6. Trend & Seasonality.ipynb

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   "source": [
    "%matplotlib inline\n",
    "from matplotlib.pylab import plt\n",
    "\n",
    "import pandas as pd\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
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   "outputs": [],
   "source": [
    "air_passengers = pd.read_csv(\"data/AirPassengers.csv\", header = 0, parse_dates = [0], names = ['Month', 'Passengers'], index_col = 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
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   },
   "outputs": [],
   "source": [
    "air_passengers.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "air_passengers = air_passengers.to_period()"
   ]
  },
  {
   "cell_type": "code",
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   "outputs": [],
   "source": [
    "air_passengers['1950-06-01':'1950-06-03']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   "outputs": [],
   "source": [
    "air_passengers.plot()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
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   "outputs": [],
   "source": [
    "# first thing we look at for a time series when we want to use common analysis techniques:\n",
    "# is it stationary: constant mean, constant variance, autocovariance does not depend on time"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# first test, informally, plot the moving average\n",
    "air_passengers.rolling(window = 20).mean().plot()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   "source": [
    "# Can you plot the variance?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
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   },
   "outputs": [],
   "source": [
    "# Can you plot the autocovariance?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
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   },
   "outputs": [],
   "source": [
    "# How should you size your window?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   },
   "outputs": [],
   "source": [
    "# next more formally apply the Dickey-Fuller test\n",
    "from statsmodels.tsa.stattools import adfuller\n",
    "\n",
    "dftest = adfuller(air_passengers.Passengers, autolag = 'AIC')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   "outputs": [],
   "source": [
    "dftest\n",
    "\n",
    "# What do these numbers mean?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
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   },
   "outputs": [],
   "source": [
    "# Can we write a function to output these #s sensibly?\n",
    "# Check out statsmodels.sourceforge.net/devel/generated/statsmodels.tsa.stattools.adfuller.html"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   "outputs": [],
   "source": [
    "# Are we stuck not doing any analysis now that our time series is not stationary? Hint: no.\n",
    "# How can we make it stationary?\n",
    "# Why is it non-stationary (2 reasons)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# First let's get rid of the 'trend' contribution to the time series not being stationary\n",
    "# One method to get rid of a trend is to do a power or log transformation that punishes larger values\n",
    "# more than smaller values\n",
    "log_passengers = air_passengers.Passengers.apply(lambda x: np.log(x))\n",
    "log_passengers.plot()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# This reduces the variance in variance as opposed to the original trend\n",
    "air_passengers.plot()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   },
   "outputs": [],
   "source": [
    "# However there is still a trend we need to remove\n",
    "# Let's calculate a rolling mean\n",
    "# Experiment with window size\n",
    "air_passengers.rolling(window = 12).mean().plot()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# What if we subtract this rolling mean from the original series?\n",
    "rolling_mean = air_passengers.rolling(window = 12).mean()\n",
    "passengers_detrended = air_passengers - rolling_mean\n",
    "passengers_detrended.plot()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Exercise: try detrending after taking the log. How does that look?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Now let's use a regression rather than a rolling mean to detrend\n",
    "from statsmodels.regression.linear_model import OLS\n",
    "model = OLS(air_passengers.Passengers.values, list(range(len(air_passengers.values))))\n",
    "result = model.fit()\n",
    "result.params\n",
    "fit = pd.Series(result.predict(list(range(len(air_passengers.values)))), index = air_passengers.index)\n",
    "\n",
    "passengers_detrended = air_passengers.Passengers - fit\n",
    "passengers_detrended.plot()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Now let's take a look at seasonality\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# When might a rolling average work better/worse than a regression?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# How might we be able to eliminate both trend and seasonality?\n",
    "# Looking for two answers..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# One common technique is differencing, let's start with log_passengers\n",
    "log_passengers_diff = log_passengers - log_passengers.shift()\n",
    "log_passengers_diff.plot()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Let's again test for stationarity with a gut level check.\n",
    "# And let's write a function to do it since this seems like something we'll have to do a lot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
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   "outputs": [],
   "source": [
    "log_passengers = log_passengers.to_timestamp()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from statsmodels.tsa.seasonal import seasonal_decompose\n",
    "\n",
    "log_passengers.interpolate(inplace = True)\n",
    "decomposition = seasonal_decompose(log_passengers)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "trend = decomposition.trend\n",
    "seasonal = decomposition.seasonal\n",
    "residual = decomposition.resid"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "(seasonal+trend).plot()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "\n",
    "plt.subplot(411)\n",
    "plt.plot(log_passengers, label='Original')\n",
    "plt.legend(loc='best')\n",
    "plt.subplot(412)\n",
    "plt.plot(trend, label='Trend')\n",
    "plt.legend(loc='best')\n",
    "plt.subplot(413)\n",
    "plt.plot(seasonal,label='Seasonality')\n",
    "plt.legend(loc='best')\n",
    "plt.subplot(414)\n",
    "plt.plot(residual, label='Residuals')\n",
    "plt.legend(loc='best')\n",
    "plt.tight_layout()"
   ]
  }
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