principal component analysis.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Principal Component Analysis\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "Principal Component Analysis is undoubtedly the most abused algorithm in this repository, lol! Since iris is my default dataset, PCA is literally used in every python notebook for visualization except the text mining algorithms. It is one of my favorite algorithms for dimension reduction and data compression. Although I am not very inclined to use it for the purpose of matching or anomaly detection. What can be better? It is so intuitive that anyone with basic understanding of linear algebra can implement the algorithm in ten minutes! \n",
    "\n",
    "Basically, the algorithm intends to find some vectors $u$ to project the original data $x$ into lower dimension subspace. The objective of the projection $x^Tu$ is to retain as much variance as possible in the original dataset. In translation to math, we would love to maximize $||x^Tu||_2$ in geometry subject to $||u||_2 = 1$. We can solve the convex optimization by Lagrangian and it is easy to see that $u$ is merely a eigenvector of the covariance matrix of $x$. After selecting top $k$ eigenvectors, we would obtain the projection $x^Tu$ in a $k$ dimensional subspace rather than the original $n$ dimensional space.\n",
    "\n",
    "Andrew Ng's material does not focus on singular value decomposition, more SVD content can be found in this tutorial on PCA by Google\n",
    "\n",
    "https://arxiv.org/pdf/1404.1100.pdf\n",
    "\n",
    "If that's not enough, feel free to read the whole textbook dedicated to PCA\n",
    "\n",
    "http://cda.psych.uiuc.edu/statistical_learning_course/Jolliffe%20I.%20Principal%20Component%20Analysis%20(2ed.,%20Springer,%202002)(518s)_MVsa_.pdf\n",
    "\n",
    "You can think of PCA as an algorithm to obtain a linear combination of the original dataset that maximizes the variability. There is also a closely related algorithm which obtains a linear combination of the original dataset that decomposes it into several independent set. Feel free to check out Independent Component Analysis\n",
    "\n",
    "https://github.com/je-suis-tm/machine-learning/blob/master/independent%20component%20analysis.ipynb\n",
    "\n",
    "PCA by SVD has a variation called Latent Semantic Indexing. It can be used on topic modelling in Natural Language Processing. Check the link below for more details\n",
    "\n",
    "https://github.com/je-suis-tm/machine-learning/blob/master/latent%20semantic%20indexing.ipynb"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sklearn.decomposition\n",
    "import sklearn.datasets\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import scipy.linalg"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Eigenvectors of Covariance Matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "#compute eigenvector of covariance matrix\n",
    "def self_pca_sigma(data,num_of_dims):\n",
    "\n",
    "    #numbers of eigenvectors do not exceed the number of features\n",
    "    assert num_of_dims<=data.shape[1],\"the transformed dimension should be smaller than the original\"\n",
    "\n",
    "    #standardize the data by minus mean and divided by std\n",
    "    #equivalent to z score in statistics\n",
    "    #equivalent to scipy.stats.zscore or sklearn.preprocessing.scale\n",
    "    z=np.divide(data-data.mean(axis=0),data.std(axis=0))\n",
    "\n",
    "    #get covariance matrix\n",
    "    #due to rounding errors\n",
    "    #it doesnt equal to np.cov(z.T)\n",
    "    sigma=z.T@z/z.shape[0]\n",
    "\n",
    "    #get eigenvectors\n",
    "    #please note eigenvectors are in the columns rather than rows\n",
    "    eigvals,eigvecs=np.linalg.eig(sigma)\n",
    "    \n",
    "    return eigvecs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Singular Value Decomposition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "#compute singular values\n",
    "def self_pca_svd(data,num_of_dims):\n",
    "\n",
    "    #numbers of eigenvectors do not exceed the number of features\n",
    "    assert num_of_dims<=data.shape[1],\"the transformed dimension should be smaller than the original\"\n",
    "\n",
    "    #standardize the data by minus mean and divided by std\n",
    "    #equivalent to z score in statistics\n",
    "    #equivalent to scipy.stats.zscore or sklearn.preprocessing.scale\n",
    "    z=np.divide(data-data.mean(axis=0),data.std(axis=0))\n",
    "\n",
    "    #the computation of scipy.linalg.svd is supposed to be faster\n",
    "    #scipy is a wrapper of LAPACK\n",
    "    #it reduces z score matrix into bidiagonal form then runs svd\n",
    "    #though np.linalg.eig is a wrapper of LAPACK as well\n",
    "    #it computes eigenvector via divide and conquer algorithm\n",
    "    U,D,Vt=scipy.linalg.svd(z)\n",
    "    \n",
    "    return U,D,Vt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "#our fav iris data\n",
    "iris=sklearn.datasets.load_iris()\n",
    "\n",
    "#initialize\n",
    "num_of_dims=2\n",
    "\n",
    "#get z score to maintain consistency with self implementations\n",
    "#although this is unnecessary for iris dataset\n",
    "#as i have never done this in other notebooks\n",
    "#because all features are in cm\n",
    "#unit variance doesnt really hurt, does it\n",
    "z=sklearn.preprocessing.StandardScaler().fit_transform(iris.data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "147 µs ± 1.32 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit self_pca_sigma(iris.data,num_of_dims)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "257 µs ± 7.32 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n"
     ]
    }
   ],
   "source": [
    "#although svd is proclaimed to be faster\n",
    "#it doesnt seem to be the case here\n",
    "#maybe for a really large matrix\n",
    "%timeit self_pca_svd(iris.data,num_of_dims)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "#self implementation\n",
    "#technically, all features are in cm\n",
    "#unit variance isnt needed\n",
    "eigvecs_sigma=self_pca_sigma(iris.data,num_of_dims)\n",
    "\n",
    "#select top k eigenvectors and transform\n",
    "self_newdims_sigma=z@eigvecs_sigma[:,:num_of_dims]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "#self implementation\n",
    "U,D,Vt=self_pca_svd(iris.data,num_of_dims)\n",
    "\n",
    "#select singular values\n",
    "self_newdims_svd=z@Vt[:num_of_dims].T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "#sklearn\n",
    "clf=sklearn.decomposition.PCA(num_of_dims)\n",
    "\n",
    "#transformation\n",
    "skl_newdims=clf.fit_transform(z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.52106591 -0.26934744  0.5804131   0.56485654]\n",
      " [-0.37741762 -0.92329566 -0.02449161 -0.06694199]]\n",
      "[[ 0.52106591 -0.26934744  0.5804131   0.56485654]\n",
      " [-0.37741762 -0.92329566 -0.02449161 -0.06694199]]\n",
      "[[ 0.52106591 -0.26934744  0.5804131   0.56485654]\n",
      " [ 0.37741762  0.92329566  0.02449161  0.06694199]]\n"
     ]
    }
   ],
   "source": [
    "#as u can see, the second eigenvector has the opposite sign\n",
    "#if u recalled what andrew ng said in the lectures\n",
    "#the basis vector can rotate freely in its own subspace\n",
    "#if u check out the code in sklearn\n",
    "#one comment is \"flip eigenvectors' sign to enforce deterministic output\"\n",
    "print(eigvecs_sigma.T[:num_of_dims])\n",
    "print(Vt[:num_of_dims])\n",
    "print(clf.components_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#given the opposite sign of the second eigenvector\n",
    "#the viz looks like vertically flipped compared to sklearn\n",
    "for i in np.unique(iris.target):\n",
    "    plt.scatter(self_newdims_sigma[:,0][iris.target==i],\n",
    "                self_newdims_sigma[:,1][iris.target==i],\n",
    "                label=f'Label {int(i)}')\n",
    "    \n",
    "plt.title('PCA by Covariance Matrix')\n",
    "plt.xlabel('Dimension 1')\n",
    "plt.ylabel('Dimension 2')\n",
    "plt.legend(loc=0)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#given the opposite sign of the secondeigen vector\n",
    "#the viz looks like vertically flipped compared to sklearn\n",
    "for i in np.unique(iris.target):\n",
    "    plt.scatter(self_newdims_svd[:,0][iris.target==i],\n",
    "                self_newdims_svd[:,1][iris.target==i],\n",
    "                label=f'Label {int(i)}')\n",
    "    \n",
    "plt.title('PCA by SVD')\n",
    "plt.xlabel('Dimension 1')\n",
    "plt.ylabel('Dimension 2')\n",
    "plt.legend(loc=0)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "for i in np.unique(iris.target):\n",
    "    plt.scatter(skl_newdims[:,0][iris.target==i],\n",
    "                skl_newdims[:,1][iris.target==i],\n",
    "                label=f'Label {int(i)}')\n",
    "    \n",
    "plt.title('PCA by sklearn')\n",
    "plt.xlabel('Dimension 1')\n",
    "plt.ylabel('Dimension 2')\n",
    "plt.legend(loc=0)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Optimal Number of Components\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "The optimal number of components is as facile as PCA itself. In our case, we already know the number because we merely want to make 2D visualization. For a more general case of dimension reduction, the optimal is determined by the explained variance (R squared). We set up a threshold (usually 95% in statistics) and find the minimum number of components to explain at least 95% of the variance.\n",
    "\n",
    "You can find the error computation on page 9\n",
    "\n",
    "https://courses.cs.washington.edu/courses/cse446/19au/10pca_annotated.pdf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "#you can think of explained variance as r squared in regression\n",
    "#the computation is essentially sum of squared regression\n",
    "#equivalent to 1-sum of squared errors/sum of squared total\n",
    "def explained_variance_general(data,newdims,eigvecs):\n",
    "\n",
    "    miu=data.mean(axis=0)\n",
    "    result=[]\n",
    "    for i in range(newdims.shape[1]):\n",
    "        \n",
    "        #construct estimation by projection and components\n",
    "        #since the components are orthogonal\n",
    "        #the inverse of an orthogonal matrix is its transpose\n",
    "        #the projection can be reverted by multiplying a transposed component\n",
    "        #and add miu since it’s the standard preprocessing\n",
    "        #if it makes you feel better\n",
    "        #u can multiply the covariance matrix as well\n",
    "        inversed=newdims[:,i].reshape(-1,1)@eigvecs[i].reshape(1,-1)+miu\n",
    "        \n",
    "        #sum of squared errors\n",
    "        numerator=np.diag((inversed-data).T@(inversed-data)).sum()**0.5\n",
    "        \n",
    "        #sum of squared total\n",
    "        denominator=np.diag((data-miu).T@(data-miu)).sum()**0.5\n",
    "        \n",
    "        #r squared\n",
    "        result.append(1-(numerator/denominator)**2)\n",
    "        \n",
    "    return result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "#svd is efficient but also convenient for explained variance computation\n",
    "#explained standard deviation is merely on the diagonal of D \n",
    "#take the squared singular values and we obtain explained variance\n",
    "def explained_variance_svd(data,D,num_of_dims):    \n",
    "    return (D**2)[:num_of_dims]/(D**2).sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.7296244541329985, 0.22850761786701734]\n",
      "[0.72962445 0.22850762]\n",
      "[0.72962445 0.22850762]\n"
     ]
    }
   ],
   "source": [
    "#explained variance of three methods are consistent\n",
    "print(explained_variance_general(z,self_newdims_sigma,\n",
    "                                 eigvecs_sigma.T[:num_of_dims]))\n",
    "\n",
    "print(explained_variance_svd(z,D,num_of_dims))\n",
    "\n",
    "print(clf.explained_variance_ratio_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "#to demonstrate the concept of the optimal number\n",
    "#we take svd as an example\n",
    "#simply becuz its r squared computation is simple\n",
    "rsquared=[]\n",
    "threshold=0.95\n",
    "\n",
    "#the number of components cannot exceed the number of features\n",
    "for num_of_dims in range(1,iris.data.shape[1]+1):\n",
    "    U,D,Vt=self_pca_svd(iris.data,num_of_dims)\n",
    "    self_newdims_svd=z@Vt[:num_of_dims].T\n",
    "    rsquared.append(sum(explained_variance_svd(z,D,num_of_dims)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "#find the optima\n",
    "optima=[i for i in rsquared if i>threshold][0]\n",
    "ind=rsquared.index(optima)+1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#viz\n",
    "plt.plot(range(1,iris.data.shape[1]+1),rsquared)\n",
    "plt.plot([ind],[optima],lw=0,marker='*',markersize=10,color='gold') \n",
    "plt.xticks(range(1,iris.data.shape[1]+1))\n",
    "plt.title('Optimal Number of Components')\n",
    "plt.xlabel('Number of Components')\n",
    "plt.ylabel('R Squared')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Low Rank Approximation\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "Since we have talked about SVD, it's harmless to mention another data compression technique called low rank approximation. This technique doesn't necessarily reduce the data dimension but it does reduce the variance of the dataset by maintaining the principal components. PCA is generally thought as low rank approximation in frobenius norm. \n",
    "\n",
    "The technique is simple. An SVD can be expressed in a linear combination of vectors, which is called outer product expansions. It is up to us to decide on how many eigenvectors to take into consideration.\n",
    "\n",
    "$A=UDV^T=\\sum_{i=1}^{n}\\lambda_iu_iv_i^T$\n",
    "\n",
    "You can check reddit for the link between PCA and Low Rank Approximation\n",
    "\n",
    "https://www.reddit.com/r/MachineLearning/comments/bygvut/d_whats_the_difference_between_low_rank\n",
    "\n",
    "If you are looking for more understanding, such as Eckart–Young Theorem, good luck on reading Wikipedia\n",
    "\n",
    "https://en.wikipedia.org/wiki/Low-rank_approximation#Proof_of_Eckart%E2%80%93Young%E2%80%93Mirsky_theorem_(for_Frobenius_norm)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#control group for comparison\n",
    "for i in np.unique(iris.target):\n",
    "    plt.scatter(z[:,0][iris.target==i],\n",
    "                z[:,1][iris.target==i],\n",
    "                label=f'Label {int(i)}')\n",
    "    \n",
    "plt.title('Raw')\n",
    "plt.xlabel(iris.feature_names[0])\n",
    "plt.ylabel(iris.feature_names[1])\n",
    "plt.legend(loc=0)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#2 eigenvectors can explain 95.8% variance\n",
    "#the matrix hasnt been compressed much\n",
    "rank2=np.sum([(U[:,i].reshape(-1,1))@(Vt[i].reshape(1,-1))*D[i] for i in range(num_of_dims)],axis=0)\n",
    "\n",
    "for i in np.unique(iris.target):\n",
    "    plt.scatter(rank2[:,0][iris.target==i],\n",
    "                rank2[:,1][iris.target==i],\n",
    "                label=f'Label {int(i)}')\n",
    "    \n",
    "plt.title('Rank 2 Approximation')\n",
    "plt.xlabel(iris.feature_names[0])\n",
    "plt.ylabel(iris.feature_names[1])\n",
    "plt.legend(loc=0)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#1 eigenvector can explain 72.7% variance\n",
    "#the matrix has been compressed a lot\n",
    "#now it looks like pca of 1 dimension, no?\n",
    "rank1=np.sum([(U[:,i].reshape(-1,1))@(Vt[i].reshape(1,-1))*D[i] for i in range(1)],axis=0)\n",
    "\n",
    "for i in np.unique(iris.target):\n",
    "    plt.scatter(rank1[:,0][iris.target==i],\n",
    "                rank1[:,1][iris.target==i],\n",
    "                label=f'Label {int(i)}')\n",
    "    \n",
    "plt.title('Rank 1 Approximation')\n",
    "plt.xlabel(iris.feature_names[0])\n",
    "plt.ylabel(iris.feature_names[1])\n",
    "plt.legend(loc=0)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.5"
  },
  "toc": {
   "base_numbering": 1,
   "nav_menu": {},
   "number_sections": true,
   "sideBar": true,
   "skip_h1_title": false,
   "title_cell": "Table of Contents",
   "title_sidebar": "Contents",
   "toc_cell": false,
   "toc_position": {},
   "toc_section_display": true,
   "toc_window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}