CH14 SVD.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# CH14 SVD"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Dependencies"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "from numpy import linalg as la\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "SVD (Singular Value Decomposition)奇异值分解\n",
    "- 隐性语义索引（Latent Semantic Indexing, LSI）或者 隐性语义分析(Latent Semantic Analysis, LSA)\n",
    "- 推荐系统\n",
    "\n",
    "\n",
    "- PCA -> 特征值\n",
    "- SVD -> 奇异值\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## MLiA"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### SVD Applications"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### LSI"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "在模式识别和信息检索中，SVD在降维方面有广泛应用，被称为是浅语义索引的基础。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### RS\n",
    "特征生成，将数据映射成重要特征。低维的数据饱含了整个数据的大部分能量"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Matrix Decomposition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "U, sigma, VT = np.linalg.svd([[1, 1],[7, 7]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "U, sigma, VT"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### CF"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "collaborative filtering"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### **Similarity**\n",
    "- 欧式距离\n",
    "    - [0, 1]\n",
    "- Jaccard相似度\n",
    "- 皮尔逊相关系数\n",
    "     - 量级不敏感\n",
    "     - [-1, 1]\n",
    "- 余弦相似度\n",
    "    - [-1, 1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def ecludSim(inA, inB):\n",
    "    return 1/(1 + la.norm(inA -inB))\n",
    "\n",
    "def pearsSim(inA, inB):\n",
    "    if len(inA) < 3:\n",
    "        return 1\n",
    "    return 0.5+0.5*np.corrcoef(inA, inB, rowvar=0)[0][1]\n",
    "\n",
    "def cosSim(inA, inB):\n",
    "    num = float(inA.T*inB)\n",
    "    denom = la.norm(inA)*la.norm(inB)\n",
    "    return 0.5+0.5*(num/denom)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "X = np.mat([[1, 1, 1, 0, 0],\n",
    "            [2, 2, 2, 0, 0],\n",
    "            [1, 1, 1, 0, 0],\n",
    "            [5, 5, 5, 0, 0],\n",
    "            [1, 1, 0, 2, 2],\n",
    "            [0, 0, 0, 3, 3],\n",
    "            [0, 0, 0, 1, 1]])\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "X[:, 0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "ecludSim(X[:, 0], X[:, 4]), ecludSim(X[:, 0], X[:, 0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "pearsSim(X[:, 0], X[:, 4]), pearsSim(X[:, 0], X[:, 0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "cosSim(X[:, 0], X[:, 4]), cosSim(X[:, 0], X[:, 0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Item based recommondation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Item or User"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- user-based CF\n",
    "- item-based CF\n",
    "\n",
    "如果用户数量很多，往往通过Item Similarity来实现推荐"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Cost\n",
    "- 打分 : RMSE\n",
    "- TopN : Precision 和 Recall\n",
    "- 覆盖率 : 熵\n",
    "- 多样性 : "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Ex : 餐饮推荐"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Scoring for unscored"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### copy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "X.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "mX = X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "mX[0,1] = 4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "X, mX"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "mX = X.copy()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "mX[0, 1] = 5\n",
    "mX, X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "mX[0, 1] = mX[0, 0]= mX[1, 0]=mX[2, 0]=4\n",
    "mX[3, 3] = 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### modified X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "mX"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "mX[:,1],mX[:,1].A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "mX[0,1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### overLaped users\n",
    "overlap means which user have scores at item and j both"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "item = 1\n",
    "j = 2\n",
    "overLap = np.nonzero(np.logical_and(mX[:, item].A>0,\n",
    "                                    mX[:, j].A>0))[0]\n",
    "mX[:, item].A,mX[:,j].A, overLap"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def standEst(dataMat, user, simMeas, item):\n",
    "    n = dataMat.shape[1]\n",
    "    simTotal = 0\n",
    "    ratSimTotal = 0\n",
    "    for j in range(n):\n",
    "        userRating = dataMat[user, j]\n",
    "        if userRating == 0:\n",
    "            continue\n",
    "        # 筛选用户： 在item 和 j两列都有值的用户\n",
    "        overLap = np.nonzero(np.logical_and(dataMat[:, item].A>0,\n",
    "                                            dataMat[:, j].A>0))[0]\n",
    "        if len(overLap) == 0:\n",
    "            similarity = 0\n",
    "        else:\n",
    "            # 通过都有值的用户数据，计算item和j列的相似度\n",
    "            similarity = simMeas(dataMat[overLap, item],\n",
    "                                 dataMat[overLap, j])\n",
    "            print(\"the %d and %d similarity is : %f\" % (item, j, similarity))\n",
    "            # item-based，基于item之间的相似度，通过其他item的打分，获取打分\n",
    "            simTotal += similarity\n",
    "            # 基于相似度，通过用户给j列的打分，预测给item的打分\n",
    "            ratSimTotal += similarity*userRating\n",
    "    return 0 if simTotal == 0 else ratSimTotal/simTotal"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### nonzero"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "mX"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "user =  1\n",
    "mX[user,:].A == 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "mX[user,:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "np.nonzero(mX[user,:].A == 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "user =  4\n",
    "mX[user,:].A == 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "np.nonzero(mX[user,:].A == 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### recommend"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def recommend(dataMat, user, N=3, simMeas=cosSim, estMethod=standEst):\n",
    "    unratedItems = np.nonzero(dataMat[user,:].A==0)[1]\n",
    "    if len(unratedItems) == 0: \n",
    "        return \"you rated everything\"\n",
    "    itemScores = [(item, estMethod(dataMat, user, simMeas, item)) for item in unratedItems]\n",
    "    return sorted(itemScores, key=lambda jj: jj[1], reverse=True)[:N]\n",
    "        "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "recommend(mX, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### SVD"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "df = pd.read_csv(\"./Data/CH14/data_14-4.csv\",index_col=0)\n",
    "df"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### SVD\n",
    "转化成低维矩阵计算相似度"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "X = np.mat(df.values)\n",
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "U, Sigma, VT = la.svd(X)\n",
    "Sigma"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "sum(Sigma),sum(Sigma)*.9"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "sum(Sigma[:2]),sum(Sigma[:3]),sum(Sigma[:4]),sum(Sigma[:5]),sum(Sigma[:6]),sum(Sigma[:7])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Sigma"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "Sig4 = np.mat(np.eye(4)*Sigma[:4])\n",
    "Sig4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "Sig5 = np.mat(np.eye(5)*Sigma[:5])\n",
    "Sig5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "Sig5.I"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Transformed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "xformedItems = X.T*U[:,:5]*Sig5.I\n",
    "xformedItems"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def svdEst(dataMat, user, simMeas, item):\n",
    "    n = dataMat.shape[1]\n",
    "    simTotal = 0\n",
    "    ratSimTotal = 0\n",
    "    U, Sigma, VT = la.svd(dataMat)\n",
    "    n_Sig = 6\n",
    "    Sig = np.mat(np.eye(n_Sig)*Sigma[:n_Sig])\n",
    "    xformedItems = dataMat.T*U[:,:n_Sig]*Sig.I\n",
    "    print(dataMat[user])\n",
    "    for j in range(n):\n",
    "        userRating = dataMat[user, j]\n",
    "        if userRating == 0 or j==item:\n",
    "            continue\n",
    "        # 通过都有值的用户数据，计算item和j列的相似度\n",
    "        similarity = simMeas(xformedItems[item,:].T,\n",
    "                             xformedItems[j,:].T)\n",
    "        print(\"the %d and %d similarity is : %f\" % (item, j, similarity))\n",
    "        # item-based，基于item之间的相似度，通过其他item的打分，获取打分\n",
    "        simTotal += similarity\n",
    "        # 基于相似度，通过用户给j列的打分，预测给item的打分\n",
    "        ratSimTotal += similarity*userRating\n",
    "    return 0 if simTotal == 0 else ratSimTotal/simTotal"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### recommend"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "recommend(X,0,estMethod=svdEst)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "recommend(X,1,estMethod=svdEst)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Summary"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- 大数据集SVD操作低频运行\n",
    "- Sparse\n",
    "- Similarity 存储\n",
    "- Cold-Start 推荐转化成搜索"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Ex : 图像压缩"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Sklearn"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from skimage import data, io, filters\n",
    "image = data.coins()\n",
    "# ... or any other NumPy array!\n",
    "edges = filters.sobel(image)\n",
    "io.imshow(edges)\n",
    "io.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## SURPRISE"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Evaluating RMSE, MAE of algorithm SVD on 5 split(s).\n",
      "\n",
      "                  Fold 1  Fold 2  Fold 3  Fold 4  Fold 5  Mean    Std     \n",
      "RMSE (testset)    0.9400  0.9384  0.9348  0.9337  0.9427  0.9379  0.0033  \n",
      "MAE (testset)     0.7408  0.7390  0.7366  0.7342  0.7434  0.7388  0.0032  \n",
      "Fit time          6.78    6.94    6.82    6.86    6.95    6.87    0.07    \n",
      "Test time         0.28    0.22    0.21    0.26    0.22    0.24    0.03    \n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "{'fit_time': (6.782602787017822,\n",
       "  6.9437360763549805,\n",
       "  6.823808670043945,\n",
       "  6.861783027648926,\n",
       "  6.952728748321533),\n",
       " 'test_mae': array([0.74076198, 0.73902482, 0.73659828, 0.73415997, 0.7433679 ]),\n",
       " 'test_rmse': array([0.93996987, 0.93836471, 0.93479145, 0.93374258, 0.94273326]),\n",
       " 'test_time': (0.2798304557800293,\n",
       "  0.223860502243042,\n",
       "  0.2138676643371582,\n",
       "  0.26283812522888184,\n",
       "  0.22286295890808105)}"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from surprise import SVD\n",
    "from surprise import Dataset\n",
    "from surprise.model_selection import cross_validate\n",
    "\n",
    "# Load the movielens-100k dataset (download it if needed).\n",
    "data = Dataset.load_builtin('ml-100k')\n",
    "\n",
    "# Use the famous SVD algorithm.\n",
    "algo = SVD()\n",
    "\n",
    "# Run 5-fold cross-validation and print results.\n",
    "cross_validate(algo, data, measures=['RMSE', 'MAE'], cv=5, verbose=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
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       " ('59', '485', 2.0, '888204466'),\n",
       " ...]"
      ]
     },
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    "data.raw_ratings"
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