{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*2018 CFA Level 2 Study Session 3 Reading 9* is an introduction to covariance, linear regression, and scatter plots. For ease I've broken these up in several parts, with part 1 exploring scatter plots and correlation coefficients. The math for this lesson can be handled using either numpy or pandas, and the plotting from matplotlib. As the complexity of the examples increase, we bring in real world data using pandas-datareader to illustrate the examples.\n",
    "\n",
    "Note: to properly view this notebook, ensure that \"show output\" is selected from the dropdown, and if you want to see the wizard behind the curtain, click \"show code\".\n",
    "\n",
    "---\n",
    "**Scatter Plots**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "from pandas_datareader import data, wb\n",
    "import numpy as np\n",
    "import matplotlib, datetime\n",
    "import matplotlib.pyplot as plt\n",
    "from pylab import *\n",
    "%matplotlib inline\n",
    "pd.set_option('display.max_rows', 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Example 1-1 concerns **scatter plots**, with a some money supply rate and an inflation rate, with the following data:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Country</th>\n",
       "      <th>InflationRate</th>\n",
       "      <th>MoneySupplyGrthRate</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>A</td>\n",
       "      <td>0.05451</td>\n",
       "      <td>0.06851</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>B</td>\n",
       "      <td>0.07761</td>\n",
       "      <td>0.11601</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>C</td>\n",
       "      <td>0.03491</td>\n",
       "      <td>0.05751</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>D</td>\n",
       "      <td>0.07351</td>\n",
       "      <td>0.10501</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>E</td>\n",
       "      <td>0.08251</td>\n",
       "      <td>0.12501</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>F</td>\n",
       "      <td>0.10761</td>\n",
       "      <td>0.13501</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "  Country  InflationRate  MoneySupplyGrthRate\n",
       "0       A        0.05451              0.06851\n",
       "1       B        0.07761              0.11601\n",
       "2       C        0.03491              0.05751\n",
       "3       D        0.07351              0.10501\n",
       "4       E        0.08251              0.12501\n",
       "5       F        0.10761              0.13501"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = pd.DataFrame({\"Country\":[\"A\",\"B\",\"C\",\"D\",\"E\",\"F\"], \"MoneySupplyGrthRate\":[0.06851,0.11601,0.05751,0.10501,0.12501,0.13501], \"InflationRate\":[0.05451,0.07761,0.03491,0.07351,0.08251,0.10761]})\n",
    "#df['InflationRate'] = df.MoneySupplyGrthRate + np.random.uniform(-1,1)\n",
    "df"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With the resulting scatter plot:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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eUJkw5gGXRkQAd6RexW5AL/AB0hQkEfEK8MoIYjEzsxEaKGF8KyLmSrosIo4exntPBTZU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\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x18f21813e48>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "figure()\n",
    "plt.scatter(df.MoneySupplyGrthRate,df.InflationRate,c='r')\n",
    "xlabel('Money Supply Growth Rate')\n",
    "ylabel('Inflation Rate')\n",
    "title('Example 1-1')\n",
    "show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Correlation analysis** describes the relationship between two data series using a * *single number* * , the **correlation coefficient**, as measured from -1 to +1, with 0 being perfectly uncorrelated. The following graphs demonstrate this effect."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x18f21cbeba8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "figure()\n",
    "x = np.random.uniform(1,10,10);\n",
    "y = x*0.7;\n",
    "plt.scatter(x,y,c='r')\n",
    "xlabel('X')\n",
    "ylabel('Y')\n",
    "title('Figure 1-2, scatter plot with correlation of +1 (y = x*0.7)')\n",
    "show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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PzcyHAb8A3tZQHZvzewuStlGNBEJmnp+Zd9e7PwD2b6KOjvzegqRtVGRmswVE/Bvwxcw8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\n1f8HgikMa5DZ8ugAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x18f21d07208>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "figure()\n",
    "x = np.random.uniform(1,10,10);\n",
    "y = x*-0.7;\n",
    "plt.scatter(x,y,c='r')\n",
    "xlabel('X')\n",
    "ylabel('Y')\n",
    "title('Figure 1-3, scatter plot with correlation of -1 (y = x*-0.7)')\n",
    "show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x18f21dea7b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "figure()\n",
    "x = np.random.uniform(0,1,100);\n",
    "y = np.random.uniform(0,1,100);\n",
    "plt.scatter(x,y,c='r')\n",
    "xlabel('X')\n",
    "ylabel('Y')\n",
    "title('Figure 1-4, scatter plot with random data, correlation of 0')\n",
    "show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that Figure 1-4 is perfectly random, no correlation, and the value of X tells us nothing about the value of Y, etc.\n",
    "\n",
    "---\n",
    "**Calculating and Interpreting Correlation Coefficient / Covariance**\n",
    "\n",
    "$$Sample \\space covariance = Cov (X,Y) = \\sum_{n}^{i=1} \\frac{(X_i - \\overline{X})(Y_i - \\overline{Y})}{(n-1)}$$\n",
    "\n",
    "where:\n",
    "n = sample size\n",
    "\n",
    "$X_i = i$th observation of variable X\n",
    "\n",
    "$\\overline{X}$ = mean observations of variable X\n",
    "\n",
    "$Y_i = i$th observation of variable Y\n",
    "\n",
    "$\\overline{Y}$ = mean observations of variable X\n",
    "\n",
    "On the **HP 12c calculator**, correlation coefficient is generated when you use the $\\boxed{\\hat{y},r}$ key after entering your data pairs into the statistics registers, but is kept in the Y register, so use the $\\boxed{X \\lessgtr Y}$ key to show the coefficient. (see the HP 12c Platinum manual pg 97)\n",
    "\n",
    "The formula for the (Pearson) **correlation coefficient** is given as follows: \n",
    "$$Correlation \\space coefficient = r = \\frac{Cov(X,Y)}{s_x s_y}$$\n",
    "where $s_x$ is the standard deviation (recall, the standard deviation is the square root of the variance) of $X$, and $s_y$ is the standard deviation of $Y$. \n",
    "In this way, you can multiply $r$ as calculated by your financial calculator by $s_x s_y$ to arrive at the covariance. \n",
    "\n",
    "Using **Pandas**, calculating the covariance, variance, and correlation coefficient is straight forward, as each series has a method for each. Using the initial dataframe as an example"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Covariance:  0.000747046666667\n"
     ]
    }
   ],
   "source": [
    "print(\"Covariance: \",df.MoneySupplyGrthRate.cov(df.InflationRate)) #covariance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "variance X:  0.0009842666666666667\n"
     ]
    }
   ],
   "source": [
    "print(\"variance X: \",df.MoneySupplyGrthRate.var()) #variance X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "variance Y:  0.0006187106666666666\n"
     ]
    }
   ],
   "source": [
    "print(\"variance Y: \",df.InflationRate.var()) #variance Y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Correlation coefficient:  0.957298860219\n"
     ]
    }
   ],
   "source": [
    "print(\"Correlation coefficient: \",df.MoneySupplyGrthRate.corr(df.InflationRate)) #correlation coefficient"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The long way to do this, as described in the text, is to innumerate the table to calculate  the cross product and the squared deviations...\n",
    "\n",
    "The (Pearson's) correlation coefficient "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Country</th>\n",
       "      <th>InflationRate</th>\n",
       "      <th>MoneySupplyGrthRate</th>\n",
       "      <th>CrossProduct</th>\n",
       "      <th>SquaredDeviationsX</th>\n",
       "      <th>SquaredDeviationsY</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>A</td>\n",
       "      <td>0.05451</td>\n",
       "      <td>0.06851</td>\n",
       "      <td>0.000564</td>\n",
       "      <td>0.001067</td>\n",
       "      <td>0.000298</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>B</td>\n",
       "      <td>0.07761</td>\n",
       "      <td>0.11601</td>\n",
       "      <td>0.000087</td>\n",
       "      <td>0.000220</td>\n",
       "      <td>0.000034</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>C</td>\n",
       "      <td>0.03491</td>\n",
       "      <td>0.05751</td>\n",
       "      <td>0.001610</td>\n",
       "      <td>0.001907</td>\n",
       "      <td>0.001359</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>D</td>\n",
       "      <td>0.07351</td>\n",
       "      <td>0.10501</td>\n",
       "      <td>0.000007</td>\n",
       "      <td>0.000015</td>\n",
       "      <td>0.000003</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>E</td>\n",
       "      <td>0.08251</td>\n",
       "      <td>0.12501</td>\n",
       "      <td>0.000256</td>\n",
       "      <td>0.000568</td>\n",
       "      <td>0.000115</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>F</td>\n",
       "      <td>0.10761</td>\n",
       "      <td>0.13501</td>\n",
       "      <td>0.001212</td>\n",
       "      <td>0.001145</td>\n",
       "      <td>0.001284</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "  Country  InflationRate  MoneySupplyGrthRate  CrossProduct  \\\n",
       "0       A        0.05451              0.06851      0.000564   \n",
       "1       B        0.07761              0.11601      0.000087   \n",
       "2       C        0.03491              0.05751      0.001610   \n",
       "3       D        0.07351              0.10501      0.000007   \n",
       "4       E        0.08251              0.12501      0.000256   \n",
       "5       F        0.10761              0.13501      0.001212   \n",
       "\n",
       "   SquaredDeviationsX  SquaredDeviationsY  \n",
       "0            0.001067            0.000298  \n",
       "1            0.000220            0.000034  \n",
       "2            0.001907            0.001359  \n",
       "3            0.000015            0.000003  \n",
       "4            0.000568            0.000115  \n",
       "5            0.001145            0.001284  "
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df['CrossProduct']=(df.MoneySupplyGrthRate - df.MoneySupplyGrthRate.mean())*(df.InflationRate-df.InflationRate.mean())\n",
    "df['SquaredDeviationsX'] = (df.MoneySupplyGrthRate - df.MoneySupplyGrthRate.mean())**2\n",
    "df['SquaredDeviationsY'] = (df.InflationRate-df.InflationRate.mean())**2\n",
    "df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Covariance:  0.0007470466666666667 \n",
      "VarX:  0.0009842666666666667 \n",
      "VarY:  0.0006187106666666666 \n",
      "Correlation Coefficient 0.9572988602190073\n"
     ]
    }
   ],
   "source": [
    "covariance = df.CrossProduct.sum()/(len(df.CrossProduct)-1) #sum of cross products / n-1\n",
    "varX = df.SquaredDeviationsX.sum()/(len(df.SquaredDeviationsX)-1) #sum of squared deviations / n-1\n",
    "varY = df.SquaredDeviationsY.sum()/(len(df.SquaredDeviationsY)-1) #sum of squared deviations / n-1\n",
    "corrcoeff = covariance/((varX**0.5)*(varY**0.5))\n",
    "print(\"Covariance: \", covariance,\"\\nVarX: \", varX,\"\\nVarY: \", varY, \"\\nCorrelation Coefficient\", corrcoeff)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "still relatively easy, but unnecessary given Pandas' built in methods. It's important to note that the correlation coefficient formula given here by CFAI is the [Pearson correlation coefficient](https://en.wikipedia.org/wiki/Pearson_correlation_coefficient). Other options available to the [Pandas method](http://pandas.pydata.org/pandas-docs/stable/generated/pandas.Series.corr.html#pandas.Series.corr) are the [Kendall Tau correlation coefficient](https://en.wikipedia.org/wiki/Kendall_rank_correlation_coefficient), and the [Spearman rank correlation coefficient](https://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient).\n",
    "\n",
    "---\n",
    "**Limitations**\n",
    "\n",
    "* It's important to understand that correlation, measured this way, measures *linear* relationships. For example, let's revisit figure 1-2 again, but also look at exponentials. In figure 1-2, I used a random sequence from numpy as the X variable, and used the equation $y = 5+3.5x$ for the Y column, which is a linear relationship. Let's make Z a quadratic, using the equation $z = (3+x)^{2}$. For simplicity we will do this in a pandas dataframe:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>X</th>\n",
       "      <th>Y</th>\n",
       "      <th>Z</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>85.284563</td>\n",
       "      <td>303.495969</td>\n",
       "      <td>7794.164003</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-12.394837</td>\n",
       "      <td>-38.381928</td>\n",
       "      <td>88.262957</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>23.291763</td>\n",
       "      <td>86.521169</td>\n",
       "      <td>691.256784</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>80.094586</td>\n",
       "      <td>285.331053</td>\n",
       "      <td>6904.710296</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-38.749728</td>\n",
       "      <td>-130.624048</td>\n",
       "      <td>1278.043045</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>7.620688</td>\n",
       "      <td>31.672409</td>\n",
       "      <td>112.799018</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>54.713177</td>\n",
       "      <td>196.496120</td>\n",
       "      <td>3330.810810</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>47.175449</td>\n",
       "      <td>170.114072</td>\n",
       "      <td>2517.575703</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>39.438109</td>\n",
       "      <td>143.033380</td>\n",
       "      <td>1800.993070</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>-67.475748</td>\n",
       "      <td>-231.165117</td>\n",
       "      <td>4157.122040</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>20 rows × 3 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "            X           Y            Z\n",
       "0   85.284563  303.495969  7794.164003\n",
       "1  -12.394837  -38.381928    88.262957\n",
       "2   23.291763   86.521169   691.256784\n",
       "3   80.094586  285.331053  6904.710296\n",
       "4  -38.749728 -130.624048  1278.043045\n",
       "..        ...         ...          ...\n",
       "15   7.620688   31.672409   112.799018\n",
       "16  54.713177  196.496120  3330.810810\n",
       "17  47.175449  170.114072  2517.575703\n",
       "18  39.438109  143.033380  1800.993070\n",
       "19 -67.475748 -231.165117  4157.122040\n",
       "\n",
       "[20 rows x 3 columns]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = np.random.uniform(-100,100,20);\n",
    "y = 5+(3.5*x);\n",
    "z = (3+x)**2\n",
    "df1_2a = pd.DataFrame({\"X\":x,\"Y\":y,\"Z\":z})\n",
    "df1_2a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "with the resulting graph:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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d7w8uLH++Ug/fdxX/TCxpjlKPzmslXVTpj9ZmZmaNbO4/nfRH0L6IeDGpA9UFpD9g3hQR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\nAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x18f21e76c18>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Correlation coefficient X to Y (red):  1.0 \n",
      " Correlation coefficient X to Z (blue):  0.438640512754\n"
     ]
    }
   ],
   "source": [
    "figure()\n",
    "plt.scatter(df1_2a.X,df1_2a.Y,c='r')\n",
    "plt.scatter(df1_2a.X,df1_2a.Z,c='b')\n",
    "xlabel('X')\n",
    "ylabel('Y (red) Z (blue)')\n",
    "title('Figure 1-2a, scatter plot showing linear and exponential relationships')\n",
    "show()\n",
    "print(\"Correlation coefficient X to Y (red): \",df1_2a.X.corr(df1_2a.Y), \n",
    "      \"\\n Correlation coefficient X to Z (blue): \",df1_2a.X.corr(df1_2a.Z))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Pearson's correlation coefficient for x:z is expected to be low, because the parabola is not easily explained with a line, as shown by the correlation coefficient near 0 (uncorrelated).\n",
    "\n",
    "*Note* if we only take the right or left half of the parabola, the correlation coefficient is quite good, as the relationship appears somewhat linear. Consider price data - the prices will always be positive. So, if using real prices in a correlation, regardless if it is an exponential it might have a linear-looking correlation. On the other hand, if the dataset contains positive and negative values (such as daily or weekly price changes), the correlation would diminish with an exponential function. For example, take only the right side (positive) data:  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x18f21e89518>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Correlation X-Z (blue dots): 0.966096670835\n"
     ]
    }
   ],
   "source": [
    "x = np.random.uniform(0,100,20);\n",
    "y = 5+(3.5*x);\n",
    "z = (3+x)**2\n",
    "df1_2b = pd.DataFrame({\"X\":x,\"Y\":y,\"Z\":z})\n",
    "figure()\n",
    "plt.scatter(df1_2b.X,df1_2b.Y,c='r')\n",
    "plt.scatter(df1_2b.X,df1_2b.Z,c='b')\n",
    "xlabel('X')\n",
    "ylabel('Y (red) Z (blue)')\n",
    "title('Figure 1-2b, scatter plot showing only the positive side of the parabola')\n",
    "show()\n",
    "print(\"Correlation X-Z (blue dots):\",df1_2b.X.corr(df1_2b.Z))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Additionally, odd numbered power functions are likely to return higher correlations, such as $z = (3+x)^{3}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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gkn5BvvtpJRHx0T76nBbg7cAdwM45OAF4Atg5D48F5heyLc9pr+Th8vRSnmW5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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x18f2237bf60>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Correlation X-Z (blue dots): 0.956656779826\n"
     ]
    }
   ],
   "source": [
    "x = np.random.uniform(-100,100,20);\n",
    "y = 5+(3.5*x);\n",
    "z = (3+x)**3\n",
    "df1_2c = pd.DataFrame({\"X\":x,\"Y\":y,\"Z\":z})\n",
    "figure()\n",
    "plt.scatter(df1_2c.X,df1_2c.Y,c='r')\n",
    "plt.scatter(df1_2c.X,df1_2c.Z,c='b')\n",
    "xlabel('X')\n",
    "ylabel('Y (red) Z (blue)')\n",
    "title('Figure 1-2c, scatter plot showing a 3rd power function, which shows a linear correlation')\n",
    "show()\n",
    "print(\"Correlation X-Z (blue dots):\",df1_2c.X.corr(df1_2c.Z))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* The text also suggests that data with outliers are unlikely to correlate well, and uses a scatter plot of monthly S&P 500 returns and monthly US inflation rate, 1990-2013. This data is also fairly easy to pull from Python using \"pandas-datareader\". We will use SPY as a proxy for the S&P 500, with data provided by Google going back to 2001:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>CPIAUCSL</th>\n",
       "      <th>SPYClose</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>DATE</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>2001-03-01</th>\n",
       "      <td>0.000568</td>\n",
       "      <td>-0.082407</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2001-04-01</th>\n",
       "      <td>0.001704</td>\n",
       "      <td>-0.083467</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2001-05-01</th>\n",
       "      <td>0.005102</td>\n",
       "      <td>0.112522</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2001-06-01</th>\n",
       "      <td>0.002256</td>\n",
       "      <td>-0.002519</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2001-07-01</th>\n",
       "      <td>-0.001688</td>\n",
       "      <td>-0.020516</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2016-08-01</th>\n",
       "      <td>0.002047</td>\n",
       "      <td>0.033441</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2016-09-01</th>\n",
       "      <td>0.002567</td>\n",
       "      <td>0.002074</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2016-10-01</th>\n",
       "      <td>0.002855</td>\n",
       "      <td>-0.007406</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2016-11-01</th>\n",
       "      <td>0.002089</td>\n",
       "      <td>-0.022106</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2016-12-01</th>\n",
       "      <td>0.002568</td>\n",
       "      <td>0.040567</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>190 rows × 2 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "            CPIAUCSL  SPYClose\n",
       "DATE                          \n",
       "2001-03-01  0.000568 -0.082407\n",
       "2001-04-01  0.001704 -0.083467\n",
       "2001-05-01  0.005102  0.112522\n",
       "2001-06-01  0.002256 -0.002519\n",
       "2001-07-01 -0.001688 -0.020516\n",
       "...              ...       ...\n",
       "2016-08-01  0.002047  0.033441\n",
       "2016-09-01  0.002567  0.002074\n",
       "2016-10-01  0.002855 -0.007406\n",
       "2016-11-01  0.002089 -0.022106\n",
       "2016-12-01  0.002568  0.040567\n",
       "\n",
       "[190 rows x 2 columns]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "start = datetime.datetime(2001,2,1) #using SPY as a proxy for S&P 500, Google's data starts Feb 5th 2001.\n",
    "end = datetime.datetime(2017,1,1)\n",
    "CPIAUSCL = data.DataReader(\"CPIAUCSL\", \"fred\", start, end)\n",
    "#note to self, the key to the data here refers to the URL at the FRED.\n",
    "#CPIAUSCL comes from this url: https://fred.stlouisfed.org/series/CPIAUCSL\n",
    "#which stands for Consumer Price Index for All Urban Consumers.\n",
    "SPY_monthly = data.get_data_google('SPY', start=start, end=end).resample('MS').first() \n",
    "#get the google data, resample the dates to the start of the month so they match the FRED data\n",
    "SPY_monthly.rename(columns={'Close':'SPYClose'}, inplace=True) #rename the column\n",
    "CPI_SPY = CPIAUSCL.join(SPY_monthly.SPYClose, how='left') #merge the two datasets\n",
    "#as percent change. \n",
    "CPI_SPYpct = CPI_SPY.pct_change().dropna() #change to percent, and drop the N/A values\n",
    "CPI_SPYpct"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
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NcTqTiRNrS3ecAaCVCmYOsImkDSWtBHwUuLhIQUmrSlqt8h/YE7ijNEkdZ7AzfTqMGrV8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\n2n1Eu7fHiagAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x18f233f99e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Pearson Correlation Coefficient: 0.259480023502\n"
     ]
    }
   ],
   "source": [
    "#plotted:\n",
    "figure()\n",
    "plt.scatter(CPI_SPYpct.CPIAUCSL,CPI_SPYpct.SPYClose,c='r')\n",
    "xlabel('Consumer Price Index for All Urban Consumers - Monthly (%) Change')\n",
    "ylabel('SPY monthly price change (%)')\n",
    "title('Figure 1-3, scatter plot of CPI to S&P 500 Monthly Returns')\n",
    "show()\n",
    "print(\"Pearson Correlation Coefficient:\",CPI_SPYpct.CPIAUCSL.corr(CPI_SPYpct.SPYClose))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The data appears to have no linear relationship, however the correlation coefficient of implies a weak relationship. The outliers are probably giving more of the appearance of a linear trend than actually exists. The text goes on to hypothesize about possible relationships between stock market returns and CPI. I would add that as CPI is a lagging indicator, it might be more fruitful to lag the CPI by one month when correlating to returns. One must be very cautious when analyzing datasets like these, however, as it can be very easy to fall prey to datamining biases, or other **spurious correlations**. \n",
    "\n",
    "Even with random data, it's possible to find correlations which are meaningless. To illustrate this point, here is a code which loops random data until it finds a correlation of greater than 0.7:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>X</th>\n",
       "      <th>Y</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>99.134156</td>\n",
       "      <td>97.346280</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-12.903767</td>\n",
       "      <td>-18.980361</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>-32.201309</td>\n",
       "      <td>47.345598</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-82.943989</td>\n",
       "      <td>-84.006622</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-49.518923</td>\n",
       "      <td>31.442232</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>-49.940656</td>\n",
       "      <td>7.841045</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>25.165204</td>\n",
       "      <td>-11.878464</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>76.105833</td>\n",
       "      <td>57.769843</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>78.111467</td>\n",
       "      <td>99.380815</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>-64.191333</td>\n",
       "      <td>-85.345219</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>20 rows × 2 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "            X          Y\n",
       "0   99.134156  97.346280\n",
       "1  -12.903767 -18.980361\n",
       "2  -32.201309  47.345598\n",
       "3  -82.943989 -84.006622\n",
       "4  -49.518923  31.442232\n",
       "..        ...        ...\n",
       "15 -49.940656   7.841045\n",
       "16  25.165204 -11.878464\n",
       "17  76.105833  57.769843\n",
       "18  78.111467  99.380815\n",
       "19 -64.191333 -85.345219\n",
       "\n",
       "[20 rows x 2 columns]"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i = 0\n",
    "while i < 1000: #simple loop which keeps generating random data until it correlates\n",
    "    x = np.random.uniform(-100,100,20)\n",
    "    y = np.random.uniform(-100,100,20)\n",
    "    df1_4 = pd.DataFrame({\"X\":x,\"Y\":y})\n",
    "    if df1_4.X.corr(df1_4.Y)>0.7 :\n",
    "        break\n",
    "df1_4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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jj5nZUKpNohwovpWdmdmcUaemVzMzs75zojQzMyvhRGlmZlbCidLMzKyEE6WZ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\nsxJOlGZmZiWcKM3MzEr8f+G4zPS82gd2AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x18f234362e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Pearson Correlation Coefficient: 0.744161187014\n"
     ]
    }
   ],
   "source": [
    "figure()\n",
    "plt.scatter(df1_4.X,df1_4.Y,c='r')\n",
    "title('Figure 1-4, scatter plot of random data showing linear correlation of >0.7')\n",
    "show()\n",
    "print(\"Pearson Correlation Coefficient:\",df1_4.X.corr(df1_4.Y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "**Testing the Significance of the Correlation Coefficient:**\n",
    "We want to know whether our correlation (null hypothesis) is actually 0 ( $H_0 : p = 0$ ) and our alternate hypothesis is the correlation is not 0 ( $H_a : p \\ne 0$ ). As we learned in level 1, the T-test is a good way to evaluate the significance of a correlation, so for this we will use a 2-tailed T-test statistic.\n",
    "\n",
    "$$t-stat = \\frac{\\overline{X}-\\mu_0}{\\frac{s}{\\sqrt{n}}} $$\n",
    "\n",
    "however, we aren't working with sample mean data in this case. The institute provides this formula: \n",
    "\n",
    "$$ t = \\frac{r \\sqrt{n-2}}{\\sqrt{1-r^2}} $$\n",
    "\n",
    "*the origin of which is somewhat uncertain.*\n",
    "\n",
    "The decision rule for using this formula is that we reject $H_0$ if the $ |\\text{t-stat}| $ exceeds $ |\\text{t-critical}| $\n",
    "\n",
    "Using this formula with our example 1-4 we come up with a test statistic of: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.7263593980386664"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(df1_4.X.corr(df1_4.Y)*(df1_4.shape[0]-2)**0.5)/(1-df1_4.X.corr(df1_4.Y)**2)**0.5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And using a student's T-table, at the 0.05 threshold our T-critical is 2.1009. Since the test statistic is greater than the critical value, we can reject the null hypothesis of no correlation at the 5% significance level.\n",
    "\n",
    "The institute notes that with a sufficient population given a certain r value, it should be able to just meet the t-critical. Let's test this by modifying our random data loop slightly to reject the null hypothesis with a low correlation coefficient, something that you wouldn't want to trade on. Let's give it a correlation coefficient of 0.35, but while rejecting the null hypothesis. We'll go with a population of n=32 (degrees of freedom = 30). At the 5% significance level (0.05), our t-critical is 2.046, so we can run the loop until our t-stat rejects the null hypothesis like so:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>X</th>\n",
       "      <th>Y</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>95.323739</td>\n",
       "      <td>-99.714708</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-84.754452</td>\n",
       "      <td>44.099240</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>55.086953</td>\n",
       "      <td>-84.587189</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-16.187571</td>\n",
       "      <td>-56.174989</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-44.815103</td>\n",
       "      <td>72.410018</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27</th>\n",
       "      <td>-1.495417</td>\n",
       "      <td>83.584759</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>-50.451061</td>\n",
       "      <td>-19.430034</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29</th>\n",
       "      <td>-5.673247</td>\n",
       "      <td>-15.951552</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>30</th>\n",
       "      <td>-24.521747</td>\n",
       "      <td>66.110319</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>31</th>\n",
       "      <td>48.693688</td>\n",
       "      <td>-59.352024</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>32 rows × 2 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "            X          Y\n",
       "0   95.323739 -99.714708\n",
       "1  -84.754452  44.099240\n",
       "2   55.086953 -84.587189\n",
       "3  -16.187571 -56.174989\n",
       "4  -44.815103  72.410018\n",
       "..        ...        ...\n",
       "27  -1.495417  83.584759\n",
       "28 -50.451061 -19.430034\n",
       "29  -5.673247 -15.951552\n",
       "30 -24.521747  66.110319\n",
       "31  48.693688 -59.352024\n",
       "\n",
       "[32 rows x 2 columns]"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i = 0\n",
    "while i < 1: #simple loop which keeps generating random data until it correlates\n",
    "    x = np.random.uniform(-100,100,32)\n",
    "    y = np.random.uniform(-100,100,32)\n",
    "    df1_5 = pd.DataFrame({\"X\":x,\"Y\":y})\n",
    "    if abs(df1_5.X.corr(df1_5.Y)) < 0.35 and abs(((df1_5.X.corr(df1_5.Y)*(df1_5.shape[0]-2)**0.5)/(1-df1_5.X.corr(df1_5.Y)**2)**0.5)) > 2.046 :\n",
    "        break\n",
    "df1_5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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2wptwcxd0bdpUmhOsW1eSWC/0kiQNicjMtmOQlm1qaipnZ2fbDkOSRkZEXJiZ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\nTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x18f21ec2ef0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Pearson Correlation Coefficient: -0.349978477298 \n",
      "T-stat :  -2.04632530115\n"
     ]
    }
   ],
   "source": [
    "figure()\n",
    "plt.scatter(df1_5.X,df1_5.Y,c='r')\n",
    "title('Figure 1-5, scatter plot of random data of n=32 which fails to reject the null hypothesis with a t-critical of 2.046')\n",
    "show()\n",
    "print(\"Pearson Correlation Coefficient:\",df1_5.X.corr(df1_5.Y), \n",
    "     \"\\nT-stat : \",(df1_5.X.corr(df1_5.Y)*(df1_5.shape[0]-2)**0.5)/(1-df1_5.X.corr(df1_5.Y)**2)**0.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As you can see, the scatter plot appears meaningless - but we've successfully rejected our null hypothesis! \n",
    "\n",
    "This goes to show that you can lie with statistics. With our two examples we have found signals in the noise which are both 'statistically significant' based on two polulations fo random data. \n",
    "\n",
    "These tools are a good starting point for evaluating relationships in datasets, but are limited and must be applied correctly to have meaning.\n",
    "\n",
    "---\n",
    "\n",
    "Going forward, it is important to remember that statistics is a very deep field of study, and python contains a great wealth of statistics packages and modules. Be sure to check the assumptions of every module before you use it.\n",
    "\n",
    "In part 2 we will look at linear regression!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
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   "language": "python",
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    "name": "ipython",
    "version": 3
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