Machine Learning_ Programming Exercise 1.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1. A simple function\n",
    "Create a 5*5 identity matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1. 0. 0. 0. 0.]\n",
      " [0. 1. 0. 0. 0.]\n",
      " [0. 0. 1. 0. 0.]\n",
      " [0. 0. 0. 1. 0.]\n",
      " [0. 0. 0. 0. 1.]]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "def iden(n):\n",
    "    print(np.identity(n))\n",
    "iden(5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2. Linear Regression with one variable"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 2.1 Plotting the Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 151,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "        vertical-align: middle;\n",
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       "\n",
       "    .dataframe tbody tr th {\n",
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>0</th>\n",
       "      <th>1</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <td>0</td>\n",
       "      <td>6.1101</td>\n",
       "      <td>17.5920</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>1</td>\n",
       "      <td>5.5277</td>\n",
       "      <td>9.1302</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>2</td>\n",
       "      <td>8.5186</td>\n",
       "      <td>13.6620</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>3</td>\n",
       "      <td>7.0032</td>\n",
       "      <td>11.8540</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>4</td>\n",
       "      <td>5.8598</td>\n",
       "      <td>6.8233</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
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      ],
      "text/plain": [
       "        0        1\n",
       "0  6.1101  17.5920\n",
       "1  5.5277   9.1302\n",
       "2  8.5186  13.6620\n",
       "3  7.0032  11.8540\n",
       "4  5.8598   6.8233"
      ]
     },
     "execution_count": 151,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import pandas as pd\n",
    "data=pd.read_csv(\"/Users/natalychacon/Documents/Machine Learning/std coursera/machine-learning-ex/ex1/ex1data1.txt\",header=None)\n",
    "data.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = data.loc[:,[0]]\n",
    "x.insert(0,\"0\",1)\n",
    "y = data.loc[:,[1]]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'Profit in $10,000s')"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "ax=data.plot(kind='scatter',x=0,y=1,color='blue',marker=\"x\",figsize=(12,8))\n",
    "ax.set_xlabel(\"Population of City in 10,000s\")\n",
    "ax.set_ylabel(\"Profit in $10,000s\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 2.2 Cost Function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "m = len(data) #Returns number of training examples, or number of rows\n",
    "iterations = 1500\n",
    "alpha = 0.01\n",
    "theta = np.matrix(np.array([[0],[0]]))\n",
    "X = np.matrix(x)\n",
    "y = np.matrix(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def computeCost(X,y,theta):\n",
    "    m = np.size(y,0)\n",
    "    resultado = (1/(2*m))*(np.sum(np.power(np.matmul(X,theta)-y,2),axis=0)) \n",
    "    return resultado.item(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "32.072733877455676"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "computeCost(X,y,theta)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "54.24245508201238"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "thetan=np.matrix(np.array([[-1],[2]]))\n",
    "computeCost(X,y,thetan)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 2.3 Gradient Descent"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def gradientDescent(X,y,theta,alpha,iteration):\n",
    "    m = np.size(y,0)\n",
    "    temporal = np.matrix(np.zeros(theta.shape))\n",
    "    J_history=[]\n",
    "    for i in range(iterations+1):\n",
    "        \n",
    "        temporal[0,:]=theta[0,:]-(alpha*1/m*(np.sum(np.multiply(np.matmul(X,theta)-y,X[:,0]))))\n",
    "        temporal[1,:]=theta[1,:]-(alpha*1/m*(np.sum(np.multiply(np.matmul(X,theta)-y,X[:,1]))))\n",
    "        theta=temporal\n",
    "       \n",
    "            \n",
    "        J_history.append(computeCost(X,y,theta))\n",
    "    return(theta)\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-3.63656668]\n",
      " [ 1.16703962]]\n"
     ]
    }
   ],
   "source": [
    "theta=gradientDescent(X,y,theta,alpha,iterations)\n",
    "print(theta)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the linear fit."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Predicted Profit vs. Population Size')"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "xaxis = np.linspace(X.min(),X.max(),100)\n",
    "yaxis = theta[0,0]+ (theta[1,0]*xaxis)\n",
    "fig,ax = plt.subplots(figsize=(12,8))\n",
    "ax.plot(xaxis,yaxis,\"r\",label='Prediction')\n",
    "ax.scatter(data[0], data[1], label='Traning Data',marker=\"x\")\n",
    "ax.set_xlabel('Population')\n",
    "ax.set_ylabel('Profit')\n",
    "ax.set_title('Predicted Profit vs. Population Size')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3. Linear Regression with multiple variables\n",
    "In this part, you will implement linear regression with multiple variables to predict the prices of houses. Suppose you are selling your house and you want to know what a good market price would be. One way to do this is to first collect information on recent houses sold and make a model of housing prices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 132,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Size</th>\n",
       "      <th>Bedrooms</th>\n",
       "      <th>Price</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <td>0</td>\n",
       "      <td>2104</td>\n",
       "      <td>3</td>\n",
       "      <td>399900</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>1</td>\n",
       "      <td>1600</td>\n",
       "      <td>3</td>\n",
       "      <td>329900</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>2</td>\n",
       "      <td>2400</td>\n",
       "      <td>3</td>\n",
       "      <td>369000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>3</td>\n",
       "      <td>1416</td>\n",
       "      <td>2</td>\n",
       "      <td>232000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>4</td>\n",
       "      <td>3000</td>\n",
       "      <td>4</td>\n",
       "      <td>539900</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   Size  Bedrooms   Price\n",
       "0  2104         3  399900\n",
       "1  1600         3  329900\n",
       "2  2400         3  369000\n",
       "3  1416         2  232000\n",
       "4  3000         4  539900"
      ]
     },
     "execution_count": 132,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data2=pd.read_csv(\"/Users/natalychacon/Documents/Machine Learning/std coursera/machine-learning-ex/ex1/ex1data2.txt\",header=None)\n",
    "data2.columns = ['Size', 'Bedrooms',\"Price\"]\n",
    "data2.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 3.1 Feature Normalization\n",
    "By looking at the values of the dataset note that house sizes are about 1000 times the number of bedrooms. When features differ by orders of magnitude, first performing feature scaling can make gradient descent converge much more quickly.\n",
    "\n",
    "1.Subtract the mean value of each feature from the dataset.\n",
    "2.After subtracting the mean, additionally scale (divide) the feature values by their respective \"standard deviations\"."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 148,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Size</th>\n",
       "      <th>Bedrooms</th>\n",
       "      <th>Price</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <td>0</td>\n",
       "      <td>0.130010</td>\n",
       "      <td>-0.223675</td>\n",
       "      <td>0.475747</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>1</td>\n",
       "      <td>-0.504190</td>\n",
       "      <td>-0.223675</td>\n",
       "      <td>-0.084074</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>2</td>\n",
       "      <td>0.502476</td>\n",
       "      <td>-0.223675</td>\n",
       "      <td>0.228626</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>3</td>\n",
       "      <td>-0.735723</td>\n",
       "      <td>-1.537767</td>\n",
       "      <td>-0.867025</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>4</td>\n",
       "      <td>1.257476</td>\n",
       "      <td>1.090417</td>\n",
       "      <td>1.595389</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "       Size  Bedrooms     Price\n",
       "0  0.130010 -0.223675  0.475747\n",
       "1 -0.504190 -0.223675 -0.084074\n",
       "2  0.502476 -0.223675  0.228626\n",
       "3 -0.735723 -1.537767 -0.867025\n",
       "4  1.257476  1.090417  1.595389"
      ]
     },
     "execution_count": 148,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data2 = (data2 - data2.mean()) / data2.std()\n",
    "data2.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 3.2 Gradient Descent"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 140,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = data2.loc[:,[\"Size\",\"Bedrooms\"]]#size of houses#number of bedrooms #features#Training data\n",
    "X.insert(0,\"0\",1)\n",
    "y = data2.loc[:,[\"Price\"]]#target variable"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 141,
   "metadata": {},
   "outputs": [],
   "source": [
    "# convert to matrices and initialize theta\n",
    "theta = np.matrix(np.array([[0],[0],[0]]))\n",
    "alpha = 0.1\n",
    "num_iters=400\n",
    "X = np.matrix(X)\n",
    "y = np.matrix(y)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 146,
   "metadata": {},
   "outputs": [],
   "source": [
    "def gradientDescent(X,y,theta,alpha,iteration):\n",
    "    m = np.size(y,0)\n",
    "    temporal = np.matrix(np.zeros(theta.shape))\n",
    "    for i in range(iterations+1):\n",
    "        mult=np.matmul(X,theta)-y\n",
    "        for j in range(X.shape[1]):\n",
    "            temporal[j,:]=theta[j,:]-(alpha*1/m*(np.sum(np.multiply(mult,X[:,j]))))\n",
    "        theta=temporal      \n",
    "    return(theta)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 152,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-7.62392513e-17]\n",
      " [ 8.84765988e-01]\n",
      " [-5.31788197e-02]]\n"
     ]
    }
   ],
   "source": [
    "theta=gradientDescent(X,y,theta,alpha,num_iters)\n",
    "print(theta)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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