DEEP LEARNING/Deep_Learning_PA2.ipynb

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    "colab": {
      "name": "Deep_Learning_PA2.ipynb",
      "provenance": [],
      "collapsed_sections": []
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
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    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "bUcXsrltYSeT"
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      "source": [
        "# **Deep Learning - Probabilistic Analysis L2**\n",
        "Date : 20.07.2021\n",
        "\n",
        "Faculty : SKM\n",
        "\n",
        "********************************************************************************\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "hYGYHgmZYNzX"
      },
      "source": [
        "**Fundamental statistics**  \n",
        "\n",
        "1. Expected Value and Mean\n",
        "2. Variance and Standard Deviation\n",
        "3. Covariance and Correlation\n",
        "4. Covariance Matrix  "
      ]
    },
    {
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      "source": [
        "1. **Expected Value and Mean**\n",
        "In probability, the average value of some random variable X is called the expected value or the expectation.  \n",
        "\n",
        "  E[X] = probability weighted sum of values\n",
        "\n"
      ]
    },
    {
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      "source": [
        "  ![image.png](data:image/png;base64,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)  \n"
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      },
      "source": [
        "![image.png](data:image/png;base64,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)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "EOYBdEAEUK6y"
      },
      "source": [
        "(probability of each event is\n",
        "just as likely.)\n",
        "\n",
        "mean,\n",
        "average, and expected value are used interchangeably"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "-99FNpVKYtD6"
      },
      "source": [
        "![image.png](data:image/png;base64,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)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "r3bA_5w0Yuhh"
      },
      "source": [
        "Where x is the vector of observations and P(x) is the calculated probability for each value."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "xqd-fbK6YIR2",
        "outputId": "461716e3-6baa-46c7-b638-95586b5bc12d"
      },
      "source": [
        "# vector mean\n",
        "from numpy import array\n",
        "from numpy import mean\n",
        "# define vector\n",
        "v = array([1,2,3,4,5,6])\n",
        "print(v)\n",
        "# calculate mean\n",
        "result = mean(v)\n",
        "print(result)"
      ],
      "execution_count": 1,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "[1 2 3 4 5 6]\n",
            "3.5\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "aF5ttSYVZL9D",
        "outputId": "d8b274cf-10b2-4f14-a376-e06b36218efc"
      },
      "source": [
        "# matrix means\n",
        "from numpy import array\n",
        "from numpy import mean\n",
        "# define matrix\n",
        "M = array([\n",
        "[1,2,3,4,5,6],\n",
        "[1,2,3,4,5,6]])\n",
        "print(M)\n",
        "# column means\n",
        "col_mean = mean(M, axis=0)\n",
        "print(col_mean)\n",
        "# row means\n",
        "row_mean = mean(M)#, axis=1)\n",
        "print(row_mean)"
      ],
      "execution_count": 2,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "[[1 2 3 4 5 6]\n",
            " [1 2 3 4 5 6]]\n",
            "[1. 2. 3. 4. 5. 6.]\n",
            "3.5\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "mBEG9svXbarx"
      },
      "source": [
        "**Variance and Standard Deviation**  \n",
        "variance of some random variable X is a measure of how much values in the\n",
        "distribution vary on average with respect to the mean.  \n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "S8KJ7Poacbl5"
      },
      "source": [
        " \n",
        "![image.png](data:image/png;base64,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)  \n",
        "\n",
        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)  \n",
        "\n",
        "![image.png](data:image/png;base64,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)\n",
        "\n",
        "![image.png](data:image/png;base64,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)  \n",
        "\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "HN4Z66qeiB_X"
      },
      "source": [
        "(the sample variance is denoted by the lower case sigma with a 2\n",
        "superscript indicating the units are squared, not that you must square the final value)"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "uhAqtqiMZT7I",
        "outputId": "cbd5b7f8-a03b-4726-cf0e-6c97e575632a"
      },
      "source": [
        "# vector variance\n",
        "from numpy import array\n",
        "from numpy import var\n",
        "# define vector\n",
        "v = array([1,2,3,4,5,6])\n",
        "print(v)\n",
        "# calculate variance\n",
        "result = var(v, ddof=1)\n",
        "print(result)"
      ],
      "execution_count": 3,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "[1 2 3 4 5 6]\n",
            "3.5\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "TzUZkrlzZYqZ",
        "outputId": "d56b1f18-1f49-4e92-9613-03104f3fd376"
      },
      "source": [
        "# matrix variances\n",
        "from numpy import array\n",
        "from numpy import var\n",
        "# define matrix\n",
        "M = array([\n",
        "[1,2,3,4,5,6],\n",
        "[1,2,3,4,5,6]])\n",
        "print(M)\n",
        "# column variances\n",
        "col_var = var(M, ddof=1, axis=0)\n",
        "print(col_var)\n",
        "# row variances\n",
        "row_var = var(M, ddof=1, axis=1)\n",
        "print(row_var)"
      ],
      "execution_count": 4,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "[[1 2 3 4 5 6]\n",
            " [1 2 3 4 5 6]]\n",
            "[0. 0. 0. 0. 0. 0.]\n",
            "[3.5 3.5]\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "xsCaZPmgZbK6",
        "outputId": "ccc3e0ca-0649-4778-fe71-d52f24cf8eb5"
      },
      "source": [
        "# matrix standard deviation\n",
        "from numpy import array\n",
        "from numpy import std\n",
        "# define matrix\n",
        "M = array([\n",
        "[1,2,3,4,5,6],\n",
        "[1,2,3,4,5,6]])\n",
        "print(M)\n",
        "# column standard deviations\n",
        "col_std = std(M, ddof=1, axis=0)\n",
        "print(col_std)\n",
        "# row standard deviations\n",
        "row_std = std(M, ddof=1, axis=1)\n",
        "print(row_std)"
      ],
      "execution_count": 5,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "[[1 2 3 4 5 6]\n",
            " [1 2 3 4 5 6]]\n",
            "[0. 0. 0. 0. 0. 0.]\n",
            "[1.87082869 1.87082869]\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Qqz-wHldnJWu"
      },
      "source": [
        "**Covariance & Correlation**  \n",
        "\n",
        "measure of the joint probability for two random variables. It\n",
        "describes how the two variables change together. It is denoted as the function cov(X; Y ), where\n",
        "X and Y are the two random variables being considered.  \n",
        "\n",
        "The sign of the covariance can be interpreted as whether the two variables increase together\n",
        "(positive) or decrease together (negative).  \n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "nzzYiVz2ZeOn",
        "outputId": "f244d5b9-32f6-4661-e2cf-254276cde7d8"
      },
      "source": [
        "# vector covariance\n",
        "from numpy import array\n",
        "from numpy import cov\n",
        "# define first vector\n",
        "x = array([1,2,3,4,5,6,7,8,9])\n",
        "#x=array([1,2,3])\n",
        "print(x)\n",
        "# define second covariance\n",
        "y = array([9,8,7,6,5,4,3,2,1])\n",
        "#y=array([4,5,6])\n",
        "print(y)\n",
        "# calculate covariance\n",
        "Sigma = cov(x,y)[0,1]\n",
        "print(Sigma)"
      ],
      "execution_count": 6,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "[1 2 3 4 5 6 7 8 9]\n",
            "[9 8 7 6 5 4 3 2 1]\n",
            "-7.5\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "oDb1AtnWrJ3e"
      },
      "source": [
        "The covariance can be normalized to a score between -1 and 1 to make the magnitude interpretable by dividing it by the standard deviation of X and Y . The result is called the correlation of the variables, also called the Pearson correlation coefficient  \n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "uP_TtDdvrbxQ"
      },
      "source": [
        "![image.png](data:image/png;base64,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)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "datb8oXUrd2a"
      },
      "source": [
        "Where r is the correlation coefficient of X and Y , cov(X; Y ) is the sample covariance of X and Y and sX and sY are the standard deviations of X and Y respectively.  \n",
        "**Correlation** is a normalized measure of the amount and direction (positive or negative) that two columns change together.  \n",
        "**Covariance** is a generalized and unnormalized version of correlation across multiple columns.  \n",
        "A **covariance matrix** is a calculation of covariance of a given matrix with covariance scores for every column with every other column, including itself."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "3wncu7IuZlS_",
        "outputId": "3a14e68e-40d7-45ed-ae01-1bf0a2684c2f"
      },
      "source": [
        "# vector correlation\n",
        "from numpy import array\n",
        "from numpy import corrcoef\n",
        "# define first vector\n",
        "x = array([1,2,3,4,5,6,7,8,9])\n",
        "print(x)\n",
        "# define second vector\n",
        "y = array([9,8,7,6,5,4,3,2,1])\n",
        "print(y)\n",
        "# calculate correlation\n",
        "corr = corrcoef(x,y)[0,1]\n",
        "print(corr)"
      ],
      "execution_count": 7,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "[1 2 3 4 5 6 7 8 9]\n",
            "[9 8 7 6 5 4 3 2 1]\n",
            "-1.0\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "VNeYxbb7tQuY"
      },
      "source": [
        "**Covariance Matrix**  \n",
        "The covariance matrix is a square and symmetric matrix that describes the covariance between two or more random variables. The diagonal of the covariance matrix are the variances of each of the random variables, as such it is often called the **variance-covariance matrix**.  \n",
        "\n",
        "\n",
        "It is a key element used in the Principal Component Analysis\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "Dm1p4WZfZpGp",
        "outputId": "ca1185a8-fddf-4d6e-f98d-458fe4f8013c"
      },
      "source": [
        "# covariance matrix\n",
        "from numpy import array\n",
        "from numpy import cov\n",
        "# define matrix of observations\n",
        "X = array([\n",
        "[1, 5, 8],\n",
        "[3, 5, 11],\n",
        "[2, 4, 9],\n",
        "[3, 6, 10],\n",
        "[1, 5, 10]])\n",
        "print(X)\n",
        "# calculate covariance matrix\n",
        "Sigma = cov(X.T)\n",
        "print(Sigma)"
      ],
      "execution_count": 8,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "[[ 1  5  8]\n",
            " [ 3  5 11]\n",
            " [ 2  4  9]\n",
            " [ 3  6 10]\n",
            " [ 1  5 10]]\n",
            "[[1.   0.25 0.75]\n",
            " [0.25 0.5  0.25]\n",
            " [0.75 0.25 1.3 ]]\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "IJow2Ww4xBxI"
      },
      "source": [
        "The covariance matrix is used widely in linear algebra and the intersection of linear algebra and statistics called multivariate analysis."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "oFT_RElDxD27"
      },
      "source": [
        "**Principal Component Analysis**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "JJxo-JMAxdjH"
      },
      "source": [
        "It is a method that uses simple matrix operations from linear algebra and\n",
        "statistics to calculate a projection of the original data into the same number or fewer dimensions."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "gxRUN-B_xfqy"
      },
      "source": [
        "PCA is an operation applied to a dataset, represented by an n * m matrix A that results in a projection of A which we will call B. Let's walk through the steps of this operation."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "h0Xx8FBRycrx"
      },
      "source": [
        "![image.png](data:image/png;base64,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)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Bs6gEPLOylwl"
      },
      "source": [
        "**PCA : Step 1**  \n",
        "calculate the mean values of each column.  \n",
        "M = mean(A)  \n",
        "**PCA : Step 2**  \n",
        "center the values in each column by subtracting the mean column value.\n",
        "C = A - M  \n",
        "**PCA : Step 3**  \n",
        "calculate the covariance matrix of the centered matrix C  \n",
        "V = cov(C)  \n",
        "**PCA : Step 4**  \n",
        "calculate the eigendecomposition of the covariance matrix V.  \n",
        "values; vectors = eig(V )  \n",
        "**PCA : Step 5**  \n",
        "select k eigenvectors, called principal components, that have the k\n",
        "largest eigenvalues.  \n",
        "B = select(values; vectors)\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "WI1kVFa11eYz"
      },
      "source": [
        "The eigenvectors represent the directions or components for the reduced subspace of B,\n",
        "whereas the eigenvalues represent the magnitudes for the directions. The eigenvectors can be\n",
        "sorted by the eigenvalues in descending order to provide a ranking of the components or axes of\n",
        "the new subspace for A.   \n",
        "**If all eigenvalues have a similar value, then we know that the existing\n",
        "representation may already be reasonably compressed or dense and that the projection may offer little.  \n",
        "If there are eigenvalues close to zero, they represent components or axes of B that may be discarded.  \n",
        "A total of m or less components must be selected to comprise the chosen subspace**."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "zYNGtb0aZ2Uc",
        "outputId": "fda7fc85-c487-4380-b99f-3b210b7bda39"
      },
      "source": [
        "# principal component analysis\n",
        "from numpy import array\n",
        "from numpy import mean\n",
        "from numpy import cov\n",
        "from numpy.linalg import eig\n",
        "# define matrix\n",
        "A = array([\n",
        "[1, 2],\n",
        "[3, 4],\n",
        "[5, 6]])\n",
        "print(A)\n",
        "# column means\n",
        "M = mean(A.T, axis=1)\n",
        "# center columns by subtracting column means\n",
        "C = A - M\n",
        "# calculate covariance matrix of centered matrix\n",
        "V = cov(C.T)\n",
        "# factorize covariance matrix\n",
        "values, vectors = eig(V)\n",
        "print(vectors)\n",
        "print(values)\n",
        "# project data\n",
        "P = vectors.T.dot(C.T)\n",
        "print(P.T)"
      ],
      "execution_count": 9,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "[[1 2]\n",
            " [3 4]\n",
            " [5 6]]\n",
            "[[ 0.70710678 -0.70710678]\n",
            " [ 0.70710678  0.70710678]]\n",
            "[8. 0.]\n",
            "[[-2.82842712  0.        ]\n",
            " [ 0.          0.        ]\n",
            " [ 2.82842712  0.        ]]\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "9Z8L6vXEZ4bh"
      },
      "source": [
        "**************************************\n",
        "Thank You!"
      ]
    }
  ]
}