6570\uff0c\u5e76\u5229\u7528<\/span>\"image.png\"<\/span>\u5f97\u5230<\/span><\/span><\/span><\/p>\n

    p<\/span>i<\/span><\/sub>q<\/span>i<\/span><\/sub>j<\/span><\/sub>=<\/span>p<\/span>j<\/span><\/sub>q<\/span>j<\/span><\/sub>i<\/span><\/sub>,<\/span> <\/span>i<\/span>,<\/span>j<\/span>\u2208<\/span>S <\/span>\uff082\uff09<\/span><\/span><\/span><\/span><\/p>\n

    \u4e8b\u5b9e\u4e0a\uff0c\u82e5<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>}<\/span><\/span>\u4e92\u901a\uff0c\u6709\u5e76\u4e14\u5b58\u5728\u5e73\u7a33\u5206\u5e03\uff0c\u5219\u7ec6\u81f4\u5e73\u8861\u6761\u4ef6\u548c\u8fd9\u4e2a\u65b9\u7a0b\u6709\u76f8\u540c\u7684\u975e\u8d1f\u89e3\uff0c\u6ee1\u8db3\u8fd9\u4e2a\u65b9\u7a0b\u7684\u6982\u7387\u5206\u5e03<\/span><\/span>p<\/span><\/span>\u4e5f\u662f<\/span><\/span>{<\/span><\/span>X<\/span><\/span>(<\/span><\/span>t<\/span><\/span>)<\/span><\/span>}<\/span><\/span>\u7684\u5e73\u7a33\u53ef\u9006\u5206\u5e03\uff0c\u6240\u4ee5\u6211\u4eec\u4e5f\u5e38\u5e38\u79f0\u516c\u5f0f\uff082\uff09\u4e3aCTMC\u7684\u7ec6\u81f4\u5e73\u8861\u6761\u4ef6\u3002<\/span><\/span><\/span><\/p>\n

    4.1.4 \u5177\u4f53\u4f8b\u5b50<\/span><\/span><\/span><\/h4>\n

    \u5bf9\u4e8e\u4e00\u4e2a\u8f6c\u79fb\u901f\u7387\u77e9\u9635\u5982\u4e0b\u7684CTMC<\/span><\/span><\/p>\n

    <\/section>\n

    \u901a\u8fc7\u89e3\u5e73\u7a33\u5206\u5e03\u4e00\u8282\u4e2d\u7ed9\u51fa\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\uff0c\u53ef\u4ee5\u5f97\u5230CTMC\u7684\u552f\u4e00\u5e73\u7a33\u5206\u5e03<\/span><\/span>\"image.png\"<\/span><\/sub>\u3002\u5e73\u7a33\u5206\u5e03\u662f\u4e0d\u662f\u53ef\u9006\u7684\u5462\uff1f\u4ee5\u72b6\u60010\u548c1\u4e3a\u4f8b\uff0c\u6709<\/span><\/span>\"image.png\"<\/span><\/sub>\uff0c\u4e0d\u6ee1\u8db3\u516c\u5f0f\uff082\uff09\u4e2d\u7684\u7ec6\u81f4\u5e73\u8861\u6761\u4ef6\uff0c\u6240\u4ee5\u8fd9\u4e2aCTMC\u662f\u4e0d\u53ef\u9006\u7684\u3002<\/span><\/span>
    <\/span><\/p>\n

    \u800c\u5bf9\u4e8e\u53e6\u4e00\u4e2a\u8f6c\u79fb\u901f\u7387\u77e9\u9635\uff1a<\/span><\/span><\/p>\n

    <\/section>\n

    \u7528\u540c\u6837\u7684\u65b9\u6cd5\u7b97\u51fa\u5b83\u7684\u5e73\u7a33\u5206\u5e03\u4e3a\u5747\u5300\u5206\u5e03<\/span><\/span>\"image.png\"<\/span><\/sub>\uff0c\u5e76\u4e14\u5b83\u6ee1\u8db3\u516c\u5f0f\uff082\uff09\u4e2d\u7684\u7ec6\u81f4\u5e73\u8861\u6761\u4ef6\uff0c\u6240\u4ee5<\/span><\/span>p<\/span><\/span>\u662f\u8be5\u8f6c\u79fb\u901f\u7387\u77e9\u9635\u7684\u5e73\u7a33\u53ef\u9006\u5206\u5e03\u3002<\/span><\/span><\/span><\/p>\n

    \u6211\u4eec\u518d\u56de\u5230\u6700\u5f00\u59cb\u7ed9\u51fa\u7684\u8d85\u5e02\u6392\u961f\u7684\u4f8b\u5b50\uff0c\u5b83\u7684\u8f6c\u79fb\u901f\u7387\u77e9\u9635\u4e3a\uff1a<\/span><\/span><\/p>\n

    <\/section>\n

    \u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u6c42\u89e3\u5e73\u7a33\u5206\u5e03\u6ee1\u8db3\u7684\u65b9\u7a0b\u7ec4\uff0c\u4f1a\u53d1\u73b0\u53ea\u6709\u5f53<\/span><\/span>\u03bb<\/span><\/span><<\/span><\/span>\u03bc<\/span><\/span>\u65f6\uff0c\u5e73\u7a33\u5206\u5e03\u7684\u65b9\u7a0b\u7ec4\u624d\u6709\u552f\u4e00\u975e\u8d1f\u89e3\uff0c\u53ef\u4ee5\u89e3\u5f97<\/span><\/span>p<\/span><\/span>n<\/span><\/span><\/sub>=<\/span><\/span>(<\/span><\/span>1<\/span><\/span>\u2212<\/span><\/span>\u03c1<\/span><\/span>)<\/span><\/span>\u03c1<\/span><\/span>n<\/span><\/span><\/sup>,<\/span><\/span>\u03c1<\/span><\/span>=<\/span><\/span>\u03bb<\/span><\/span>\/<\/span><\/span>\u03bc<\/span><\/span>\uff0c\u5f53<\/span><\/span>\u03c1<\/span><\/span>\u2265<\/span><\/span>1<\/span><\/span>\u65f6\uff0c\u961f\u4f0d\u4f1a\u65e0\u9650\u589e\u957f\uff0c\u4e0d\u5b58\u5728\u7a33\u6001\u5206\u5e03\u3002\u8fd9\u4e2a\u5e73\u7a33\u5206\u5e03\u662f\u5426\u662f\u53ef\u9006\u7684\u5462\uff1f\u7531\u4e8e\u8fd9\u4e2aCTMC\u7684\u72b6\u6001\u662f\u5c40\u90e8\u8f6c\u79fb\u7684\uff0c\u6211\u4eec\u53ef\u4ee5\u5bf9\u516c\u5f0f\uff082\uff09\u4e2d\u7684\u7ec6\u81f4\u5e73\u8861\u6761\u4ef6\u8fdb\u884c\u5c40\u90e8\u9a8c\u8bc1\uff1a<\/span><\/span>p<\/span><\/span>n<\/span><\/span><\/sub>\u03bb<\/span><\/span>=<\/span><\/span>p<\/span><\/span>n<\/span><\/sub><\/span>+<\/span><\/span><\/sub>1<\/span><\/sub><\/span>\u03bc<\/span><\/span>\uff0c\u4ee3\u5165<\/span><\/span>p<\/span><\/span>n<\/span><\/span><\/sub>=<\/span><\/span>(<\/span><\/span>1<\/span><\/span>\u2212<\/span><\/span>\u03c1<\/span><\/span>)<\/span><\/span>\u03c1<\/span><\/span>n<\/span><\/span><\/sup>\u53ef\u77e5\u7b49\u5f0f\u6210\u7acb\uff0c\u6240\u4ee5\u5b83\u7684\u5e73\u7a33\u5206\u5e03<\/span><\/span>p<\/span><\/span>\u662f\u53ef\u9006\u7684\u3002<\/span><\/span><\/span><\/p>\n

    4.3 \u8fde\u7eed\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\u7684\u7c97\u7c92\u5316<\/span><\/span><\/span><\/h3>\n

    \u5728\u8fde\u7eed\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\u4e2d\u7c97\u7c92\u5316\u7684\u672c\u8d28\u662f\u6784\u9020\u6620\u5c04<\/span><\/span>\u03a6<\/span><\/span>:<\/span><\/span>S<\/span><\/span>\u2192<\/span><\/span>S<\/span><\/span>\u2032<\/span><\/span>\uff0c\u5c06\u539f\u59cb\u72b6\u6001\u7a7a\u95f4<\/span><\/span>S<\/span><\/span>\u5212\u5206\u4e3a\u4e92\u4e0d\u76f8\u4ea4\u7684\u533a\u5757<\/span><\/span>{<\/span><\/span>B<\/span><\/span>1<\/span><\/span><\/sub>,<\/span><\/span>B<\/span><\/span>2<\/span><\/span><\/sub>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>B<\/span><\/span>m<\/span><\/span><\/sub>}<\/span><\/span>\uff0c\u5e76\u5b9a\u4e49\u65b0\u7684\u8f6c\u79fb\u901f\u7387\u77e9\u9635<\/span><\/span>Q<\/span><\/span>\u2032<\/span><\/span>\u63cf\u8ff0\u533a\u5757\u95f4\u7684\u8f6c\u79fb\u52a8\u529b\u5b66\u3002\u7406\u60f3\u7684\u7c97\u7c92\u5316\u9700\u8981\u6ee1\u8db3\u4e09\u4e2a\u8981\u6c42\uff1a1. \u7ef4\u5ea6\u538b\u7f29\uff08<\/span><\/span>|<\/span><\/span>S<\/span><\/span>\u2032<\/span><\/span>|<\/span><\/span><<\/span><\/span>|<\/span><\/span>S<\/span><\/span>|<\/span><\/span>\uff09\uff1b2. \u4fe1\u606f\u4fdd\u7559\uff08\u5173\u952e\u7edf\u8ba1\u91cf\u4e0e\u52a8\u529b\u5b66\u7279\u5f81\u4e0d\u53d8\uff09\uff1b3. \u7ed3\u6784\u4e00\u81f4\u6027\uff08\u7c97\u7c92\u5ea6\u8fc7\u7a0b\u4ecd\u7136\u4e3aCTMC\uff09\u3002\u5177\u4f53\u5b9e\u73b0\u4e2d\uff0c\u4e0d\u540c\u65b9\u6cd5\u4fa7\u91cd\u4e0d\u540c\u76ee\u6807\uff0c\u5f62\u6210\u591a\u6837\u5316\u7684\u6280\u672f\u8def\u7ebf\u3002<\/span><\/span><\/span><\/p>\n

    4.3.1 \u5f3a\u6210\u5757\u6027<\/span><\/span><\/span><\/h4>\n

    \u5f53CTMC\u6ee1\u8db3<\/span><\/span>\u5f3a\u6210\u5757\u6027<\/span><\/span><\/b>\u65f6\uff0c\u5373\u5b58\u5728\u72b6\u6001\u5212\u5206\u4f7f\u5f97\u4efb\u610f\u533a\u5757<\/span><\/span>B<\/span><\/span>k<\/span><\/span><\/sub>\u5185\u7684\u72b6\u6001\u5bf9\u5916\u7684\u96c6\u4f53\u8f6c\u79fb\u901f\u7387\u4e00\u81f4\u3002\u6570\u5b66\u4e0a\uff0c\u5212\u5206<\/span><\/span>{<\/span><\/span>B<\/span><\/span>k<\/span><\/span><\/sub>}<\/span><\/span>\u9700\u6ee1\u8db3\uff1a<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u6b64\u65f6\u7c97\u7c92\u5316\u540e\u7684\u8f6c\u79fb\u901f\u7387\u77e9\u9635<\/span><\/span>Q<\/span><\/span>\u2032<\/span><\/span>\u7684\u5143\u7d20\u4e3a\uff1a<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u5f3a\u6210\u5757\u6027\u786e\u4fdd\u4e86\u533a\u5757\u5185\u72b6\u6001\u7684\u4e0d\u53ef\u533a\u5206\u6027\uff0c\u7c97\u7c92\u5316\u8fc7\u7a0b\u4e25\u683c\u4fdd\u6301\u77ac\u6001\u4e0e\u7a33\u6001\u5206\u5e03<\/span><\/span>[12]<\/span><\/span>\u3002<\/span><\/span><\/span><\/p>\n

    4.3.2 \u65f6\u95f4\u5c3a\u5ea6\u5206\u89e3<\/span><\/span><\/span><\/h4>\n


    <\/span><\/span><\/h4>\n

    \u5bf9\u4e8e\u6709\u5feb\u6162\u5206\u79bb\u7279\u6027\u7684\u7cfb\u7edf\uff0c\u5feb\u53d8\u91cf\u53ef\u5f88\u5feb\u8fbe\u5230\u51c6\u7a33\u6001\uff0c\u6162\u53d8\u91cf\u4e3b\u5bfc\u5b8f\u89c2\u52a8\u529b\u5b66\u3002\u8bbe\u5feb\u53d8\u91cf\u72b6\u6001\u96c6<\/span><\/span>S<\/span><\/span>f<\/span><\/span><\/sub>\u548c\u6162\u53d8\u91cf\u72b6\u6001\u96c6<\/span><\/span>S<\/span><\/span>s<\/span><\/span><\/sub>\uff0c\u7c97\u7c92\u5316\u540e\u7684\u8f6c\u79fb\u901f\u7387\u77e9\u9635<\/span><\/span>Q<\/span><\/span>\u2032<\/span><\/span>\u7684\u5143\u7d20\u8ba1\u7b97\u4e3a\uff1a<\/span><\/span><\/span><\/p>\n

    <\/section>\n

    \u5176\u4e2d<\/span><\/span>\"image.png\"<\/span><\/sub>\u4e3a\u533a\u5757<\/span><\/span>B<\/span><\/span>k<\/span><\/span><\/sub>\u5185\u72b6\u6001\u7684\u51c6\u7a33\u6001\u5206\u5e03\uff0c\u8be5\u65b9\u6cd5\u5728\u751f\u7269\u5316\u5b66\u7f51\u7edc\u548c\u901a\u4fe1\u534f\u8bae\u4e2d\u5e7f\u6cdb\u5e94\u7528<\/span><\/span>[13]<\/span><\/span>\u3002<\/span><\/span><\/span><\/p>\n


    <\/span><\/p>\n

    \n
    \n
    \n

    \u53c2\u8003\u6587\u732e<\/span><\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n

    \n
    \n
    \n
    \n


    <\/span><\/p>\n

      <\/ol>\n


      <\/span><\/p>\n

        \n
      1. \n

         \u674e\u4e1c\u98ce. \u5e94\u7528\u968f\u673a\u8fc7\u7a0b, 2024, 163-215. https:\/\/www.math.pku.edu.cn\/teachers\/lidf\/course\/stochproc\/stochprocnotes\/html\/_book\/StocProc.pdf<\/span><\/span><\/p>\n<\/li>\n

      2. \n

         \u9646\u5927\u91d1\uff0c\u5f20\u98a2. \u968f\u673a\u8fc7\u7a0b\u53ca\u5176\u5e94\u7528, \u6e05\u534e\u51fa\u7248\u793e\uff0c2012, 176-255.<\/span><\/span><\/p>\n<\/li>\n

      3. \n

         Markov A A\uff0eRasprostranenie zakona bol\u2019shih chisel na velichiny, zavisyaschie drug ot druga\uff0eIzvestiya Fiziko-matematicheskogo obschestva pri Kazanskom universitete, 1906, 15: 135-156.<\/span><\/span><\/p>\n<\/li>\n

      4. \n

         Markov A A. Issledovanie statisti\u010deskich zakonov vjumel\u2019nogo jazika (Analysis of statistical laws in the verse language). Trudy Imperatorskogo S.-Peterburgskogo Mineralogi\u010deskogo Ob\u0161\u010destva, 2nd series, 1913, 42:153\u2013162.<\/span><\/span><\/p>\n<\/li>\n

      5. \n

         Kendall D G, Batchelor G K, Bingham N H, Hayman W K, Hyland J M E, Lorentz G G, Moffatt H K, Parry W, Razborov A A, Robinson C A and Whittle P. Andrei Nikolaevich Kolmogorov (1903\u20131987). Bulletin of the London Mathematical Society, 1990, 22(1): 31-100.<\/span><\/span><\/p>\n<\/li>\n

      6. \n

         Metropolis N, Rosenbluth A W, Rosenbluth M N, Teller A H and Teller E. Equation of state calculations by fast computing machines. The journal of chemical physics, 1953, 21(6): 1087-1092.<\/span><\/span><\/p>\n<\/li>\n

      7. \n

         Baum L E and Petrie T. Statistical inference for probabilistic functions of finite state Markov chains. The annals of mathematical statistics, 1966, 37(6): 1554-1563.<\/span><\/span><\/p>\n<\/li>\n

      8. \n

         Kindermann R, Snell J L. Markov random fields and their applications. American mathematical society, 1980.<\/span><\/span><\/p>\n<\/li>\n

      9. \n

         http:\/\/www.stat.yale.edu\/~pollard\/Courses\/251.spring2013\/Handouts\/Chang-MarkovChains.pdf<\/span><\/span><\/p>\n<\/li>\n

      10. \n

         Zhang J, Tao R Y, Leong H K, Yang M Z, Yuan B. Dynamical reversibility and a new theory of causal emergence based on SVD. npj Complexity, 2025, 2(3): 1-11.<\/span><\/span><\/p>\n<\/li>\n

      11. \n

         Hoel E P, Albantakis L, Tononi G. Quantifying causal emergence shows that macro can beat micro. Proceedings of the National Academy of Sciences, 2013, 110(49): 19790-19795.<\/span><\/span><\/p>\n<\/li>\n

      12. \n

         Luca Cardelli and Radu Grosu and Kim G. Larsen and Mirco Tribastone and Max Tschaikowski and Andrea Vandin. Lumpability for Uncertain Continuous-Time Markov Chains. QEST, 2021, 391-409.<\/span><\/span><\/p>\n<\/li>\n

      13. \n

         Daria Stepanova, Helen M. Byrne, Philip K. Maini, and Tom\u00e1s Alarc\u00f3n. A Method to Coarse-Grain Multiagent Stochastic Systems with Regions of Multistability. Multiscale Model. Simul. 2022, 20(1):404-432.<\/span><\/span><\/p>\n<\/li>\n<\/ol>\n


        <\/span><\/span><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

        \n
        \n

        (\u53c2\u8003\u6587\u732e\u53ef<\/span>\u4e0a\u4e0b\u6ed1\u52a8<\/span><\/strong>\u67e5\u770b)<\/span><\/p>\n<\/section>\n<\/section>\n


        <\/span><\/p>\n


        <\/span><\/p>\n

        \u4f5c\u8005\u7b80\u4ecb<\/span><\/span><\/section>\n
        \"\u56fe\u7247\"<\/section>\n
        <\/section>\n

        <\/span><\/section>\n
        \n

        \u672c\u8bcd\u6761\u7531\u96c6\u667a\u4ff1\u4e50\u90e8\u4f17\u5305\u751f\u4ea7\uff0c\u96be\u514d\u5b58\u5728\u7eb0\u6f0f\u548c\u95ee\u9898\uff0c\u6b22\u8fce\u5927\u5bb6\u7559\u8a00\u53cd\u9988\uff0c\u4e00\u7ecf\u91c7\u7eb3\uff0c\u53ef\u4ee5\u83b7\u5f97\u5bf9\u5e94\u7684\u79ef\u5206\u5956\u52b1\u5662\uff01<\/span><\/p>\n<\/blockquote>\n


        <\/span><\/span><\/section>\n

        <\/span><\/section>\n
        \n
        \n
        \n
        \n

        \u52a0\u5165\u6211\u4eec<\/span><\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n


        <\/span><\/section>\n

        \u4eb2\u7231\u7684\u793e\u533a\u4f19\u4f34\u4e0e\u77e5\u8bc6\u63a2\u7d22\u8005\uff1a<\/span><\/span><\/strong><\/p>\n

        \u6211\u4eec\u8bda\u631a\u9080\u8bf7\u70ed\u7231\u77e5\u8bc6\u5206\u4eab\u7684\u60a8\uff0c\u52a0\u5165<\/span><\/span>\u96c6\u667a\u767e\u79d1\u8bcd\u6761\u7f16\u5199\u5fd7\u613f\u56e2\u961f<\/span><\/span><\/strong>\uff01\u65e0\u8bba\u60a8\u662f\u9886\u57df\u4e13\u5bb6\uff0c\u8fd8\u662f\u5bf9\u7279\u5b9a\u4e3b\u9898\u5145\u6ee1\u70ed\u5ff1\u7684\u5b66\u4e60\u8005\uff0c\u8fd9\u91cc\u90fd\u6709\u60a8\u7684\u821e\u53f0\u3002\u901a\u8fc7\u7f16\u5199\u767e\u79d1\u8bcd\u6761\uff0c\u60a8\u5c06\u4e3a\u5168\u7403\u8bfb\u8005\u4f20\u9012\u6743\u5a01\u77e5\u8bc6\uff0c\u540c\u65f6\u83b7\u5f97<\/span><\/span>\u4e13\u5bb6\u6307\u5bfc<\/span><\/span><\/strong>\u4e0e<\/span><\/span>\u4e2a\u4eba\u80fd\u529b<\/span><\/span><\/strong>\u8dc3\u5347\u7684\u53cc\u91cd\u6210\u957f\u3002<\/span><\/span><\/p>\n


        <\/span><\/p>\n

        \ud83d\udcdd <\/span><\/span>\u5fd7\u613f\u8005\u804c\u8d23<\/span><\/span><\/strong><\/p>\n

        \u521b\u4f5c\u65b0\u8bcd\u6761\uff1a<\/span><\/span><\/strong>\u8986\u76d6\u590d\u6742\u7cfb\u7edf\u3001\u4eba\u5de5\u667a\u80fd\u7b49\u524d\u6cbf\u9886\u57df<\/span><\/span><\/p>\n

        \u8fed\u4ee3\u7ecf\u5178\u5185\u5bb9<\/span><\/span><\/strong>\uff1a\u66f4\u65b0\u73b0\u6709\u8bcd\u6761\uff0c\u5b88\u62a4\u77e5\u8bc6\u7684\u51c6\u786e\u6027\u4e0e\u65f6\u6548\u6027<\/span><\/span><\/p>\n

        \u8d28\u91cf\u5b88\u62a4\u8005<\/span><\/span><\/strong>\uff1a\u53c2\u4e0e\u5185\u5bb9\u6821\u5bf9\u5ba1\u6838\uff0c\u5171\u5efa\u7cbe\u54c1\u77e5\u8bc6\u5e93<\/span><\/span><\/p>\n


        <\/span><\/p>\n

        \ud83c\udf1f <\/span><\/span>\u6211\u4eec\u671f\u5f85\u60a8<\/span><\/span><\/strong><\/p>\n

        \u96c6\u667a\u8bfb\u4e66\u4f1a\u6210\u5458\uff08\u9700\u5b8c\u6210\u5171\u521b\u4efb\u52a1\u5e76\u83b7\u5f97\u9000\u8d39\u8d44\u683c\uff09<\/span><\/span><\/p>\n

        \u62e5\u6709\u6e05\u6670\u8868\u8fbe\u590d\u6742\u6982\u5ff5\u7684\u5199\u4f5c\u80fd\u529b<\/span><\/span><\/p>\n

        \u5bf9\u7279\u5b9a\u9886\u57df\u6709\u6df1\u5ea6\u7814\u7a76\u6216\u5f3a\u70c8\u5174\u8da3<\/span><\/span><\/p>\n

        \u5177\u5907\u4fe1\u606f\u68c0\u7d22\u4e0e\u6574\u5408\u7d20\u517b<\/span><\/span><\/p>\n

        \u6000\u63e3\u8d23\u4efb\u611f\u4e0e\u534f\u4f5c\u7cbe\u795e\uff0c\u613f\u4e3a\u77e5\u8bc6\u5171\u4eab\u8d4b\u80fd<\/span><\/span><\/p>\n


        <\/span><\/p>\n

        \ud83c\udf81 <\/span><\/span>\u60a8\u5c06\u6536\u83b7<\/span><\/span><\/strong><\/p>\n

        \u767e\u79d1\u79ef\u5206\uff08\u652f\u6301\u5151\u6362\u96c6\u667a\u4ff1\u4e50\u90e8\u5468\u8fb9\uff1a\u6587\u5316\u886b\u3001\u590d\u6742\u79d1\u5b66\u77e5\u8bc6\u5361\u7b49\uff09<\/span><\/span><\/p>\n

        \u96c6\u667a\u4ff1\u4e50\u90e8\u521b\u59cb\u4eba\u5f20\u6c5f\u6559\u6388\u4eb2\u81ea\u6307\u5bfc\u5199\u4f5c<\/span><\/span><\/p>\n

        \u79d1\u7814\u52a9\u7406\u664b\u5347\u901a\u9053\uff1a\u8868\u73b0\u4f18\u5f02\u8005\u53ef\u52a0\u5165\u5f20\u6c5f\u6559\u6388\u79d1\u7814\u56e2\u961f<\/span><\/span><\/p>\n

        \"\u56fe\u7247\"<\/section>\n
        <\/section>\n

        \u52a0\u5165\u5f20\u6c5f\u7ec4\u53d1\u8d77\u7684\u4efb\u4f55\u4e00\u671f\u8bfb\u4e66\u4f1a\u5e76\u5b8c\u6210\u5171\u521b\u4efb\u52a1\u5373\u53ef \u8ba9\u6211\u4eec\u4e00\u8d77\uff0c\u7528\u77e5\u8bc6\u8fde\u63a5\u4e16\u754c\uff01<\/span><\/span><\/p>\n


        <\/span>
        <\/span><\/section>\n
        \n
        \n
        \n
        \n

        \u56e0\u679c\u6d8c\u73b0\u8bfb\u4e66\u4f1a\u7b2c\u516d\u5b63<\/span><\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n


        <\/span><\/section>\n

        \u5728\u9713\u8679\u706f\u7684\u95ea\u70c1\u3001\u8681\u7fa4\u7684\u7cbe\u5bc6\u534f\u4f5c\u3001\u4eba\u7c7b\u610f\u8bc6\u7684\u8bde\u751f\u4e2d\uff0c\u9690\u85cf\u7740\u5fae\u89c2\u4e0e\u5b8f\u89c2\u4e4b\u95f4\u6df1\u523b\u7684\u56e0\u679c\u5173\u8054\u2014\u2014\u8fd9\u4e9b\u770b\u4f3c\u7b80\u5355\u7684\u4e2a\u4f53\u884c\u4e3a\uff0c\u5982\u4f55\u8de8\u8d8a\u5c3a\u5ea6\uff0c\u6d8c\u73b0\u51fa\u4ee4\u4eba\u60ca\u53f9\u7684\u590d\u6742\u73b0\u8c61\uff1f\u56e0\u679c\u6d8c\u73b0\u7406\u8bba\u4e3a\u6211\u4eec\u63ed\u793a\u4e86\u7b54\u6848\uff1a\u590d\u6742\u7cfb\u7edf\u7684\u5b8f\u89c2\u7279\u5f81\u65e0\u6cd5\u901a\u8fc7\u5fae\u89c2\u5143\u7d20\u7684\u7b80\u5355\u53e0\u52a0\u89e3\u91ca\uff0c\u800c\u662f\u6e90\u4e8e\u591a\u5c3a\u5ea6\u52a8\u6001\u4ea4\u4e92\u4e2d\u6d8c\u73b0\u7684\u56e0\u679c\u7ed3\u6784\u3002\u4ece\u5947\u5f02\u503c\u5206\u89e3\uff08SVD\uff09\u9a71\u52a8\u7684\u52a8\u6001\u53ef\u9006\u6027\u5206\u6790\uff0c\u5230\u56e0\u679c\u62bd\u8c61\u4e0e\u4fe1\u606f\u5206\u89e3\u7684\u91cf\u5316\u5de5\u5177\uff0c\u7814\u7a76\u8005\u4eec\u6b63\u9010\u6b65\u6784\u5efa\u8d77\u4e00\u5957\u8de8\u8d8a\u6570\u5b66\u3001\u7269\u7406\u4e0e\u4fe1\u606f\u79d1\u5b66\u7684\u7406\u8bba\u6846\u67b6\uff0c\u8bd5\u56fe\u89e3\u7801\u590d\u6742\u7cfb\u7edf\u7684\u201c\u6d8c\u73b0\u5bc6\u7801\u201d\u3002<\/span><\/span><\/p>\n


        <\/span><\/p>\n

        \u4e3a\u4e86\u7cfb\u7edf\u68b3\u7406\u56e0\u679c\u6d8c\u73b0\u6700\u65b0\u8fdb\u5c55\uff0c\u5317\u4eac\u5e08\u8303\u5927\u5b66\u7cfb\u7edf\u79d1\u5b66\u5b66\u9662\u6559\u6388\u3001\u96c6\u667a\u4ff1\u4e50\u90e8\u521b\u59cb\u4eba\u5f20\u6c5f\u8001\u5e08\u9886\u8854\u53d1\u8d77\u300c\u56e0\u679c\u6d8c\u73b0\u7b2c\u516d\u5b63\u300d\u8bfb\u4e66\u4f1a<\/a>\uff0c\u7ec4\u7ec7\u5bf9\u672c\u8bdd\u9898\u611f\u5174\u8da3\u7684\u670b\u53cb\uff0c\u6df1\u5165\u7814\u8bfb\u76f8\u5173\u6587\u732e\uff0c\u6fc0\u53d1\u79d1\u7814\u7075\u611f\u3002<\/span><\/span><\/p>\n


        <\/span><\/p>\n

        \u8bfb\u4e66\u4f1a\u5c06\u4ece2025\u5e743\u670816\u65e5\u5f00\u59cb\uff0c\u6bcf\u5468\u65e5\u65e99:00-11:00\uff0c\u6301\u7eed\u65f6\u95f4\u9884\u8ba110\u5468\u5de6\u53f3\u3002\u6bcf\u5468\u8fdb\u884c\u7ebf\u4e0a\u4f1a\u8bae\uff0c\u4e0e\u4e3b\u8bb2\u4eba\u7b49\u793e\u533a\u6210\u5458\u5f53\u9762\u4ea4\u6d41\uff0c\u4e4b\u540e\u53ef\u4ee5\u83b7\u5f97\u89c6\u9891\u56de\u653e\u6301\u7eed\u5b66\u4e60\u3002\u8bda\u631a\u9080\u8bf7\u9886\u57df\u5185\u7814\u7a76\u8005\u3001\u5bfb\u6c42\u8de8\u9886\u57df\u878d\u5408\u7684\u7814\u7a76\u8005\u52a0\u5165\uff0c\u5171\u540c\u63a2\u8ba8\u3002<\/span><\/span><\/p>\n


        <\/span><\/span><\/p>\n

        \"\u56fe\u7247\"<\/span><\/a><\/section>\n
        \n


        <\/span><\/span><\/p>\n

        \u8be6\u60c5\u8bf7\u89c1\uff1a\u56e0\u679c\u6d8c\u73b0\u7b2c\u516d\u5b63\u2014\u2014\u52a8\u529b\u5b66\u3001\u56e0\u679c\u62bd\u8c61\u4e0e\u4fe1\u606f\u5206\u89e3<\/a><\/span><\/span><\/p>\n<\/section>\n


        <\/span><\/p>\n

        \u63a8\u8350\u9605\u8bfb<\/span><\/span><\/strong><\/strong><\/strong><\/p>\n

        1. \u4ec0\u4e48\u6837\u7684\u9a6c\u5c14\u79d1\u592b\u94fe\u53ef\u4ee5\u88ab\u65e0\u635f\u5730\u7c97\u7c92\u5316\uff1f\uff5c\u96c6\u667a\u767e\u79d1<\/span><\/a><\/span><\/strong><\/section>\n
        2. \u57fa\u4e8e\u4fe1\u606f\u5206\u89e3\u7684\u56e0\u679c\u6d8c\u73b0\u7406\u8bba\uff5c\u96c6\u667a\u767e\u79d1<\/span><\/a><\/span><\/strong>
        <\/span><\/strong><\/span><\/section>\n
        3. \u56e0\u679c\u6548\u5e94\u5ea6\u91cf\uff5c\u96c6\u667a\u767e\u79d1<\/span><\/a><\/span><\/strong>
        <\/span><\/strong><\/span><\/section>\n
        4. <\/span>\u6d8c\u73b0\u52a8\u529b\u5b66\u5982\u4f55\u7528\u6765\u5206\u6790\u590d\u6742\u7cfb\u7edf\uff1f | \u65b0\u8bfe\u4e0a\u7ebf<\/span><\/a><\/span><\/strong><\/span>
        <\/span><\/strong><\/strong><\/span><\/section>\n
        <\/strong>5. <\/span><\/strong>\u96c6\u667a\u5b66\u56ed\u7cbe\u54c1\u8bfe\u7a0b\u514d\u8d39\u5f00\u653e\uff0c\u89e3\u9501\u7cfb\u7edf\u79d1\u5b66\u4e0e AI \u65b0\u4e16\u754c<\/span><\/a><\/span><\/strong><\/section>\n

        6. \u9ad8\u8003\u5206\u6570\u53ea\u662f\u5f20\u5165\u573a\u5238\uff0c\u4f60\u7684\u79d1\u7814\u5192\u9669\u5728\u8fd9\u91cc\u542f\u822a\uff01<\/span><\/a><\/span><\/strong><\/p>\n

        7.<\/span><\/span><\/strong><\/strong><\/span> <\/span><\/span><\/strong><\/strong><\/span>\u7cfb\u7edf\u79d1\u5b66\u524d\u6cbf\u5341\u8bb2\uff1a\u7406\u89e3\u81ea\u7136\u3001\u751f\u547d\u4e0e\u793e\u4f1a\u7684\u590d\u6742\u6027<\/a><\/span><\/strong><\/strong><\/span><\/p>\n


        <\/span><\/strong><\/span><\/section>\n


        <\/span><\/p>\n

        \u70b9\u51fb\u201c\u9605\u8bfb\u539f\u6587\u201d\uff0c<\/span>\u62a5\u540d\u8bfb\u4e66\u4f1a<\/span><\/strong><\/span><\/strong><\/span><\/strong><\/p>\n<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

        \u5bfc\u8bed \u9a6c\u5c14\u79d1\u592b\u94fe\u662f\u6307\u4e00\u7c7b\u5b9a\u4e49\u5728\u53ef\u5217\u96c6\u5408\u4e0a\u3001\u5177\u6709\u9a6c\u5c14\u79d1\u592b\u6027\u7684\u968f\u673a\u8fc7\u7a0b\u3002\u5176\u4e2d\uff0c\u8fd9\u4e2a\u96c6\u5408\u88ab\u79f0\u4e3a\u662f\u72b6\u6001\u7a7a\u95f4\uff0c\u800c\u9a6c\u5c14\u79d1\u592b\u6027\u662f\u6307\u5728\u7ed9\u5b9a\u7cfb\u7edf\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u5bf9\u4e8e\u672a\u6765\u72b6\u6001\u7684\u9884\u6d4b\u4e0d\u518d\u4f9d\u8d56\u4e8e\u8fc7\u53bb\u72b6\u6001\u3002\u5b83\u63cf\u8ff0\u4e86\u968f\u7740\u65f6\u95f4\u7684\u63a8\u79fb\uff0c\u7cfb\u7edf\u72b6\u6001\u7684\u53d8\u5316\u8fc7\u7a0b\u3002\u8fdb\u4e00\u6b65\uff0c\u6839\u636e\u65f6\u95f4\u53c2\u6570\u7684\u79bb\u6563\u6216\u8fde\u7eed\u7279\u6027\uff0c\u5b83\u53ef\u4ee5\u5177\u4f53\u5212\u5206\u4e3a\u79bb\u6563\u65f6\u95f4\u9a6c\u5c14\u79d1\u592b\u94fe\u548c\u8fde\u7eed\u65f6…<\/p>\n","protected":false},"author":0,"featured_media":63318,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[1],"tags":[],"special":[],"_links":{"self":[{"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/posts\/63433"}],"collection":[{"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"replies":[{"embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=63433"}],"version-history":[{"count":0,"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/posts\/63433\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/media\/63318"}],"wp:attachment":[{"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=63433"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=63433"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=63433"},{"taxonomy":"special","embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Fspecial&post=63433"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}