{"id":4858,"date":"2017-04-12T22:12:14","date_gmt":"2017-04-12T14:12:14","guid":{"rendered":"https:\/\/lttt.blog.ustc.edu.cn\/?p=4858"},"modified":"2017-04-12T22:12:14","modified_gmt":"2017-04-12T14:12:14","slug":"jiyuxuexiquxiandeyigexiaoshouyucemoxing","status":"publish","type":"post","link":"https:\/\/lttt.vanabel.cn\/?p=4858","title":{"rendered":"\u57fa\u4e8e\u5b66\u4e60\u66f2\u7ebf\u7684\u4e00\u4e2a\u9500\u552e\u9884\u6d4b\u6a21\u578b"},"content":{"rendered":"

\\section{\u95ee\u9898\u7684\u63cf\u8ff0}
\n\u5047\u8bbe\u4f60\u60f3\u5411\u5ba2\u6237\u63a8\u8350\u4e00\u4e2a\u4ea7\u54c1, \u4f60\u5bf9\u8be5\u4ea7\u54c1\u4e5f\u662f\u7b2c\u4e00\u6b21\u63a5\u89e6. \u5728\u4e0d\u65ad\u7684\u63a8\u9500\u8fc7\u7a0b\u4e2d, \u4f60\u4e5f\u5728\u4e0d\u65ad\u5b66\u4e60\u63a8\u9500\u8be5\u4ea7\u54c1\u7684\u6280\u5de7. \u5f88\u660e\u663e, \u5bf9\u4ea7\u54c1\u8d8a\u719f\u6089, \u90a3\u4e48\u4f60\u63a8\u9500\u7684\u6210\u529f\u7387(\u7528\u6237\u8d2d\u4e70\u7b49)\u8d8a\u9ad8. \u90a3\u4e48\u95ee\u9898\u662f\u5982\u4f55\u7528\u6570\u5b66\u6765\u523b\u753b\u8fd9\u4ef6\u4e8b\u5462?
\n\\section{\u5b66\u4e60\u66f2\u7ebf\u7684\u523b\u753b}
\n\u6211\u4eec\u77e5\u9053, \u5b66\u4e60\u662f\u4e00\u4e2a\u8bb0\u5fc6–\u9057\u5fd8–\u8bb0\u5fc6\u7684\u5faa\u73af\u8fc7\u7a0b, \u5173\u4e8e\u8be5\u8fc7\u7a0b\u7684\u5206\u6790\u7559\u5f85\u4e0b\u6b21\u5177\u4f53\u8ba8\u8bba. \u4e00\u822c\u800c\u8a00, \u8be5\u8fc7\u7a0b\u5173\u4e8e\u65f6\u95f4\u7684\u66f2\u7ebf\u662f\u6240\u8c13\u7684S-\u578b\u66f2\u7ebf. \u6211\u4eec\u8fd9\u91cc\u5047\u8bbe$\\lambda(t)$\u8868\u793a$t$\u65f6\u523b\u638c\u63e1\u7684\u6280\u80fd\u7684\u767e\u5206\u6bd4(\u5b8c\u5168\u638c\u63e1\u65f6$\\lambda=1$). \u800c\u6700\u7b80\u5355\u7684S-\u66f2\u7ebf\u5c31\u662f\u6240\u8c13\u7684Logistics\u66f2\u7ebf, \u5b83\u542b\u6709\u4e00\u4e2a\u53c2\u6570$k$, \u8868\u793a\u6837\u672c(\u9500\u552e\u91cf)\u5bf9\u5b66\u4e60(\u63a8\u65b0\u4ea7\u54c1\u7684\u6280\u5de7)\u901f\u5ea6\u7684\u5f71\u54cd\u56e0\u5b50\\footnote{\u4e8b\u5b9e\u4e0a, $\\lambda(t)$\u6240\u6ee1\u8db3\u7684\u5e38\u5fae\u5206\u65b9\u7a0b\u4e3a
\n$$
\n\\lambda'(t)=k\\lambda(t)(1-\\lambda(t)),\\quad \\lambda(0)=\\frac{1}{2}.
\n$$
\n\u6c42\u89e3\u540e\u5f97\u5230
\n$$
\n\\lambda(t)=\\frac{1}{1+\\exp(-k t)}.
\n$$}, \u8fd9\u6837\u6211\u4eec$\\lambda(t)$\u7684\u8868\u8fbe\u5f0f\u4e3a:
\n$$
\n\\lambda(t)=\\frac{1}{1+\\exp(-k t)}.
\n$$
\n\u4e3a\u4e86\u76f4\u89c2\u8d77\u89c1, \u6211\u7528Mathematica\u4f5c\u56fe\u5982\u4e0b:<\/p>\n

\"logistics\"<\/a><\/p>\n

<\/p>\n

\u9644\u4e0a\u4f5c\u56fe\u4ee3\u7801:<\/p>\n

d = 60; l = 0.8;(*60\u5929\u638c\u63e180%\u7684\u6280\u80fd*)\n d0 = 3 d;(*\u521d\u59cb\u6280\u80fd\u4e3a-3d\u65f6\u7684\u503c*)\n LearnCurve[t_, k_] := 1\/(1 + Exp[-k t])\n Solve[LearnCurve[d, k] == l, k, Reals];\n \\[Kappa] = k \/. % \/\/ First; (*\u5b66\u4e60\u66f2\u7ebf\u7684\u53c2\u6570*)\n \\[Lambda]0 = LearnCurve[-d0, \\[Kappa]];(*\u5b66\u4e60\u80fd\u529b\u7684\u521d\u59cb\u503c*)\n Print[\"\\!\\(\\*SubscriptBox[\\(\\[Lambda]\\), \\(0\\)]\\)=\", \\[Lambda]0, \"\\n\\\n \\[Kappa]=\", \\[Kappa]]<\/pre>\n
\\section{\u9500\u552e\u91cf\u7684\u6570\u5b66\u523b\u753b}
\n\u6211\u4eec\u5c06\u9996\u5148\u7ed9\u51fa\u9500\u552e\u91cf\u4e0e\u9500\u552e\u6280\u80fd\u7684\u57fa\u672c\u5173\u7cfb, \u5b83\u662f\u7528\u4e00\u4e2a\u5e38\u5fae\u5206\u65b9\u7a0b\u6765\u523b\u753b\u7684.<\/p>\n
\u6211\u4eec\u5047\u8bbe$x(t)$\u4e3a$t$\u65f6\u523b\u7684\u9500\u552e\u91cf, \u5e76\u7ed9\u51fa\u6211\u4eec\u7684\u57fa\u672c\u5047\u8bbe:<\/p>\n
a. \u9500\u552e\u901f\u7387$x'(t)$\u4e0e\u9500\u552e\u6280\u80fd$\\lambda(t)$\u6210\u6b63\u6bd4, \u4e0e\u5b66\u4e60\u6837\u672c$x(t)$\u4e5f\u6210\u6b63\u6bd4<\/p><\/blockquote>\n
\u6b64\u5916, \u6211\u4eec\u8fd8\u7ed9\u51fa\u53e6\u5916\u4e24\u4e2a\u5047\u8bbe:<\/p>\n
b. \u5f00\u59cb\u65f6, \u9500\u552e\u6280\u80fd$\\lambda(0)$\u4e3a\u96f6. (\u5176\u5b9e, \u5728\u540e\u9762\u7684\u6a21\u578b\u6c42\u89e3\u662f\u6211\u4eec\u5047\u8bbe$\\lambda_0=\\lambda(0)=\\lambda(-3 d)$\u4ed6\u975e\u5e38\u63a5\u8fd1\u4e8e0, \u8fd9\u91cc$d$\u662f\u638c\u63e180%\u7684\u6280\u80fd\u6240\u9700\u7684\u65f6\u95f4)<\/p><\/blockquote>\n
c. \u638c\u63e180%\u7684\u6280\u80fd\u6240\u9700\u7684\u65f6\u95f4\u4e3a$d=60$\u5929<\/p><\/blockquote>\n
\\section{\u6a21\u578b\u7684\u6c42\u89e3}
\n\\subsection{\u53c2\u6570$k$\u7684\u6c42\u89e3}
\n\u6839\u636e\u5047\u8bbec, \u6211\u4eec\u6c42\u89e3$k$\u5982\u4e0b:
\n$$
\nd=60,\\quad \\lambda(d)=\\frac{1}{1+\\exp(-k d)}=0.8\\implies k=0.0231049
\n$$
\n\u5176\u6b21, \u6211\u4eec\u6839\u636e\u521d\u503c\u7684\u5047\u5b9ab, $\\lambda_0=\\lambda(-3d)=0.0153846$.<\/p>\n
\u4e0a\u8ff0\u6c42\u89e3\u8fc7\u7a0b\u7684Mathematica\u4ee3\u7801\u4e3a:<\/p>\n
d = 60; l = 0.8;(*60\u5929\u638c\u63e180%\u7684\u6280\u80fd*)\nd0 = 3 d;(*\u521d\u59cb\u6280\u80fd\u4e3a-3d\u65f6\u7684\u503c*)\nLearnCurve[t_, k_] := 1\/(1 + Exp[-k t])\nSolve[LearnCurve[d, k] == l, k, Reals];\n\\[Kappa] = k \/. % \/\/ First; (*\u5b66\u4e60\u66f2\u7ebf\u7684\u53c2\u6570*)\n\\[Lambda]0 = LearnCurve[-d0, \\[Kappa]];(*\u5b66\u4e60\u80fd\u529b\u7684\u521d\u59cb\u503c*)\nPrint[\"\\!\\(\\*SubscriptBox[\\(\\[Lambda]\\), \\(0\\)]\\)=\", \\[Lambda]0, \"\\n\\\n\\[Kappa]=\", \\[Kappa]]<\/span><\/span><\/pre>\n
\u8fd9\u6837, \u6211\u4eec\u5f97\u5230
\n$$
\n\\lambda(t)=\\frac{1}{1+\\exp(-0.0231049 t)}.
\n$$
\n\\subsection{\u6c42\u89e3\u9500\u552e\u91cf$x(t)$}
\n\u9996\u5148, \u66f4\u52a0\u5047\u8bbea, \u6211\u4eec\u5f97\u5230$x(t)$\u6240\u6ee1\u8db3\u7684\u65b9\u7a0b\u4e3a:
\n$$
\nx'(t)=x(t)(\\lambda(t)-\\lambda_0).
\n$$
\n\u4e0a\u8ff0\u65b9\u7a0b\u662f\u4e00\u9636ODE, \u9700\u8981\u7ed9\u5b9a\u4e00\u4e2a\u521d\u503c, \u8fd9\u91cc\u6211\u5047\u8bbe$x(1)=34$. \u5219\u6c42\u89e3\u4e0a\u8ff0\u5e38\u5fae\u5206\u65b9\u7a0b\u5f97\u5230
\n$$
\nx(t)=1.954\\times 10^{-12} e^{-0.0153846 t} \\left(1+e^{0.0231049 t}\\right)^{43.2809}
\n$$
\n\u6c42\u89e3\u7684Mathematica\u4ee3\u7801:<\/p>\n
 eq = x'[t] == x[t] (LearnCurve[t, \\[Kappa]] - \\[Lambda]0);\n DSolve[eq && x[1] == 34, x[t], t]\n Datalist = {34, 55};\n Show[Plot[x[t] \/. %%, {t, 0, 5}], ListPlot[Datalist], PlotRange -> All]\n Table[x[t] \/. %%% \/\/ First \/\/ Floor, {t, 5}]<\/pre>\n
\u8fd9\u91cc, \u6211\u8fd8\u9a8c\u8bc1\u4e86\u539f\u59cb\u6570\u636e$x(1)=34$, $x(2)=55$. \u5e76\u9884\u6d4b\u4e86\u524d\u4e94\u5929\u7684\u9500\u552e\u91cf:
\n$${34, 55, 91, 151, 253}$$
\n\u56fe\u5f62\u5316\u663e\u793a\u4e3a\"xiaoshouyuce\"<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"
\\section{\u95ee\u9898\u7684\u63cf\u8ff0} \u5047\u8bbe\u4f60\u60f3\u5411\u5ba2\u6237\u63a8\u8350\u4e00\u4e2a\u4ea7\u54c1, \u4f60\u5bf9\u8be5\u4ea7\u54c1\u4e5f\u662f\u7b2c\u4e00 …<\/p>\n
  \u57fa\u4e8e\u5b66\u4e60\u66f2\u7ebf\u7684\u4e00\u4e2a\u9500\u552e\u9884\u6d4b\u6a21\u578b<\/span> \u9605\u8bfb\u66f4\u591a »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[544,95,722,445,388],"class_list":["post-4858","post","type-post","status-publish","format-standard","hentry","category-net","tag-logistics","tag-xuexiquxian","tag-moxing","tag-xiaoshou","tag-yuce"],"_links":{"self":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/4858","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4858"}],"version-history":[{"count":0,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/4858\/revisions"}],"wp:attachment":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4858"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4858"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4858"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}