poms/model/RFM-PY36.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "fecd214a",
   "metadata": {},
   "source": [
    "python 3.6.13\n",
    "matplotlib 3.3.4\n",
    "numpy 1.19.2\n",
    "pandas 1.1.5\n",
    "scikit-learn 0.24.2\n",
    "seaborn 0.11.2\n",
    "sqlalchemy 1.4.22"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "12d24a6c",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.cluster import KMeans\n",
    "from sklearn.metrics import silhouette_score\n",
    "from sqlalchemy import create_engine\n",
    "from urllib.parse import quote_plus as urlquote\n",
    "import datetime\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import seaborn as sns\n",
    "import time"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "121a3f95",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-05T07:53:31.824351Z",
     "start_time": "2022-01-05T07:53:31.780403Z"
    }
   },
   "outputs": [],
   "source": [
    "db_password = 'POMS_129@MySQL'\n",
    "db_engine = create_engine(f'mysql+pymysql://poms:{urlquote(db_password)}@101.133.164.208:3306/dev_poms?charset=utf8')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "faa12069",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-05T07:53:32.314051Z",
     "start_time": "2022-01-05T07:53:31.827278Z"
    }
   },
   "outputs": [],
   "source": [
    "sql = \"SELECT `create_user` AS `account`, `point_uuid`, `point`, `create_time` FROM `sys_point_record` WHERE `audited` = 1\"\n",
    "raw_data = pd.read_sql(sql, db_engine)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "8120b8c2",
   "metadata": {},
   "outputs": [
    {
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       "      <th></th>\n",
       "      <th>account</th>\n",
       "      <th>point_uuid</th>\n",
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       "      <td>2021-12-29</td>\n",
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       "  </tbody>\n",
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       "<p>76050 rows × 4 columns</p>\n",
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      ],
      "text/plain": [
       "       account                        point_uuid  point create_time\n",
       "0      1951134  681cb6da8c9a4984ab6bf5c39bc9cfdc      9  2021-01-01\n",
       "1      2051111  8312dac5b3cf5d93b511b1111a9aad30      6  2021-01-01\n",
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       "3      2051110  afe319c5b9cec39700d219bc2cb55137      4  2021-01-01\n",
       "4      1951122  afe319c5b9cec39700d219bc2cb55137      2  2021-01-01\n",
       "...        ...                               ...    ...         ...\n",
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       "76046  2051107  a373d0b08d8f3698f94c4a81d070bc3d      9  2021-12-29\n",
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       "76049  1951129  136e05d5470d9e843d699e1bc10bd17a      5  2021-12-29\n",
       "\n",
       "[76050 rows x 4 columns]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 看一眼\n",
    "raw_data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "c1954360",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-05T07:53:32.439640Z",
     "start_time": "2022-01-05T07:53:32.365840Z"
    }
   },
   "outputs": [],
   "source": [
    "# 去除缺失值\n",
    "df2 = raw_data.copy().fillna(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "1f387b2f",
   "metadata": {},
   "outputs": [
    {
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       "      <th>account</th>\n",
       "      <th>point_uuid</th>\n",
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       "      <td>2021-12-29</td>\n",
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       "<p>76050 rows × 4 columns</p>\n",
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      ],
      "text/plain": [
       "       account                        point_uuid  point create_time\n",
       "0      1951134  681cb6da8c9a4984ab6bf5c39bc9cfdc      9  2021-01-01\n",
       "1      2051111  8312dac5b3cf5d93b511b1111a9aad30      6  2021-01-01\n",
       "2      1951122  b2f8e1aa52697d4088f0c3d27cf5fd01      2  2021-01-01\n",
       "3      2051110  afe319c5b9cec39700d219bc2cb55137      4  2021-01-01\n",
       "4      1951122  afe319c5b9cec39700d219bc2cb55137      2  2021-01-01\n",
       "...        ...                               ...    ...         ...\n",
       "76045  5396174  b2f8e1aa52697d4088f0c3d27cf5fd01      7  2021-12-29\n",
       "76046  2051107  a373d0b08d8f3698f94c4a81d070bc3d      9  2021-12-29\n",
       "76047  2051102  dde9d4a8db4ec74a39e3ceae0734815f      4  2021-12-29\n",
       "76048  1951130  6f99a016eb26d149cc3eb2ef4ea5bba8      3  2021-12-29\n",
       "76049  1951129  136e05d5470d9e843d699e1bc10bd17a      5  2021-12-29\n",
       "\n",
       "[76050 rows x 4 columns]"
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     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 看一眼\n",
    "df2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "c423e07e",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-05T07:53:41.383720Z",
     "start_time": "2022-01-05T07:53:32.508456Z"
    }
   },
   "outputs": [],
   "source": [
    "## Recent【最近一次消费数据】\n",
    "now = pd.to_datetime(datetime.datetime.now())\n",
    "# 添加时间差数据\n",
    "df2[\"recent\"] = df2[\"create_time\"].map(lambda x: (now - pd.to_datetime(x)).days)\n",
    "# 用户最近一次购买商品的时间\n",
    "recent_df = df2.groupby(by=\"account\", as_index=False).agg({\"recent\": \"min\"})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "dc957d34",
   "metadata": {},
   "outputs": [],
   "source": [
    "## Frequency【一段时间内的消费次数】\n",
    "frequency_df = df2.groupby(by=\"account\", as_index=False).agg({\"point_uuid\": \"count\"})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "308bc08d",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-05T07:53:41.636044Z",
     "start_time": "2022-01-05T07:53:41.593163Z"
    }
   },
   "outputs": [],
   "source": [
    "## Mount【一段时间内的消费总金额】\n",
    "mount_df = df2.groupby(by=\"account\", as_index=False).agg({\"point\": \"sum\"})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "de978fb4",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-05T07:53:41.760711Z",
     "start_time": "2022-01-05T07:53:41.671948Z"
    }
   },
   "outputs": [],
   "source": [
    "## RFM模型\n",
    "# 根据 account 合并数据\n",
    "rfm_df = recent_df.merge(frequency_df, on=\"account\", how=\"left\").merge(mount_df, on=\"account\", how=\"left\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "dc4af967",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-05T07:53:41.806587Z",
     "start_time": "2022-01-05T07:53:41.794621Z"
    }
   },
   "outputs": [],
   "source": [
    "rfm_df2 = rfm_df.copy()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "0c145514",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-05T07:53:41.883382Z",
     "start_time": "2022-01-05T07:53:41.840497Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0        0\n",
       "0.2        2\n",
       "0.4        4\n",
       "0.6        6\n",
       "0.8        8\n",
       "1.0    13057\n",
       "Name: point, dtype: int64"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rfm_df2[\"point\"].quantile(q=np.linspace(0, 1, num=6), interpolation='nearest')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "3d49d903",
   "metadata": {},
   "outputs": [
    {
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      ],
      "text/plain": [
       "      account  recent  point_uuid  point\n",
       "0     1001611     464           1      4\n",
       "1     1005374     589           1      7\n",
       "2     1005407     624           1      5\n",
       "3     1010153     756           1     10\n",
       "4     1010528     467           1      7\n",
       "...       ...     ...         ...    ...\n",
       "5881  9990197     809           1      8\n",
       "5882  9990991     573           1      4\n",
       "5883  9992257     666           1      0\n",
       "5884  9993826     498           1      0\n",
       "5885  9998323     734           1      7\n",
       "\n",
       "[5886 rows x 4 columns]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rfm_df2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "605888d7",
   "metadata": {},
   "outputs": [],
   "source": [
    "# 根据分位数分层，划分为5个等级\n",
    "recent_labels = [5, 4, 3, 2, 1]\n",
    "mount_labels = [1, 2, 3, 4, 5]\n",
    "frequency_bins = [1, 3, 5, 12]\n",
    "frequency_labels = [1, 2, 3]\n",
    "\n",
    "m_bins = rfm_df2[\"point\"].quantile(q=np.linspace(0, 1, num=6), interpolation='nearest')\n",
    "r_bins = rfm_df2[\"recent\"].quantile(q=np.linspace(0, 1, num=6), interpolation='nearest')\n",
    "rfm_df2[\"R\"] = pd.cut(rfm_df2[\"recent\"], bins=r_bins, labels=recent_labels, include_lowest=True)\n",
    "rfm_df2[\"F\"] = pd.cut(rfm_df2[\"point_uuid\"], bins=frequency_bins, labels=frequency_labels, include_lowest=True)\n",
    "rfm_df2[\"M\"] = pd.cut(rfm_df2[\"point\"], bins=m_bins, labels=mount_labels, include_lowest=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "6dbd63f1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>account</th>\n",
       "      <th>R</th>\n",
       "      <th>F</th>\n",
       "      <th>M</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1001611</td>\n",
       "      <td>5</td>\n",
       "      <td>1</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1005374</td>\n",
       "      <td>4</td>\n",
       "      <td>1</td>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1005407</td>\n",
       "      <td>3</td>\n",
       "      <td>1</td>\n",
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       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1010153</td>\n",
       "      <td>1</td>\n",
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       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1010528</td>\n",
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       "      <td>4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
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       "      <td>...</td>\n",
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       "    <tr>\n",
       "      <th>5881</th>\n",
       "      <td>9990197</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5882</th>\n",
       "      <td>9990991</td>\n",
       "      <td>4</td>\n",
       "      <td>1</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5883</th>\n",
       "      <td>9992257</td>\n",
       "      <td>3</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5884</th>\n",
       "      <td>9993826</td>\n",
       "      <td>5</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5885</th>\n",
       "      <td>9998323</td>\n",
       "      <td>2</td>\n",
       "      <td>1</td>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>5886 rows × 4 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "      account  R  F  M\n",
       "0     1001611  5  1  2\n",
       "1     1005374  4  1  4\n",
       "2     1005407  3  1  3\n",
       "3     1010153  1  1  5\n",
       "4     1010528  5  1  4\n",
       "...       ... .. .. ..\n",
       "5881  9990197  1  1  4\n",
       "5882  9990991  4  1  2\n",
       "5883  9992257  3  1  1\n",
       "5884  9993826  5  1  1\n",
       "5885  9998323  2  1  4\n",
       "\n",
       "[5886 rows x 4 columns]"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 删除不需要的列\n",
    "rfm_df2 = rfm_df2.drop(['recent', 'point_uuid', 'point'], axis=1)\n",
    "rfm_df2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "c81fdc73",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-05T07:53:42.199538Z",
     "start_time": "2022-01-05T07:53:42.187570Z"
    }
   },
   "outputs": [],
   "source": [
    "rfm_df2 = rfm_df2.fillna(3)\n",
    "rfm_df2 = rfm_df2.astype(int)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "8e19592c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "      <th>R</th>\n",
       "      <th>F</th>\n",
       "      <th>M</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
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       "    <tr>\n",
       "      <th>1</th>\n",
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       "      <th>3</th>\n",
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       "      <th>5882</th>\n",
       "      <td>9990991</td>\n",
       "      <td>4</td>\n",
       "      <td>1</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5883</th>\n",
       "      <td>9992257</td>\n",
       "      <td>3</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5884</th>\n",
       "      <td>9993826</td>\n",
       "      <td>5</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
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       "      <th>5885</th>\n",
       "      <td>9998323</td>\n",
       "      <td>2</td>\n",
       "      <td>1</td>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>5886 rows × 4 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "      account  R  F  M\n",
       "0     1001611  5  1  2\n",
       "1     1005374  4  1  4\n",
       "2     1005407  3  1  3\n",
       "3     1010153  1  1  5\n",
       "4     1010528  5  1  4\n",
       "...       ... .. .. ..\n",
       "5881  9990197  1  1  4\n",
       "5882  9990991  4  1  2\n",
       "5883  9992257  3  1  1\n",
       "5884  9993826  5  1  1\n",
       "5885  9998323  2  1  4\n",
       "\n",
       "[5886 rows x 4 columns]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rfm_df2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "1a033585",
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "# 对数据进行提取选取我们需要的列\n",
    "rfm = rfm_df2[['R', 'F', 'M']]\n",
    "#使用lambda函数对数据进行归一化处理\n",
    "normalization_rfm = rfm.apply(lambda x: (x - np.min(x)) / (np.max(x) - np.min(x)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "1b20e949",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "  <thead>\n",
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       "      <th></th>\n",
       "      <th>R</th>\n",
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       "      <th>5881</th>\n",
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       "    <tr>\n",
       "      <th>5882</th>\n",
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       "    <tr>\n",
       "      <th>5883</th>\n",
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       "      <td>0.00</td>\n",
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       "    <tr>\n",
       "      <th>5884</th>\n",
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       "      <td>0.0</td>\n",
       "      <td>0.00</td>\n",
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       "    <tr>\n",
       "      <th>5885</th>\n",
       "      <td>0.25</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.75</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>5886 rows × 3 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "         R    F     M\n",
       "0     1.00  0.0  0.25\n",
       "1     0.75  0.0  0.75\n",
       "2     0.50  0.0  0.50\n",
       "3     0.00  0.0  1.00\n",
       "4     1.00  0.0  0.75\n",
       "...    ...  ...   ...\n",
       "5881  0.00  0.0  0.75\n",
       "5882  0.75  0.0  0.25\n",
       "5883  0.50  0.0  0.00\n",
       "5884  1.00  0.0  0.00\n",
       "5885  0.25  0.0  0.75\n",
       "\n",
       "[5886 rows x 3 columns]"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "normalization_rfm = normalization_rfm.fillna(0)\n",
    "normalization_rfm"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3a13399c",
   "metadata": {},
   "source": [
    "$$\n",
    "SSE = \\sum^{k}_{i=1}\\sum_{p\\in{C_i}}|p-m_i|^2\n",
    "$$\n",
    "$C_i$是第$i$个簇，$p$是$C_i$中的样本点，$m_i$是$C_i$的质心（$C_i$中所有样本的均值），$SSE$是所有样本的聚类误差，代表了聚类效果的好坏。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "422c054b",
   "metadata": {},
   "outputs": [],
   "source": [
    "clusters = [2, 3, 4, 5, 6, 7, 8]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "a0c1889b",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[934.2103968347631, 597.3608447740422, 384.3004298136629, 317.9862632164778, 271.5227306631748, 231.6340074056924, 196.15821396905733]\n"
     ]
    }
   ],
   "source": [
    "# 手肘法来确定最优K值\n",
    "SSE = []  # 误差平方和\n",
    "SC = []  # 轮廓系数\n",
    "for i in clusters:\n",
    "    kmeans = KMeans(n_clusters=i)\n",
    "    kmeans.fit(normalization_rfm)\n",
    "    SSE.append(kmeans.inertia_)\n",
    "print(SSE)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "a50606ef",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'SSE')"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sns.lineplot(x=clusters, y=SSE)\n",
    "plt.xlabel('K')\n",
    "plt.ylabel('SSE')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "714adb7d",
   "metadata": {},
   "outputs": [],
   "source": [
    "# 计算曲率\n",
    "x_t = np.gradient(clusters)\n",
    "y_t = np.gradient(SSE)\n",
    "\n",
    "vel = np.array([[x_t[i], y_t[i]] for i in range(x_t.size)])\n",
    "\n",
    "speed = np.sqrt(x_t * x_t + y_t * y_t)\n",
    "\n",
    "tangent = np.array([1 / speed] * 2).transpose() * vel\n",
    "\n",
    "tangent_x = tangent[:, 0]\n",
    "tangent_y = tangent[:, 1]\n",
    "\n",
    "d_tangent_x = np.gradient(tangent_x)\n",
    "d_tangent_y = np.gradient(tangent_y)\n",
    "\n",
    "dT_dt = np.array([[d_tangent_x[i], d_tangent_y[i]] for i in range(d_tangent_x.size)])\n",
    "\n",
    "length_dT_dt = np.sqrt(d_tangent_x * d_tangent_x + d_tangent_y * d_tangent_y)\n",
    "\n",
    "normal = np.array([1 / length_dT_dt] * 2).transpose() * dT_dt\n",
    "\n",
    "d2s_dt2 = np.gradient(speed)\n",
    "d2x_dt2 = np.gradient(x_t)\n",
    "d2y_dt2 = np.gradient(y_t)\n",
    "\n",
    "curvature = np.abs(d2x_dt2 * y_t - x_t * d2y_dt2) / (x_t * x_t + y_t * y_t) ** 1.5\n",
    "t_component = np.array([d2s_dt2] * 2).transpose()\n",
    "n_component = np.array([curvature * speed * speed] * 2).transpose()\n",
    "\n",
    "acceleration = t_component * tangent + n_component * normal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "eadb907c",
   "metadata": {},
   "outputs": [],
   "source": [
    "normalization_acceleration = acceleration\n",
    "normalization_acceleration[:, 1] = (acceleration[:, 1] - np.min(acceleration[:, 1]))/(np.max(acceleration[:, 1]) - np.min(acceleration[:,1]))\n",
    "normalization_curvature = normalization_acceleration[:, 1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "b2d383e8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'normalization curvature')"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sns.lineplot(x=clusters, y=normalization_curvature)\n",
    "plt.xlabel('K')\n",
    "plt.ylabel('normalization curvature')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "08948ffb",
   "metadata": {},
   "source": [
    "$$\n",
    "S = \\frac{b - a}{\\max{(a, b)}}\n",
    "$$\n",
    "$S$样本点$X_i$的轮廓系数，$a$是$X_i$与同簇的其他样本的平均距离，称为凝聚度，$b$是$X_i$与最近簇中所有样本的平均距离，称为分离度。\n",
    "$$\n",
    "C_j = \\arg{\\min_{C_i}\\frac{1}{n}\\sum_{p\\in{C_i}}|p-X_i|^2}\n",
    "$$\n",
    "$C_j$是最近簇，其中p是某个簇$C_k$中的样本。事实上，简单点讲，就是用$X_i$到某个簇所有样本平均距离作为衡量该点到该簇的距离后，选择离$X_i$最近的一个簇作为最近簇。求出所有样本的轮廓系数后再求平均值就得到了平均轮廓系数。平均轮廓系数的取值范围为[-1,1]，且簇内样本的距离越近，簇间样本距离越远，平均轮廓系数越大，聚类效果越好。那么，很自然地，平均轮廓系数最大的k便是最佳聚类数。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "321eceaf",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.3926532652562246, 0.4165845111425452, 0.45642014396553005, 0.4581536763935582, 0.46087941918329456, 0.4601172904178992, 0.4810293417489168]\n"
     ]
    }
   ],
   "source": [
    "# 轮廓系数来确定最优K值，数据量大时效率不高\n",
    "SCORE = []\n",
    "for i in clusters:\n",
    "    kmeans = KMeans(n_clusters=i)\n",
    "    result = kmeans.fit_predict(normalization_rfm)\n",
    "    score = silhouette_score(normalization_rfm, result)\n",
    "    SCORE.append(score)\n",
    "print(SCORE)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "dbf44f9e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'SCORE')"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sns.lineplot(x=clusters, y=SCORE)\n",
    "plt.xlabel('K')\n",
    "plt.ylabel('SCORE')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "85850690",
   "metadata": {},
   "outputs": [],
   "source": [
    "tup_list = []\n",
    "for i in range(7):\n",
    "    item = (clusters[i], normalization_curvature[i] * 0.1 + SCORE[i] * 0.9)\n",
    "    tup_list.append(item)\n",
    "tup_list = sorted(tup_list, key=lambda x: x[1], reverse=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "7ce62d18",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "clf_KMeans聚类中心\n",
      " [[ 7.54696970e-01 -1.38777878e-17  9.34848485e-02]\n",
      " [ 7.45000000e-01  1.47368421e-02  7.53289474e-01]\n",
      " [ 1.28099174e-01  8.67361738e-19  9.94031221e-02]\n",
      " [ 1.22093023e-01 -1.73472348e-18  7.59222133e-01]]\n"
     ]
    }
   ],
   "source": [
    "# 模型训练。得到预测值。\n",
    "k = tup_list[0][0]\n",
    "k_model = KMeans(n_clusters=k)\n",
    "k_model.fit(normalization_rfm)\n",
    "print(\"clf_KMeans聚类中心\\n\", k_model.cluster_centers_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "e5365af7",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "cluster2聚类数量\n",
      " 1    1900\n",
      "0    1650\n",
      "3    1247\n",
      "2    1089\n",
      "dtype: int64\n"
     ]
    }
   ],
   "source": [
    "quantity = pd.Series(k_model.labels_).value_counts()\n",
    "print(\"cluster2聚类数量\\n\", quantity)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "592e7328",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "UPDATE `sys_user` SET `persona` = 1 WHERE `account` IN ('1234567', '1951119', '1951122', '1951123', '1951124', '1951125', '1951126', '1951127', '1951128', '1951129', '1951130', '1951131', '1951132', '1951133', '1951134', '1951135', '2051101', '2051102', '2051103', '2051104', '2051105', '2051106', '2051107', '2051108', '2051109', '2051110', '2051111', '2052520')\n"
     ]
    }
   ],
   "source": [
    "# 获取聚类之后每个聚类中心的数据\n",
    "res0Series = pd.Series(k_model.labels_)\n",
    "message = [\"\", \"\", \"据分析，您近期的表现不佳，请注意！\", \"据分析，您近期的表现存在严重问题，请及时整改！\"]\n",
    "query_sql = \"SELECT `account` FROM `sys_user`\"\n",
    "query_data = pd.read_sql(query_sql, db_engine)\n",
    "for i in range(0, k):\n",
    "    res0 = res0Series[res0Series.values == i]\n",
    "    temp_list = []\n",
    "    for account in rfm_df2['account'].iloc[res0.index].values:\n",
    "        if str(account) in query_data[\"account\"].values:\n",
    "            temp_list.append(str(account))\n",
    "            if i >= 2:\n",
    "                insert_sql = \"INSERT INTO `sys_message` (`account`, `title`, `content`, `create_user`, `create_time`) VALUES ('\" + str(account) + \"', '分析结果提醒', '\" + message[i] + \"', 'admin', '\" + str(time.strftime(\"%Y-%m-%d %H:%M:%S\", time.localtime())) + \"')\"\n",
    "                db_engine.execute(insert_sql)\n",
    "    if len(temp_list) > 0:\n",
    "        update_sql = \"UPDATE `sys_user` SET `persona` = \" + str(i) + \" WHERE `account` IN (\" + str(temp_list).replace(\"[\", \"\").replace(\"]\", \"\") + \")\"\n",
    "        db_engine.execute(update_sql)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "06dd4a51",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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