{"id":519,"date":"2021-12-22T16:27:48","date_gmt":"2021-12-22T08:27:48","guid":{"rendered":"http:\/\/139.9.1.231\/?p=519"},"modified":"2021-12-22T16:27:50","modified_gmt":"2021-12-22T08:27:50","slug":"k-means","status":"publish","type":"post","link":"http:\/\/139.9.1.231\/index.php\/2021\/12\/22\/k-means\/","title":{"rendered":"k-means"},"content":{"rendered":"\n<div class=\"wp-block-image is-style-default\"><figure class=\"aligncenter size-large\"><img src=\"https:\/\/cdn.pixabay.com\/photo\/2021\/12\/08\/10\/45\/christmas-6855327_1280.jpg\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>    k-means\u7b97\u6cd5\u7b97\u662f\u7ecf\u5178\u7684\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\uff0c\u8bb0\u5f97\u5e94\u8be5\u662f\u5927\u4e09\u7684\u8bfe\u4e0a\u5b66\u8fc7\uff0c\u540e\u9762\u5c31\u4e00\u76f4\u6ca1\u5728\u63a5\u89e6\u8fc7\u8be5\u7b97\u6cd5\u5bf9\u5e94\u7684\u95ee\u9898\uff0c\u73b0\u5728\u5c31\u6765\u56de\u987e\u8bb0\u5f55\u4e00\u4e0bkmeans\u7b97\u6cd5\u3002<\/p>\n\n\n\n<p>K-means \u6709\u4e00\u4e2a\u8457\u540d\u7684\u89e3\u91ca\uff1a\u7267\u5e08\u2014\u6751\u6c11\u6a21\u578b\uff1a<\/p>\n\n\n\n<blockquote class=\"wp-block-quote\"><p>\u6709\u56db\u4e2a\u7267\u5e08\u53bb\u90ca\u533a\u5e03\u9053\uff0c\u4e00\u5f00\u59cb\u7267\u5e08\u4eec\u968f\u610f\u9009\u4e86\u51e0\u4e2a\u5e03\u9053\u70b9\uff0c\u5e76\u4e14\u628a\u8fd9\u51e0\u4e2a\u5e03\u9053\u70b9\u7684\u60c5\u51b5\u516c\u544a\u7ed9\u4e86\u90ca\u533a\u6240\u6709\u7684\u6751\u6c11\uff0c\u4e8e\u662f\u6bcf\u4e2a\u6751\u6c11\u5230\u79bb\u81ea\u5df1\u5bb6\u6700\u8fd1\u7684\u5e03\u9053\u70b9\u53bb\u542c\u8bfe\u3002<br>\u542c\u8bfe\u4e4b\u540e\uff0c\u5927\u5bb6\u89c9\u5f97\u8ddd\u79bb\u592a\u8fdc\u4e86\uff0c\u4e8e\u662f\u6bcf\u4e2a\u7267\u5e08\u7edf\u8ba1\u4e86\u4e00\u4e0b\u81ea\u5df1\u7684\u8bfe\u4e0a\u6240\u6709\u7684\u6751\u6c11\u7684\u5730\u5740\uff0c\u642c\u5230\u4e86\u6240\u6709\u5730\u5740\u7684\u4e2d\u5fc3\u5730\u5e26\uff0c\u5e76\u4e14\u5728\u6d77\u62a5\u4e0a\u66f4\u65b0\u4e86\u81ea\u5df1\u7684\u5e03\u9053\u70b9\u7684\u4f4d\u7f6e\u3002<br>\u7267\u5e08\u6bcf\u4e00\u6b21\u79fb\u52a8\u4e0d\u53ef\u80fd\u79bb\u6240\u6709\u4eba\u90fd\u66f4\u8fd1\uff0c\u6709\u7684\u4eba\u53d1\u73b0A\u7267\u5e08\u79fb\u52a8\u4ee5\u540e\u81ea\u5df1\u8fd8\u4e0d\u5982\u53bbB\u7267\u5e08\u5904\u542c\u8bfe\u66f4\u8fd1\uff0c\u4e8e\u662f\u6bcf\u4e2a\u6751\u6c11\u53c8\u53bb\u4e86\u79bb\u81ea\u5df1\u6700\u8fd1\u7684\u5e03\u9053\u70b9\u2026\u2026<br>\u5c31\u8fd9\u6837\uff0c\u7267\u5e08\u6bcf\u4e2a\u793c\u62dc\u66f4\u65b0\u81ea\u5df1\u7684\u4f4d\u7f6e\uff0c\u6751\u6c11\u6839\u636e\u81ea\u5df1\u7684\u60c5\u51b5\u9009\u62e9\u5e03\u9053\u70b9\uff0c\u6700\u7ec8\u7a33\u5b9a\u4e86\u4e0b\u6765\u3002<\/p><\/blockquote>\n\n\n\n<p>\u6211\u4eec\u53ef\u4ee5\u770b\u5230\u8be5\u7267\u5e08\u7684\u76ee\u7684\u662f\u4e3a\u4e86\u8ba9\u6bcf\u4e2a\u6751\u6c11\u5230\u5176\u6700\u8fd1\u4e2d\u5fc3\u70b9\u7684\u8ddd\u79bb\u548c\u6700\u5c0f\u3002<\/p>\n\n\n\n<p>\u6240\u4ee5 K-means \u7684\u7b97\u6cd5\u6b65\u9aa4\u4e3a:<\/p>\n\n\n\n<ol><li>\u9009\u62e9\u521d\u59cb\u5316\u7684\\( \\mathrm{k} \\)\u4e2a\u6837\u672c\u4f5c\u4e3a\u521d\u59cb\u805a\u7c7b\u4e2d\u5fc3 \\(a=a_{1}, a_{2}, \\ldots a_{k}\\) \uff1b<\/li><li>\u9488\u5bf9\u6570\u636e\u96c6\u4e2d\u6bcf\u4e2a\u6837\u672c \\(x_{i}\\) \u8ba1\u7b97\u5b83\u5230 \\(\\mathrm{k}\\) \u4e2a\u805a\u7c7b\u4e2d\u5fc3\u7684\u8ddd\u79bb\u5e76\u5c06\u5176\u5206\u5230\u8ddd\u79bb\u6700\u5c0f\u7684\u805a\u7c7b\u4e2d\u5fc3\u6240\u5bf9 \u5e94\u7684\u7c7b\u4e2d\uff1b<\/li><li>\u9488\u5bf9\u6bcf\u4e2a\u7c7b\u522b \\(a_{j}\\) \uff0c\u91cd\u65b0\u8ba1\u7b97\u5b83\u7684\u805a\u7c7b\u4e2d\u5fc3 \\(a_{j}=\\frac{1}{\\left|c_{i}\\right|} \\sum_{x \\in c_{i}} x\\) (\u5373\u5c5e\u4e8e\u8be5\u7c7b\u7684\u6240\u6709\u6837\u672c\u7684\u8d28\u5fc3\uff09\uff1b<\/li><li>\u91cd\u590d\u4e0a\u9762 23 \u4e24\u6b65\u64cd\u4f5c\uff0c\u76f4\u5230\u8fbe\u5230\u67d0\u4e2a\u4e2d\u6b62\u6761\u4ef6\uff08\u8fed\u4ee3\u6b21\u6570\u3001\u6700\u5c0f\u8bef\u5dee\u53d8\u5316\u7b49\uff09\u3002<\/li><\/ol>\n\n\n\n<p>K-means\u7b97\u6cd5Python\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code># -*- coding:utf-8 -*-\r\nimport numpy as np\r\nfrom matplotlib import pyplot\r\n\r\n\r\nclass K_Means(object):\r\n    # k\u662f\u5206\u7ec4\u6570\uff1btolerance\u2018\u4e2d\u5fc3\u70b9\u8bef\u5dee\u2019\uff1bmax_iter\u662f\u8fed\u4ee3\u6b21\u6570\r\n    def __init__(self, k=2, tolerance=0.0001, max_iter=300):\r\n        self.k_ = k\r\n        self.tolerance_ = tolerance\r\n        self.max_iter_ = max_iter\r\n\r\n    def fit(self, data):\r\n        self.centers_ = {}\r\n        for i in range(self.k_):\r\n            self.centers_&#91;i] = data&#91;i]\r\n\r\n        for i in range(self.max_iter_):\r\n            self.clf_ = {}\r\n            for i in range(self.k_):\r\n                self.clf_&#91;i] = &#91;]\r\n            # print(\"\u8d28\u70b9:\",self.centers_)\r\n            for feature in data:\r\n                # distances = &#91;np.linalg.norm(feature-self.centers&#91;center]) for center in self.centers]\r\n                distances = &#91;]\r\n                for center in self.centers_:\r\n                    # \u6b27\u62c9\u8ddd\u79bb\r\n                    # np.sqrt(np.sum((features-self.centers_&#91;center])**2))\r\n                    distances.append(np.linalg.norm(feature - self.centers_&#91;center]))\r\n                classification = distances.index(min(distances))\r\n                self.clf_&#91;classification].append(feature)\r\n\r\n            # print(\"\u5206\u7ec4\u60c5\u51b5:\",self.clf_)\r\n            prev_centers = dict(self.centers_)\r\n            for c in self.clf_:\r\n                self.centers_&#91;c] = np.average(self.clf_&#91;c], axis=0)\r\n\r\n            # '\u4e2d\u5fc3\u70b9'\u662f\u5426\u5728\u8bef\u5dee\u8303\u56f4\r\n            optimized = True\r\n            for center in self.centers_:\r\n                org_centers = prev_centers&#91;center]\r\n                cur_centers = self.centers_&#91;center]\r\n                if np.sum((cur_centers - org_centers) \/ org_centers * 100.0) > self.tolerance_:\r\n                    optimized = False\r\n            if optimized:\r\n                break\r\n\r\n    def predict(self, p_data):\r\n        distances = &#91;np.linalg.norm(p_data - self.centers_&#91;center]) for center in self.centers_]\r\n        index = distances.index(min(distances))\r\n        return index\r\n\r\n\r\nif __name__ == '__main__':\r\n    x = np.array(&#91;&#91;1, 2], &#91;1.5, 1.8], &#91;5, 8], &#91;8, 8], &#91;1, 0.6], &#91;9, 11]])\r\n    k_means = K_Means(k=2)\r\n    k_means.fit(x)\r\n    print(k_means.centers_)\r\n    for center in k_means.centers_:\r\n        pyplot.scatter(k_means.centers_&#91;center]&#91;0], k_means.centers_&#91;center]&#91;1], marker='*', s=150)\r\n\r\n    for cat in k_means.clf_:\r\n        for point in k_means.clf_&#91;cat]:\r\n            pyplot.scatter(point&#91;0], point&#91;1], c=('r' if cat == 0 else 'b'))\r\n\r\n    predict = &#91;&#91;2, 1], &#91;6, 9]]\r\n    for feature in predict:\r\n        cat = k_means.predict(predict)\r\n        pyplot.scatter(feature&#91;0], feature&#91;1], c=('r' if cat == 0 else 'b'), marker='x')\r\n\r\n    pyplot.show()\r\n<\/code><\/pre>\n\n\n\n<h3>2.1 \u4f18\u70b9<\/h3>\n\n\n\n<ul><li>\u5bb9\u6613\u7406\u89e3\uff0c\u805a\u7c7b\u6548\u679c\u4e0d\u9519\uff0c\u867d\u7136\u662f\u5c40\u90e8\u6700\u4f18\uff0c \u4f46\u5f80\u5f80\u5c40\u90e8\u6700\u4f18\u5c31\u591f\u4e86\uff1b<\/li><li>\u5904\u7406\u5927\u6570\u636e\u96c6\u7684\u65f6\u5019\uff0c\u8be5\u7b97\u6cd5\u53ef\u4ee5\u4fdd\u8bc1\u8f83\u597d\u7684\u4f38\u7f29\u6027\uff1b<\/li><li>\u5f53\u7c07\u8fd1\u4f3c\u9ad8\u65af\u5206\u5e03\u7684\u65f6\u5019\uff0c\u6548\u679c\u975e\u5e38\u4e0d\u9519\uff1b<\/li><li>\u7b97\u6cd5\u590d\u6742\u5ea6\u4f4e\u3002<\/li><\/ul>\n\n\n\n<h3>2.2 \u7f3a\u70b9<\/h3>\n\n\n\n<ul><li>K \u503c\u9700\u8981\u4eba\u4e3a\u8bbe\u5b9a\uff0c\u4e0d\u540c K \u503c\u5f97\u5230\u7684\u7ed3\u679c\u4e0d\u4e00\u6837\uff1b<\/li><li>\u5bf9\u521d\u59cb\u7684\u7c07\u4e2d\u5fc3\u654f\u611f\uff0c\u4e0d\u540c\u9009\u53d6\u65b9\u5f0f\u4f1a\u5f97\u5230\u4e0d\u540c\u7ed3\u679c\uff1b<\/li><li>\u5bf9\u5f02\u5e38\u503c\u654f\u611f\uff1b<\/li><li>\u6837\u672c\u53ea\u80fd\u5f52\u4e3a\u4e00\u7c7b\uff0c\u4e0d\u9002\u5408\u591a\u5206\u7c7b\u4efb\u52a1\uff1b<\/li><li>\u4e0d\u9002\u5408\u592a\u79bb\u6563\u7684\u5206\u7c7b\u3001\u6837\u672c\u7c7b\u522b\u4e0d\u5e73\u8861\u7684\u5206\u7c7b\u3001\u975e\u51f8\u5f62\u72b6\u7684\u5206\u7c7b\u3002<\/li><\/ul>\n\n\n\n<p>\u9488\u5bf9 K-means \u7b97\u6cd5\u7684\u7f3a\u70b9\uff0c\u6211\u4eec\u53ef\u4ee5\u6709\u5f88\u591a\u79cd\u8c03\u4f18\u65b9\u5f0f\uff1a\u5982\u6570\u636e\u9884\u5904\u7406\uff08\u53bb\u9664\u5f02\u5e38\u70b9\uff09\uff0c\u5408\u7406\u9009\u62e9 K \u503c\uff0c\u9ad8\u7ef4\u6620\u5c04\u7b49\u3002<\/p>\n\n\n\n<h3>\u6570\u636e\u9884\u5904\u7406<\/h3>\n\n\n\n<p>K-means \u7684\u672c\u8d28\u662f\u57fa\u4e8e\u6b27\u5f0f\u8ddd\u79bb\u7684\u6570\u636e\u5212\u5206\u7b97\u6cd5\uff0c\u5747\u503c\u548c\u65b9\u5dee\u5927\u7684\u7ef4\u5ea6\u5c06\u5bf9\u6570\u636e\u7684\u805a\u7c7b\u4ea7\u751f\u51b3\u5b9a\u6027\u5f71\u54cd\u3002\u6240\u4ee5\u672a\u505a\u5f52\u4e00\u5316\u5904\u7406\u548c\u7edf\u4e00\u5355\u4f4d\u7684\u6570\u636e\u662f\u65e0\u6cd5\u76f4\u63a5\u53c2\u4e0e\u8fd0\u7b97\u548c\u6bd4\u8f83\u7684\u3002\u5e38\u89c1\u7684\u6570\u636e\u9884\u5904\u7406\u65b9\u5f0f\u6709\uff1a\u6570\u636e\u5f52\u4e00\u5316\uff0c\u6570\u636e\u6807\u51c6\u5316\u3002<\/p>\n\n\n\n<p>\u6b64\u5916\uff0c\u79bb\u7fa4\u70b9\u6216\u8005\u566a\u58f0\u6570\u636e\u4f1a\u5bf9\u5747\u503c\u4ea7\u751f\u8f83\u5927\u7684\u5f71\u54cd\uff0c\u5bfc\u81f4\u4e2d\u5fc3\u504f\u79fb\uff0c\u56e0\u6b64\u6211\u4eec\u8fd8\u9700\u8981\u5bf9\u6570\u636e\u8fdb\u884c\u5f02\u5e38\u70b9\u68c0\u6d4b\u3002<\/p>\n\n\n\n<h3>\u5408\u7406\u9009\u62e9 K \u503c<\/h3>\n\n\n\n<p>K \u503c\u7684\u9009\u53d6\u5bf9 K-means \u5f71\u54cd\u5f88\u5927\uff0c\u8fd9\u4e5f\u662f K-means \u6700\u5927\u7684\u7f3a\u70b9\uff0c\u5e38\u89c1\u7684\u9009\u53d6 K \u503c\u7684\u65b9\u6cd5\u6709\uff1a\u624b\u8098\u6cd5\u3001Gap statistic \u65b9\u6cd5\u3002<\/p>\n\n\n\n<p>\u624b\u8098\u6cd5 (Elbow Method) \u672c\u8d28\u4e0a\u4e5f\u662f\u4e00\u79cd\u95f4\u63a5\u7684\u89c2\u5bdf\u6cd5\u3002\u5f53\u6211\u4eec\u5bf9\u6570\u636e \\(\\left\\{x_{1}, \\ldots, x_{N}\\right\\}\\) \u8fdb\u884cK\u5747\u503c\u805a \u7c7b\u540e\uff0c\u6211\u4eec\u5c06\u5f97\u5230 \\(K\\) \u4e2a\u805a\u7c7b\u7684\u4e2d\u5fc3\u70b9 \\(\\mu_{k}, k=1,2, \\ldots, K\\) \uff0c\u4ee5\u53ca\u6570\u636e\u70b9 \\(x_{i}\\) \u6240\u5bf9\u5e94\u7684\u7c07 \\(C_{k}, k=1,2, \\ldots, K\\) \uff0c\u6bcf\u4e2a\u7c07\u4e2d\u6709 \\(n_{k}\\) \u4e2a\u6570\u636e\u70b9\u3002\u6211\u4eec\u5b9a\u4e49\u6bcf\u4e2a\u7c07\u4e2d\u7684\u6240\u6709\u70b9\u76f8\u4e92\u4e4b\u95f4\u7684\u8ddd\u79bb\u7684\u548c \u4e3a<\/p>\n\n\n\n<p>$$D_{k}=\\sum_{x_{i} \\in C_{k}} \\sum_{x_{j} \\in C_{k}}\\left\\|x_{i}-x_{j}\\right\\|^{2}$$<\/p>\n\n\n\n<p>\u5176\u4e2d \\(\\|\\cdot\\|\\) \u4e3a2\u8303\u6570\u3002\u5219\u5bf9\u4e8e\u805a\u7c7b\u4e2a\u6570\u4e3a \\(K\\) \u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5b9a\u4e49\u4e00\u4e2a\u6d4b\u5ea6\u91cf\u4e3a<\/p>\n\n\n\n<p>$$W_{K}=\\sum_{k=1}^{K} \\frac{1}{2 n_{k}} D_{k}$$<\/p>\n\n\n\n<p>\u5bf9\u4e8e\u4e0d\u540c\u7684   \\(K\\) \uff0c\u7ecf\u8fc7 \\(\\mathrm{K}-\\mathrm{means}\\) \u7b97\u6cd5\u540e\u6211\u4eec\u4f1a\u5f97\u5230\u4e0d\u540c\u7684\u4e2d\u5fc3\u70b9\u548c\u6570\u636e\u70b9\u6240\u5c5e\u7684\u7c07\uff0c\u4ece\u800c\u5f97\u5230\u4e0d\u540c\u7684 \u5ea6\u91cf \\(W_{K}\\) \u3002\u5c06\u805a\u7c7b\u4e2a\u6570 \\(K\\) \u4f5c\u4e3a\u6a2a\u5750\u6807\uff0c \\(W_{K}\\) \u4f5c\u4e3a\u7eb5\u5750\u6807\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u7c7b\u4f3c\u4e0b\u9762\u7684\u56fe\u50cf:<\/p>\n\n\n\n<div class=\"wp-block-image is-style-default\"><figure class=\"aligncenter size-large\"><img src=\"https:\/\/www.scutmath.com\/k_means_choose_k\/elbow.png\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>\u56fe\u50cf\u4e2d\u56fe\u5f62\u5f88\u50cf\u4eba\u7684\u624b\u8098\uff0c\u8be5\u65b9\u6cd5\u7684\u547d\u540d\u5c31\u662f\u4ece\u8fd9\u800c\u6765\u3002\u4ece\u56fe\u50cf\u4e2d\u6211\u4eec\u53ef\u4ee5\u89c2\u5bdf\u5230\uff0cK=1\u00a0\u5230\u00a04\u00a0\u4e0b\u964d\u5f88\u5feb\uff0cK=4\u00a0\u4e4b\u540e\u8d8b\u4e8e\u5e73\u7a33\uff0c\u56e0\u6b64\u00a0K=4\u00a0\u662f\u4e00\u4e2a\u62d0\u70b9\uff0c\u624b\u8098\u6cd5\u8ba4\u4e3a\u8fd9\u4e2a\u62d0\u70b9\u5c31\u662f\u6700\u4f73\u7684\u805a\u7c7b\u4e2a\u6570\u00a0K\u3002<\/p>\n\n\n\n<p>\u00a0Gap statistic \u65b9\u6cd5\uff0c\u8fd9\u4e2a\u65b9\u6cd5\u51fa\u81ea\u65af\u5766\u798f\u5927\u5b66\u7684\u51e0\u4e2a\u5b66\u8005\u7684\u8bba\u6587\uff1a<a rel=\"noreferrer noopener\" href=\"https:\/\/statweb.stanford.edu\/~gwalther\/gap\" target=\"_blank\">Estimating the number of clusters in a data set via the gap statistic<\/a><\/p>\n\n\n\n<p>$$<br>\\operatorname{Gap}(K)=\\mathrm{E}\\left(\\log D_{k}\\right)-\\log D_{k}<br>$$<br>\u5176\u4e2d \\(D_{k}\\) \u4e3a\u635f\u5931\u51fd\u6570\uff0c\u8fd9\u91cc\\(E\\left(\\log D_{k}\\right)\\) \u6307\u7684\u662f \\(\\log D_{k}\\) \u7684\u671f\u671b\u3002\u8fd9\u4e2a\u6570\u503c\u901a\u5e38\u901a\u8fc7\u8499\u7279\u5361\u6d1b \u6a21\u62df\u4ea7\u751f\uff0c\u6211\u4eec\u5728\u6837\u672c\u91cc\u6240\u5728\u7684\u533a\u57df\u4e2d\u6309\u7167\u5747\u5300\u5206\u5e03\u968f\u673a\u4ea7\u751f\u548c\u539f\u59cb\u6837\u672c\u6570\u4e00\u6837\u591a\u7684\u968f\u673a\u6837\u672c\uff0c\u5e76 \u5bf9\u8fd9\u4e2a\u968f\u673a\u6837\u672c\u505a K-Means\uff0c\u4ece\u800c\u5f97\u5230\u4e00\u4e2a \\(D_{k}\\) \u3002\u5982\u6b64\u5f80\u590d\u591a\u6b21\uff0c\u901a\u5e38 20 \u6b21\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230 20 \u4e2a \\(\\log D_{k}\\)\u3002\u5bf9\u8fd9 20 \u4e2a\u6570\u503c\u6c42\u5e73\u5747\u503c\uff0c\u5c31\u5f97\u5230\u4e86 \\(E\\left(\\log D_{k}\\right)\\) \u7684\u8fd1\u4f3c\u503c\u3002\u6700\u7ec8\u53ef\u4ee5\u8ba1\u7b97 Gap Statisitc\u3002\u800c Gap statistic \u53d6\u5f97\u6700\u5927\u503c\u6240\u5bf9\u5e94\u7684 K \u5c31\u662f\u6700\u4f73\u7684 K\u3002<\/p>\n\n\n\n<p>python\u5b9e\u73b0gap\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>import numpy as np\n\n\ndef calculate_Wk(data, centroids, cluster):\n    K = centroids.shape&#91;0]\n    wk = 0.0\n    for k in range(K):\n        data_in_cluster = data&#91;cluster == k, :]\n        center = centroids&#91;k, :]\n        num_points = data_in_cluster.shape&#91;0]\n        for i in range(num_points):\n            wk = wk + np.linalg.norm(data_in_cluster&#91;i, :]-center, ord=2) ** 2\n\n    return wk\n\n\ndef bounding_box(data):\n    dim = data.shape&#91;1]\n    boxes = &#91;]\n    for i in range(dim):\n        data_min = np.amin(data&#91;:, i])\n        data_max = np.amax(data&#91;:, i])\n        boxes.append((data_min, data_max))\n\n    return boxes\n\n\ndef gap_statistic(data, max_K, B, cluster_algorithm):\n    num_points, dim = data.shape\n    K_range = np.arange(1, max_K, dtype=int)\n    num_K = len(K_range)\n    boxes = bounding_box(data)\n    data_generate = np.zeros((num_points, dim))\n\n    ''' \u5199\u6cd51\n    log_Wks = np.zeros(num_K)\n    gaps = np.zeros(num_K)\n    sks = np.zeros(num_K)\n    for ind_K, K in enumerate(K_range):\n        cluster_centers, labels, _ = cluster_algorithm(data, K)\n        log_Wks&#91;ind_K] = np.log(calculate_Wk(data, cluster_centers, labels))\n\n        # generate B reference data sets\n        log_Wkbs = np.zeros(B)\n        for b in range(B):\n            for i in range(num_points):\n                for j in range(dim):\n                    data_generate&#91;i]&#91;j] = \\\n                        np.random.uniform(boxes&#91;j]&#91;0], boxes&#91;j]&#91;1])\n            cluster_centers, labels, _ = cluster_algorithm(data_generate, K)\n            log_Wkbs&#91;b] = \\\n                np.log(calculate_Wk(data_generate, cluster_centers, labels))\n        gaps&#91;ind_K] = np.mean(log_Wkbs) - log_Wks&#91;ind_K]\n        sks&#91;ind_K] = np.std(log_Wkbs) * np.sqrt(1 + 1.0 \/ B)\n    '''\n\n    ''' \u5199\u6cd52\n    '''\n    log_Wks = np.zeros(num_K)\n    for indK, K in enumerate(K_range):\n        cluster_centers, labels, _ = cluster_algorithm(data, K)\n        log_Wks&#91;indK] = np.log(calculate_Wk(data, cluster_centers, labels))\n\n    gaps = np.zeros(num_K)\n    sks = np.zeros(num_K)\n    log_Wkbs = np.zeros((B, num_K))\n\n    # generate B reference data sets\n    for b in range(B):\n        for i in range(num_points):\n            for j in range(dim):\n                data_generate&#91;i, j] = \\\n                    np.random.uniform(boxes&#91;j]&#91;0], boxes&#91;j]&#91;1])\n        for indK, K in enumerate(K_range):\n            cluster_centers, labels, _ = cluster_algorithm(data_generate, K)\n            log_Wkbs&#91;b, indK] = \\\n                np.log(calculate_Wk(data_generate, cluster_centers, labels))\n\n    for k in range(num_K):\n        gaps&#91;k] = np.mean(log_Wkbs&#91;:, k]) - log_Wks&#91;k]\n        sks&#91;k] = np.std(log_Wkbs&#91;:, k]) * np.sqrt(1 + 1.0 \/ B)\n\n    return gaps, sks, log_Wks\n\n\n\u91c7\u7528\u6838\u51fd\u6570\n\u57fa\u4e8e\u6b27\u5f0f\u8ddd\u79bb\u7684 K-means \u5047\u8bbe\u4e86\u4e86\u5404\u4e2a\u6570\u636e\u7c07\u7684\u6570\u636e\u5177\u6709\u4e00\u6837\u7684\u7684\u5148\u9a8c\u6982\u7387\u5e76\u5448\u73b0\u7403\u5f62\u5206\u5e03\uff0c\u4f46\u8fd9\u79cd\u5206\u5e03\u5728\u5b9e\u9645\u751f\u6d3b\u4e2d\u5e76\u4e0d\u5e38\u89c1\u3002\u9762\u5bf9\u975e\u51f8\u7684\u6570\u636e\u5206\u5e03\u5f62\u72b6\u65f6\u6211\u4eec\u53ef\u4ee5\u5f15\u5165\u6838\u51fd\u6570\u6765\u4f18\u5316\uff0c\u8fd9\u65f6\u7b97\u6cd5\u53c8\u79f0\u4e3a\u6838 K-means \u7b97\u6cd5\uff0c\u662f\u6838\u805a\u7c7b\u65b9\u6cd5\u7684\u4e00\u79cd\u3002\u6838\u805a\u7c7b\u65b9\u6cd5\u7684\u4e3b\u8981\u601d\u60f3\u662f\u901a\u8fc7\u4e00\u4e2a\u975e\u7ebf\u6027\u6620\u5c04\uff0c\u5c06\u8f93\u5165\u7a7a\u95f4\u4e2d\u7684\u6570\u636e\u70b9\u6620\u5c04\u5230\u9ad8\u4f4d\u7684\u7279\u5f81\u7a7a\u95f4\u4e2d\uff0c\u5e76\u5728\u65b0\u7684\u7279\u5f81\u7a7a\u95f4\u4e2d\u8fdb\u884c\u805a\u7c7b\u3002\u975e\u7ebf\u6027\u6620\u5c04\u589e\u52a0\u4e86\u6570\u636e\u70b9\u7ebf\u6027\u53ef\u5206\u7684\u6982\u7387\uff0c\u4ece\u800c\u5728\u7ecf\u5178\u7684\u805a\u7c7b\u7b97\u6cd5\u5931\u6548\u7684\u60c5\u51b5\u4e0b\uff0c\u901a\u8fc7\u5f15\u5165\u6838\u51fd\u6570\u53ef\u4ee5\u8fbe\u5230\u66f4\u4e3a\u51c6\u786e\u7684\u805a\u7c7b\u7ed3\u679c\u3002\n\n<\/code><\/pre>\n\n\n\n<p>\u00a0K-means++\u7b97\u6cd5 \uff1a\u6539\u8fdb\u7684\u7b97\u6cd5<\/p>\n\n\n\n<p>\u539f\u59cbK-means\u7b97\u6cd5\u6700\u5f00\u59cb\u968f\u673a\u9009\u53d6\u6570\u636e\u96c6\u4e2dK\u4e2a\u70b9\u4f5c\u4e3a\u805a\u7c7b\u4e2d\u5fc3\uff0c\u800cK-means++\u6309\u7167\u5982\u4e0b\u7684\u601d\u60f3\u9009\u53d6K\u4e2a\u805a\u7c7b\u4e2d\u5fc3\uff1a\u5047\u8bbe\u5df2\u7ecf\u9009\u53d6\u4e86n\u4e2a\u521d\u59cb\u805a\u7c7b\u4e2d\u5fc3(0&lt;n&lt;K)\uff0c\u5219\u5728\u9009\u53d6\u7b2cn+1\u4e2a\u805a\u7c7b\u4e2d\u5fc3\u65f6\uff1a\u8ddd\u79bb\u5f53\u524dn\u4e2a\u805a\u7c7b\u4e2d\u5fc3\u8d8a\u8fdc\u7684\u70b9\u4f1a\u6709\u66f4\u9ad8\u7684\u6982\u7387\u88ab\u9009\u4e3a\u7b2cn+1\u4e2a\u805a\u7c7b\u4e2d\u5fc3\u3002\u5728\u9009\u53d6\u7b2c\u4e00\u4e2a\u805a\u7c7b\u4e2d\u5fc3(n=1)\u65f6\u540c\u6837\u901a\u8fc7\u968f\u673a\u7684\u65b9\u6cd5\u3002\u53ef\u4ee5\u8bf4\u8fd9\u4e5f\u7b26\u5408\u6211\u4eec\u7684\u76f4\u89c9\uff1a\u805a\u7c7b\u4e2d\u5fc3\u5f53\u7136\u662f\u4e92\u76f8\u79bb\u5f97\u8d8a\u8fdc\u8d8a\u597d\u3002\u8fd9\u4e2a\u6539\u8fdb\u867d\u7136\u76f4\u89c2\u7b80\u5355\uff0c\u4f46\u662f\u5374\u975e\u5e38\u5f97\u6709\u6548\u3002<\/p>\n\n\n\n<ol><li>\u968f\u673a\u9009\u53d6\u4e00\u4e2a\u4e2d\u5fc3\u70b9 \\(a_{1}\\) \uff1b<\/li><li>\u8ba1\u7b97\u6570\u636e\u5230\u4e4b\u524d \\(\\mathrm{n}\\) \u4e2a\u805a\u7c7b\u4e2d\u5fc3\u6700\u8fdc\u7684\u8ddd\u79bb \\(D(x)\\) \uff0c\u5e76\u4ee5\u4e00\u5b9a\u6982\u7387 \\(\\frac{D(x)^{2}}{\\sum D(x)^{2}}\u9009\u62e9\u65b0\u4e2d\u5fc3\u70b9 a_{i} ;\\)<\/li><li>\u91cd\u590d\u7b2c\u4e8c\u6b65\u3002<\/li><\/ol>\n\n\n\n<p>\u4f46\u662f\u8fd9\u4e2a\u7b97\u6cd5\u7684\u7f3a\u70b9\u5728\u4e8e\uff0c\u96be\u4ee5\u5e76\u884c\u5316\u3002\u6240\u4ee5 k-means II \u6539\u53d8\u53d6\u6837\u7b56\u7565\uff0c\u5e76\u975e\u6309\u7167 k-means++ \u90a3\u6837\u6bcf\u6b21\u904d\u5386\u53ea\u53d6\u6837\u4e00\u4e2a\u6837\u672c\uff0c\u800c\u662f\u6bcf\u6b21\u904d\u5386\u53d6\u6837 k \u4e2a\uff0c\u91cd\u590d\u8be5\u53d6\u6837\u8fc7\u7a0b\u00a0log(n)\u6b21\uff0c\u5219\u5f97\u5230\u00a0klog(n)\u4e2a\u6837\u672c\u70b9\u7ec4\u6210\u7684\u96c6\u5408\uff0c\u7136\u540e\u4ece\u8fd9\u4e9b\u70b9\u4e2d\u9009\u53d6 k \u4e2a\u3002\u5f53\u7136\u4e00\u822c\u4e5f\u4e0d\u9700\u8981\u00a0log(n)\u6b21\u53d6\u6837\uff0c5 \u6b21\u5373\u53ef\u3002<\/p>\n\n\n\n<p><strong>K-means\u4e0eISODATA\uff1a<\/strong>ISODATA\u7684\u5168\u79f0\u662f\u8fed\u4ee3\u81ea\u7ec4\u7ec7\u6570\u636e\u5206\u6790\u6cd5\u3002\u5728K-means\u4e2d\uff0cK\u7684\u503c\u9700\u8981\u9884\u5148\u4eba\u4e3a\u5730\u786e\u5b9a\uff0c\u5e76\u4e14\u5728\u6574\u4e2a\u7b97\u6cd5\u8fc7\u7a0b\u4e2d\u65e0\u6cd5\u66f4\u6539\u3002\u800c\u5f53\u9047\u5230\u9ad8\u7ef4\u5ea6\u3001\u6d77\u91cf\u7684\u6570\u636e\u96c6\u65f6\uff0c\u4eba\u4eec\u5f80\u5f80\u5f88\u96be\u51c6\u786e\u5730\u4f30\u8ba1\u51faK\u7684\u5927\u5c0f\u3002ISODATA\u5c31\u662f\u9488\u5bf9\u8fd9\u4e2a\u95ee\u9898\u8fdb\u884c\u4e86\u6539\u8fdb\uff0c\u5b83\u7684\u601d\u60f3\u4e5f\u5f88\u76f4\u89c2\uff1a\u5f53\u5c5e\u4e8e\u67d0\u4e2a\u7c7b\u522b\u7684\u6837\u672c\u6570\u8fc7\u5c11\u65f6\u628a\u8fd9\u4e2a\u7c7b\u522b\u53bb\u9664\uff0c\u5f53\u5c5e\u4e8e\u67d0\u4e2a\u7c7b\u522b\u7684\u6837\u672c\u6570\u8fc7\u591a\u3001\u5206\u6563\u7a0b\u5ea6\u8f83\u5927\u65f6\u628a\u8fd9\u4e2a\u7c7b\u522b\u5206\u4e3a\u4e24\u4e2a\u5b50\u7c7b\u522b\u3002<\/p>\n\n\n\n<p><strong>K-means\u4e0eKernel K-means\uff1a<\/strong>\u4f20\u7edfK-means\u91c7\u7528\u6b27\u5f0f\u8ddd\u79bb\u8fdb\u884c\u6837\u672c\u95f4\u7684\u76f8\u4f3c\u5ea6\u5ea6\u91cf\uff0c\u663e\u7136\u5e76\u4e0d\u662f\u6240\u6709\u7684\u6570\u636e\u96c6\u90fd\u9002\u7528\u4e8e\u8fd9\u79cd\u5ea6\u91cf\u65b9\u5f0f\u3002\u53c2\u7167\u652f\u6301\u5411\u91cf\u673a\u4e2d\u6838\u51fd\u6570\u7684\u601d\u60f3\uff0c\u5c06\u6240\u6709\u6837\u672c\u6620\u5c04\u5230\u53e6\u5916\u4e00\u4e2a\u7279\u5f81\u7a7a\u95f4\u4e2d\u518d\u8fdb\u884c\u805a\u7c7b\uff0c\u5c31\u6709\u53ef\u80fd\u6539\u5584\u805a\u7c7b\u6548\u679c\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>k-means\u7b97\u6cd5\u7b97\u662f\u7ecf\u5178\u7684\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\uff0c\u8bb0\u5f97\u5e94\u8be5\u662f\u5927\u4e09\u7684\u8bfe\u4e0a\u5b66\u8fc7\uff0c\u540e\u9762\u5c31\u4e00\u76f4\u6ca1\u5728\u63a5\u89e6\u8fc7\u8be5\u7b97\u6cd5\u5bf9\u5e94\u7684\u95ee\u9898\uff0c\u73b0\u5728 &hellip; <a href=\"http:\/\/139.9.1.231\/index.php\/2021\/12\/22\/k-means\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">k-means<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[4],"tags":[],"_links":{"self":[{"href":"http:\/\/139.9.1.231\/index.php\/wp-json\/wp\/v2\/posts\/519"}],"collection":[{"href":"http:\/\/139.9.1.231\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/139.9.1.231\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/139.9.1.231\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/139.9.1.231\/index.php\/wp-json\/wp\/v2\/comments?post=519"}],"version-history":[{"count":21,"href":"http:\/\/139.9.1.231\/index.php\/wp-json\/wp\/v2\/posts\/519\/revisions"}],"predecessor-version":[{"id":540,"href":"http:\/\/139.9.1.231\/index.php\/wp-json\/wp\/v2\/posts\/519\/revisions\/540"}],"wp:attachment":[{"href":"http:\/\/139.9.1.231\/index.php\/wp-json\/wp\/v2\/media?parent=519"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/139.9.1.231\/index.php\/wp-json\/wp\/v2\/categories?post=519"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/139.9.1.231\/index.php\/wp-json\/wp\/v2\/tags?post=519"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}