{"id":827,"date":"2023-07-28T14:53:35","date_gmt":"2023-07-28T14:53:35","guid":{"rendered":"https:\/\/statorials.org\/uk\/mahalanobis-%d0%b2%d1%96%d0%b4%d0%b4%d0%b0%d0%bb%d0%b5%d0%bd%d0%b8%d0%b8-%d0%bf%d1%96%d1%82%d0%be%d0%bd\/"},"modified":"2023-07-28T14:53:35","modified_gmt":"2023-07-28T14:53:35","slug":"mahalanobis-%d0%b2%d1%96%d0%b4%d0%b4%d0%b0%d0%bb%d0%b5%d0%bd%d0%b8%d0%b8-%d0%bf%d1%96%d1%82%d0%be%d0%bd","status":"publish","type":"post","link":"https:\/\/statorials.org\/uk\/mahalanobis-%d0%b2%d1%96%d0%b4%d0%b4%d0%b0%d0%bb%d0%b5%d0%bd%d0%b8%d0%b8-%d0%bf%d1%96%d1%82%d0%be%d0%bd\/","title":{"rendered":"\u042f\u043a \u043e\u0431\u0447\u0438\u0441\u043b\u0438\u0442\u0438 \u0432\u0456\u0434\u0441\u0442\u0430\u043d\u044c \u043c\u0430\u0445\u0430\u043b\u0430\u043d\u043e\u0431\u0456\u0441\u0430 \u0432 python"},"content":{"rendered":"<p><\/p>\n<hr>\n<p><span style=\"color: #000000;\"><strong>\u0412\u0456\u0434\u0441\u0442\u0430\u043d\u044c \u041c\u0430\u0445\u0430\u043b\u0430\u043d\u043e\u0431\u0456\u0441\u0430<\/strong> \u2014 \u0446\u0435 \u0432\u0456\u0434\u0441\u0442\u0430\u043d\u044c \u043c\u0456\u0436 \u0434\u0432\u043e\u043c\u0430 \u0442\u043e\u0447\u043a\u0430\u043c\u0438 \u0432 \u0431\u0430\u0433\u0430\u0442\u043e\u0432\u0438\u043c\u0456\u0440\u043d\u043e\u043c\u0443 \u043f\u0440\u043e\u0441\u0442\u043e\u0440\u0456. \u0419\u043e\u0433\u043e \u0447\u0430\u0441\u0442\u043e \u0432\u0438\u043a\u043e\u0440\u0438\u0441\u0442\u043e\u0432\u0443\u044e\u0442\u044c \u0434\u043b\u044f \u0432\u0438\u044f\u0432\u043b\u0435\u043d\u043d\u044f \u0432\u0438\u043a\u0438\u0434\u0456\u0432 \u0443 \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u0447\u043d\u043e\u043c\u0443 \u0430\u043d\u0430\u043b\u0456\u0437\u0456, \u0449\u043e \u0432\u043a\u043b\u044e\u0447\u0430\u0454 \u043a\u0456\u043b\u044c\u043a\u0430 \u0437\u043c\u0456\u043d\u043d\u0438\u0445.<\/span><\/p>\n<p> <span style=\"color: #000000;\">\u0426\u0435\u0439 \u043f\u043e\u0441\u0456\u0431\u043d\u0438\u043a \u043f\u043e\u044f\u0441\u043d\u044e\u0454, \u044f\u043a \u043e\u0431\u0447\u0438\u0441\u043b\u0438\u0442\u0438 \u0432\u0456\u0434\u0441\u0442\u0430\u043d\u044c \u041c\u0430\u0445\u0430\u043b\u0430\u043d\u043e\u0431\u0456\u0441\u0430 \u0432 Python.<\/span><\/p>\n<h3> <span style=\"color: #000000;\"><strong>\u041f\u0440\u0438\u043a\u043b\u0430\u0434: \u0412\u0456\u0434\u0441\u0442\u0430\u043d\u044c \u041c\u0430\u0445\u0430\u043b\u0430\u043d\u043e\u0431\u0456\u0441\u0430 \u0432 Python<\/strong><\/span><\/h3>\n<p> <span style=\"color: #000000;\">\u0412\u0438\u043a\u043e\u043d\u0430\u0439\u0442\u0435 \u043d\u0430\u0441\u0442\u0443\u043f\u043d\u0456 \u0434\u0456\u0457, \u0449\u043e\u0431 \u043e\u0431\u0447\u0438\u0441\u043b\u0438\u0442\u0438 \u0432\u0456\u0434\u0441\u0442\u0430\u043d\u044c \u041c\u0430\u0445\u0430\u043b\u0430\u043d\u043e\u0431\u0456\u0441\u0430 \u0434\u043b\u044f \u043a\u043e\u0436\u043d\u043e\u0433\u043e \u0441\u043f\u043e\u0441\u0442\u0435\u0440\u0435\u0436\u0435\u043d\u043d\u044f \u0432 \u043d\u0430\u0431\u043e\u0440\u0456 \u0434\u0430\u043d\u0438\u0445 \u0443 Python.<\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>\u041a\u0440\u043e\u043a 1: \u0421\u0442\u0432\u043e\u0440\u0456\u0442\u044c \u043d\u0430\u0431\u0456\u0440 \u0434\u0430\u043d\u0438\u0445.<\/strong><\/span><\/p>\n<p> <span style=\"color: #000000;\">\u041f\u043e-\u043f\u0435\u0440\u0448\u0435, \u043c\u0438 \u0441\u0442\u0432\u043e\u0440\u0438\u043c\u043e \u043d\u0430\u0431\u0456\u0440 \u0434\u0430\u043d\u0438\u0445, \u044f\u043a\u0438\u0439 \u0432\u0456\u0434\u043e\u0431\u0440\u0430\u0436\u0430\u0442\u0438\u043c\u0435 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u0438 \u0456\u0441\u043f\u0438\u0442\u0456\u0432 20 \u0441\u0442\u0443\u0434\u0435\u043d\u0442\u0456\u0432, \u0430 \u0442\u0430\u043a\u043e\u0436 \u043a\u0456\u043b\u044c\u043a\u0456\u0441\u0442\u044c \u0433\u043e\u0434\u0438\u043d, \u044f\u043a\u0456 \u0432\u043e\u043d\u0438 \u0432\u0438\u0442\u0440\u0430\u0442\u0438\u043b\u0438 \u043d\u0430 \u043d\u0430\u0432\u0447\u0430\u043d\u043d\u044f, \u043a\u0456\u043b\u044c\u043a\u0456\u0441\u0442\u044c \u043f\u0440\u0430\u043a\u0442\u0438\u0447\u043d\u0438\u0445 \u0456\u0441\u043f\u0438\u0442\u0456\u0432, \u044f\u043a\u0456 \u0432\u043e\u043d\u0438 \u0441\u043a\u043b\u0430\u043b\u0438, \u0456 \u0457\u0445\u043d\u044e \u043f\u043e\u0442\u043e\u0447\u043d\u0443 \u043e\u0446\u0456\u043d\u043a\u0443 \u0437\u0430 \u043a\u0443\u0440\u0441:<\/span><\/p>\n<pre style=\"background-color: #ececec; font-size: 15px;\"> <strong><span style=\"color: #107d3f;\">import<\/span> numpy <span style=\"color: #107d3f;\">as<\/span> np\n<span style=\"color: #107d3f;\">import<\/span> pandas <span style=\"color: #107d3f;\">as<\/span> pd<span style=\"color: #107d3f;\">\nimport<\/span> scipy <span style=\"color: #107d3f;\">as<\/span> stats\n\ndata = {'score': [91, 93, 72, 87, 86, 73, 68, 87, 78, 99, 95, 76, 84, 96, 76, 80, 83, 84, 73, 74],\n        'hours': [16, 6, 3, 1, 2, 3, 2, 5, 2, 5, 2, 3, 4, 3, 3, 3, 4, 3, 4, 4],\n        'prep': [3, 4, 0, 3, 4, 0, 1, 2, 1, 2, 3, 3, 3, 2, 2, 2, 3, 3, 2, 2],\n        'grade': [70, 88, 80, 83, 88, 84, 78, 94, 90, 93, 89, 82, 95, 94, 81, 93, 93, 90, 89, 89]\n        }\n\ndf = pd.DataFrame(data,columns=['score', 'hours', 'prep','grade'])\ndf.head()\n\n score hours prep grade\n0 91 16 3 70\n1 93 6 4 88\n2 72 3 0 80\n3 87 1 3 83\n4 86 2 4 88\n<\/strong><\/pre>\n<p> <span style=\"color: #000000;\"><strong>\u041a\u0440\u043e\u043a 2: \u041e\u0431\u0447\u0438\u0441\u043b\u0456\u0442\u044c \u0432\u0456\u0434\u0441\u0442\u0430\u043d\u044c \u041c\u0430\u0445\u0430\u043b\u0430\u043d\u043e\u0431\u0456\u0441\u0430 \u0434\u043b\u044f \u043a\u043e\u0436\u043d\u043e\u0433\u043e \u0441\u043f\u043e\u0441\u0442\u0435\u0440\u0435\u0436\u0435\u043d\u043d\u044f.<\/strong><\/span><\/p>\n<p> <span style=\"color: #000000;\">\u0414\u0430\u043b\u0456 \u043c\u0438 \u043d\u0430\u043f\u0438\u0448\u0435\u043c\u043e \u043a\u043e\u0440\u043e\u0442\u043a\u0443 \u0444\u0443\u043d\u043a\u0446\u0456\u044e \u0434\u043b\u044f \u043e\u0431\u0447\u0438\u0441\u043b\u0435\u043d\u043d\u044f \u0432\u0456\u0434\u0441\u0442\u0430\u043d\u0456 \u041c\u0430\u0445\u0430\u043b\u0430\u043d\u043e\u0431\u0456\u0441\u0430.<\/span><\/p>\n<pre style=\"background-color: #ececec; font-size: 15px;\"> <strong><span style=\"color: #107d3f;\"><span style=\"color: #008080;\">#create function to calculate Mahalanobis distance<\/span>\ndef<\/span> mahalanobis(x= <span style=\"color: #107d3f;\">None<\/span> , data= <span style=\"color: #107d3f;\">None<\/span> , cov= <span style=\"color: #107d3f;\">None<\/span> ):\n\n    x_mu = x - np.mean(data)\n    if not cov:\n        cov = np.cov(data.values.T)\n    inv_covmat = np.linalg.inv(cov)\n    left = np.dot(x_mu, inv_covmat)\n    mahal = np.dot(left, x_mu.T)\n    return mahal.diagonal()\n\n<span style=\"color: #107d3f;\"><span style=\"color: #008080;\">#create new column in dataframe that contains Mahalanobis distance for each row<\/span><\/span>\ndf['mahalanobis'] = mahalanobis(x=df, data=df[['score', 'hours', 'prep', 'grade']])\n\n<span style=\"color: #008080;\">#display first five rows of dataframe\n<\/span>df.head()\n\n score hours prep grade mahalanobis\n0 91 16 3 70 16.501963\n1 93 6 4 88 2.639286\n2 72 3 0 80 4.850797\n3 87 1 3 83 5.201261\n4 86 2 4 88 3.828734\n<\/strong><\/pre>\n<p> <span style=\"color: #000000;\"><strong>\u041a\u0440\u043e\u043a 3: \u041e\u0431\u0447\u0438\u0441\u043b\u0456\u0442\u044c p-\u0437\u043d\u0430\u0447\u0435\u043d\u043d\u044f \u0434\u043b\u044f \u043a\u043e\u0436\u043d\u043e\u0457 \u0432\u0456\u0434\u0441\u0442\u0430\u043d\u0456 \u041c\u0430\u0445\u0430\u043b\u0430\u043d\u043e\u0431\u0456\u0441\u0430.<\/strong><\/span><\/p>\n<p> <span style=\"color: #000000;\">\u041c\u0438 \u0431\u0430\u0447\u0438\u043c\u043e, \u0449\u043e \u0434\u0435\u044f\u043a\u0456 \u0432\u0456\u0434\u0441\u0442\u0430\u043d\u0456 \u041c\u0430\u0445\u0430\u043b\u0430\u043d\u043e\u0431\u0456\u0441\u0430 \u043d\u0430\u0431\u0430\u0433\u0430\u0442\u043e \u0431\u0456\u043b\u044c\u0448\u0456, \u043d\u0456\u0436 \u0456\u043d\u0448\u0456. \u0429\u043e\u0431 \u0432\u0438\u0437\u043d\u0430\u0447\u0438\u0442\u0438, \u0447\u0438 \u0454 \u0431\u0443\u0434\u044c-\u044f\u043a\u0430 \u0437 \u0432\u0456\u0434\u0441\u0442\u0430\u043d\u0435\u0439 \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u0447\u043d\u043e \u0437\u043d\u0430\u0447\u0443\u0449\u043e\u044e, \u043d\u0430\u043c \u043f\u043e\u0442\u0440\u0456\u0431\u043d\u043e \u043e\u0431\u0447\u0438\u0441\u043b\u0438\u0442\u0438 \u0457\u0445\u043d\u0456 p-\u0437\u043d\u0430\u0447\u0435\u043d\u043d\u044f.<\/span><\/p>\n<p> <span style=\"color: #000000;\">P-\u0437\u043d\u0430\u0447\u0435\u043d\u043d\u044f \u0434\u043b\u044f \u043a\u043e\u0436\u043d\u043e\u0457 \u0432\u0456\u0434\u0441\u0442\u0430\u043d\u0456 \u0440\u043e\u0437\u0440\u0430\u0445\u043e\u0432\u0443\u0454\u0442\u044c\u0441\u044f \u044f\u043a p-\u0437\u043d\u0430\u0447\u0435\u043d\u043d\u044f, \u044f\u043a\u0435 \u0432\u0456\u0434\u043f\u043e\u0432\u0456\u0434\u0430\u0454 \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u0446\u0456 \u0445\u0456-\u043a\u0432\u0430\u0434\u0440\u0430\u0442 \u0432\u0456\u0434\u0441\u0442\u0430\u043d\u0456 \u041c\u0430\u0445\u0430\u043b\u0430\u043d\u043e\u0431\u0456\u0441\u0430 \u0437 k-1 \u0441\u0442\u0443\u043f\u0435\u043d\u044f\u043c\u0438 \u0441\u0432\u043e\u0431\u043e\u0434\u0438, \u0434\u0435 k = \u043a\u0456\u043b\u044c\u043a\u0456\u0441\u0442\u044c \u0437\u043c\u0456\u043d\u043d\u0438\u0445. \u0422\u043e\u043c\u0443 \u0432 \u0446\u044c\u043e\u043c\u0443 \u0432\u0438\u043f\u0430\u0434\u043a\u0443 \u043c\u0438 \u0431\u0443\u0434\u0435\u043c\u043e \u0432\u0438\u043a\u043e\u0440\u0438\u0441\u0442\u043e\u0432\u0443\u0432\u0430\u0442\u0438 \u0441\u0442\u0443\u043f\u0435\u043d\u0456 \u0441\u0432\u043e\u0431\u043e\u0434\u0438 4-1 = 3.<\/span><\/p>\n<pre style=\"background-color: #ececec; font-size: 15px;\"> <strong><span style=\"color: #107d3f;\">from<\/span> scipy.stats <span style=\"color: #107d3f;\">import<\/span> chi2\n\n<span style=\"color: #008080;\">#calculate p-value for each mahalanobis distance<\/span> \ndf['p'] = 1 - chi2.cdf(df['mahalanobis'], 3)\n\n<span style=\"color: #008080;\">#display p-values for first five rows in dataframe<\/span>\ndf.head()\n\n score hours prep grade mahalanobis p\n0 91 16 3 70 16.501963 0.000895\n1 93 6 4 88 2.639286 0.450644\n2 72 3 0 80 4.850797 0.183054\n3 87 1 3 83 5.201261 0.157639\n4 86 2 4 88 3.828734 0.280562\n<\/strong><\/pre>\n<p> <span style=\"color: #000000;\">\u042f\u043a \u043f\u0440\u0430\u0432\u0438\u043b\u043e, p-\u0437\u043d\u0430\u0447\u0435\u043d\u043d\u044f <strong>\u043c\u0435\u043d\u0448\u0435 0,001<\/strong> \u0432\u0432\u0430\u0436\u0430\u0454\u0442\u044c\u0441\u044f \u0432\u0438\u043a\u0438\u0434\u043e\u043c.<\/span> <span style=\"color: #000000;\">\u041c\u0438 \u0431\u0430\u0447\u0438\u043c\u043e, \u0449\u043e \u043f\u0435\u0440\u0448\u0435 \u0441\u043f\u043e\u0441\u0442\u0435\u0440\u0435\u0436\u0435\u043d\u043d\u044f \u0454 \u0432\u0438\u043a\u0438\u0434\u043e\u043c \u0443 \u043d\u0430\u0431\u043e\u0440\u0456 \u0434\u0430\u043d\u0438\u0445, \u043e\u0441\u043a\u0456\u043b\u044c\u043a\u0438 \u0432\u043e\u043d\u043e \u043c\u0430\u0454 p-\u0437\u043d\u0430\u0447\u0435\u043d\u043d\u044f \u043c\u0435\u043d\u0448\u0435 0,001.<\/span><\/p>\n<p> <span style=\"color: #000000;\">\u0417\u0430\u043b\u0435\u0436\u043d\u043e \u0432\u0456\u0434 \u043a\u043e\u043d\u0442\u0435\u043a\u0441\u0442\u0443 \u043f\u0440\u043e\u0431\u043b\u0435\u043c\u0438 \u0432\u0438 \u043c\u043e\u0436\u0435\u0442\u0435 \u0432\u0438\u0440\u0456\u0448\u0438\u0442\u0438 \u0432\u0438\u0434\u0430\u043b\u0438\u0442\u0438 \u0446\u0435 \u0441\u043f\u043e\u0441\u0442\u0435\u0440\u0435\u0436\u0435\u043d\u043d\u044f \u0437 \u043d\u0430\u0431\u043e\u0440\u0443 \u0434\u0430\u043d\u0438\u0445, \u043e\u0441\u043a\u0456\u043b\u044c\u043a\u0438 \u0432\u043e\u043d\u043e \u0454 \u0432\u0438\u043a\u0438\u0434\u043e\u043c \u0456 \u043c\u043e\u0436\u0435 \u0432\u043f\u043b\u0438\u043d\u0443\u0442\u0438 \u043d\u0430 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u0438 \u0430\u043d\u0430\u043b\u0456\u0437\u0443.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u0412\u0456\u0434\u0441\u0442\u0430\u043d\u044c \u041c\u0430\u0445\u0430\u043b\u0430\u043d\u043e\u0431\u0456\u0441\u0430 \u2014 \u0446\u0435 \u0432\u0456\u0434\u0441\u0442\u0430\u043d\u044c \u043c\u0456\u0436 \u0434\u0432\u043e\u043c\u0430 \u0442\u043e\u0447\u043a\u0430\u043c\u0438 \u0432 \u0431\u0430\u0433\u0430\u0442\u043e\u0432\u0438\u043c\u0456\u0440\u043d\u043e\u043c\u0443 \u043f\u0440\u043e\u0441\u0442\u043e\u0440\u0456. \u0419\u043e\u0433\u043e \u0447\u0430\u0441\u0442\u043e \u0432\u0438\u043a\u043e\u0440\u0438\u0441\u0442\u043e\u0432\u0443\u044e\u0442\u044c \u0434\u043b\u044f \u0432\u0438\u044f\u0432\u043b\u0435\u043d\u043d\u044f \u0432\u0438\u043a\u0438\u0434\u0456\u0432 \u0443 \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u0447\u043d\u043e\u043c\u0443 \u0430\u043d\u0430\u043b\u0456\u0437\u0456, \u0449\u043e \u0432\u043a\u043b\u044e\u0447\u0430\u0454 \u043a\u0456\u043b\u044c\u043a\u0430 \u0437\u043c\u0456\u043d\u043d\u0438\u0445. \u0426\u0435\u0439 \u043f\u043e\u0441\u0456\u0431\u043d\u0438\u043a \u043f\u043e\u044f\u0441\u043d\u044e\u0454, \u044f\u043a \u043e\u0431\u0447\u0438\u0441\u043b\u0438\u0442\u0438 \u0432\u0456\u0434\u0441\u0442\u0430\u043d\u044c \u041c\u0430\u0445\u0430\u043b\u0430\u043d\u043e\u0431\u0456\u0441\u0430 \u0432 Python. \u041f\u0440\u0438\u043a\u043b\u0430\u0434: \u0412\u0456\u0434\u0441\u0442\u0430\u043d\u044c \u041c\u0430\u0445\u0430\u043b\u0430\u043d\u043e\u0431\u0456\u0441\u0430 \u0432 Python 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