distance_metrics_in_machine_learning.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Distance Metrics in Machine Learning"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Continuous or numerical variables:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Euclidean Distance\n",
    "Represents the shortest distance between two points."
   ]
  },
  {
   "attachments": {
    "Euclidean%20Distance.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![Euclidean%20Distance.png](attachment:Euclidean%20Distance.png)\n",
    "This calculation is related to the L2 vector norm."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "((6, 5, 4), (3, 2, 1))"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Importing the library\n",
    "from scipy.spatial import distance\n",
    "\n",
    "# Defining the points\n",
    "point_1 = (6, 5, 4)\n",
    "point_2 = (3, 2, 1)\n",
    "point_1, point_2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Euclidean distance between (6, 5, 4) and (3, 2, 1) is: 5.196152422706632\n"
     ]
    }
   ],
   "source": [
    "# Computing the Euclidean distance\n",
    "euclidean_distance = distance.euclidean(point_1, point_2)\n",
    "print('Euclidean distance between', point_1, 'and', point_2, 'is:', euclidean_distance)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Function to calculate Euclidean distance between two vectors\n",
    "def euclidean_distance(row1, row2):\n",
    "    distance = 0.0\n",
    "    for i in range(len(row1)-1):\n",
    "        distance += (row1[i] - row2[i])**2\n",
    "    return sqrt(distance)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Manhattan Distance\n",
    "Sum of absolute differences between points across all the dimensions (taxicab geometry)."
   ]
  },
  {
   "attachments": {
    "Manhattan%20Distance.png": {
     "image/png": "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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![Manhattan%20Distance.png](attachment:Manhattan%20Distance.png)\n",
    "The Manhattan distance is related to the L1 vector norm.\n",
    "\n",
    "And it is preferred over the Euclidean distance metric as the dimension of the data increases. This occurs due to something known as the ‘curse of dimensionality’."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.spatial.distance import cityblock"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Manhattan distance between (6, 5, 4) and (3, 2, 1) is: 9\n"
     ]
    }
   ],
   "source": [
    "# Computing the Manhattan distance\n",
    "manhattan_distance = distance.cityblock(point_1, point_2)\n",
    "print('Manhattan distance between', point_1, 'and', point_2, 'is:', manhattan_distance)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Minkowski Distance\n",
    "The generalized form of Euclidean and Manhattan Distance."
   ]
  },
  {
   "attachments": {
    "Minkowski%20Distance.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![Minkowski%20Distance.png](attachment:Minkowski%20Distance.png)\n",
    "Where “p” is the order parameter.\n",
    "When p is set to 1, the calculation is the same as the Manhattan distance. When p is set to 2, it is the same as the Euclidean distance.\n",
    "\n",
    "The Minkowski distance is related to the Lp vector norm.\n",
    "\n",
    "It is common to use Minkowski distance when implementing a machine learning algorithm that uses distance measures as it gives control over the type of distance measure used for real-valued vectors via a hyperparameter “p” that can be tuned."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.spatial.distance import cityblock"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Minkowski distance between (6, 5, 4) and (3, 2, 1) is: 4.3267487109222245\n"
     ]
    }
   ],
   "source": [
    "# Computing the Minkowski distance\n",
    "minkowski_distance = distance.minkowski(point_1, point_2, p=3)\n",
    "print('Minkowski distance between', point_1, 'and', point_2, 'is:', minkowski_distance)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Vector Norm\n",
    "The length of a vector is a non-negative number that describes the extent of the vector in space, and is sometimes referred to as the vector’s magnitude or the norm."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Vector L1 Norm**\n",
    "\n",
    "The length of a vector can be calculated using the L1 norm. The L1 norm, represented as ||v||1 is calculated as the sum of the absolute vector values, where the absolute value of a scalar uses the notation |a1|. Clearly, the norm is a calculation of the Manhattan distance from the origin of the vector space."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "||v||1 = |a1| + |a2| + |a3|"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1 2 3]\n",
      "6.0\n"
     ]
    }
   ],
   "source": [
    "# L1 norm of a vector\n",
    "from numpy import array\n",
    "from numpy.linalg import norm\n",
    "a = array([1, 2, 3])\n",
    "print(a)\n",
    "l1 = norm(a, 1)\n",
    "print(l1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Vector L2 Norm**\n",
    "\n",
    "The L2 norm, represented as ||v||2 is calculated as the square root of the sum of the squared vector values.Clearly, the norm is a calculation of the Euclidean distance from the origin of the vector space."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "||v||2 = sqrt(a1² + a2² + a3²)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1 2 3]\n",
      "3.7416573867739413\n"
     ]
    }
   ],
   "source": [
    "# L2 norm of a vector\n",
    "from numpy import array\n",
    "from numpy.linalg import norm\n",
    "a = array([1, 2, 3])\n",
    "print(a)\n",
    "l2 = norm(a)\n",
    "print(l2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Vector Lp Norm**\n",
    "\n",
    "The Lp norm, represented as ||v||p is a calculation of the Minkowski distance from the origin of the vector space."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "||v||p = (a1^p + a2^p + a3^p)^(1/p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Categorical variables:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Hamming Distance\n",
    "Measures the similarity between two strings of the same length. The Hamming Distance between two strings of the same length is the number of positions at which the corresponding characters are different."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.spatial.distance import hamming\n",
    "\n",
    "# Defining two strings\n",
    "string_1 = 'euclidean'\n",
    "string_2 = 'manhattan'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Hamming Distance between euclidean and manhattan is:  7.0\n"
     ]
    }
   ],
   "source": [
    "# Computing the harmming distance\n",
    "hamming_distance = distance.hamming(list(string_1), list(string_2)) * len(string_1)\n",
    "print('Hamming Distance between', string_1, 'and', string_2, 'is: ', hamming_distance)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hamming distance only works when we have strings or arrays of the same length."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Cosine Distance & Cosine Similarity\n",
    "Cosine similarity is a metric used to measure how similar the documents are irrespective of their size. Mathematically, it measures the cosine of the angle between two vectors projected in a multi-dimensional space. The cosine similarity is advantageous because even if the two similar documents are far apart by the Euclidean distance (due to the size of the document), chances are they may still be oriented closer together. The smaller the angle, higher the cosine similarity."
   ]
  },
  {
   "attachments": {
    "Cosine%20similarity%20formula.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![Cosine%20similarity%20formula.png](attachment:Cosine%20similarity%20formula.png)\n",
    "Cosine similarity is given by Cos θ, and cosine distance is 1- Cos θ.\n",
    "\n",
    "Cosine distance metric is mainly used to find the amount of similarity"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "documents = (\n",
    "\"The sky is blue\",\n",
    "\"The sun is bright\",\n",
    "\"The sun in the sky is bright\",\n",
    "\"We can see the shining sun, the bright sun\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(4, 11)\n"
     ]
    }
   ],
   "source": [
    "import pandas as pd\n",
    "from sklearn.feature_extraction.text import CountVectorizer\n",
    "count_vectorizer = CountVectorizer()\n",
    "count_matrix = count_vectorizer.fit_transform(documents)\n",
    "print(count_matrix.shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1.        , 0.5       , 0.66666667, 0.2773501 ]])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn.metrics.pairwise import cosine_similarity\n",
    "cosine_similarity(count_matrix[0:1], count_matrix)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The first value of the array is 1.0 because it is the Cosine Similarity between the first document with itself. Due to the presence of similar words on the third document (“The sun in the sky is bright”), it achieved a better score."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Bonus**\n",
    "\n",
    "To understand the CountVectorizer"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  (0, 0)\t1\n",
      "  (0, 4)\t1\n",
      "  (0, 7)\t1\n",
      "  (0, 9)\t1\n",
      "  (1, 1)\t1\n",
      "  (1, 4)\t1\n",
      "  (1, 8)\t1\n",
      "  (1, 9)\t1\n",
      "  (2, 1)\t1\n",
      "  (2, 3)\t1\n",
      "  (2, 4)\t1\n",
      "  (2, 7)\t1\n",
      "  (2, 8)\t1\n",
      "  (2, 9)\t2\n",
      "  (3, 1)\t1\n",
      "  (3, 2)\t1\n",
      "  (3, 5)\t1\n",
      "  (3, 6)\t1\n",
      "  (3, 8)\t2\n",
      "  (3, 9)\t2\n",
      "  (3, 10)\t1\n",
      "\n",
      "[[1 0 0 0 1 0 0 1 0 1 0]\n",
      " [0 1 0 0 1 0 0 0 1 1 0]\n",
      " [0 1 0 1 1 0 0 1 1 2 0]\n",
      " [0 1 1 0 0 1 1 0 2 2 1]]\n",
      "\n",
      "   blue  bright  can  in  is  see  shining  sky  sun  the  we\n",
      "0     1       0    0   0   1    0        0    1    0    1   0\n",
      "1     0       1    0   0   1    0        0    0    1    1   0\n",
      "2     0       1    0   1   1    0        0    1    1    2   0\n",
      "3     0       1    1   0   0    1        1    0    2    2   1\n",
      "\n",
      "{'the': 9, 'sky': 7, 'is': 4, 'blue': 0, 'sun': 8, 'bright': 1, 'in': 3, 'we': 10, 'can': 2, 'see': 5, 'shining': 6}\n"
     ]
    }
   ],
   "source": [
    "print(count_matrix)\n",
    "print()\n",
    "count_array = count_matrix.toarray()\n",
    "df = pd.DataFrame(data=count_array,columns = count_vectorizer.get_feature_names())\n",
    "print(count_array)\n",
    "print()\n",
    "print(df)\n",
    "print()\n",
    "# Position of the words in the matrix\n",
    "print(count_vectorizer.vocabulary_)"
   ]
  }
 ],
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