Notebooks/Numpy/00_Numpy_Preprocessing.ipynb

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   "source": [
    "# [Preprocessing for Deep Learning](https://hadrienj.github.io/posts/Preprocessing-for-deep-learning/)"
   ]
  },
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   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns"
   ]
  },
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   "execution_count": 2,
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    "plt.style.use('ggplot')\n",
    "plt.rcParams['axes.facecolor']='w'\n",
    "plt.rcParams['axes.edgecolor'] = '#D6D6D6'\n",
    "plt.rcParams['axes.linewidth'] = 2"
   ]
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   "source": [
    "$$\n",
    "\\newcommand\\bs[1]{\\boldsymbol{#1}}\n",
    "$$"
   ]
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  {
   "cell_type": "code",
   "execution_count": 3,
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    "%%html\n",
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    "# Preprocessing for deep learning: from covariance matrix to image whitening\n",
    "\n",
    "A version of this notebook has been published [here](https://hadrienj.github.io/posts/Preprocessing-for-deep-learning/).\n",
    "\n",
    "The goal of this post/notebook is to go from the basics of data preprocessing to modern techniques used in deep learning. My point is that we can use code (Python/Numpy etc.) to better understand abstract mathematical notions! Thinking by coding! 💥\n",
    "\n",
    "We will start with basic but very useful concepts in data science and machine learning/deep learning like variance and covariance matrix and we will go further to some preprocessing techniques used to feed images into neural networks. We will try to get more concrete insights using code to actually see what each equation is doing!\n",
    "\n",
    "We call preprocessing all transformations on the raw data before it is fed to the machine learning or deep learning algorithm. For instance, training a convolutional neural network on raw images will probably lead to bad classification performances ([Pal & Sudeep, 2016](https://ieeexplore.ieee.org/document/7808140/)). The preprocessing is also important to speed up training (for instance, centering and scaling techniques, see [Lecun et al., 2012; see 4.3](http://yann.lecun.com/exdb/publis/pdf/lecun-98b.pdf)).\n",
    "\n",
    "Here is the syllabus of this tutorial:\n",
    "\n",
    "1. **Background**: In the first part, we will get some reminders about variance and covariance and see how to generate and plot fake data to get a better understanding of these concepts.\n",
    "\n",
    "2. **Preprocessing**: In the second part, we will see the basics of some preprocessing techniques that can be applied to any kind of data: mean normalization, standardisation and whitening.\n",
    "\n",
    "3. **Whitening images**: In the third part, we will use the tools and concepts gained in 1. and 2. to do a special kind of whitening called Zero Component Analysis (ZCA). It can be used to preprocess images for deep learning. This part will be very practical and fun ☃️!\n",
    "\n",
    "Feel free to fork the notebook. For instance, check the shapes of the matrices each time you have a doubt :)"
   ]
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    "# 1. Background\n",
    "\n",
    "## A. Variance and covariance\n",
    "\n",
    "The variance of a variable describes how much the values are spread. The covariance is a measure that tells the amount of dependency between two variables. A positive covariance means that values of the first variable are large when values of the second variables are also large. A negative covariance means the opposite: large values from one variable are associated with small values of the other. The covariance value depends on the scale of the variable so it is hard to analyse it. It is possible to use the correlation coefficient that is easier to interpret. It is just the covariance normalized.\n",
    "\n",
    "<img src=\"./Images/negative-and-positive-covariance.png\" width=\"500\" alt=\"Intuition about the covariance between two variables\" title=\"Representation of the covariance between two variables.\">\n",
    "<em>A positive covariance means that large values of one variable are associated with big values from the other (left). A negative covariance means that large values of one variable are associated with small values of the other one (right).</em>\n",
    "\n",
    "The covariance matrix is a matrix that summarizes the variances and covariances of a set of vectors and it can tell a lot of things about your variables. The diagonal corresponds to the variance of each vector:\n",
    "\n",
    "<img src=\"./Images/covariance1.png\" width=\"400\" alt=\"Variance in the matrix of covariance\" title=\"Variance in the matrix of covariance is on the diagonal\">\n",
    "<em>A matrix $\\bs{A}$ and its matrix of covariance. The diagonal corresponds to the variance of each column vector.</em>\n",
    "\n",
    "Let's just check with the formula of the variance:\n",
    "\n",
    "$$\n",
    "V(\\bs{X}) = \\frac{1}{n}\\sum_{i=1}^{n}(x_i-\\bar{x})^2\n",
    "$$\n",
    "\n",
    "with $n$ the length of the vector, and $\\bar{x}$ the mean of the vector. For instance, the variance of the first column vector of $\\bs{A}$ is:\n",
    "\n",
    "$$\n",
    "V(\\bs{A}_{:,1}) = \\frac{(1-3)^2+(5-3)^2+(3-3)^2}{3} = 2.67\n",
    "$$"
   ]
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    "This is the first cell of our covariance matrix. The second element on the diagonal corresponds of the variance of the second column vector from $\\bs{A}$ and so on.\n",
    "\n",
    "*Note*: the vectors extracted from the matrix $\\bs{A}$ correspond to the columns of $\\bs{A}$.\n",
    "\n",
    "The other cells correspond to the covariance between two column vectors from $\\bs{A}$. For instance, the covariance between the first and the third column is located in the covariance matrix as the column 1 and the row 3 (or the column 3 and the row 1).\n",
    "\n",
    "<img src=\"./Images/covariance2.png\" width=\"400\" alt=\"Covariance in the matrix of covariance\" title=\"The position in the covariance matrix.\">\n",
    "<em>The position in the covariance matrix. Column corresponds to the first variable and row to the second (or the opposite). The covariance between the first and the third column vector of $\\bs{A}$ is the element in column 1 and row 3 (or the opposite = same value).</em>\n"
   ]
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    "Let's check that the covariance between the first and the third column vector of $\\bs{A}$ is equal to $-2.67$. The formula of the covariance between two variables $\\bs{X}$ and $\\bs{Y}$ is:\n",
    "\n",
    "$$\n",
    "cov(\\bs{X},\\bs{Y}) = \\frac{1}{n} \\sum_{i=1}^{n}(x_i-\\bar{x})(y_i-\\bar{y})\n",
    "$$\n",
    "\n",
    "The variables $\\bs{X}$ and $\\bs{Y}$ are the first and the third column vectors in the last example. Let's split this formula to be sure that it is crystal clear:\n",
    "\n",
    "1. $(x_1-\\bar{x})$. The sum symbol means that we will iterate on the elements of the vectors. We will start with the first element ($i=1$) and calculate the first element of $\\bs{X}$ minus the mean of the vector $\\bs{X}$.\n",
    "2. $(x_1-\\bar{x})(y_1-\\bar{y})$. Multiply the result with the first element of $\\bs{Y}$ minus the mean of the vector $\\bs{Y}$.\n",
    "3. $\\sum_{i=1}^{n}(x_i-\\bar{x})(y_i-\\bar{y})$. Reiterate the process for each element of the vectors and calculate the sum of all results.\n",
    "4. $\\frac{1}{n} \\sum_{i=1}^{n}(x_i-\\bar{x})(y_i-\\bar{y})$. Divide by the number of elements in the vector.\n",
    "\n",
    "\n",
    "#### Example 1.\n",
    "\n",
    "Let's start with the matrix $\\bs{A}$:\n",
    "\n",
    "$$\n",
    "\\boldsymbol{A}=\n",
    "\\begin{bmatrix}\n",
    "    1 & 3 & 5\\\\\\\\\n",
    "    5 & 4 & 1\\\\\\\\\n",
    "    3 & 8 & 6\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "We will calculate the covariance between the first and the third column vectors:\n",
    "\n",
    "$$\n",
    "\\boldsymbol{X} = \\begin{bmatrix}\n",
    "    1\\\\\\\\\n",
    "    5\\\\\\\\\n",
    "    3\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "and\n",
    "\n",
    "$$\\boldsymbol{Y} = \\begin{bmatrix}\n",
    "    5\\\\\\\\\n",
    "    1\\\\\\\\\n",
    "    6\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "$\\boldsymbol{\\bar{x}}=3$, $\\boldsymbol{\\bar{y}}=4$ and $n=3$ so we have:\n",
    "\n",
    "$$\n",
    "cov(X,Y) = \\frac{(1-3)(5-4)+(5-3)(1-4)+(3-3)(6-4)}{3}=\\frac{-8}{3}=-2.67\n",
    "$$\n",
    "\n",
    "Ok, great! That the value of the covariance matrix.\n",
    "\n",
    "Now the easy way! With Numpy, the covariance matrix can be calculated with the function `np.cov`. It is worth noting that if you want Numpy to use the columns as vectors, the parameter `rowvar=False` has to be used. Also, `bias=True` allows to divide by $n$ and not by $n-1$.\n",
    "\n",
    "Let's create the array first:"
   ]
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    {
     "data": {
      "text/plain": [
       "array([[1, 3, 5],\n",
       "       [5, 4, 1],\n",
       "       [3, 8, 6]])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
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   "source": [
    "A = np.array([[1, 3, 5], [5, 4, 1], [3, 8, 6]])\n",
    "A"
   ]
  },
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   "source": [
    "Now we will calculate the covariance with the Numpy function:"
   ]
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   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 2.66666667,  0.66666667, -2.66666667],\n",
       "       [ 0.66666667,  4.66666667,  2.33333333],\n",
       "       [-2.66666667,  2.33333333,  4.66666667]])"
      ]
     },
     "execution_count": 5,
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   ],
   "source": [
    "np.cov(A, rowvar=False, bias=True)"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "Looks good! "
   ]
  },
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   "source": [
    "### Finding the covariance matrix with the dot product\n",
    "\n",
    "There is another way to compute the covariance matrix of $\\bs{A}$. You can center $\n",
    "\\bs{A}$ around 0 (subtract the mean of the vector to each element of the vector to have a vector of mean equal to 0, *cf*. below), multiply it with its own transpose and divide by the number of observations. Let's start with an implementation and then we'll try to understand the link with the previous equation:"
   ]
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   "source": [
    "def calculateCovariance(X):\n",
    "    meanX = np.mean(X, axis = 0)\n",
    "    lenX = X.shape[0]\n",
    "    X = X - meanX\n",
    "    covariance = X.T.dot(X)/lenX\n",
    "    return covariance"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "Let's test it on our matrix $\\boldsymbol{A}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 2.66666667,  0.66666667, -2.66666667],\n",
       "       [ 0.66666667,  4.66666667,  2.33333333],\n",
       "       [-2.66666667,  2.33333333,  4.66666667]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "calculateCovariance(A)"
   ]
  },
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   "source": [
    "We end up with the same result as before!\n",
    "\n",
    "The explanation is simple. The dot product between two vectors can be expressed:\n",
    "\n",
    "$$\n",
    "\\bs{X^\\text{T}Y}= \\sum_{i=1}^{n}(x_i)(y_i)\n",
    "$$\n",
    "\n",
    "That's right, it is the sum of the products of each element of the vectors:\n",
    "\n",
    "<img src=\"./Images/dot-product.png\" width=\"400\" alt=\"The dot product corresponds to the sum of the products of each elements of the vectors\" title=\"The dot product.\">\n",
    "<em>The dot product corresponds to the sum of the products of each element of the vectors.</em>\n",
    "\n",
    "If $n$ is the number of elements in our vectors and that we divide by $n$:\n",
    "\n",
    "$$\n",
    "\\frac{1}{n}\\bs{X^\\text{T}Y}= \\frac{1}{n}\\sum_{i=1}^{n}(x_i)(y_i)\n",
    "$$\n",
    "\n",
    "You can note that this is not too far from the formula of the covariance we have seen above:\n",
    "\n",
    "$$\n",
    "cov(\\bs{X},\\bs{Y}) = \\frac{1}{n} \\sum_{i=1}^{n}(x_i-\\bar{x})(y_i-\\bar{y})\n",
    "$$\n",
    "\n",
    "The only difference is that in the covariance formula we subtract the mean of a vector to each of its elements. This is why we need to center the data before doing the dot product.\n",
    "\n",
    "Now if we have a matrix $\\bs{A}$, the dot product between $\\bs{A}$ and its transpose will give you a new matrix:\n",
    "\n",
    "<img src=\"./Images/covariance-dot-product.png\" width=\"400\" alt=\"Covariance matrix and dot product\" title=\"Covariance matrix and dot product.\">\n",
    "<em>If you start with a zero-centered matrix, the dot product between this matrix and its transpose will give you the variance of each vector and covariance between them, that is to say the covariance matrix.</em>\n",
    "\n",
    "This is the covariance matrix! 🌵"
   ]
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   "source": [
    "## B. Visualize data and covariance matrices\n",
    "\n",
    "In order to get more insights about the covariance matrix and how it can be useful, we will create a function used to visualize it along with 2D data. You will be able to see the link between the covariance matrix and the data.\n",
    "\n",
    "This function will calculate the covariance matrix as we have seen above. It will create two subplots: one for the covariance matrix and one for the data. The `heatmap` function from Seaborn is used to create gradients of color: small values will be colored in light green and large values in dark blue. The data is represented as a scatterplot. We choose one of our palette colors, but you may prefer other colors 🌈."
   ]
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   "cell_type": "code",
   "execution_count": 8,
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   "source": [
    "def plotDataAndCov(data):\n",
    "    ACov = np.cov(data, rowvar=False, bias=True)\n",
    "    print('Covariance matrix:\\n', ACov)\n",
    "\n",
    "    fig, ax = plt.subplots(nrows=1, ncols=2)\n",
    "    fig.set_size_inches(10, 10)\n",
    "\n",
    "    ax0 = plt.subplot(2, 2, 1)\n",
    "    \n",
    "    # Choosing the colors\n",
    "    cmap = sns.color_palette(\"GnBu\", 10)\n",
    "    sns.heatmap(ACov, cmap=cmap, vmin=0)\n",
    "\n",
    "    ax1 = plt.subplot(2, 2, 2)\n",
    "    \n",
    "    # data can include the colors\n",
    "    if data.shape[1]==3:\n",
    "        c=data[:,2]\n",
    "    else:\n",
    "        c=\"#0A98BE\"\n",
    "    ax1.scatter(data[:,0], data[:,1], c=c, s=40)\n",
    "    \n",
    "    # Remove the top and right axes from the data plot\n",
    "    ax1.spines['right'].set_visible(False)\n",
    "    ax1.spines['top'].set_visible(False)"
   ]
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   "source": [
    "## C. Simulating data\n",
    "\n",
    "### Uncorrelated data\n",
    "\n",
    "Now that we have the plot function, we will generate some random data to visualize what the covariance matrice can tell us. We will start with some data drawn from a normal distribution with the Numpy function `np.random.normal()`.\n",
    "\n",
    "<img src=\"./Images/np.random.normal.png\" width=\"400\" alt=\"Parameters of numpy random normal\" title=\"Parameters of Numpy random normal.\">\n",
    "<em>Drawing sample from a normal distribution with Numpy.</em>\n",
    "\n",
    "This function needs the mean, the standard deviation and the number of observations of the distribution as input. We will create two random variables of 300 observations with a standard deviation of 1. The first will have a mean of 1 and the second a mean of 2. If we draw two times 300 observations from a normal distribution, both vectors will be uncorrelated."
   ]
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  {
   "cell_type": "code",
   "execution_count": 9,
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "(300, 2)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.seed(1234)\n",
    "a1 = np.random.normal(2, 1, 300)\n",
    "a2 = np.random.normal(1, 1, 300)\n",
    "A = np.array([a1, a2]).T\n",
    "A.shape"
   ]
  },
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   "source": [
    "*Note 1*:  We transpose the data with `.T` because the original shape is (2, 300) and we want the number of observations as rows (so with shape (300, 2)).\n",
    "\n",
    "*Note 2*: We use `np.random.seed` function for reproducibility. The same random number will be used the next time we run the cell!\n",
    "\n",
    "Let's check how the data looks like:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 2.47143516,  1.52704645],\n",
       "       [ 0.80902431,  1.7111124 ],\n",
       "       [ 3.43270697,  0.78245452],\n",
       "       [ 1.6873481 ,  3.63779121],\n",
       "       [ 1.27941127, -0.74213763],\n",
       "       [ 2.88716294,  0.90556519],\n",
       "       [ 2.85958841,  2.43118375],\n",
       "       [ 1.3634765 ,  1.59275845],\n",
       "       [ 2.01569637,  1.1702969 ],\n",
       "       [-0.24268495, -0.75170595]])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A[:10,:]"
   ]
  },
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   "source": [
    "Nice, we have our two columns vectors.\n",
    "\n",
    "Now, we can check that the distributions are normal:"
   ]
  },
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   "cell_type": "code",
   "execution_count": 11,
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    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sns.distplot(A[:,0], color=\"#53BB04\")\n",
    "sns.distplot(A[:,1], color=\"#0A98BE\")\n",
    "plt.show()\n",
    "plt.close()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "Looks good! We can see that the distributions have equivalent standard deviations but different means (1 and 2). So that's exactly what we have asked for!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "Now we can plot our dataset and its covariance matrix with our function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false,
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Covariance matrix:\n",
      " [[ 0.95171641 -0.0447816 ]\n",
      " [-0.0447816   0.87959853]]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plotDataAndCov(A)\n",
    "plt.show()\n",
    "#plt.close()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "We can see on the scatterplot that the two dimensions are uncorrelated. Note that we have one dimension with a mean of 1 and the other with the mean of 2. Also, the covariance matrix shows that the variance of each variable is very large (around 1) and the covariance of columns 1 and 2 is very small (around 0). Since we insured that the two vectors are independent this is coherent (the opposite is not necessarily true: a covariance of 0 doesn't guaranty independency (see [here](https://stats.stackexchange.com/questions/12842/covariance-and-independence))."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "### Correlated data\n",
    "\n",
    "Now, let's construct dependent data by specifying one column from the other one."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Covariance matrix:\n",
      " [[0.95171641 0.92932561]\n",
      " [0.92932561 1.12683445]]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "np.random.seed(1234)\n",
    "b1 =  np.random.normal(3, 1, 300)\n",
    "b2 = b1 + np.random.normal(7, 1, 300)/2.\n",
    "B = np.array([b1, b2]).T\n",
    "plotDataAndCov(B)\n",
    "plt.show()\n",
    "plt.close()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "The correlation between the two dimensions is visible on the scatter plot. We can see that a line could be drawn and used to predict $\\boldsymbol{y}$ from $\\boldsymbol{x}$ and vice versa. The covariance matrix is not diagonal (there are non-zero cells outside of the diagonal). That means that the covariance between dimensions is non-zero.\n",
    "\n",
    "That's great! ⚡️ We now have all the tools to see different preprocessing techniques."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "# 2. Preprocessing\n",
    "\n",
    "## A. Mean normalization\n",
    "\n",
    "Mean normalization is just removing the mean from each observation.\n",
    "\n",
    "$$\n",
    "\\bs{X'} = \\bs{X} - \\bar{x}\n",
    "$$\n",
    "\n",
    "where $\\bs{X'}$ is the normalized dataset, $\\bs{X}$ the original dataset and $\\bar{x}$ the mean of $\\bs{X}$.\n",
    "\n",
    "It will have the effect of centering the data around 0. We will create the function `center()` to do that:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "code_folding": [],
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [],
   "source": [
    "def center(X):\n",
    "    newX = X - np.mean(X, axis = 0)\n",
    "    return newX"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "Let's give it a try with the matrix $\\bs{B}$ we have created above:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Before:\n",
      "\n",
      "\n",
      "Covariance matrix:\n",
      " [[0.95171641 0.92932561]\n",
      " [0.92932561 1.12683445]]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "After:\n",
      "\n",
      "\n",
      "Covariance matrix:\n",
      " [[0.95171641 0.92932561]\n",
      " [0.92932561 1.12683445]]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "BCentered = center(B)\n",
    "\n",
    "print('Before:\\n\\n')\n",
    "\n",
    "plotDataAndCov(B)\n",
    "plt.show()\n",
    "plt.close()\n",
    "\n",
    "print('After:\\n\\n')\n",
    "\n",
    "plotDataAndCov(BCentered)\n",
    "plt.show()\n",
    "plt.close()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "The first plot shows again the original data $\\bs{B}$ and the second plot shows the centered data (look at the scale)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "## B. Standardization\n",
    "\n",
    "The standardization is used to put all features on the same scale. The way to do it is to divide each zero-centered dimension by its standard deviation.\n",
    "\n",
    "$$\n",
    "\\bs{X'} = \\frac{\\bs{X} - \\bar{x}}{\\sigma_{\\bs{X}}}\n",
    "$$\n",
    "\n",
    "where $\\bs{X'}$ is the standardized dataset, $\\bs{X}$ the original dataset, $\\bar{x}$ the mean of $\\bs{X}$ and $\\sigma_{\\bs{X}}$ the standard deviation of $\\bs{X}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [],
   "source": [
    "def standardize(X):\n",
    "    newX = center(X)/np.std(X, axis = 0)\n",
    "    return newX"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "Let's create another dataset with a different scale to check that it is working."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Covariance matrix:\n",
      " [[0.95171641 0.83976242]\n",
      " [0.83976242 6.22529922]]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "np.random.seed(1234)\n",
    "c1 =  np.random.normal(3, 1, 300)\n",
    "c2 = c1 + np.random.normal(7, 5, 300)/2.\n",
    "C = np.array([c1, c2]).T\n",
    "\n",
    "plotDataAndCov(C)\n",
    "plt.xlim(0, 15)\n",
    "plt.ylim(0, 15)\n",
    "plt.show()\n",
    "plt.close()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "We can see that the scales of $x$ and $y$ are different. Note also that the correlation seems smaller because of the scale differences. Now let's standardise it:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Covariance matrix:\n",
      " [[1.         0.34500274]\n",
      " [0.34500274 1.        ]]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "CStandardized = standardize(C)\n",
    "\n",
    "plotDataAndCov(CStandardized)\n",
    "plt.show()\n",
    "plt.close()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "Looks good! You can see that the scales are the same and that the dataset is zero-centered according to both axes. Now, have a look at the covariance matrix: you can see that the variance of each coordinate (the top-left cell and the bottom-right cell) is equal to 1. By the way, this new covariance matrix is actually the correlation matrix!💥 The Pearson correlation coefficient between the two variables ($\\bs{c1}$ and $\\bs{c2}$) is 0.54220151."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "## C. Whitening\n",
    "\n",
    "Whitening or sphering data means that we want to transform it in a way to have a covariance matrix that is the identity matrix (1 in the diagonal and 0 for the other cells; [more details on the identity matrix](https://hadrienj.github.io/posts/Deep-Learning-Book-Series-2.3-Identity-and-Inverse-Matrices/)). It is called whitening in reference to white noise.\n",
    "\n",
    "Whitening is a bit more complicated but we now have all the tools that we need to do it. It involves the following steps:\n",
    "\n",
    "    1- Zero-center the data\n",
    "    2- Decorrelate the data\n",
    "    3- Rescale the data\n",
    "    \n",
    "Let's take again $\\bs{C}$ and try to do these steps.\n",
    "\n",
    "#### 1. Zero-centering"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Covariance matrix:\n",
      " [[0.95171641 0.83976242]\n",
      " [0.83976242 6.22529922]]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "CCentered = center(C)\n",
    "\n",
    "plotDataAndCov(CCentered)\n",
    "plt.show()\n",
    "plt.close()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "#### 2. Decorrelate\n",
    "\n",
    "At this point, we need to decorrelate our data. Intuitively, it means that we want to rotate the data until there is no correlation anymore. Look at the following cartoon to see what I mean:\n",
    "\n",
    "<img src=\"./Images/rotation.png\" width=\"500\" alt=\"Rotation to decorrelate the data\" title=\"The rotation can decorrelate the data.\">\n",
    "<em>The left plot shows correlated data. For instance, if you take a data point with a big $x$ value, chances are that $y$ will also be quite big. Now take all data points and do a rotation (maybe around 45 degrees counterclockwise): the new data (plotted on the right) is not correlated anymore.</em>\n",
    "\n",
    "The question is: how could we find the right rotation in order to get the uncorrelated data? Actually, it is exactly what the **eigenvectors** of the covariance matrix do: they indicate the direction where the spread of the data is at its maximum:\n",
    "\n",
    "<img src=\"./Images/maxVar.png\" width=\"300\" alt=\"Direction where the variance is maximum\" title=\"There is one direction where the variance is maximum.\">\n",
    "<em>The eigenvectors of the covariance matrix give you the direction that maximizes the variance. The direction of the green line is where the variance is maximum. Just look at the smallest and largest point projected on this line: the spread is big. Compare that with the projection on the orange line: the spread is very small.</em>\n",
    "\n",
    "For more details about the eigendecomposition, see [this post](https://hadrienj.github.io/posts/Deep-Learning-Book-Series-2.7-Eigendecomposition/).\n",
    "\n",
    "So we can decorrelate the data by projecting it on the eigenvectors basis. This will have the effect to apply the rotation needed and remove correlations between the dimensions. Here are the steps:\n",
    "\n",
    "    1- Calculate the covariance matrix\n",
    "    2- Calculate the eigenvectors of the covariance matrix\n",
    "    3- Apply the matrix of eigenvectors to the data (this will apply the rotation)\n",
    "    \n",
    "Let's pack that into a function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [],
   "source": [
    "def decorrelate(X):\n",
    "    newX = center(X)\n",
    "    cov = X.T.dot(X)/float(X.shape[0])\n",
    "    # Calculate the eigenvalues and eigenvectors of the covariance matrix\n",
    "    eigVals, eigVecs = np.linalg.eig(cov)\n",
    "    # Apply the eigenvectors to X\n",
    "    decorrelated = X.dot(eigVecs)\n",
    "    return decorrelated"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "Let's try to decorrelate our zero-centered matrix $\\bs{C}$ to see it in action:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Covariance matrix:\n",
      " [[0.95171641 0.83976242]\n",
      " [0.83976242 6.22529922]]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Covariance matrix:\n",
      " [[8.21222171e-01 8.28966525e-17]\n",
      " [8.28966525e-17 6.35579346e+00]]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plotDataAndCov(C)\n",
    "plt.show()\n",
    "plt.close()\n",
    "\n",
    "CDecorrelated = decorrelate(CCentered)\n",
    "plotDataAndCov(CDecorrelated)\n",
    "plt.xlim(-5,5)\n",
    "plt.ylim(-5,5)\n",
    "plt.show()\n",
    "plt.close()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "Nice! This is working 🎄\n",
    "\n",
    "We can see that the correlation is not here anymore and that the covariance matrix (now a diagonal matrix) confirms that the covariance between the two dimensions is equal to 0."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "#### 3. Rescale the data\n",
    "\n",
    "The next step is to scale the uncorrelated matrix in order to obtain a covariance matrix corresponding to the identity matrix (ones on the diagonal and zeros on the other cells). To do that we scale our decorrelated data by dividing each dimension by the square-root of its corresponding eigenvalue."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [],
   "source": [
    "def whiten(X):\n",
    "    newX = center(X)\n",
    "    cov = X.T.dot(X)/float(X.shape[0])\n",
    "    # Calculate the eigenvalues and eigenvectors of the covariance matrix\n",
    "    eigVals, eigVecs = np.linalg.eig(cov)\n",
    "    # Apply the eigenvectors to X\n",
    "    decorrelated = X.dot(eigVecs)\n",
    "    # Rescale the decorrelated data\n",
    "    whitened = decorrelated / np.sqrt(eigVals + 1e-5)\n",
    "    return whitened"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "*Note:* we add a small value (here $10^{-5}$) to avoid the division by $0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Covariance matrix:\n",
      " [[9.99987823e-01 2.51650552e-17]\n",
      " [2.51650552e-17 9.99998427e-01]]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "CWhitened = whiten(CCentered)\n",
    "\n",
    "plotDataAndCov(CWhitened)\n",
    "plt.xlim(-5,5)\n",
    "plt.ylim(-5,5)\n",
    "plt.show()\n",
    "plt.close()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "Hooray! We can see that with the covariance matrix that this is all good. We have something that really looks to the identity matrix ($1$ on the diagonal and $0$ elsewhere). 🌵"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "# 3. Image whitening\n",
    "\n",
    "We will see how whitening can be applied to preprocess image dataset. To do so we will use the paper of [Pal & Sudeep (2016)](https://ieeexplore.ieee.org/document/7808140/) where they give some details about the process. This preprocessing technique is called Zero component analysis (ZCA).\n",
    "\n",
    "Check out the paper, but here is the kind of result they got:\n",
    "\n",
    "<img src=\"./Images/whitening-images-cifar10-pal-sudeep.png\" width=\"800\" alt=\"Whitening images from the CIFAR10 dataset. Results from the paper of Pal & Sudeep (2016).\">\n",
    "<em>Whitening images from the CIFAR10 dataset. Results from the paper of Pal & Sudeep (2016). The original images (left) and the images after the ZCA (right) are shown.</em>\n",
    "\n",
    "First thing first: we will load images from the CIFAR dataset. This dataset is available from Keras but you can also download it [here](https://www.cs.toronto.edu/~kriz/cifar.html)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Using TensorFlow backend.\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(50000, 32, 32, 3)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from keras.datasets import cifar10\n",
    "\n",
    "(X_train, y_train), (X_test, y_test) = cifar10.load_data()\n",
    "\n",
    "X_train.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "The training set of the CIFAR10 dataset contains 50000 images. The shape of `X_train` is (50000, 32, 32, 3). Each image is 32px by 32px and each pixel contains 3 dimensions (R, G, B). Each value is the brightness of the corresponding color between 0 and 255.\n",
    "\n",
    "We will start by selecting only a subset of the images, let's say 1000:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(1000, 32, 32, 3)\n"
     ]
    }
   ],
   "source": [
    "X = X_train[:1000]\n",
    "print(X.shape)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "That's better! Now we will reshape the array to have flat image data with one image per row. Each image will be (1, 3072) because $32 \\times 32 \\times 3 = 3072$. Thus, the array containing all images will be (1000, 3072):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false,
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(1000, 3072)\n"
     ]
    }
   ],
   "source": [
    "X = X.reshape(X.shape[0], X.shape[1]*X.shape[2]*X.shape[3])\n",
    "print(X.shape)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "The next step is to be able to see the images. The function `imshow()` from Matplotlib ([doc](https://matplotlib.org/api/_as_gen/matplotlib.pyplot.imshow.html)) can be used to show images. It needs images with the shape ($M \\times N \\times 3$) so let's create a function to reshape the images and be able to visualize them from the shape (1, 3072)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [],
   "source": [
    "def plotImage(X):\n",
    "    plt.figure(figsize=(1.5, 1.5))\n",
    "    plt.imshow(X.reshape(32,32,3))\n",
    "    plt.grid(None)\n",
    "    plt.show()\n",
    "    plt.close()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "For instance, let's plot one of the images we have loaded:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 108x108 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plotImage(X[12, :])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "Cute! 🌵"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "We can now implement the whitening of the images. [Pal & Sudeep (2016)](https://ieeexplore.ieee.org/document/7808140/) describe the process:\n",
    "\n",
    "1. The first step is to rescale the images to obtain the range [0, 1] by dividing by 255 (the maximum value of the pixels).\n",
    "\n",
    "Remind that the formula to obtain the range [0, 1] is:\n",
    "\n",
    "$$\\frac{data - min(data)}{max(data) - min(data)}$$\n",
    "\n",
    "but here, the minimum value is 0, so this leads to:\n",
    "\n",
    "$$\\frac{data}{max(data)} = \\frac{data}{255}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "X.min() 0.0\n",
      "X.max() 1.0\n"
     ]
    }
   ],
   "source": [
    "X_norm = X / 255.\n",
    "print('X.min()', X_norm.min())\n",
    "print('X.max()', X_norm.max())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "### Mean subtraction: per-pixel or per-image?\n",
    "\n",
    "Ok cool, the range of our pixel values is between 0 and 1 now. The next step is:\n",
    "\n",
    "2. Subtract the mean from all image.\n",
    "\n",
    "Be careful here 🌪🎢:\n",
    "\n",
    "One way to do it is to take each image and remove the mean of this image from every pixel ([Jarrett et al., 2009](https://www.computer.org/csdl/proceedings/iccv/2009/4420/00/05459469.pdf)). The intuition behind this process is that it centers the pixels of each image around 0.\n",
    "\n",
    "Another way to do it is to take each of the 3072 pixels that we have (32 by 32 pixels for R, G and B) for every image and subtract the mean of that pixel across all images. This is called per-pixel mean subtraction. This time, each pixel will be centered around 0 *according to all images*. When you will feed your network with the images, each pixel is considered as a different feature. With the per-pixel mean subtraction, we have centered each feature (pixel) around 0. This technique is commonly used (e.g [Wan et al., 2013](http://proceedings.mlr.press/v28/wan13.html)).\n",
    "\n",
    "We will now do the per-pixel mean subtraction from our 1000 images. Our data are organized with these dimensions (images, pixels). It was (1000, 3072) because there are 1000 images with $32 \\times 32 \\times 3 = 3072$ pixels. The mean per-pixel can thus be obtained from the first axis:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3072,)"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X_norm.mean(axis=0).shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "This gives us 3072 values which is the number of means: one per pixel. Let's see the kind of values we have:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.5234    , 0.54323137, 0.5274    , ..., 0.50369804, 0.50011765,\n",
       "       0.45227451])"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X_norm.mean(axis=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "This is near 0.5 because we already have normalized to the range [0, 1]. However, we still need to remove the mean from each pixel:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [],
   "source": [
    "X_norm = X_norm - X_norm.mean(axis=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "Just to convince ourselves that it worked, we will compute the mean of the first pixel. Let's hope that it is 0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-5.30575583e-16, -5.98021632e-16, -4.23439062e-16, ...,\n",
       "       -1.81965554e-16, -2.49800181e-16,  3.98570066e-17])"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X_norm.mean(axis=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "This is not exactly 0 but it is small enough that we can consider that it worked! 🌵"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "Now we want to calculate the covariance matrix of the zero-centered data. Like we have seen above, we can calculate it with the `np.cov()` function from Numpy.\n",
    "\n",
    "There are two possible correlation matrices that we can calculate from the matrix $\\bs{X}$: either the correlation between rows or between columns. In our case, each row of the matrix $\\bs{X}$ is an image, so the rows of the matrix correspond to the observations and the columns of the matrix corresponds to the features (the images pixels). We want to calculate the correlation between the pixels because the goal of the whitening is to remove these correlations to force the algorithm to focus on higher-order relations.\n",
    "\n",
    "To do so, we will tell this to Numpy with the parameter `rowvar=False` (see the [doc](https://docs.scipy.org/doc/numpy/reference/generated/numpy.cov.html)): it will use the columns as variables (or features) and the rows as observations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [],
   "source": [
    "cov = np.cov(X_norm, rowvar=False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The covariance matrix should have a shape of 3072 by 3072 to represent the correlation between each pair of pixels (and there are 3072 pixels):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3072, 3072)"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cov.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "Now the magic part: we will calculate the singular values and vectors of the covariance matrix and use them to rotate our dataset. Have a look at [my post](https://hadrienj.github.io/posts/Deep-Learning-Book-Series-2.8-Singular-Value-Decomposition/) on the singular value decomposition if you need more details!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [],
   "source": [
    "U,S,V = np.linalg.svd(cov)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "In the paper, they used the following equation:\n",
    "\n",
    "$$\n",
    "\\bs{X}_{ZCA} = \\bs{U}.diag(\\frac{1}{\\sqrt{diag(\\bs{S}) + \\epsilon}}).\\bs{U^\\text{T}.X}\n",
    "$$\n",
    "\n",
    "with $\\bs{U}$ the left singular vectors, and $\\bs{S}$ the singular values of the covariance of the initial normalized dataset of images and $\\bs{X}$ the normalized dataset. $\\epsilon$ (*epsilon*) is an hyper-parameter called the whitening coefficient. $diag(a)$ corresponds to a matrix with the vector $a$ as a diagonal and 0 in all other cells.\n",
    "\n",
    "We will try to implement this equation. Let's start by checking the dimensions of the SVD:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(3072, 3072) (3072,)\n"
     ]
    }
   ],
   "source": [
    "print(U.shape, S.shape)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "$\\bs{S}$ is a vector containing 3072 elements (the singular values). $diag(\\bs{S})$ will thus be of shape (3072, 3072) with $\\bs{S}$ as the diagonal:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[5.46967832e+01 0.00000000e+00 0.00000000e+00 ... 0.00000000e+00\n",
      "  0.00000000e+00 0.00000000e+00]\n",
      " [0.00000000e+00 2.02861434e+01 0.00000000e+00 ... 0.00000000e+00\n",
      "  0.00000000e+00 0.00000000e+00]\n",
      " [0.00000000e+00 0.00000000e+00 1.24476904e+01 ... 0.00000000e+00\n",
      "  0.00000000e+00 0.00000000e+00]\n",
      " ...\n",
      " [0.00000000e+00 0.00000000e+00 0.00000000e+00 ... 5.20778268e-15\n",
      "  0.00000000e+00 0.00000000e+00]\n",
      " [0.00000000e+00 0.00000000e+00 0.00000000e+00 ... 0.00000000e+00\n",
      "  3.21171593e-15 0.00000000e+00]\n",
      " [0.00000000e+00 0.00000000e+00 0.00000000e+00 ... 0.00000000e+00\n",
      "  0.00000000e+00 1.53626844e-15]]\n",
      "\n",
      "shape: (3072, 3072)\n"
     ]
    }
   ],
   "source": [
    "print(np.diag(S))\n",
    "print('\\nshape:', np.diag(S).shape)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "$diag(\\frac{1}{\\sqrt{diag(\\bs{S}) + \\epsilon}})$ is also of shape (3072, 3072) as well as $\\bs{U}$ and $\\bs{U^{\\text{T}}}$. We have seen also that $\\bs{X}$ has the shape (1000, 3072) and we need to transpose it to have (3072, 1000). The shape of $\\bs{X}_{ZCA}$ is thus:\n",
    "\n",
    "$$\n",
    "(3072, 3072) . (3072, 3072) . (1000, 3072)^{\\text{T}} = (3072, 3072) . (3072, 1000) = (3072, 1000)\n",
    "$$\n",
    "\n",
    "which corresponds to the shape of the initial dataset after transposition. Nice!\n",
    "\n",
    "We have:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [],
   "source": [
    "epsilon = 0.1\n",
    "X_ZCA = U.dot(np.diag(1.0/np.sqrt(S + epsilon))).dot(U.T).dot(X_norm.T).T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's rescale the images:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "min: 0.0\n",
      "max: 1.0\n"
     ]
    }
   ],
   "source": [
    "X_ZCA_rescaled = (X_ZCA - X_ZCA.min()) / (X_ZCA.max() - X_ZCA.min())\n",
    "print('min:', X_ZCA_rescaled.min())\n",
    "print('max:', X_ZCA_rescaled.max())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we can look at the effect of whitening by comparing an image before and after whitening:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 108x108 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 108x108 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plotImage(X[12, :])\n",
    "plotImage(X_ZCA_rescaled[12, :])"
   ]
  },
  {
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    "hideCode": false,
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   "source": [
    "Hooray! That's great!⚡️It looks like the images from the paper [Pal & Sudeep (2016)](https://ieeexplore.ieee.org/document/7808140/). They used `epsilon = 0.1`. You can try other values to see the effect on the image."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
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   },
   "source": [
    "That's all! 🌴\n",
    "\n",
    "I hope that you found something interesting in this notebook! You can find it in a nicer layout on my [blog](https://hadrienj.github.io/posts/Preprocessing-data-for-machine-learning-and-deep-learning/), along with other articles!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
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   },
   "source": [
    "# References\n",
    "\n",
    "[K. Jarrett, K. Kavukcuoglu, M. Ranzato, and Y. LeCun, “What is the best multi-stage architecture for object recognition?,” in 2009 IEEE 12th International Conference on Computer Vision, 2009, pp. 2146–2153.\n",
    "](https://www.computer.org/csdl/proceedings/iccv/2009/4420/00/05459469.pdf)\n",
    "\n",
    "[A. Krizhevsky, “Learning Multiple Layers of Features from Tiny Images,” Master’s thesis, University of Tront, 2009.\n",
    "](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.222.9220&rep=rep1&type=pdf)\n",
    "\n",
    "[Y. A. LeCun, L. Bottou, G. B. Orr, and K.-R. Müller, “Efficient BackProp,” in Neural Networks: Tricks of the Trade, Springer, Berlin, Heidelberg, 2012, pp. 9–48.\n",
    "](http://yann.lecun.com/exdb/publis/pdf/lecun-98b.pdf)\n",
    "\n",
    "[K. K. Pal and K. S. Sudeep, “Preprocessing for image classification by convolutional neural networks,” in 2016 IEEE International Conference on Recent Trends in Electronics, Information Communication Technology (RTEICT), 2016, pp. 1778–1781.\n",
    "](https://ieeexplore.ieee.org/document/7808140/)\n",
    "\n",
    "[L. Wan, M. Zeiler, S. Zhang, Y. L. Cun, and R. Fergus, “Regularization of Neural Networks using DropConnect,” in International Conference on Machine Learning, 2013, pp. 1058–1066.\n",
    "](http://proceedings.mlr.press/v28/wan13.html)\n",
    "\n",
    "And also these great resources and QA:\n",
    "\n",
    "[Wikipedia - Whitening transformation](https://en.wikipedia.org/wiki/Whitening_transformation)\n",
    "\n",
    "[CS231 - Convolutional Neural Networks for Visual Recognition](http://cs231n.github.io/neural-networks-2/)\n",
    "\n",
    "[Dustin Stansbury - The Clever Machine](https://theclevermachine.wordpress.com/2013/03/30/the-statistical-whitening-transform/)\n",
    "\n",
    "[Some details about the covariance matrix](http://www.visiondummy.com/2014/04/geometric-interpretation-covariance-matrix/)\n",
    "\n",
    "[SO - Image whitening in Python](https://stackoverflow.com/questions/41635737/is-this-the-correct-way-of-whitening-an-image-in-python)\n",
    "\n",
    "[Mean normalization per image or from the entire dataset](http://ufldl.stanford.edu/wiki/index.php/Data_Preprocessing)\n",
    "\n",
    "[Mean subtraction - all images or per image?](https://stackoverflow.com/questions/29743523/subtract-mean-from-image)\n",
    "\n",
    "[Why centering is important - See section 4.3](http://yann.lecun.com/exdb/publis/pdf/lecun-98b.pdf)\n",
    "\n",
    "[Kaggle kernel on ZCA](https://www.kaggle.com/nicw102168/exploring-zca-and-color-image-whitening/notebook)\n",
    "\n",
    "[How ZCA is implemented in Keras](https://github.com/keras-team/keras-preprocessing/blob/b9d142456a64ef228475f07cb2f2d38fd05bd249/keras_preprocessing/image.py#L1254:L1257)"
   ]
  }
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