Assignment_01_OLS.ipynb

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   "source": [
    "**Machine Learning  \n",
    "Robert Knox**"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assignment details:\n",
    "\n",
    "Use Linear equation normal equation to predict water temperature T_degC\n",
    "\n",
    "1) Only use 'Salnty', 'STheta' for predictors\n",
    "\n",
    "2) Remove NaN / NA values from dataset (prior to building train/test sets). \n",
    "\n",
    "3) Solve for rmse, variance explained, and r-squared."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Step 1: Import"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from numpy import linalg as LA\n",
    "import pandas as pd\n",
    "import os\n",
    "from sklearn.linear_model import LinearRegression\n",
    "import sklearn.metrics as metrics\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Step 2: Read in Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "with open('bottle.csv') as f:\n",
    "    line = f.readline()\n",
    "line_column_names = line.split(',')\n",
    "\n",
    "#47 (MeanAq) & 73 (pH1) have mixed types so I only use the desired columns\n",
    "\n",
    "desired_cols = ['T_degC','Salnty','STheta']\n",
    "df = pd.read_csv('bottle.csv',sep=',',header='infer',index_col=None,usecols=desired_cols)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>T_degC</th>\n",
       "      <th>Salnty</th>\n",
       "      <th>STheta</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>10.50</td>\n",
       "      <td>33.440</td>\n",
       "      <td>25.649</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>10.46</td>\n",
       "      <td>33.440</td>\n",
       "      <td>25.656</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>10.46</td>\n",
       "      <td>33.437</td>\n",
       "      <td>25.654</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>10.45</td>\n",
       "      <td>33.420</td>\n",
       "      <td>25.643</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>10.45</td>\n",
       "      <td>33.421</td>\n",
       "      <td>25.643</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
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      "text/plain": [
       "   T_degC  Salnty  STheta\n",
       "0   10.50  33.440  25.649\n",
       "1   10.46  33.440  25.656\n",
       "2   10.46  33.437  25.654\n",
       "3   10.45  33.420  25.643\n",
       "4   10.45  33.421  25.643"
      ]
     },
     "execution_count": 3,
     "metadata": {},
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   ],
   "source": [
    "df.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Step 3: Handle Nulls & NAN"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "df.dropna(axis=0,inplace=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "T_degC    False\n",
       "Salnty    False\n",
       "STheta    False\n",
       "dtype: bool"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#make sure it worked\n",
    "df.isna().any()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "df[\"Intercept\"] = np.ones(len(df))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Step 4: Convert to arrays & solve using the Normal Equation\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "theta_best:  [89.7647 -0.0555 -2.9838]\n",
      "\n",
      "RMSE:\t\t\t 2.3595\n",
      "Variance Explained:\t 0.6875\n",
      "R-Squared:\t\t 0.6875\n"
     ]
    }
   ],
   "source": [
    "T_degC = df['T_degC'].values\n",
    "Intercept = df['Intercept'].values\n",
    "Salnty = df['Salnty'].values\n",
    "STheta = df['STheta'].values\n",
    "\n",
    "#build our data matrix A\n",
    "X = np.column_stack((Intercept,Salnty,STheta))\n",
    "\n",
    "#create the transpose of the data matrix for ease of calculation\n",
    "Xtran = np.transpose(X)\n",
    "\n",
    "#define our y_true as T_degC\n",
    "y_true = T_degC.copy()\n",
    "#print(\"y_true: \",y_true)\n",
    "\n",
    "theta_best = LA.inv((Xtran.dot(X))).dot(Xtran).dot(y_true)\n",
    "yhat = X.dot(theta_best)\n",
    "\n",
    "#Make the output look nice with array_str\n",
    "print(\"theta_best: \",np.array_str(theta_best,precision=4,suppress_small=True))\n",
    "\n",
    "ybar = sum(y_true)/len(y_true)\n",
    "#print(\"ybar\",ybar)\n",
    "\n",
    "#alternative calculations\n",
    "#SST = sum((y_true-ybar)**2)\n",
    "#SSM = sum((yhat-ybar)**2)\n",
    "#SSE = sum((y_true-yhat)**2)\n",
    "\n",
    "#Calculate Variance Explained\n",
    "var_explained = 1-np.cov(y_true-yhat)/np.cov(y_true)\n",
    "\n",
    "#Calculate root mean square error\n",
    "rmse = (sum((yhat-y_true)**2)/len(y_true))**0.5\n",
    "\n",
    "#Calculate R-Squared (not adjusted)\n",
    "r_squared = 1-((y_true-yhat)**2).sum()/(len(y_true)*(y_true.std()**2))\n",
    "\n",
    "print(\"\\nRMSE:\\t\\t\\t\",np.round(rmse,4))\n",
    "print(\"Variance Explained:\\t\",np.round(var_explained,4))\n",
    "print(\"R-Squared:\\t\\t\",np.round(r_squared,4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Check using Scikit learn linear model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Intercept:  89.76473480839081 \n",
      "Thetas:  [ 0.         -0.05547897 -2.98377603]\n",
      "R-Squared: 0.6875216833872659\n",
      "RMSE 2.3595303631129534\n"
     ]
    }
   ],
   "source": [
    "lin_reg = LinearRegression()\n",
    "reg = lin_reg.fit(X, y_true)\n",
    "print(\"Intercept: \",reg.intercept_,\"\\nThetas: \", reg.coef_)\n",
    "print(\"R-Squared:\",reg.score(X,y_true))\n",
    "yhat_sk = reg.predict(X)\n",
    "print(\"RMSE\",metrics.mean_squared_error(y_true,yhat_sk)**0.5)"
   ]
  }
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