{"id":57792,"date":"2025-08-15T19:50:27","date_gmt":"2025-08-15T11:50:27","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/57792.html"},"modified":"2025-08-15T19:50:27","modified_gmt":"2025-08-15T11:50:27","slug":"2-%e7%bb%9f%e8%ae%a1%e5%ad%a6%e4%b9%a0%e7%ac%94%e8%ae%b0%e4%b9%8b%e3%80%90%e5%b2%ad%e5%9b%9e%e5%bd%92%e4%b8%8e%e5%b9%bf%e4%b9%89%e5%b2%ad%e5%9b%9e%e5%bd%92%e6%80%a7%e8%b4%a8%e8%af%81%e6%98%8e","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/57792.html","title":{"rendered":"2. \u7edf\u8ba1\u5b66\u4e60\u7b14\u8bb0\u4e4b\u3010\u5cad\u56de\u5f52\u4e0e\u5e7f\u4e49\u5cad\u56de\u5f52\u6027\u8d28\u8bc1\u660e\u3011"},"content":{"rendered":"<p>\u672c\u535a\u6587\u5df2\u540c\u6b65\u4e8e\u77e5\u4e4e\u3002<\/p>\n<hr \/>\n<p>\u5728\u591a\u5143\u7ebf\u6027\u56de\u5f52\u5206\u6790\u4e2d&#xff0c;\u5f53\u89e3\u91ca\u53d8\u91cf\u5b58\u5728\u591a\u91cd\u5171\u7ebf\u6027\u65f6&#xff0c;\u666e\u901a\u6700\u5c0f\u4e8c\u4e58\u4f30\u8ba1\u7684\u6548\u679c\u4f1a\u53d7\u5230\u5f71\u54cd&#xff0c;\u800c\u5cad\u56de\u5f52\u548c\u5e7f\u4e49\u5cad\u56de\u5f52\u4f5c\u4e3a\u5e38\u7528\u7684\u6709\u504f\u4f30\u8ba1\u65b9\u6cd5&#xff0c;\u80fd\u6709\u6548\u6539\u5584\u8fd9\u4e00\u95ee\u9898\u3002<\/p>\n<p>\u5728\u4e2d\u56fd\u4eba\u6c11\u5927\u5b66\u7248\u672c\u7684\u300a\u5e94\u7528\u56de\u5f52\u5206\u6790\u300b\u4e2d&#xff0c;\u53ea\u5217\u51fa\u4e86\u5cad\u56de\u5f52\u7684\u76f8\u5173\u6027\u8d28\u800c\u6ca1\u6709\u8bc1\u660e&#xff0c;\u5bf9\u5e7f\u4e49\u5cad\u56de\u5f52\u7684\u6a21\u578b\u548c\u6027\u8d28\u4e5f\u6ca1\u6709\u6df1\u5165\u63a2\u8ba8\u3002\u76f8\u5173\u5b66\u4e60\u7f51\u7ad9\u4e0a\u4e5f\u7f3a\u4e4f\u76f8\u5173\u8bc1\u660e\u3002<\/p>\n<p>\u4ee5\u4e0b\u5c06\u6574\u7406\u5cad\u56de\u5f52\u548c\u5e7f\u4e49\u5cad\u56de\u5f52\u4e3b\u8981\u6027\u8d28\u7684\u8bc1\u660e\u8fc7\u7a0b&#xff0c;\u4ee5\u660e\u6670\u5176\u7406\u8bba\u4f9d\u636e\u3002<\/p>\n<hr \/>\n<h2>\u4e00\u3001\u5cad\u56de\u5f52<\/h2>\n<p>\u5cad\u56de\u5f52\u7684\u53c2\u6570\u4f30\u8ba1\u4e3a&#xff1a;<img decoding=\"async\" alt=\"\\\\hat{\\\\beta}(k) &#061; (X&#039;X &#043; kI)^{-1}X&#039;y\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115017-689f1ef9f3af1.png\" \/>\u3002<\/p>\n<h3>1\u3001\u7ebf\u6027\u6027<\/h3>\n<p>\u5728<img decoding=\"async\" alt=\"k\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115018-689f1efac2f14.png\" \/>\u4e3a\u5e38\u6570\u65f6&#xff0c;<img decoding=\"async\" alt=\"\\\\hat{\\\\beta}(k)&#061;(X&#039;X&#043;kI)^{-1}X&#039;y\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115019-689f1efb72743.png\" \/>\u662f\u6700\u5c0f\u4e8c\u4e58\u4f30\u8ba1<img decoding=\"async\" alt=\"\\\\hat{\\\\beta}\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115020-689f1efc0cddf.png\" \/>\u7684\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362&#xff0c;\u4e5f\u662f<img decoding=\"async\" alt=\"y\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115020-689f1efc7650e.png\" \/>\u7684\u51fd\u6570\u3002<\/p>\n<p>\u5b9e\u9645\u60c5\u51b5\u4e0b&#xff0c;<img decoding=\"async\" alt=\"k\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115018-689f1efac2f14.png\" \/>\u7684\u9009\u53d6\u53d6\u51b3\u4e8e<img decoding=\"async\" alt=\"X\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115021-689f1efd066c9.png\" \/>\u548c<img decoding=\"async\" alt=\"y\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115020-689f1efc7650e.png\" \/>&#xff0c;\u4ece\u800c\u7ebf\u6027\u6027\u4e0d\u6210\u7acb\u3002<\/p>\n<h3>2\u3001\u6709\u504f\u6027<\/h3>\n<p><img decoding=\"async\" alt=\"\\\\hat{\\\\beta}(k)\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115021-689f1efd5d81f.png\" \/>\u662f<img decoding=\"async\" alt=\"\\\\beta\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115021-689f1efdd63a0.png\" \/>\u7684\u6709\u504f\u4f30\u8ba1\u3002<\/p>\n<p><img decoding=\"async\" alt=\"E(\\\\hat{\\\\beta}(k)) &#061; (X&#039;X &#043; kI)^{-1}X&#039;X\\\\beta\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115022-689f1efe41c29.png\" \/><\/p>\n<p><img decoding=\"async\" alt=\"Cov(\\\\hat{\\\\beta}(k)) &#061; \\\\sigma^2(X&#039;X &#043; kI)^{-1}X&#039;X(X&#039;X &#043; kI)^{-1}\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115022-689f1efe9eaef.png\" \/><\/p>\n<p style=\"text-align:right\"><span style=\"color:#7b7f82\">\u524d\u4e24\u6761\u6027\u8d28\u5728\u6559\u6750\u90fd\u6709\u8bc1\u660e\u8fc7\u7a0b&#xff0c;\u5728\u6b64\u7701\u7565\u3002\u7b14\u8005\u6ce8\u3002<\/span><\/p>\n<h3>3\u3001\u538b\u7f29\u6027<\/h3>\n<p>\u5bf9\u4e8e\u4efb\u610f<img decoding=\"async\" alt=\"k&gt;0\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115023-689f1eff2f8a3.png\" \/>\u548c<img decoding=\"async\" alt=\"\\\\|\\\\hat{\\\\beta}\\\\|\\\\neq0\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115023-689f1eff9d894.png\" \/>&#xff0c;\u603b\u6709<img decoding=\"async\" alt=\"\\\\|\\\\hat{\\\\beta}(k)\\\\|&lt;\\\\|\\\\hat{\\\\beta}\\\\|\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115023-689f1effda79c.png\" \/>\u3002\u8fd9\u4e2a\u6027\u8d28\u8868\u660e<img decoding=\"async\" alt=\"\\\\hat{\\\\beta}(k)\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115021-689f1efd5d81f.png\" \/>\u53ef\u4ee5\u770b\u4f5c<img decoding=\"async\" alt=\"\\\\hat{\\\\beta}\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115020-689f1efc0cddf.png\" \/>\u8fdb\u884c\u67d0\u79cd\u5411\u539f\u70b9\u7684\u538b\u7f29\u3002<\/p>\n<p style=\"text-align:right\"><span style=\"color:#7b7f82\">\u8fd9\u4e5f\u662f\u5cad\u56de\u5f52\u53ef\u4ee5\u8fdb\u884c\u53d8\u91cf\u9009\u62e9\u7684\u7406\u8bba\u4f9d\u636e\u3002\u7b14\u8005\u6ce8\u3002<\/span><\/p>\n<p>\u8bc1\u660e&#xff1a;<\/p>\n<p>\u8003\u8651\u56de\u5f52\u6a21\u578b<img decoding=\"async\" alt=\"Y &#061; X\\\\beta&#043;\\\\varepsilon\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115024-689f1f00402c0.png\" \/>\u3002<\/p>\n<p>\u5c06\u5176\u5199\u4e3a\u5178\u5219\u5f62\u5f0f&#xff1a;<img decoding=\"async\" alt=\"Y &#061; Z\\\\alpha &#043; \\\\varepsilon...(1)\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115024-689f1f006dc23.png\" \/>\u3002\u5176\u4e2d&#xff0c;<img decoding=\"async\" alt=\"Z &#061; X\\\\Phi,\\\\quad\\\\alpha&#061;\\\\Phi&#039;\\\\beta\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115024-689f1f00cc99c.png\" \/>\u3002<\/p>\n<p><img decoding=\"async\" alt=\"\\\\Phi\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115025-689f1f013ee60.png\" \/>\u4e3a\u6b63\u4ea4\u77e9\u9635&#xff0c;\u662f\u8bbe\u8ba1\u77e9\u9635\u4ea4\u53c9\u4e58\u79ef\u9635<img decoding=\"async\" alt=\"X&#039;X\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115025-689f1f0175156.png\" \/>\u7684\u7279\u5f81\u5411\u91cf\u7ec4\u6210\u7684\u77e9\u9635\u3002<\/p>\n<p>\u7531\u4e8e<img decoding=\"async\" alt=\"Z&#039;Z&#061;\\\\Phi&#039;X&#039;X\\\\Phi&#061;\\\\Lambda\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115025-689f1f01b0d99.png\" \/>&#xff0c;<\/p>\n<p>\u5219\u7531<img decoding=\"async\" alt=\"(1)\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115026-689f1f02066e9.png\" \/>\u5f0f\u5bfc\u51fa\u7684<img decoding=\"async\" alt=\"\\\\alpha\" class=\"mathcode\" src=\"2025-08-15gu2t253yte0.png\" \/>\u7684\u6700\u5c0f\u4e8c\u4e58\u4f30\u8ba1\u4e3a&#xff1a;<img decoding=\"async\" alt=\"\\\\hat{\\\\alpha} &#061; (Z&#039;Z)^{-1}Z&#039;Y &#061; \\\\Lambda^{-1}Z&#039;Y\" class=\"mathcode\" src=\"2025-08-15t05sutkf5be.png\" \/>\u3002<\/p>\n<p>\u53ef\u4ee5\u8bc1\u660e&#xff0c;<img decoding=\"async\" alt=\"\\\\beta\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115021-689f1efdd63a0.png\" \/>\u7684\u6700\u5c0f\u4e8c\u4e58\u4f30\u8ba1<img decoding=\"async\" alt=\"\\\\hat{\\\\beta}\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115020-689f1efc0cddf.png\" \/>\u4e0e<img decoding=\"async\" alt=\"\\\\hat{\\\\alpha}\" class=\"mathcode\" src=\"2025-08-154g22zkyrpcw.png\" \/>\u6709\u5982\u4e0b\u5173\u7cfb&#xff1a;<img decoding=\"async\" alt=\"\\\\hat{\\\\alpha}&#061;\\\\Phi&#039;\\\\hat{\\\\beta}\" class=\"mathcode\" src=\"2025-08-15eghyicv1rel.png\" \/>&#xff0c;\u7b49\u4ef7\u5730&#xff0c;<img decoding=\"async\" alt=\"\\\\hat{\\\\beta}&#061;\\\\Phi\\\\hat{\\\\alpha}\" class=\"mathcode\" src=\"2025-08-15ealdk1of34j.png\" \/>\u3002<\/p>\n<p>\u76f8\u5e94\u7684\u5cad\u4f30\u8ba1\u5206\u522b\u4e3a&#xff1a;<img decoding=\"async\" alt=\"\\\\hat{\\\\alpha}(k) &#061; (\\\\Lambda &#043; k\\\\cdot I)^{-1}Z&#039;Y\" class=\"mathcode\" src=\"https:\/\/latex.csdn.net\/eq?%5Chat%7B%5Calpha%7D%28k%29%20%3D%20%28%5CLambda%20&amp;plus;%20k%5Ccdot%20I%29%5E%7B-1%7DZ%27Y\" \/>\u548c<img decoding=\"async\" alt=\"\\\\hat{\\\\beta}(k) &#061; \\\\Phi\\\\hat{\\\\alpha}(k)\" class=\"mathcode\" src=\"2025-08-155zyxbhogg4k.png\" \/>\u3002<\/p>\n<p>\u56e0\u6b64\u6709<img decoding=\"async\" alt=\"\\\\|\\\\hat{\\\\beta}(k)\\\\| &#061; \\\\|\\\\hat{\\\\alpha}(k)\\\\| &#061; \\\\left\\\\|(\\\\Lambda &#043; k\\\\cdot I)^{-1}\\\\Lambda\\\\hat{\\\\alpha}\\\\right\\\\| &lt; \\\\|\\\\hat{\\\\alpha}\\\\| &#061; \\\\|\\\\hat{\\\\beta}\\\\|\" class=\"mathcode\" src=\"2025-08-15dawwyz0pfgl.png\" \/>\u3002\u5f97\u8bc1\u3002<\/p>\n<p style=\"text-align:right\">\u25a1<\/p>\n<h3>4\u3001\u66f4\u4f18\u6027<\/h3>\n<p>\u5fc5\u5b58\u5728\u4e00\u4e2a<img decoding=\"async\" alt=\"k&gt;0\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115023-689f1eff2f8a3.png\" \/>&#xff0c;\u4f7f\u5f97<img decoding=\"async\" alt=\"MSE(\\\\hat{\\\\beta}(k)) &lt; MSE(\\\\hat{\\\\beta}(0))\" class=\"mathcode\" src=\"2025-08-15scysf5wessk.png\" \/>\u3002<\/p>\n<p>\u8fd9\u4e2a\u6027\u8d28\u8868\u660e\u5cad\u4f30\u8ba1\u53ef\u901a\u8fc7\u9009\u62e9\u5408\u9002\u7684<img decoding=\"async\" alt=\"k\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115018-689f1efac2f14.png\" \/>&#xff0c;\u5728 \u201c\u5f15\u5165\u5c11\u91cf\u504f\u5dee\u201d \u548c \u201c\u663e\u8457\u964d\u4f4e\u65b9\u5dee\u201d \u4e4b\u95f4\u53d6\u5f97\u5e73\u8861&#xff0c;\u6700\u7ec8\u5b9e\u73b0\u66f4\u4f18\u7684\u6574\u4f53\u4f30\u8ba1\u6548\u679c&#xff08;\u4ee5 MSE \u8861\u91cf&#xff09;&#xff0c;\u4e3a\u5cad\u4f30\u8ba1\u5728\u591a\u91cd\u5171\u7ebf\u6027\u95ee\u9898\u4e2d\u7684\u5e94\u7528\u63d0\u4f9b\u4e86\u5173\u952e\u7406\u8bba\u652f\u6491\u3002<\/p>\n<p>\u8bc1\u660e&#xff1a;<\/p>\n<p>\u6839\u636e<img decoding=\"async\" alt=\"(2)\" class=\"mathcode\" src=\"2025-08-154lacqlrfapj.png\" \/>\u5f0f&#xff0c;\u6709<img decoding=\"async\" alt=\"MSE(\\\\hat{\\\\beta}(k))&#061;MSE(\\\\hat{\\\\alpha}(k))\" class=\"mathcode\" src=\"2025-08-15nzaioh14zfu.png\" \/>.<\/p>\n<p>\u56e0\u6b64\u53ea\u9700\u8bc1\u660e\u5b58\u5728<img decoding=\"async\" alt=\"k&gt;0\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115023-689f1eff2f8a3.png\" \/>&#xff0c;\u4f7f\u5f97<img decoding=\"async\" alt=\"MSE(\\\\hat{\\\\alpha}(k)) &lt; MSE(\\\\hat{\\\\alpha}(0))\" class=\"mathcode\" src=\"2025-08-15gsyl5wpcdsh.png\" \/>\u3002<\/p>\n<p>\u5176\u4e2d&#xff0c;<img decoding=\"async\" alt=\"Cov(\\\\hat{\\\\alpha}(k)) &#061; \\\\sigma^2(\\\\Lambda &#043; kI)^{-1}\\\\Lambda(\\\\Lambda &#043; kI)^{-1}\" class=\"mathcode\" src=\"2025-08-15duzbu312a35.png\" \/>&#xff0c;<\/p>\n<p><img decoding=\"async\" alt=\"E(\\\\hat{\\\\alpha}(k)) &#061; (\\\\Lambda &#043; kI)^{-1}Z&#039;Z\\\\alpha &#061; (\\\\Lambda &#043; kI)^{-1}\\\\Lambda\\\\alpha\" class=\"mathcode\" src=\"2025-08-1502c0ifqyqjp.png\" \/>\u3002<\/p>\n<p>\u56e0\u6b64&#xff0c;\u5747\u65b9\u8bef\u5dee\u53ef\u5206\u89e3\u4e3a&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"\\\\begin{aligned} MSE(\\\\hat{\\\\alpha}(k)) &amp;&#061; trCov(\\\\hat{\\\\alpha}(k)) &#043; \\\\|E[\\\\hat{\\\\alpha}(k)] - \\\\alpha\\\\|^2 \\\\\\\\ &amp;&#061; \\\\sigma^2\\\\sum_{i&#061;1}^{p}\\\\frac{\\\\lambda_i}{(\\\\lambda_i &#043; k)^2} &#043; k^2\\\\sum_{i&#061;1}^{p}\\\\frac{\\\\alpha_i^2}{(\\\\lambda_i &#043; k)^2} \\\\\\\\ &amp;\\\\triangleq g_1(k) &#043; g_2(k) \\\\triangleq g(k) \\\\end{aligned}\" class=\"mathcode\" src=\"2025-08-15ymmoqshqnih.png\" \/><\/p>\n<p>\u5bf9\u4e0a\u8ff0\u51fd\u6570\u5173\u4e8e<img decoding=\"async\" alt=\"k\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115018-689f1efac2f14.png\" \/>\u6c42\u5bfc&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"g_1&#039;(k) &#061; -2\\\\sigma^2\\\\sum_{i&#061;1}^{p}\\\\frac{\\\\lambda_i}{(\\\\lambda_i &#043; k)^3}\" class=\"mathcode\" src=\"2025-08-1522iqjajbiop.png\" \/>&#xff0c;<img decoding=\"async\" alt=\"g_2&#039;(k) &#061; 2k\\\\sum_{i&#061;1}^{p}\\\\frac{\\\\lambda_i\\\\alpha_i^2}{(\\\\lambda_i &#043; k)^3}\" class=\"mathcode\" src=\"2025-08-15dmov4rrudis.png\" \/>\u3002<\/p>\n<p>\u7531\u4e8e<img decoding=\"async\" alt=\"g_1&#039;(0)&lt;0\" class=\"mathcode\" src=\"2025-08-15zd3o2cccr0t.png\" \/>&#xff0c;<img decoding=\"async\" alt=\"g_2&#039;(0)&lt;0\" class=\"mathcode\" src=\"2025-08-15ti3if5a0v4q.png\" \/>&#xff0c;\u6240\u4ee5<img decoding=\"async\" alt=\"g&#039;(0)&lt;0\" class=\"mathcode\" src=\"2025-08-15j5ecqjmtfvb.png\" \/>\u3002<\/p>\n<p>\u53c8\u56e0\u4e3a<img decoding=\"async\" alt=\"g_1(k)\" class=\"mathcode\" src=\"2025-08-15zok2kyg2ofm.png\" \/>\u548c<img decoding=\"async\" alt=\"g_2&#039;(k)\" class=\"mathcode\" src=\"2025-08-15jukkuo1iyam.png\" \/>\u5728<img decoding=\"async\" alt=\"k\\\\geqslant 0\" class=\"mathcode\" src=\"2025-08-15zcvfvdu0ljb.png\" \/>\u4e0a\u8fde\u7eed&#xff0c;<\/p>\n<p>\u56e0\u6b64\u5f53<img decoding=\"async\" alt=\"k&gt;0\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115023-689f1eff2f8a3.png\" \/>\u4e14\u5145\u5206\u5c0f\u65f6&#xff0c;\u6709<img decoding=\"async\" alt=\"\" class=\"mathcode\" src=\"https:\/\/latex.csdn.net\/eq?\" \/><img decoding=\"async\" alt=\"g&#039;(k)&#061;g_1&#039;(k)&#043;g_2&#039;(k)&lt;0\" class=\"mathcode\" src=\"https:\/\/latex.csdn.net\/eq?g%27%28k%29%3Dg_1%27%28k%29&amp;plus;g_2%27%28k%29%3C0\" \/>&#xff0c;<\/p>\n<p>\u5373<img decoding=\"async\" alt=\"g(k)&#061;MSE(\\\\alpha^{k})\" class=\"mathcode\" src=\"2025-08-15hv5cnu2sg2u.png\" \/>\u662f\u51cf\u51fd\u6570\u3002<\/p>\n<p>\u6545\u5b58\u5728<img decoding=\"async\" alt=\"k&gt;0\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115023-689f1eff2f8a3.png\" \/>&#xff0c;\u4f7f\u5f97<img decoding=\"async\" alt=\"g(k)&lt;g(0)\" class=\"mathcode\" src=\"2025-08-15qveoobghk1z.png\" \/>&#xff0c;\u5373<img decoding=\"async\" alt=\"MSE(\\\\hat{\\\\alpha}(k)) &lt; MSE(\\\\hat{\\\\alpha}(0))\" class=\"mathcode\" src=\"2025-08-15gsyl5wpcdsh.png\" \/>\u3002\u5f97\u8bc1\u3002<\/p>\n<p style=\"text-align:right\">\u25a1<\/p>\n<h3>5\u3001\u5bf9\u8bc1\u660e\u7684\u8bf4\u660e<\/h3>\n<p>Q1&#xff1a;\u4e3a\u4ec0\u4e48\u8981\u4f7f\u7528\u5178\u5219\u5f62\u5f0f&#xff1f;<\/p>\n<p>\u5178\u5219\u5f62\u5f0f\u662f\u5c06\u539f\u591a\u5143\u7ebf\u6027\u56de\u5f52\u6a21\u578b\u901a\u8fc7\u6b63\u4ea4\u53d8\u6362\u8f6c\u5316\u5f97\u5230\u7684\u7b80\u5316\u5f62\u5f0f\u3002<\/p>\n<p style=\"text-align:right\"><span style=\"color:#7b7f82\">\u3010\u5178\u5219\u5f62\u5f0f\u3011\u8fd9\u4e2a\u540d\u79f0\u4e0d\u662f\u7ebf\u6027\u56de\u5f52\u72ec\u6709\u7684&#xff0c;\u6bd4\u5982\u6700\u4f18\u5316\u95ee\u9898\u4e5f\u6709\u5b83\u7684\u3010\u5178\u5219\u5f62\u5f0f\u3011&#xff0c;\u5728\u8bed\u4e49\u4e0a\u53ef\u4ee5\u7b49\u540c\u4e8e\u3010\u6807\u51c6\u5f62\u5f0f\u3011\u3002\u6709\u65f6\u662f\u5b9a\u4e49\u7684\u5dee\u522b&#xff0c;\u6709\u65f6\u4ec5\u662f\u7ffb\u8bd1\u7684\u95ee\u9898\u3002\u7b14\u8005\u6ce8\u3002<\/span><\/p>\n<p>\u4f7f\u7528\u5178\u5219\u5f62\u5f0f\u7684\u6838\u5fc3\u539f\u56e0\u662f\u7b80\u5316&#xff08;\u4e00\u822c\u5316&#xff09;\u8fd0\u7b97&#xff1a;<\/p>\n<p>\u539f\u6a21\u578b\u4e2d<img decoding=\"async\" alt=\"X&#039;X\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115025-689f1f0175156.png\" \/>\u53ef\u80fd\u662f\u75c5\u6001\u77e9\u9635&#xff08;\u7279\u5f81\u503c\u5dee\u5f02\u5927&#xff09;&#xff0c;\u800c\u5178\u5219\u5f62\u5f0f\u4e2d<img decoding=\"async\" alt=\"Z&#039;Z &#061; \\\\Lambda\" class=\"mathcode\" src=\"2025-08-15e5ub2orxmsx.png\" \/>\u4e3a\u5bf9\u89d2\u9635&#xff0c;\u5176\u9006\u77e9\u9635\u3001\u8ff9\u7b49\u8fd0\u7b97\u66f4\u7b80\u5355&#xff0c;\u4fbf\u4e8e\u63a8\u5bfc\u4f30\u8ba1\u91cf\u7684\u6027\u8d28&#xff08;\u5982\u5cad\u4f30\u8ba1\u7684\u8303\u6570\u538b\u7f29\u6027\u3001\u5747\u65b9\u8bef\u5dee\u7b49&#xff09;\u3002<\/p>\n<p>Q2&#xff1a;\u5cad\u56de\u5f52\u5c31\u662f\u5cad\u4f30\u8ba1\u5417&#xff1f;<\/p>\n<p>\u8fd9\u4e2a\u95ee\u9898\u53ef\u80fd\u4f1a\u666e\u904d\u5730\u56f0\u6270\u4e00\u4e9b\u521d\u5b66\u8005&#xff0c;\u5c24\u5176\u662f\u5bf9\u6982\u5ff5\u5bb9\u6613\u6df7\u6dc6\u7684\u540c\u5b66\u3002<\/p>\n<p>\u5cad\u4f30\u8ba1\u662f\u3010\u53c2\u6570\u7684\u4f30\u8ba1\u65b9\u6cd5 \/ \u7ed3\u679c\u3011&#xff0c;\u5cad\u56de\u5f52\u662f\u3010\u57fa\u4e8e\u8be5\u4f30\u8ba1\u6784\u5efa\u7684\u56de\u5f52\u65b9\u7a0b\u3011\u3002<\/p>\n<p>\u4e8c\u8005\u662f\u4e0d\u540c\u5c42\u9762\u7684\u6982\u5ff5&#xff0c;\u4f46\u7684\u786e\u662f\u540c\u4e00\u5957\u6b65\u9aa4\u548c\u7406\u8bba\u6846\u67b6\u4e0b\u7684\u3002<\/p>\n<p style=\"text-align:right\"><span style=\"color:#7b7f82\">\u5728\u5927\u591a\u6570\u6559\u6750\u4e2d&#xff0c;\u8fd9\u4e00\u90e8\u5206\u77e5\u8bc6\u90fd\u53eb\u505a\u3010\u5cad\u56de\u5f52\u3011&#xff0c;\u4f46\u51e0\u4e4e\u90fd\u53ea\u505c\u7559\u5728\u3010\u5cad\u4f30\u8ba1\u3011\u7684\u9636\u6bb5&#xff0c;\u8fd9\u662f\u56e0\u4e3a\u5728\u5b8c\u6210\u5cad\u4f30\u8ba1\u540e\u7684\u6b65\u9aa4\u548c\u7ebf\u6027\u56de\u5f52\u5b8c\u5168\u76f8\u540c\u3002\u7c7b\u4f3c\u7684\u60c5\u51b5\u8fd8\u6709\u3010\u53c2\u6570\u4f30\u8ba1\u3011\u548c\u3010\u7ecf\u9a8c\u56de\u5f52\u65b9\u7a0b\u3011\u3001\u3010\u4e3b\u6210\u5206\u5206\u89e3\/\u5206\u6790\u3011\u548c\u3010\u4e3b\u6210\u5206\u56de\u5f52\u3011\u7b49\u3002\u7b14\u8005\u6ce8\u3002<\/span><\/p>\n<hr \/>\n<h2>\u4e8c\u3001\u5e7f\u4e49\u5cad\u56de\u5f52<\/h2>\n<p>Hoerl \u548c Kennard (1970) \u63d0\u51fa\u4e86\u5cad\u4f30\u8ba1\u7684\u4e00\u79cd\u63a8\u5e7f\u5f62\u5f0f&#xff0c; \u79f0\u4e3a\u5e7f\u4e49\u5cad\u4f30\u8ba1 (Generalized ridge estimate)\u3002<\/p>\n<p>\u666e\u901a\u7684\u5cad\u56de\u5f52\u4f30\u8ba1\u662f\u7ed9\u6837\u672c\u76f8\u5173\u9635\u7684\u4e3b\u5bf9\u89d2\u7ebf\u52a0\u4e0a\u76f8\u540c\u7684\u5e38\u6570<img decoding=\"async\" alt=\"k\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115018-689f1efac2f14.png\" \/>&#xff0c;\u5e7f\u4e49\u5cad\u56de\u5f52\u662f\u7ed9\u6837\u672c\u76f8\u5173\u9635\u7684\u4e3b\u5bf9\u89d2\u7ebf\u52a0\u4e0a\u5404\u4e0d\u76f8\u540c\u7684\u5e38\u6570\u00a0<img decoding=\"async\" alt=\"k_j\" class=\"mathcode\" src=\"2025-08-150pc5yhn1z2r.png\" \/>\u00a0\u3002<\/p>\n<p>\u5e7f\u4e49\u5cad\u56de\u5f52\u7684\u53c2\u6570\u4f30\u8ba1\u4e3a&#xff1a;<img decoding=\"async\" alt=\"\\\\hat{\\\\beta}(K) &#061;L(\\\\Lambda &#043; K)^{-1}L&#039;X&#039;y&#061; (X&#039;X &#043; LKL&#039;)^{-1}X&#039;y\" class=\"mathcode\" src=\"https:\/\/latex.csdn.net\/eq?%5Chat%7B%5Cbeta%7D%28K%29%20%3DL%28%5CLambda%20&amp;plus;%20K%29%5E%7B-1%7DL%27X%27y%3D%20%28X%27X%20&amp;plus;%20LKL%27%29%5E%7B-1%7DX%27y\" \/>\u3002<\/p>\n<p style=\"text-align:right\"><span style=\"color:#7b7f82\">\u524d\u4e00\u4e2a\u7b49\u53f7\u4e3a\u5b9a\u4e49&#xff08;\u901a\u8fc7\u5178\u5219\u5f62\u5f0f\u8868\u793a&#xff09;&#xff0c;\u540e\u4e00\u4e2a\u7b49\u53f7\u89c1\u6027\u8d281\u3002\u7b14\u8005\u6ce8\u3002<\/span><\/p>\n<p>\u5f53<img decoding=\"async\" alt=\"K&#061;kI\" class=\"mathcode\" src=\"2025-08-154juhymtc2aa.png\" \/>\u7684\u65f6\u5019&#xff0c;\u5e7f\u4e49\u5cad\u4f30\u8ba1\u5c31\u662f\u5cad\u4f30\u8ba1\u7b49\u4ef7\u3002<\/p>\n<h3>1\u3001\u7ebf\u6027\u6027<\/h3>\n<p>\u5e7f\u4e49\u5cad\u4f30\u8ba1\u53ef\u5199\u4f5c&#xff1a;<img decoding=\"async\" alt=\"\\\\hat{\\\\beta}(K) &#061; B_k\\\\hat{\\\\beta}\" class=\"mathcode\" src=\"2025-08-15jaagyredmf5.png\" \/>&#xff0c;\u5176\u4e2d<img decoding=\"async\" alt=\"B_k&#061;(X&#039;X &#043; LKL&#039;)^{-1}(X&#039;X)\" class=\"mathcode\" src=\"https:\/\/latex.csdn.net\/eq?B_k%3D%28X%27X%20&amp;plus;%20LKL%27%29%5E%7B-1%7D%28X%27X%29\" \/>&#xff0c;\u5373\u5e7f\u4e49\u5cad\u4f30\u8ba1\u662f\u6700\u5c0f\u4e8c\u4e58\u4f30\u8ba1\u7684\u7ebf\u6027\u53d8\u6362\u3002<\/p>\n<p>\u8bc1\u660e&#xff1a;<\/p>\n<p>\u5e7f\u4e49\u5cad\u56de\u5f52\u5b9a\u4e49\u4e3a&#xff1a;<img decoding=\"async\" alt=\"\\\\hat{\\\\beta}(K) &#061; L(\\\\Lambda &#043; K)^{-1}L&#039;X&#039;y\" class=\"mathcode\" src=\"https:\/\/latex.csdn.net\/eq?%5Chat%7B%5Cbeta%7D%28K%29%20%3D%20L%28%5CLambda%20&amp;plus;%20K%29%5E%7B-1%7DL%27X%27y\" \/><\/p>\n<p>\u800c\u6700\u5c0f\u4e8c\u4e58\u4f30\u8ba1<img decoding=\"async\" alt=\"\\\\hat{\\\\beta}&#061;(X&#039;X)^{-1}X&#039;y\" class=\"mathcode\" src=\"2025-08-15v4fs0jg50c3.png\" \/>&#xff0c;\u5373<img decoding=\"async\" alt=\"X&#039;y &#061; X&#039;X\\\\hat{\\\\beta}\" class=\"mathcode\" src=\"2025-08-15uttc5itkulx.png\" \/>\u3002\u4ee3\u5165\u4e0a\u5f0f\u5f97&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"\\\\hat{\\\\beta}(K) &#061; L(\\\\Lambda &#043; K)^{-1}L&#039;X&#039;X\\\\hat{\\\\beta}\" class=\"mathcode\" src=\"https:\/\/latex.csdn.net\/eq?%5Chat%7B%5Cbeta%7D%28K%29%20%3D%20L%28%5CLambda%20&amp;plus;%20K%29%5E%7B-1%7DL%27X%27X%5Chat%7B%5Cbeta%7D\" \/>\u3002<\/p>\n<p>\u53c8\u56e0\u4e3a<img decoding=\"async\" alt=\"X&#039;X &#061; L\\\\Lambda L&#039;\" class=\"mathcode\" src=\"2025-08-15omhkzrdhr1j.png\" \/>&#xff0c;\u5219&#xff1a;<img decoding=\"async\" alt=\"\\\\hat{\\\\beta}(K) &#061; L(\\\\Lambda &#043; K)^{-1}L&#039; \\\\cdot L\\\\Lambda L&#039; \\\\cdot \\\\hat{\\\\beta} &#061; L(\\\\Lambda &#043; K)^{-1}\\\\Lambda L&#039; \\\\hat{\\\\beta}\" class=\"mathcode\" src=\"https:\/\/latex.csdn.net\/eq?%5Chat%7B%5Cbeta%7D%28K%29%20%3D%20L%28%5CLambda%20&amp;plus;%20K%29%5E%7B-1%7DL%27%20%5Ccdot%20L%5CLambda%20L%27%20%5Ccdot%20%5Chat%7B%5Cbeta%7D%20%3D%20L%28%5CLambda%20&amp;plus;%20K%29%5E%7B-1%7D%5CLambda%20L%27%20%5Chat%7B%5Cbeta%7D\" \/>\u3002<\/p>\n<p>\u4ee4<img decoding=\"async\" alt=\"B_k &#061; L(\\\\Lambda &#043; K)^{-1}\\\\Lambda L&#039; &#061; (X&#039;X &#043; LKL&#039;)^{-1}(X&#039;X)\" class=\"mathcode\" src=\"https:\/\/latex.csdn.net\/eq?B_k%20%3D%20L%28%5CLambda%20&amp;plus;%20K%29%5E%7B-1%7D%5CLambda%20L%27%20%3D%20%28X%27X%20&amp;plus;%20LKL%27%29%5E%7B-1%7D%28X%27X%29\" \/>&#xff0c;\u53ef\u5f97<img decoding=\"async\" alt=\"\\\\hat{\\\\beta}(K) &#061; B_k\\\\hat{\\\\beta}\" class=\"mathcode\" src=\"2025-08-15jaagyredmf5.png\" \/>\u3002\u5f97\u8bc1\u3002<\/p>\n<p style=\"text-align:right\">\u25a1<\/p>\n<h3>2\u3001\u6709\u504f\u6027<\/h3>\n<p>\u5e7f\u4e49\u5cad\u4f30\u8ba1<img decoding=\"async\" alt=\"\\\\hat{\\\\beta}(K)\" class=\"mathcode\" src=\"2025-08-15xedbtqxef4z.png\" \/>\u4e5f\u662f<img decoding=\"async\" alt=\"\\\\beta\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115021-689f1efdd63a0.png\" \/>\u7684\u6709\u504f\u4f30\u8ba1\u3002<\/p>\n<p>\u8bc1\u660e&#xff1a;<\/p>\n<p><img decoding=\"async\" alt=\"E\\\\hat{\\\\beta}(K) &#061; E(B_k\\\\hat{\\\\beta}) &#061; B_kE(\\\\hat{\\\\beta}) &#061; B_k\\\\beta\" class=\"mathcode\" src=\"2025-08-15ptwhsmwpbvg.png\" \/>&#xff0c;\u5373\u53ea\u8981<img decoding=\"async\" alt=\"B_k \\\\neq I\" class=\"mathcode\" src=\"2025-08-15lnk4epspibf.png\" \/>&#xff08;\u7b49\u4ef7\u4e8e<img decoding=\"async\" alt=\"K \\\\neq 0\" class=\"mathcode\" src=\"2025-08-15bl0k4tqeu12.png\" \/>&#xff09;&#xff0c;\u5e7f\u4e49\u5cad\u4f30\u8ba1\u5c31\u662f\u6709\u504f\u4f30\u8ba1\u3002\u5f97\u8bc1\u3002<\/p>\n<p style=\"text-align:right\">\u25a1<\/p>\n<h3>3\u3001\u538b\u7f29\u6027<\/h3>\n<p>\u5bf9\u4e8e\u4efb\u610f\u7684<img decoding=\"async\" alt=\"K &#061; diag(k_1, ..., k_p)\\\\),\\\\(k_i \\\\geq 0\\\\),\\\\(\\\\|\\\\hat{\\\\beta}\\\\| &gt; 0\" class=\"mathcode\" src=\"2025-08-15qm0yfoy3wui.png\" \/>&#xff0c;\u603b\u6709<img decoding=\"async\" alt=\"\\\\|\\\\hat{\\\\beta}(K)\\\\| &lt; \\\\|\\\\hat{\\\\beta}\\\\|\" class=\"mathcode\" src=\"2025-08-15k4ltdfprgqr.png\" \/>\u3002<\/p>\n<p>\u8bc1\u660e&#xff1a;<\/p>\n<p>\u7531\u5e7f\u4e49\u5cad\u4f30\u8ba1\u7684\u5b9a\u4e49<img decoding=\"async\" alt=\"\\\\hat{\\\\beta}(K) &#061; L\\\\hat{\\\\alpha}(K)\" class=\"mathcode\" src=\"2025-08-15gcqcjpu2s2n.png\" \/>&#xff0c;\u5176\u4e2d<img decoding=\"async\" alt=\"\\\\hat{\\\\alpha}(K) &#061; (\\\\Lambda &#043; K)^{-1}Z&#039;Y\" class=\"mathcode\" src=\"https:\/\/latex.csdn.net\/eq?%5Chat%7B%5Calpha%7D%28K%29%20%3D%20%28%5CLambda%20&amp;plus;%20K%29%5E%7B-1%7DZ%27Y\" \/>\u3002<\/p>\n<p>\u800c<img decoding=\"async\" alt=\"\\\\hat{\\\\alpha} &#061; \\\\Lambda^{-1}Z&#039;Y\" class=\"mathcode\" src=\"2025-08-15qwnmruaum0e.png\" \/>&#xff0c;\u6545<img decoding=\"async\" alt=\"\\\\hat{\\\\alpha}(K) &#061; (\\\\Lambda &#043; K)^{-1}\\\\Lambda \\\\hat{\\\\alpha}\" class=\"mathcode\" src=\"https:\/\/latex.csdn.net\/eq?%5Chat%7B%5Calpha%7D%28K%29%20%3D%20%28%5CLambda%20&amp;plus;%20K%29%5E%7B-1%7D%5CLambda%20%5Chat%7B%5Calpha%7D\" \/>\u3002<\/p>\n<p>\u56e0\u6b64&#xff1a;<img decoding=\"async\" alt=\"\\\\|\\\\hat{\\\\beta}(K)\\\\| &#061; \\\\|\\\\hat{\\\\alpha}(K)\\\\| &#061; \\\\left\\\\|(\\\\Lambda &#043; K)^{-1}\\\\Lambda \\\\hat{\\\\alpha}\\\\right\\\\|\" class=\"mathcode\" src=\"https:\/\/latex.csdn.net\/eq?%5C%7C%5Chat%7B%5Cbeta%7D%28K%29%5C%7C%20%3D%20%5C%7C%5Chat%7B%5Calpha%7D%28K%29%5C%7C%20%3D%20%5Cleft%5C%7C%28%5CLambda%20&amp;plus;%20K%29%5E%7B-1%7D%5CLambda%20%5Chat%7B%5Calpha%7D%5Cright%5C%7C\" \/>\u3002<\/p>\n<p>\u7531\u4e8e<img decoding=\"async\" alt=\"(\\\\Lambda &#043; K)^{-1}\\\\Lambda\" class=\"mathcode\" src=\"https:\/\/latex.csdn.net\/eq?%28%5CLambda%20&amp;plus;%20K%29%5E%7B-1%7D%5CLambda\" \/>\u662f\u5bf9\u89d2\u5143\u5c0f\u4e8e 1 \u7684\u5bf9\u89d2\u77e9\u9635&#xff08;\u56e0<img decoding=\"async\" alt=\"k_i \\\\geq 0\" class=\"mathcode\" src=\"2025-08-15ameefg5nljo.png\" \/>&#xff09;&#xff0c;<\/p>\n<p>\u6545<img decoding=\"async\" alt=\"\\\\left\\\\|(\\\\Lambda &#043; K)^{-1}\\\\Lambda \\\\hat{\\\\alpha}\\\\right\\\\| &lt; \\\\|\\\\hat{\\\\alpha}\\\\| &#061; \\\\|\\\\hat{\\\\beta}\\\\|\" class=\"mathcode\" src=\"https:\/\/latex.csdn.net\/eq?%5Cleft%5C%7C%28%5CLambda%20&amp;plus;%20K%29%5E%7B-1%7D%5CLambda%20%5Chat%7B%5Calpha%7D%5Cright%5C%7C%20%3C%20%5C%7C%5Chat%7B%5Calpha%7D%5C%7C%20%3D%20%5C%7C%5Chat%7B%5Cbeta%7D%5C%7C\" \/>&#xff0c;\u5373<img decoding=\"async\" alt=\"\\\\|\\\\hat{\\\\beta}(K)\\\\| &lt; \\\\|\\\\hat{\\\\beta}\\\\|\" class=\"mathcode\" src=\"2025-08-15k4ltdfprgqr.png\" \/>\u3002\u5f97\u8bc1\u3002<\/p>\n<p style=\"text-align:right\">\u25a1<\/p>\n<h3>4\u3001\u66f4\u4f18\u6027<\/h3>\n<p>\u5b58\u5728<img decoding=\"async\" alt=\"K &#061; diag(k_1, ..., k_p) &gt; 0\" class=\"mathcode\" src=\"2025-08-154v50hg1annv.png\" \/>&#xff0c;\u4f7f\u5f97<img decoding=\"async\" alt=\"MSE(\\\\hat{\\\\beta}(K)) &lt; MSE(\\\\hat{\\\\beta})\" class=\"mathcode\" src=\"2025-08-15o414zwrltoi.png\" \/>\u3002<\/p>\n<p>\u8bc1\u660e&#xff1a;<\/p>\n<p>\u5747\u65b9\u8bef\u5dee<img decoding=\"async\" alt=\"MSE(\\\\hat{\\\\beta}(K)) &#061; trCov(\\\\hat{\\\\beta}(K)) &#043; \\\\|E\\\\hat{\\\\beta}(K) - \\\\beta\\\\|^2\" class=\"mathcode\" src=\"https:\/\/latex.csdn.net\/eq?MSE%28%5Chat%7B%5Cbeta%7D%28K%29%29%20%3D%20trCov%28%5Chat%7B%5Cbeta%7D%28K%29%29%20&amp;plus;%20%5C%7CE%5Chat%7B%5Cbeta%7D%28K%29%20-%20%5Cbeta%5C%7C%5E2\" \/>\u3002<\/p>\n<p>\u5176\u4e2d&#xff0c;<img decoding=\"async\" alt=\"Cov(\\\\hat{\\\\beta}(K)) &#061; \\\\sigma^2B_k(X&#039;X)^{-1}B_k&#039;\" class=\"mathcode\" src=\"2025-08-15kwnfqk2psct.png\" \/>&#xff0c;\u504f\u5dee\u9879<img decoding=\"async\" alt=\"\\\\|E\\\\hat{\\\\beta}(K) - \\\\beta\\\\|^2 &#061; \\\\|(B_k - I)\\\\beta\\\\|^2\" class=\"mathcode\" src=\"2025-08-15tfdub5lplh0.png\" \/>\u3002<\/p>\n<p>\u5f53<img decoding=\"async\" alt=\"K\" class=\"mathcode\" src=\"2025-08-15tdcsiaewhfp.png\" \/>\u5145\u5206\u5c0f\u65f6&#xff0c;\u534f\u65b9\u5dee\u9879\u7684\u51cf\u5c11\u91cf\u4f1a\u5927\u4e8e\u504f\u5dee\u9879\u7684\u589e\u52a0\u91cf\u3002<\/p>\n<p>\u4eff\u7167\u5cad\u56de\u5f52\u6027\u8d28 4 \u7684\u8bc1\u660e\u601d\u8def&#xff0c;\u901a\u8fc7\u5bf9MSE\u5173\u4e8e<img decoding=\"async\" alt=\"K\" class=\"mathcode\" src=\"2025-08-15tdcsiaewhfp.png\" \/>\u7684\u5bfc\u6570\u5206\u6790\u53ef\u77e5&#xff0c;<\/p>\n<p>\u5b58\u5728<img decoding=\"async\" alt=\"K&gt;0\" class=\"mathcode\" src=\"2025-08-153bx3bpc2sn3.png\" \/>\u4f7f\u5f97<img decoding=\"async\" alt=\"MSE(\\\\hat{\\\\beta}(K)) &lt; MSE(\\\\hat{\\\\beta})\" class=\"mathcode\" src=\"2025-08-15o414zwrltoi.png\" \/>\u3002\u5f97\u8bc1\u3002<\/p>\n<p style=\"text-align:right\">\u25a1<\/p>\n<h2>\u4e09\u3001\u5e7f\u4e49\u5cad\u56de\u5f52\u7684\u610f\u4e49<\/h2>\n<p>\u5e7f\u4e49\u5cad\u56de\u5f52\u4f5c\u4e3a\u5cad\u56de\u5f52\u7684\u63a8\u5e7f&#xff0c;\u901a\u8fc7\u66f4\u7075\u6d3b\u7684\u53c2\u6570\u8bbe\u7f6e\u5b9e\u73b0\u5bf9\u591a\u91cd\u5171\u7ebf\u6027\u95ee\u9898\u7684\u66f4\u7cbe\u51c6\u5904\u7406&#xff0c;\u540c\u65f6\u8fdb\u4e00\u6b65\u4f18\u5316\u4f30\u8ba1\u6027\u80fd&#xff0c;\u5177\u4f53\u610f\u4e49\u53ef\u4ece\u4ee5\u4e0b\u4e24\u4e2a\u65b9\u9762\u5c55\u5f00&#xff1a;<\/p>\n<h3>1. \u63d0\u5347\u53c2\u6570\u538b\u7f29\u7684\u7075\u6d3b\u6027<\/h3>\n<p>\u5cad\u56de\u5f52\u7684\u6838\u5fc3\u662f\u5f15\u5165\u5355\u4e00\u5cad\u53c2\u6570\u00a0<img decoding=\"async\" alt=\"k\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115018-689f1efac2f14.png\" \/>&#xff08;\u5373<img decoding=\"async\" alt=\"K&#061;kI\" class=\"mathcode\" src=\"2025-08-154juhymtc2aa.png\" \/>&#xff0c;I\u4e3a\u5355\u4f4d\u77e9\u9635&#xff09;&#xff0c;\u5bf9\u6240\u6709\u56de\u5f52\u7cfb\u6570&#xff08;\u6216\u5178\u5219\u7cfb\u6570\u00a0<img decoding=\"async\" alt=\"\\\\alpha\" class=\"mathcode\" src=\"2025-08-15gu2t253yte0.png\" \/>\u00a0\u8fdb\u884c\u7edf\u4e00\u7a0b\u5ea6\u7684\u538b\u7f29\u3002<\/p>\n<p>\u8fd9\u79cd\u65b9\u5f0f\u65e0\u6cd5\u533a\u5206\u4e0d\u540c\u89e3\u91ca\u53d8\u91cf\u7684\u591a\u91cd\u5171\u7ebf\u6027\u7a0b\u5ea6\u5dee\u5f02 \u2014\u2014 \u4f8b\u5982&#xff0c;\u5bf9\u7279\u5f81\u503c\u6781\u5c0f&#xff08;\u53d7\u591a\u91cd\u5171\u7ebf\u6027\u5f71\u54cd\u4e25\u91cd&#xff09;\u7684\u5178\u5219\u7cfb\u6570\u4e0e\u7279\u5f81\u503c\u8f83\u5927&#xff08;\u53d7\u5f71\u54cd\u8f83\u5c0f&#xff09;\u7684\u5178\u5219\u7cfb\u6570&#xff0c;\u538b\u7f29\u529b\u5ea6\u5b8c\u5168\u4e00\u81f4&#xff0c;\u53ef\u80fd\u5bfc\u81f4 \u3010\u8fc7\u5ea6\u538b\u7f29\u6709\u7528\u4fe1\u606f\u3011\u6216\u3010\u538b\u7f29\u4e0d\u8db3\u4ecd\u5b58\u5171\u7ebf\u6027\u3011\u7684\u95ee\u9898\u3002<\/p>\n<p>\u800c\u5e7f\u4e49\u5cad\u56de\u5f52\u5c06\u5355\u4e00\u53c2\u6570<img decoding=\"async\" alt=\"k\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115018-689f1efac2f14.png\" \/>\u63a8\u5e7f\u4e3a\u5bf9\u89d2\u77e9\u9635\u00a0<img decoding=\"async\" alt=\"K&#061;diag(k_1,k_2,\\\\dots,k_p),(k_i\\\\geq0)\" class=\"mathcode\" src=\"2025-08-15ku0b3cd35li.png\" \/>\u00a0\u4e3a\u5bf9\u5e94\u7b2c\u00a0<img decoding=\"async\" alt=\"i\" class=\"mathcode\" src=\"2025-08-15o1exw0gmkof.png\" \/>\u00a0\u4e2a\u5178\u5219\u7cfb\u6570\u7684\u5cad\u53c2\u6570&#xff09;&#xff0c;\u53ef\u9488\u5bf9\u6bcf\u4e2a\u5178\u5219\u7cfb\u6570\u00a0<img decoding=\"async\" alt=\"\\\\alpha_i\" class=\"mathcode\" src=\"2025-08-151bs00dnel4d.png\" \/>&#xff08;\u5bf9\u5e94\u539f\u6a21\u578b\u4e2d\u4e0d\u540c\u89e3\u91ca\u53d8\u91cf\u7684 \u3010\u6709\u6548\u4fe1\u606f\u6210\u5206\u3011&#xff09;\u8bbe\u7f6e\u72ec\u7acb\u7684\u538b\u7f29\u5f3a\u5ea6\u3002<\/p>\n<p>\u4f8b\u5982&#xff0c;\u5bf9\u7279\u5f81\u503c\u00a0<img decoding=\"async\" alt=\"\\\\lambda_i\\\\approx0\" class=\"mathcode\" src=\"2025-08-15l1cwmyvsavw.png\" \/>&#xff08;\u5171\u7ebf\u6027\u4e25\u91cd&#xff09;\u7684\u5178\u5219\u7cfb\u6570&#xff0c;\u53ef\u8bbe\u7f6e\u66f4\u5927\u7684\u00a0<img decoding=\"async\" alt=\"k_i\" class=\"mathcode\" src=\"2025-08-15sai10oiknlw.png\" \/>\u00a0\u4ee5\u66f4\u5f3a\u538b\u7f29\u65b9\u5dee&#xff1b;\u5bf9\u00a0<img decoding=\"async\" alt=\"\\\\lambda_i\" class=\"mathcode\" src=\"2025-08-15kz1nuzkkmur.png\" \/>\u00a0\u8f83\u5927&#xff08;\u5171\u7ebf\u6027\u8f7b\u5fae&#xff09;\u7684\u5178\u5219\u7cfb\u6570&#xff0c;\u8bbe\u7f6e\u66f4\u5c0f\u7684<img decoding=\"async\" alt=\"k_i\" class=\"mathcode\" src=\"2025-08-15sai10oiknlw.png\" \/>\u4ee5\u51cf\u5c11\u4e0d\u5fc5\u8981\u7684\u504f\u5dee&#xff0c;\u5b9e\u73b0 \u201c\u6309\u9700\u538b\u7f29\u201d&#xff0c;\u66f4\u8d34\u5408\u5b9e\u9645\u6570\u636e\u4e2d\u4e0d\u540c\u53d8\u91cf\u7684\u5171\u7ebf\u6027\u5dee\u5f02\u3002<\/p>\n<p style=\"text-align:right\"><span style=\"color:#7b7f82\">\u5b9e\u9645\u4e0a&#xff0c;<img decoding=\"async\" alt=\"k_i\" class=\"mathcode\" src=\"2025-08-15sai10oiknlw.png\" \/>\u7684\u9009\u62e9\u65b9\u6cd5\u548c\u5cad\u56de\u5f52\u662f\u76f8\u4f3c\u7684\u3002\u7b14\u8005\u6ce8\u3002<\/span><\/p>\n<h3>2. \u8fdb\u4e00\u6b65\u4f18\u5316\u5747\u65b9\u8bef\u5dee&#xff08;MSE&#xff09;<\/h3>\n<p>\u4f30\u8ba1\u91cf\u7684\u5747\u65b9\u8bef\u5dee\u7531 \u201c\u65b9\u5dee\u201d \u548c \u201c\u504f\u5dee\u5e73\u65b9\u201d \u5171\u540c\u51b3\u5b9a&#xff0c;\u5cad\u56de\u5f52\u901a\u8fc7\u5f15\u5165\u5355\u4e00\u00a0<img decoding=\"async\" alt=\"k\" class=\"mathcode\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815115018-689f1efac2f14.png\" \/>\u00a0\u5b9e\u73b0 \u201c\u65b9\u5dee\u964d\u4f4e\u5e45\u5ea6\u8986\u76d6\u504f\u5dee\u589e\u52a0\u5e45\u5ea6\u201d&#xff0c;\u800c\u5e7f\u4e49\u5cad\u56de\u5f52\u51ed\u501f\u591a\u53c2\u6570\u00a0<img decoding=\"async\" alt=\"k_i\" class=\"mathcode\" src=\"2025-08-15sai10oiknlw.png\" \/>\u00a0\u7684\u7075\u6d3b\u6027&#xff0c;\u80fd\u66f4\u7cbe\u7ec6\u5730\u5e73\u8861\u4e8c\u8005\u5173\u7cfb&#xff1a;<\/p>\n<ul>\n<li><span style=\"color:#494949\">\u5bf9\u53d7\u591a\u91cd\u5171\u7ebf\u6027\u5f71\u54cd\u5927\u7684\u53c2\u6570&#xff08;\u5bf9\u5e94\u5c0f\u00a0<img decoding=\"async\" alt=\"\\\\lambda_i\" class=\"mathcode\" src=\"2025-08-15kz1nuzkkmur.png\" \/>\u00a0&#xff09;&#xff0c;\u8f83\u5927\u7684\u00a0<img decoding=\"async\" alt=\"k_i\" class=\"mathcode\" src=\"2025-08-15sai10oiknlw.png\" \/>\u00a0\u53ef\u663e\u8457\u964d\u4f4e\u5176\u65b9\u5dee&#xff0c;\u4e14\u4ec5\u5f15\u5165\u5c11\u91cf\u504f\u5dee&#xff08;\u56e0\u8be5\u53c2\u6570\u672c\u8eab\u4f30\u8ba1\u7a33\u5b9a\u6027\u5dee&#xff0c;\u9002\u5ea6\u504f\u5dee\u5bf9\u6574\u4f53 MSE \u5f71\u54cd\u5c0f&#xff09;&#xff1b;<\/span><\/li>\n<li><span style=\"color:#494949\">\u5bf9\u53d7\u591a\u91cd\u5171\u7ebf\u6027\u5f71\u54cd\u5c0f\u7684\u53c2\u6570&#xff08;\u5bf9\u5e94\u5927\u00a0<img decoding=\"async\" alt=\"\\\\lambda_i\" class=\"mathcode\" src=\"2025-08-15kz1nuzkkmur.png\" \/>&#xff09;&#xff0c;\u8f83\u5c0f\u7684\u00a0<img decoding=\"async\" alt=\"k_i\" class=\"mathcode\" src=\"2025-08-15sai10oiknlw.png\" \/>\u00a0\u53ef\u51e0\u4e4e\u4fdd\u6301\u5176\u539f\u6709\u4f4e\u504f\u5dee\u7279\u6027&#xff0c;\u540c\u65f6\u907f\u514d\u65b9\u5dee\u8fc7\u5ea6\u589e\u52a0\u3002<\/span><\/li>\n<\/ul>\n<p>\u8fd9\u79cd\u3010\u9488\u5bf9\u6027\u8c03\u8282\u3011\u4f7f\u5e7f\u4e49\u5cad\u56de\u5f52\u7406\u8bba\u4e0a\u80fd\u5b9e\u73b0\u6bd4\u5cad\u56de\u5f52\u66f4\u5c0f\u7684 MSE&#xff0c;\u8fdb\u4e00\u6b65\u63d0\u5347\u4f30\u8ba1\u7684\u6574\u4f53\u7cbe\u5ea6&#xff0c;\u5c24\u5176\u5728\u89e3\u91ca\u53d8\u91cf\u5171\u7ebf\u6027\u7a0b\u5ea6\u5dee\u5f02\u8f83\u5927\u7684\u573a\u666f\u4e2d&#xff0c;\u4f18\u52bf\u66f4\u660e\u663e\u3002<\/p>\n<hr 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