{"id":50036,"date":"2025-07-30T20:56:50","date_gmt":"2025-07-30T12:56:50","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/50036.html"},"modified":"2025-07-30T20:56:50","modified_gmt":"2025-07-30T12:56:50","slug":"%e9%80%bb%e8%be%91%e5%9b%9e%e5%bd%92%ef%bc%9a%e4%bb%8e%e7%ba%bf%e6%80%a7%e5%9b%9e%e5%bd%92%e5%88%b0%e5%88%86%e7%b1%bb%e5%86%b3%e7%ad%96%e7%9a%84%e6%bc%94%e5%8c%96","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/50036.html","title":{"rendered":"\u903b\u8f91\u56de\u5f52\uff1a\u4ece\u7ebf\u6027\u56de\u5f52\u5230\u5206\u7c7b\u51b3\u7b56\u7684\u6f14\u5316"},"content":{"rendered":"<p>\u7ed3\u5408\u903b\u8f91\u56de\u5f52\u7684\u6838\u5fc3\u539f\u7406\u3001\u6570\u5b66\u63a8\u5bfc\u4e0e\u5de5\u7a0b\u5b9e\u8df5\u8fdb\u884c\u7cfb\u7edf\u5316\u9610\u8ff0&#xff1a;<\/p>\n<hr \/>\n<h4>\u903b\u8f91\u56de\u5f52&#xff1a;\u4ece\u7ebf\u6027\u56de\u5f52\u5230\u5206\u7c7b\u51b3\u7b56\u7684\u6f14\u5316<\/h4>\n<p>\u200b\u4e3a\u4f55\u540d\u4e3a\u201c\u56de\u5f52\u201d\u5374\u7528\u4e8e\u5206\u7c7b&#xff1f;\u63ed\u79d8Sigmoid\u51fd\u6570\u7684\u6865\u6881\u4f5c\u7528\u4e0e\u51b3\u7b56\u8fb9\u754c\u672c\u8d28\u200b<\/p>\n<hr \/>\n<h5>\u4e00\u3001\u95ee\u9898\u8d77\u6e90&#xff1a;\u7ebf\u6027\u56de\u5f52\u4e3a\u4f55\u4e0d\u80fd\u76f4\u63a5\u7528\u4e8e\u5206\u7c7b&#xff1f;<\/h5>\n<li>\n<p>\u200b\u56de\u5f52 vs \u5206\u7c7b\u7684\u672c\u8d28\u5dee\u5f02\u200b<\/p>\n<ul>\n<li>\u7ebf\u6027\u56de\u5f52&#xff1a;\u8f93\u51fa\u8fde\u7eed\u503c&#xff08;\u5982\u623f\u4ef7\u9884\u6d4b&#xff09;&#xff0c;\u62df\u5408\u76f4\u7ebf\u6700\u5c0f\u5316\u8bef\u5dee&#xff08;\u5982\u5747\u65b9\u8bef\u5dee&#xff09;<\/li>\n<li>\u5206\u7c7b\u4efb\u52a1&#xff1a;\u8f93\u51fa\u79bb\u6563\u7c7b\u522b&#xff08;\u5982\u7ea2\/\u7eff\u70b9\u5206\u7c7b&#xff09;&#xff0c;\u9700\u627e\u5230\u51b3\u7b56\u8fb9\u754c\u5206\u79bb\u6570\u636e<\/li>\n<li>\u5173\u952e\u77db\u76fe&#xff1a;\u7ebf\u6027\u56de\u5f52\u7684\u8f93\u51fa\u8303\u56f4\u662f(\u2212\u221e,&#043;\u221e)(-\\\\infty, &#043;\\\\infty)(\u2212\u221e,&#043;\u221e)&#xff0c;\u65e0\u6cd5\u76f4\u63a5\u6620\u5c04\u5230\u7c7b\u522b\u6807\u7b7e&#xff08;0\/1&#xff09;<\/li>\n<\/ul>\n<\/li>\n<li>\n<p>\u200b\u76f4\u63a5\u4f7f\u7528\u7ebf\u6027\u56de\u5f52\u7684\u7f3a\u9677\u200b<\/p>\n<ul>\n<li>\u82e5\u5f3a\u884c\u8bbe\u5b9a\u9608\u503c&#xff08;\u5982 y&gt;0.5y&gt;0.5y&gt;0.5 \u5224\u4e3a\u6b63\u7c7b&#xff09;&#xff0c;\u5b58\u5728\u4e24\u5927\u95ee\u9898&#xff1a;\n<ul>\n<li>\u8f93\u51fa\u503c\u65e0\u6982\u7387\u610f\u4e49&#xff08;\u53ef\u80fd\u8d85\u51fa[0,1]\u8303\u56f4&#xff09;<\/li>\n<li>\u5206\u6bb5\u51fd\u6570\u4e0d\u53ef\u5bfc&#xff0c;\u65e0\u6cd5\u4f18\u5316\u53c2\u6570<br \/>\n\u4f8b&#xff1a;\u4e8c\u7ef4\u7a7a\u95f4\u4e2d&#xff0c;\u7ebf\u6027\u65b9\u7a0b y&#061;\u03b80&#043;\u03b81&#215;1&#043;\u03b82x2y &#061; \\\\theta_0 &#043; \\\\theta_1x_1 &#043; \\\\theta_2x_2y&#061;\u03b80\u200b&#043;\u03b81\u200bx1\u200b&#043;\u03b82\u200bx2\u200b \u65e0\u6cd5\u5c06\u7ea2\u7eff\u70b9\u5206\u5230\u76f4\u7ebf\u4e24\u4fa7<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<hr \/>\n<h5>\u4e8c\u3001\u6838\u5fc3\u7a81\u7834&#xff1a;Sigmoid\u51fd\u6570\u5982\u4f55\u8fde\u63a5\u56de\u5f52\u4e0e\u5206\u7c7b&#xff1f;<\/h5>\n<li>\n<p>\u200bSigmoid\u51fd\u6570\u7684\u6570\u5b66\u9b54\u529b\u200b<br \/>\n\u03c3(z)&#061;11&#043;e\u2212z\u5176\u4e2d\u00a0z&#061;\u03b8Tx\\\\sigma(z) &#061; \\\\frac{1}{1 &#043; e^{-z}} \\\\quad \\\\text{\u5176\u4e2d } z &#061; \\\\theta^Tx\u03c3(z)&#061;1&#043;e\u2212z1\u200b\u5176\u4e2d\u00a0z&#061;\u03b8Tx<\/p>\n<ul>\n<li>\u200b\u503c\u57df\u538b\u7f29\u200b&#xff1a;\u5c06\u4efb\u610f\u5b9e\u6570 zzz \u6620\u5c04\u5230 (0,1)(0,1)(0,1) \u533a\u95f4&#xff0c;\u8f93\u51fa\u53ef\u89e3\u91ca\u4e3a\u6982\u7387<\/li>\n<li>\u200b\u53ef\u5bfc\u6027\u200b&#xff1a;\u5bfc\u6570 \u03c3\u2032(z)&#061;\u03c3(z)(1\u2212\u03c3(z))\\\\sigma&#039;(z) &#061; \\\\sigma(z)(1-\\\\sigma(z))\u03c3\u2032(z)&#061;\u03c3(z)(1\u2212\u03c3(z))&#xff0c;\u4fbf\u4e8e\u68af\u5ea6\u4e0b\u964d\u4f18\u5316<\/li>\n<li>\u200b\u51b3\u7b56\u89c4\u5219\u200b&#xff1a;<br \/>\ny^&#061;{1if\u00a0\u03c3(z)\u22650.50otherwise\\\\hat{y} &#061; \\\\begin{cases} 1 &amp; \\\\text{if } \\\\sigma(z) \\\\geq 0.5 \\\\\\\\ 0 &amp; \\\\text{otherwise} \\\\end{cases}y^\u200b&#061;{10\u200bif\u00a0\u03c3(z)\u22650.5otherwise\u200b<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"943\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/07\/20250730125648-688a169069556.png\" width=\"1408\" \/><\/p>\n<\/li>\n<li>\n<p>\u200b\u4e3a\u4f55\u4e0d\u7528\u5206\u6bb5\u51fd\u6570&#xff1f;\u200b\u200b<\/p>\n<ul>\n<li>\u5206\u6bb5\u51fd\u6570&#xff08;\u5982 z&gt;0z&gt;0z&gt;0 \u8f93\u51fa1&#xff09;\u4e0d\u53ef\u5bfc&#xff0c;\u65e0\u6cd5\u901a\u8fc7\u68af\u5ea6\u4e0b\u964d\u6c42\u89e3\u53c2\u6570 \u03b8\\\\theta\u03b8<\/li>\n<li>Sigmoid\u63d0\u4f9b\u5e73\u6ed1\u8fd1\u4f3c&#xff0c;\u4fdd\u7559\u53ef\u5bfc\u6027\u4e14\u903c\u8fd1\u9636\u8dc3\u51fd\u6570<\/li>\n<\/ul>\n<\/li>\n<hr \/>\n<h5>\u4e09\u3001\u6570\u5b66\u63a8\u5bfc&#xff1a;\u4ece\u6982\u7387\u5efa\u6a21\u5230\u53c2\u6570\u4f18\u5316<\/h5>\n<li>\n<p>\u200b\u6982\u7387\u5efa\u6a21\u200b<\/p>\n<ul>\n<li>\u6b63\u7c7b\u6982\u7387&#xff1a; P(y&#061;1\u2223x)&#061;\u03c3(\u03b8Tx)&#061;11&#043;e\u2212\u03b8TxP(y&#061;1|x) &#061; \\\\sigma(\\\\theta^Tx) &#061; \\\\frac{1}{1&#043;e^{-\\\\theta^Tx}}P(y&#061;1\u2223x)&#061;\u03c3(\u03b8Tx)&#061;1&#043;e\u2212\u03b8Tx1\u200b<\/li>\n<li>\u8d1f\u7c7b\u6982\u7387&#xff1a; P(y&#061;0\u2223x)&#061;1\u2212P(y&#061;1\u2223x)P(y&#061;0|x) &#061; 1 &#8211; P(y&#061;1|x)P(y&#061;0\u2223x)&#061;1\u2212P(y&#061;1\u2223x)<\/li>\n<\/ul>\n<\/li>\n<li>\n<p>\u200b\u635f\u5931\u51fd\u6570&#xff1a;\u4ea4\u53c9\u71b5&#xff08;Cross-Entropy&#xff09;\u200b\u200b<\/p>\n<ul>\n<li>\u901a\u8fc7\u6781\u5927\u4f3c\u7136\u4f30\u8ba1\u63a8\u5bfc&#xff1a;<br \/>\nL(\u03b8)&#061;\u220fi&#061;1nP(yi\u2223xi;\u03b8)&#061;\u220fi&#061;1ny^iyi(1\u2212y^i)1\u2212yiL(\\\\theta) &#061; \\\\prod_{i&#061;1}^n P(y_i|x_i;\\\\theta) &#061; \\\\prod_{i&#061;1}^n \\\\hat{y}_i^{y_i} (1-\\\\hat{y}_i)^{1-y_i}L(\u03b8)&#061;\u220fi&#061;1n\u200bP(yi\u200b\u2223xi\u200b;\u03b8)&#061;\u220fi&#061;1n\u200by^\u200biyi\u200b\u200b(1\u2212y^\u200bi\u200b)1\u2212yi\u200b<\/li>\n<li>\u53d6\u8d1f\u5bf9\u6570\u4f3c\u7136\u5f97\u635f\u5931\u51fd\u6570&#xff1a;<br \/>\nJ(\u03b8)&#061;\u22121n\u2211i&#061;1n[yiln\u2061(y^i)&#043;(1\u2212yi)ln\u2061(1\u2212y^i)]J(\\\\theta) &#061; -\\\\frac{1}{n} \\\\sum_{i&#061;1}^n \\\\left[ y_i \\\\ln(\\\\hat{y}_i) &#043; (1-y_i) \\\\ln(1-\\\\hat{y}_i) \\\\right]J(\u03b8)&#061;\u2212n1\u200b\u2211i&#061;1n\u200b[yi\u200bln(y^\u200bi\u200b)&#043;(1\u2212yi\u200b)ln(1\u2212y^\u200bi\u200b)]<br \/>\n\u6ce8&#xff1a;\u6b64\u51fd\u6570\u51f8\u6027\u4fdd\u8bc1\u68af\u5ea6\u4e0b\u964d\u6536\u655b\u5230\u5168\u5c40\u6700\u4f18<\/li>\n<\/ul>\n<\/li>\n<li>\n<p>\u200b\u68af\u5ea6\u4e0b\u964d\u4f18\u5316\u200b<\/p>\n<ul>\n<li>\u68af\u5ea6\u8ba1\u7b97&#xff1a;<br \/>\n\u2207\u03b8J&#061;1nXT(\u03c3(X\u03b8)\u2212y)\\\\nabla_\\\\theta J &#061; \\\\frac{1}{n} X^T (\\\\sigma(X\\\\theta) &#8211; y)\u2207\u03b8\u200bJ&#061;n1\u200bXT(\u03c3(X\u03b8)\u2212y)<\/li>\n<li>\u53c2\u6570\u66f4\u65b0&#xff1a;<br \/>\n\u03b8:&#061;\u03b8\u2212\u03b1\u2207\u03b8J\\\\theta :&#061; \\\\theta &#8211; \\\\alpha \\\\nabla_\\\\theta J\u03b8:&#061;\u03b8\u2212\u03b1\u2207\u03b8\u200bJ<br \/>\n&#xff08;\u03b1\\\\alpha\u03b1 \u4e3a\u5b66\u4e60\u7387&#xff09;<\/li>\n<\/ul>\n<\/li>\n<hr \/>\n<h5>\u56db\u3001\u51b3\u7b56\u8fb9\u754c&#xff1a;\u7ebf\u6027\u4e0e\u975e\u7ebf\u6027\u62d3\u5c55<\/h5>\n<li>\n<p>\u200b\u7ebf\u6027\u51b3\u7b56\u8fb9\u754c\u200b<\/p>\n<ul>\n<li>\u5f53 \u03c3(z)&#061;0.5\\\\sigma(z)&#061;0.5\u03c3(z)&#061;0.5 \u65f6 z&#061;0z&#061;0z&#061;0&#xff0c;\u5373 \u03b8Tx&#061;0\\\\theta^Tx &#061; 0\u03b8Tx&#061;0 \u4e3a\u5206\u7c7b\u8d85\u5e73\u9762<br \/>\n\u4f8b&#xff1a;\u4e8c\u7ef4\u7a7a\u95f4\u4e2d\u4e3a\u76f4\u7ebf&#xff0c;\u4e09\u7ef4\u4e3a\u5e73\u9762<\/li>\n<\/ul>\n<\/li>\n<li>\n<p>\u200b\u975e\u7ebf\u6027\u8fb9\u754c\u62d3\u5c55\u200b<\/p>\n<ul>\n<li>\u901a\u8fc7\u7279\u5f81\u5de5\u7a0b\u5f15\u5165\u591a\u9879\u5f0f\u9879&#xff08;\u5982 x12,x1x2x_1^2, x_1x_2x12\u200b,x1\u200bx2\u200b)<\/li>\n<li>\u6838\u51fd\u6570&#xff08;Kernel Trick&#xff09;\u6620\u5c04\u9ad8\u7ef4\u7a7a\u95f4&#xff08;\u9700\u914d\u5408\u6b63\u5219\u5316\u9632\u8fc7\u62df\u5408&#xff09;<\/li>\n<\/ul>\n<\/li>\n<hr \/>\n<h5>\u4e94\u3001\u4ee3\u7801\u5b9e\u73b0&#xff1a;Python\u5b9e\u6218\u793a\u4f8b<\/h5>\n<p>import numpy as np<\/p>\n<p>class LogisticRegression:<br \/>\n    def __init__(self, lr&#061;0.01, n_iters&#061;1000):<br \/>\n        self.lr &#061; lr          # \u5b66\u4e60\u7387<br \/>\n        self.n_iters &#061; n_iters # \u8fed\u4ee3\u6b21\u6570<br \/>\n        self.weights &#061; None    # \u6743\u91cd\u53c2\u6570<br \/>\n        self.bias &#061; None       # \u504f\u7f6e\u9879<\/p>\n<p>    def _sigmoid(self, z):<br \/>\n        return 1 \/ (1 &#043; np.exp(-z))  # Sigmoid\u6fc0\u6d3b\u51fd\u6570<\/p>\n<p>    def fit(self, X, y):<br \/>\n        n_samples, n_features &#061; X.shape<br \/>\n        self.weights &#061; np.zeros(n_features)<br \/>\n        self.bias &#061; 0<\/p>\n<p>        # \u68af\u5ea6\u4e0b\u964d\u4f18\u5316<br \/>\n        for _ in range(self.n_iters):<br \/>\n            z &#061; np.dot(X, self.weights) &#043; self.bias<br \/>\n            y_pred &#061; self._sigmoid(z)<\/p>\n<p>            # \u8ba1\u7b97\u68af\u5ea6<br \/>\n            dw &#061; (1 \/ n_samples) * np.dot(X.T, (y_pred &#8211; y))<br \/>\n            db &#061; (1 \/ n_samples) * np.sum(y_pred &#8211; y)<\/p>\n<p>            # \u66f4\u65b0\u53c2\u6570<br \/>\n            self.weights -&#061; self.lr * dw<br \/>\n            self.bias -&#061; self.lr * db<\/p>\n<p>    def predict(self, X, threshold&#061;0.5):<br \/>\n        z &#061; np.dot(X, self.weights) &#043; self.bias<br \/>\n        y_prob &#061; self._sigmoid(z)<br \/>\n        return (y_prob &gt;&#061; threshold).astype(int)  # \u6309\u9608\u503c\u5206\u7c7b<\/p>\n<p>\u200b\u5173\u952e\u6b65\u9aa4\u8bf4\u660e\u200b&#xff1a;<\/p>\n<ul>\n<li>fit() \u4e2d\u68af\u5ea6\u8ba1\u7b97\u5bf9\u5e94\u516c\u5f0f \u2207\u03b8J\\\\nabla_\\\\theta J\u2207\u03b8\u200bJ<\/li>\n<li>\u9884\u6d4b\u65f6\u4ee50.5\u4e3a\u9ed8\u8ba4\u9608\u503c<\/li>\n<\/ul>\n<hr 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