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ML-For-Beginners/TimeSeries/2-ARIMA/solution/notebook.ipynb

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{
"cells": [
{
"source": [
"# Time series forecasting with ARIMA\n",
"\n",
"In this notebook, we demonstrate how to:\n",
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"- prepare time series data for training an ARIMA time series forecasting model\n",
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"- implement a simple ARIMA model to forecast the next HORIZON steps ahead (time *t+1* through *t+HORIZON*) in the time series\n",
"- evaluate the model \n",
"\n",
"\n",
"The data in this example is taken from the GEFCom2014 forecasting competition<sup>1</sup>. It consists of 3 years of hourly electricity load and temperature values between 2012 and 2014. The task is to forecast future values of electricity load. In this example, we show how to forecast one time step ahead, using historical load data only.\n",
"\n",
"<sup>1</sup>Tao Hong, Pierre Pinson, Shu Fan, Hamidreza Zareipour, Alberto Troccoli and Rob J. Hyndman, \"Probabilistic energy forecasting: Global Energy Forecasting Competition 2014 and beyond\", International Journal of Forecasting, vol.32, no.3, pp 896-913, July-September, 2016."
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],
"cell_type": "markdown",
"metadata": {}
},
{
"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"Requirement already satisfied: statsmodels in /Library/Frameworks/Python.framework/Versions/3.7/lib/python3.7/site-packages (0.12.2)\n",
"Requirement already satisfied: numpy>=1.15 in /Library/Frameworks/Python.framework/Versions/3.7/lib/python3.7/site-packages (from statsmodels) (1.19.2)\n",
"Requirement already satisfied: pandas>=0.21 in /Library/Frameworks/Python.framework/Versions/3.7/lib/python3.7/site-packages (from statsmodels) (1.1.2)\n",
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"Requirement already satisfied: scipy>=1.1 in /Library/Frameworks/Python.framework/Versions/3.7/lib/python3.7/site-packages (from statsmodels) (1.4.1)\n",
"Requirement already satisfied: patsy>=0.5 in /Library/Frameworks/Python.framework/Versions/3.7/lib/python3.7/site-packages (from statsmodels) (0.5.1)\n",
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"Requirement already satisfied: pytz>=2017.2 in /Library/Frameworks/Python.framework/Versions/3.7/lib/python3.7/site-packages (from pandas>=0.21->statsmodels) (2019.1)\n",
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"Requirement already satisfied: python-dateutil>=2.7.3 in /Library/Frameworks/Python.framework/Versions/3.7/lib/python3.7/site-packages (from pandas>=0.21->statsmodels) (2.8.0)\n",
"Requirement already satisfied: six in /Users/jenlooper/Library/Python/3.7/lib/python/site-packages (from patsy>=0.5->statsmodels) (1.12.0)\n",
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"\u001b[33mWARNING: You are using pip version 20.2.3; however, version 21.1.1 is available.\n",
"You should consider upgrading via the '/Library/Frameworks/Python.framework/Versions/3.7/bin/python3.7 -m pip install --upgrade pip' command.\u001b[0m\n",
"Note: you may need to restart the kernel to use updated packages.\n"
]
}
],
"source": [
"pip install statsmodels"
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]
},
{
"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
"outputs": [],
"source": [
"import os\n",
"import warnings\n",
"import matplotlib.pyplot as plt\n",
"import numpy as np\n",
"import pandas as pd\n",
"import datetime as dt\n",
"import math\n",
"\n",
"from pandas.plotting import autocorrelation_plot\n",
"from statsmodels.tsa.statespace.sarimax import SARIMAX\n",
"from sklearn.preprocessing import MinMaxScaler\n",
"from common.utils import load_data, mape\n",
"from IPython.display import Image\n",
"\n",
"%matplotlib inline\n",
"pd.options.display.float_format = '{:,.2f}'.format\n",
"np.set_printoptions(precision=2)\n",
"warnings.filterwarnings(\"ignore\") # specify to ignore warning messages\n"
]
},
{
"cell_type": "code",
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"execution_count": 3,
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"metadata": {},
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": [
" load\n",
"2012-01-01 00:00:00 2,698.00\n",
"2012-01-01 01:00:00 2,558.00\n",
"2012-01-01 02:00:00 2,444.00\n",
"2012-01-01 03:00:00 2,402.00\n",
"2012-01-01 04:00:00 2,403.00\n",
"2012-01-01 05:00:00 2,453.00\n",
"2012-01-01 06:00:00 2,560.00\n",
"2012-01-01 07:00:00 2,719.00\n",
"2012-01-01 08:00:00 2,916.00\n",
"2012-01-01 09:00:00 3,105.00"
],
"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>load</th>\n </tr>\n </thead>\n <tbody>\n <tr>\n <th>2012-01-01 00:00:00</th>\n <td>2,698.00</td>\n </tr>\n <tr>\n <th>2012-01-01 01:00:00</th>\n <td>2,558.00</td>\n </tr>\n <tr>\n <th>2012-01-01 02:00:00</th>\n <td>2,444.00</td>\n </tr>\n <tr>\n <th>2012-01-01 03:00:00</th>\n <td>2,402.00</td>\n </tr>\n <tr>\n <th>2012-01-01 04:00:00</th>\n <td>2,403.00</td>\n </tr>\n <tr>\n <th>2012-01-01 05:00:00</th>\n <td>2,453.00</td>\n </tr>\n <tr>\n <th>2012-01-01 06:00:00</th>\n <td>2,560.00</td>\n </tr>\n <tr>\n <th>2012-01-01 07:00:00</th>\n <td>2,719.00</td>\n </tr>\n <tr>\n <th>2012-01-01 08:00:00</th>\n <td>2,916.00</td>\n </tr>\n <tr>\n <th>2012-01-01 09:00:00</th>\n <td>3,105.00</td>\n </tr>\n </tbody>\n</table>\n</div>"
},
"metadata": {},
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"execution_count": 3
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}
],
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"source": [
"energy = load_data('./data')[['load']]\n",
"energy.head(10)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Plot all available load data (January 2012 to Dec 2014)"
]
},
{
"cell_type": "code",
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"execution_count": 4,
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"metadata": {},
"outputs": [
{
"output_type": "display_data",
"data": {
"text/plain": "<Figure size 1080x576 with 1 Axes>",
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},
"metadata": {
"needs_background": "light"
}
}
],
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"source": [
"energy.plot(y='load', subplots=True, figsize=(15, 8), fontsize=12)\n",
"plt.xlabel('timestamp', fontsize=12)\n",
"plt.ylabel('load', fontsize=12)\n",
"plt.show()"
]
},
{
"source": [
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"## Create training and testing data sets\n"
],
"cell_type": "markdown",
"metadata": {}
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},
{
"cell_type": "code",
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"execution_count": 5,
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"metadata": {},
"outputs": [],
"source": [
"train_start_dt = '2014-11-01 00:00:00'\n",
"test_start_dt = '2014-12-30 00:00:00' "
]
},
{
"cell_type": "code",
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"execution_count": 6,
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"metadata": {},
"outputs": [
{
"output_type": "display_data",
"data": {
"text/plain": "<Figure size 1080x576 with 1 Axes>",
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time series work
4 years ago
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},
"metadata": {
"needs_background": "light"
}
}
],
time series 1
4 years ago
"source": [
"energy[(energy.index < test_start_dt) & (energy.index >= train_start_dt)][['load']].rename(columns={'load':'train'}) \\\n",
" .join(energy[test_start_dt:][['load']].rename(columns={'load':'test'}), how='outer') \\\n",
" .plot(y=['train', 'test'], figsize=(15, 8), fontsize=12)\n",
"plt.xlabel('timestamp', fontsize=12)\n",
"plt.ylabel('load', fontsize=12)\n",
"plt.show()"
]
},
{
"cell_type": "code",
time series 2
4 years ago
"execution_count": 7,
time series work
4 years ago
"metadata": {},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"Training data shape: (1416, 1)\nTest data shape: (48, 1)\n"
]
}
],
time series 1
4 years ago
"source": [
"train = energy.copy()[(energy.index >= train_start_dt) & (energy.index < test_start_dt)][['load']]\n",
"test = energy.copy()[energy.index >= test_start_dt][['load']]\n",
"\n",
"print('Training data shape: ', train.shape)\n",
"print('Test data shape: ', test.shape)"
]
},
{
"cell_type": "code",
time series 2
4 years ago
"execution_count": 8,
time series work
4 years ago
"metadata": {},
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": [
" load\n",
"2014-11-01 00:00:00 0.10\n",
"2014-11-01 01:00:00 0.07\n",
"2014-11-01 02:00:00 0.05\n",
"2014-11-01 03:00:00 0.04\n",
"2014-11-01 04:00:00 0.06\n",
"2014-11-01 05:00:00 0.10\n",
"2014-11-01 06:00:00 0.19\n",
"2014-11-01 07:00:00 0.31\n",
"2014-11-01 08:00:00 0.40\n",
"2014-11-01 09:00:00 0.48"
],
"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>load</th>\n </tr>\n </thead>\n <tbody>\n <tr>\n <th>2014-11-01 00:00:00</th>\n <td>0.10</td>\n </tr>\n <tr>\n <th>2014-11-01 01:00:00</th>\n <td>0.07</td>\n </tr>\n <tr>\n <th>2014-11-01 02:00:00</th>\n <td>0.05</td>\n </tr>\n <tr>\n <th>2014-11-01 03:00:00</th>\n <td>0.04</td>\n </tr>\n <tr>\n <th>2014-11-01 04:00:00</th>\n <td>0.06</td>\n </tr>\n <tr>\n <th>2014-11-01 05:00:00</th>\n <td>0.10</td>\n </tr>\n <tr>\n <th>2014-11-01 06:00:00</th>\n <td>0.19</td>\n </tr>\n <tr>\n <th>2014-11-01 07:00:00</th>\n <td>0.31</td>\n </tr>\n <tr>\n <th>2014-11-01 08:00:00</th>\n <td>0.40</td>\n </tr>\n <tr>\n <th>2014-11-01 09:00:00</th>\n <td>0.48</td>\n </tr>\n </tbody>\n</table>\n</div>"
},
"metadata": {},
time series 2
4 years ago
"execution_count": 8
time series work
4 years ago
}
],
time series 1
4 years ago
"source": [
"scaler = MinMaxScaler()\n",
"train['load'] = scaler.fit_transform(train)\n",
"train.head(10)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Original vs scaled data:"
]
},
{
"cell_type": "code",
time series 2
4 years ago
"execution_count": 9,
time series work
4 years ago
"metadata": {},
"outputs": [
{
"output_type": "display_data",
"data": {
"text/plain": "<Figure size 432x288 with 1 Axes>",
time series 2
4 years ago
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time series work
4 years ago
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time series 2
4 years ago
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time series work
4 years ago
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAYEAAAD7CAYAAACMlyg3AAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4xLjAsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy+17YcXAAAaCklEQVR4nO3de5RV5Znn8e+PSyjlIhEK1GBZagJeQBQL7ehSM9EWO2omhjjRqOOltaIOsTNZzdLJ0oTYtunOmpl0JEokbTAg2iRpcKKxXYKXVmNPQsE0ERQ0LsHQogIJSCF3n/ljn9JjUUXtU5x9Tp3av89aZ8m+nf285+B5ePfz7ncrIjAzs3zqU+0AzMysepwEzMxyzEnAzCzHnATMzHLMScDMLMf6VTuAUgwfPjwaGxurHYaZWU1ZsmTJhoio72hbTSWBxsZGWlpaqh2GmVlNkbSms22+HGRmlmNOAmZmOeYkYGaWYzVVEzCz2rJr1y7Wrl3L9u3bqx1KLtTV1TFq1Cj69++f+hgnATPLzNq1axk8eDCNjY1IqnY4vVpEsHHjRtauXcuRRx6Z+jhfDjKzzGzfvp1hw4Y5AVSAJIYNG1Zyr8tJwMwy5QRQOd35rJ0EzMxyzDUBM6uYxlt+Vdb3W/1355f1/To8x+rVXHDBBSxfvjz1MVdddRUXXHABX/rSl1Ktr0RMnXESMMuB4h/fSvxwWu3w5SAz67W2bt3K+eefz/jx4xk7dizz5s0DYPHixZx22mmMHz+eU045hS1btrB69WrOOOMMJkyYwIQJE3jhhRf2er89e/YwdepUJk6cyAknnMC9994LJCNzpkyZwpgxYzjnnHN45513uoztySef5KSTTmLcuHFcc8017NixA4Dbb7+diRMnMnbsWJqbm2l7+uOSJUsYP34848eP5+677y7XR1SZJCDpU5K2S3qgaN1XJK2RtFXSw5IOrkQsZpYfjz/+OIcddhjLli1j+fLlnHfeeezcuZMvf/nL/OAHP2DZsmUsWrSIAw44gBEjRrBw4UKWLl3KvHnzuOmmm/Z6v/vuu4+DDjqIxYsXs3jxYn784x/z+uuvs2DBAlatWsVLL73E7NmzO0wgxbZv385VV13FvHnzePHFF9m9ezczZswAYMqUKSxevJjly5ezbds2Hn30UQCuvvpqpk+fzrJly8r6GVWqJ3A3sLhtQdLxwL3AFcBI4D3gngrFYmY5MW7cOBYuXMjNN9/Mc889x0EHHcSqVas49NBDmThxIgBDhgyhX79+7Nq1i+uuu45x48Zx8cUX89JLL+31fk888QSzZ8/mxBNP5NRTT2Xjxo28+uqrPPvss1x66aX07duXww47jM9+9rP7jGvVqlUceeSRjB49GoArr7ySZ599FoCnn36aU089lXHjxvHUU0+xYsUKNm3axKZNmzjzzDMBuOKKK8r2GWVeE5B0CbAJeAH4ZGH1ZcAjEfFsYZ/bgJclDY6ILVnHZGb5MHr0aJYuXcpjjz3Grbfeytlnn81FF13U4b7f//73GTlyJMuWLeP999+nrq5ur30igunTpzNp0qSPrH/sscfKEu/27du58cYbaWlp4fDDD2fatGmZ322daU9A0hDgduAb7TYdD3zQp4mI14CdwOgO3qNZUouklvXr12cZrlnuNN7yqw9evdGbb77JgQceyOWXX87UqVNZunQpY8aMYd26dSxenFyc2LJlC7t372bz5s0ceuih9OnThzlz5rBnz5693m/SpEnMmDGDXbt2AfDKK6+wdetWzjzzTObNm8eePXtYt24dTz/99D7jGjNmDKtXr+b3v/89AHPmzOGss8764Ad/+PDhtLa28otf/AKAoUOHMnToUJ5//nkA5s6dW54PiOx7An8D3BcRa9vdxDAI2Nxu383A4PZvEBEzgZkATU1NkVGcZlYBlR6Z9OKLLzJ16lT69OlD//79mTFjBh/72MeYN28eX/va19i2bRsHHHAAixYt4sYbb2Ty5MnMnj2b8847j4EDB+71ftdeey2rV69mwoQJRAT19fU8/PDDXHTRRTz11FMcd9xxNDQ08OlPf3qfcdXV1TFr1iwuvvhidu/ezcSJE7n++usZMGAA1113HWPHjuWQQw754JIVwKxZs7jmmmuQxLnnnlu2z0htledyk3QiMBc4KSJ2SpoGfDIiLpf0f4BfR8T3ivbfAnwmIpZ09p5NTU3hh8qYla6zIaJZDx19+eWXOfbYY8v+vta5jj5zSUsioqmj/bPsCXwGaATeKPQCBgF9JR0HPA6MLwrwKGAA8EqG8ZiZWTtZJoGZwD8VLf81SVK4ARgB/JukM4ClJHWD+S4Km5lVVmZJICLeIxn6CYCkVmB7RKwH1ku6nuRy0TBgEXB1VrGYWfVEhCeRq5DuXN6v2LQRETGt3fKDwIOVOr+ZVV5dXR0bN270dNIV0PY8gY6Gtu6L5w4ys8yMGjWKtWvX4uHdldH2ZLFSOAmYWWb69+9f0lOurPI8gZyZWY45CZiZ5ZiTgJlZjjkJmJnlmJOAmVmOOQmYmeWYk4CZWY45CZiZ5ZiTgJlZjjkJmJnlmJOAmVmOee4gsx4g6yd8mXXGPQEzsxzLNAlIekDSOknvSnpF0rWF9Y2SQlJr0eu2LGMxM7O9ZX056LvAX0bEDknHAM9I+n/AxsL2oRGxO+MYzMysE5n2BCJiRUTsaFssvI7O8pxmZpZe5jUBSfdIeg9YCawDHivavEbSWkmzJA3v5PhmSS2SWvx0IjOz8so8CUTEjcBg4AxgPrAD2ABMBI4ATi5sn9vJ8TMjoikimurr67MO18wsVyoyOigi9kTE88Ao4IaIaI2IlojYHRFvA1OAcyUNrkQ8ZmaWqPQQ0X50XBOIwn89ZNXMrIIy+9GVNELSJZIGSeoraRJwKfCkpFMljZHUR9Iw4C7gmYjYnFU8Zma2tyyHiAZwA/AjkmSzBvh6RPxS0qXAncAI4F1gIUmCMLMeoPgOZvBdzL1ZZkkgItYDZ3Wy7SHgoazObWZm6fgavJlZjjkJmJnlmJOAmVmOeSppM+uSp7ruvdwTMDPLMScBM7MccxIwM8sxJwEzsxxzYdgsYy6qWk/mnoCZWY45CZiZ5ZiTgJlZjjkJmJnlmJOAmVmOOQmYmeVYpklA0gOS1kl6V9Irkq4t2na2pJWS3pP0tKQjsozFzMz2lnVP4LtAY0QMAT4P3CHpZEnDgfnAbcDBQAswL+NYzMysnUxvFouIFcWLhdfRwMnAioj4OYCkacAGScdExMosYzIzsw9lXhOQdI+k94CVwDrgMeB4YFnbPhGxFXitsL798c2SWiS1rF+/PutwzcxyJfMkEBE3AoOBM0guAe0ABgGb2+26ubBf++NnRkRTRDTV19dnHa6ZWa5UZHRQROyJiOeBUcANQCswpN1uQ4AtlYjHzMwSlR4i2o+kJrACGN+2UtLAovVmZlYhmSUBSSMkXSJpkKS+kiYBlwJPAguAsZImS6oDvgX8zkVhM7PKynJ0UJBc+vkRSbJZA3w9In4JIGky8EPgAeA3wCUZxmLWI2Q9rbSnrbZSZZYEImI9cNY+ti8Cjsnq/GZm1jVPG2FmlmNOAmZmOeYkYGaWY37GsFkP4+KuVZJ7AmZmOeYkYGaWY04CZmY55iRgZpZjLgybWUlcuO5d3BMwM8sxJwEzsxxzEjAzy7FUSUDSuKwDMTOzyktbGL5H0gDgfmBuRLR/NKSZZcBFWMtaqp5ARJwBXAYcDiyR9KCkP880MjMzy1zqmkBEvArcCtxM8pyAuyStlPTFjvaXNEDSfZLWSNoi6d8l/UVhW6OkkNRa9LqtHA0yM7P0Ul0OknQCcDVwPrAQuDAilko6DPg3YH4n7/0HkoTxBvA54Gft6gtDI2L3fsRvZmb7IW1NYDrwj8A3I2Jb28qIeFPSrR0dEBFbgWlFqx6V9DpwMrCke+GamVk5pU0C5wPbImIPgKQ+QF1EvBcRc9K8gaSRwGhgRdHqNZKCpHcxNSI2pA/dzMz2V9oksAg4B2gtLB8IPAGcluZgSf2BucBPI2KlpEHARODfgWHA3YXtkzo
},
"metadata": {
"needs_background": "light"
}
}
],
time series 1
4 years ago
"source": [
"energy[(energy.index >= train_start_dt) & (energy.index < test_start_dt)][['load']].rename(columns={'load':'original load'}).plot.hist(bins=100, fontsize=12)\n",
"train.rename(columns={'load':'scaled load'}).plot.hist(bins=100, fontsize=12)\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's also scale the test data"
]
},
{
"cell_type": "code",
time series 2
4 years ago
"execution_count": 10,
time series work
4 years ago
"metadata": {},
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": [
" load\n",
"2014-12-30 00:00:00 0.33\n",
"2014-12-30 01:00:00 0.29\n",
"2014-12-30 02:00:00 0.27\n",
"2014-12-30 03:00:00 0.27\n",
"2014-12-30 04:00:00 0.30"
],
"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>load</th>\n </tr>\n </thead>\n <tbody>\n <tr>\n <th>2014-12-30 00:00:00</th>\n <td>0.33</td>\n </tr>\n <tr>\n <th>2014-12-30 01:00:00</th>\n <td>0.29</td>\n </tr>\n <tr>\n <th>2014-12-30 02:00:00</th>\n <td>0.27</td>\n </tr>\n <tr>\n <th>2014-12-30 03:00:00</th>\n <td>0.27</td>\n </tr>\n <tr>\n <th>2014-12-30 04:00:00</th>\n <td>0.30</td>\n </tr>\n </tbody>\n</table>\n</div>"
},
"metadata": {},
time series 2
4 years ago
"execution_count": 10
time series work
4 years ago
}
],
time series 1
4 years ago
"source": [
"test['load'] = scaler.transform(test)\n",
"test.head()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Implement ARIMA method"
]
},
{
"cell_type": "code",
time series 2
4 years ago
"execution_count": 11,
time series work
4 years ago
"metadata": {},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"Forecasting horizon: 3 hours\n"
]
}
],
time series 1
4 years ago
"source": [
"# Specify the number of steps to forecast ahead\n",
"HORIZON = 3\n",
"print('Forecasting horizon:', HORIZON, 'hours')"
]
},
{
"cell_type": "code",
time series 2
4 years ago
"execution_count": 29,
time series work
4 years ago
"metadata": {},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
time series 2
4 years ago
" SARIMAX Results \n==========================================================================================\nDep. Variable: load No. Observations: 1416\nModel: SARIMAX(4, 1, 0)x(1, 1, 0, 24) Log Likelihood 3477.239\nDate: Fri, 14 May 2021 AIC -6942.477\nTime: 17:05:41 BIC -6911.050\nSample: 11-01-2014 HQIC -6930.725\n - 12-29-2014 \nCovariance Type: opg \n==============================================================================\n coef std err z P>|z| [0.025 0.975]\n------------------------------------------------------------------------------\nar.L1 0.8403 0.016 52.226 0.000 0.809 0.872\nar.L2 -0.5220 0.034 -15.388 0.000 -0.588 -0.456\nar.L3 0.1536 0.044 3.470 0.001 0.067 0.240\nar.L4 -0.0778 0.036 -2.158 0.031 -0.148 -0.007\nar.S.L24 -0.2327 0.024 -9.718 0.000 -0.280 -0.186\nsigma2 0.0004 8.32e-06 47.358 0.000 0.000 0.000\n===================================================================================\nLjung-Box (L1) (Q): 0.05 Jarque-Bera (JB): 1464.60\nProb(Q): 0.83 Prob(JB): 0.00\nHeteroskedasticity (H): 0.84 Skew: 0.14\nProb(H) (two-sided): 0.07 Kurtosis: 8.02\n===================================================================================\n\nWarnings:\n[1] Covariance matrix calculated using the outer product of gradients (complex-step).\n"
time series work
4 years ago
]
}
],
time series 1
4 years ago
"source": [
"order = (4, 1, 0)\n",
"seasonal_order = (1, 1, 0, 24)\n",
"\n",
"model = SARIMAX(endog=train, order=order, seasonal_order=seasonal_order)\n",
"results = model.fit()\n",
"\n",
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"print(results.summary())\n"
time series 1
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]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Next we display the distribution of residuals. A zero mean in the residuals may indicate that there is no bias in the prediction. "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Evaluate the model"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Create a test data point for each HORIZON step."
]
},
{
"cell_type": "code",
time series 2
4 years ago
"execution_count": 13,
time series work
4 years ago
"metadata": {},
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": [
" load load+1 load+2\n",
"2014-12-30 00:00:00 0.33 0.29 0.27\n",
"2014-12-30 01:00:00 0.29 0.27 0.27\n",
"2014-12-30 02:00:00 0.27 0.27 0.30\n",
"2014-12-30 03:00:00 0.27 0.30 0.41\n",
"2014-12-30 04:00:00 0.30 0.41 0.57"
],
"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>load</th>\n <th>load+1</th>\n <th>load+2</th>\n </tr>\n </thead>\n <tbody>\n <tr>\n <th>2014-12-30 00:00:00</th>\n <td>0.33</td>\n <td>0.29</td>\n <td>0.27</td>\n </tr>\n <tr>\n <th>2014-12-30 01:00:00</th>\n <td>0.29</td>\n <td>0.27</td>\n <td>0.27</td>\n </tr>\n <tr>\n <th>2014-12-30 02:00:00</th>\n <td>0.27</td>\n <td>0.27</td>\n <td>0.30</td>\n </tr>\n <tr>\n <th>2014-12-30 03:00:00</th>\n <td>0.27</td>\n <td>0.30</td>\n <td>0.41</td>\n </tr>\n <tr>\n <th>2014-12-30 04:00:00</th>\n <td>0.30</td>\n <td>0.41</td>\n <td>0.57</td>\n </tr>\n </tbody>\n</table>\n</div>"
},
"metadata": {},
time series 2
4 years ago
"execution_count": 13
time series work
4 years ago
}
],
time series 1
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"source": [
"test_shifted = test.copy()\n",
"\n",
"for t in range(1, HORIZON):\n",
" test_shifted['load+'+str(t)] = test_shifted['load'].shift(-t, freq='H')\n",
" \n",
"test_shifted = test_shifted.dropna(how='any')\n",
"test_shifted.head(5)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Make predictions on the test data"
]
},
{
"cell_type": "code",
time series 2
4 years ago
"execution_count": 14,
time series 1
4 years ago
"metadata": {
"scrolled": true
},
time series work
4 years ago
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"2014-12-30 00:00:00\n",
"1 : predicted = [0.32 0.29 0.28] expected = [0.32945389435989236, 0.2900626678603402, 0.2739480752014323]\n",
"2014-12-30 01:00:00\n",
"2 : predicted = [0.3 0.29 0.3 ] expected = [0.2900626678603402, 0.2739480752014323, 0.26812891674127126]\n",
"2014-12-30 02:00:00\n",
"3 : predicted = [0.27 0.28 0.32] expected = [0.2739480752014323, 0.26812891674127126, 0.3025962399283795]\n",
"2014-12-30 03:00:00\n",
"4 : predicted = [0.28 0.32 0.42] expected = [0.26812891674127126, 0.3025962399283795, 0.40823634735899716]\n",
"2014-12-30 04:00:00\n",
"5 : predicted = [0.3 0.39 0.54] expected = [0.3025962399283795, 0.40823634735899716, 0.5689346463742166]\n",
"2014-12-30 05:00:00\n",
"6 : predicted = [0.4 0.55 0.66] expected = [0.40823634735899716, 0.5689346463742166, 0.6799462846911368]\n",
"2014-12-30 06:00:00\n",
"7 : predicted = [0.57 0.68 0.75] expected = [0.5689346463742166, 0.6799462846911368, 0.7309758281110115]\n",
"2014-12-30 07:00:00\n",
"8 : predicted = [0.68 0.75 0.8 ] expected = [0.6799462846911368, 0.7309758281110115, 0.7511190689346463]\n",
"2014-12-30 08:00:00\n",
"9 : predicted = [0.75 0.8 0.82] expected = [0.7309758281110115, 0.7511190689346463, 0.7636526410026856]\n",
"2014-12-30 09:00:00\n",
"10 : predicted = [0.77 0.78 0.78] expected = [0.7511190689346463, 0.7636526410026856, 0.7381378692927483]\n",
"2014-12-30 10:00:00\n",
"11 : predicted = [0.76 0.75 0.74] expected = [0.7636526410026856, 0.7381378692927483, 0.7188898836168307]\n",
"2014-12-30 11:00:00\n",
"12 : predicted = [0.77 0.76 0.75] expected = [0.7381378692927483, 0.7188898836168307, 0.7090420769919425]\n",
"2014-12-30 12:00:00\n",
"13 : predicted = [0.7 0.68 0.69] expected = [0.7188898836168307, 0.7090420769919425, 0.7081468218442255]\n",
"2014-12-30 13:00:00\n",
"14 : predicted = [0.72 0.73 0.76] expected = [0.7090420769919425, 0.7081468218442255, 0.7385854968666068]\n",
"2014-12-30 14:00:00\n",
"15 : predicted = [0.71 0.73 0.86] expected = [0.7081468218442255, 0.7385854968666068, 0.8478066248880931]\n",
"2014-12-30 15:00:00\n",
"16 : predicted = [0.73 0.85 0.97] expected = [0.7385854968666068, 0.8478066248880931, 0.9516562220232765]\n",
"2014-12-30 16:00:00\n",
"17 : predicted = [0.87 0.99 0.97] expected = [0.8478066248880931, 0.9516562220232765, 0.934198746642793]\n",
"2014-12-30 17:00:00\n",
"18 : predicted = [0.94 0.92 0.86] expected = [0.9516562220232765, 0.934198746642793, 0.8876454789615038]\n",
"2014-12-30 18:00:00\n",
"19 : predicted = [0.94 0.89 0.82] expected = [0.934198746642793, 0.8876454789615038, 0.8294538943598924]\n",
"2014-12-30 19:00:00\n",
"20 : predicted = [0.88 0.82 0.71] expected = [0.8876454789615038, 0.8294538943598924, 0.7197851387645477]\n",
"2014-12-30 20:00:00\n",
"21 : predicted = [0.83 0.72 0.58] expected = [0.8294538943598924, 0.7197851387645477, 0.5747538048343777]\n",
"2014-12-30 21:00:00\n",
"22 : predicted = [0.72 0.58 0.47] expected = [0.7197851387645477, 0.5747538048343777, 0.4592658907788718]\n",
"2014-12-30 22:00:00\n",
"23 : predicted = [0.58 0.47 0.39] expected = [0.5747538048343777, 0.4592658907788718, 0.3858549686660697]\n",
"2014-12-30 23:00:00\n",
"24 : predicted = [0.46 0.38 0.34] expected = [0.4592658907788718, 0.3858549686660697, 0.34377797672336596]\n",
"2014-12-31 00:00:00\n",
"25 : predicted = [0.38 0.34 0.33] expected = [0.3858549686660697, 0.34377797672336596, 0.32542524619516544]\n",
"2014-12-31 01:00:00\n",
"26 : predicted = [0.36 0.34 0.34] expected = [0.34377797672336596, 0.32542524619516544, 0.33034914950760963]\n",
"2014-12-31 02:00:00\n",
"27 : predicted = [0.32 0.32 0.35] expected = [0.32542524619516544, 0.33034914950760963, 0.3706356311548791]\n",
"2014-12-31 03:00:00\n",
"28 : predicted = [0.32 0.36 0.47] expected = [0.33034914950760963, 0.3706356311548791, 0.470008952551477]\n",
"2014-12-31 04:00:00\n",
"29 : predicted = [0.37 0.48 0.65] expected = [0.3706356311548791, 0.470008952551477, 0.6145926589077886]\n",
"2014-12-31 05:00:00\n",
"30 : predicted = [0.48 0.64 0.75] expected = [0.470008952551477, 0.6145926589077886, 0.7247090420769919]\n",
"2014-12-31 06:00:00\n",
"31 : predicted = [0.63 0.73 0.79] expected = [0.6145926589077886, 0.7247090420769919, 0.786034019695613]\n",
"2014-12-31 07:00:00\n",
"32 : predicted = [0.71 0.76 0.79] expected = [0.7247090420769919, 0.786034019695613, 0.8012533572068039]\n",
"2014-12-31 08:00:00\n",
"33 : predicted = [0.79 0.82 0.83] expected = [0.786034019695613, 0.8012533572068039, 0.7994628469113696]\n",
"2014-12-31 09:00:00\n",
"34 : predicted = [0.82 0.83 0.81] expected = [0.8012533572068039, 0.7994628469113696, 0.780214861235452]\n",
"2014-12-31 10:00:00\n",
"35 : predicted = [0.8 0.78 0.76] expected = [0.7994628469113696, 0.780214861235452, 0.7587287376902416]\n",
"2014-12-31 11:00:00\n",
"36 : predicted = [0.77 0.75 0.74] expected = [0.780214861235452, 0.7587287376902416, 0.7367949865711727]\n",
"2014-12-31 12:00:00\n",
"37 : predicted = [0.77 0.76 0.76] expected = [0.7587287376902416, 0.7367949865711727, 0.7188898836168307]\n",
"2014-12-31 13:00:00\n",
"38 : predicted = [0.75 0.75 0.78] expected = [0.7367949865711727, 0.7188898836168307, 0.7273948075201431]\n",
"2014-12-31 14:00:00\n",
"39 : predicted = [0.73 0.75 0.87] expected = [0.7188898836168307, 0.7273948075201431, 0.8299015219337511]\n",
"2014-12-31 15:00:00\n",
"40 : predicted = [0.74 0.85 0.96] expected = [0.7273948075201431, 0.8299015219337511, 0.909579230080573]\n",
"2014-12-31 16:00:00\n",
"41 : predicted = [0.83 0.94 0.93] expected = [0.8299015219337511, 0.909579230080573, 0.855863921217547]\n",
"2014-12-31 17:00:00\n",
"42 : predicted = [0.94 0.93 0.88] expected = [0.909579230080573, 0.855863921217547, 0.7721575649059982]\n",
"2014-12-31 18:00:00\n",
"43 : predicted = [0.87 0.82 0.77] expected = [0.855863921217547, 0.7721575649059982, 0.7023276633840643]\n",
"2014-12-31 19:00:00\n",
"44 : predicted = [0.79 0.73 0.63] expected = [0.7721575649059982, 0.7023276633840643, 0.6195165622202325]\n",
"2014-12-31 20:00:00\n",
"45 : predicted = [0.7 0.59 0.46] expected = [0.7023276633840643, 0.6195165622202325, 0.5425246195165621]\n",
"2014-12-31 21:00:00\n",
"46 : predicted = [0.6 0.47 0.36] expected = [0.6195165622202325, 0.5425246195165621, 0.4735899731423454]\n",
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"CPU times: user 13min 24s, sys: 2min 53s, total: 16min 17s\n",
"Wall time: 2min 53s\n"
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]
}
],
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"source": [
"%%time\n",
"training_window = 720 # dedicate 30 days (720 hours) for training\n",
"\n",
"train_ts = train['load']\n",
"test_ts = test_shifted\n",
"\n",
"history = [x for x in train_ts]\n",
"history = history[(-training_window):]\n",
"\n",
"predictions = list()\n",
"\n",
"# let's user simpler model for demonstration\n",
"order = (2, 1, 0)\n",
"seasonal_order = (1, 1, 0, 24)\n",
"\n",
"for t in range(test_ts.shape[0]):\n",
" model = SARIMAX(endog=history, order=order, seasonal_order=seasonal_order)\n",
" model_fit = model.fit()\n",
" yhat = model_fit.forecast(steps = HORIZON)\n",
" predictions.append(yhat)\n",
" obs = list(test_ts.iloc[t])\n",
" # move the training window\n",
" history.append(obs[0])\n",
" history.pop(0)\n",
" print(test_ts.index[t])\n",
" print(t+1, ': predicted =', yhat, 'expected =', obs)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Compare predictions to actual load"
]
},
{
"cell_type": "code",
time series 2
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"execution_count": 15,
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"metadata": {},
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": [
" timestamp h prediction actual\n",
"0 2014-12-30 00:00:00 t+1 3,008.74 3,023.00\n",
"1 2014-12-30 01:00:00 t+1 2,955.53 2,935.00\n",
"2 2014-12-30 02:00:00 t+1 2,900.17 2,899.00\n",
"3 2014-12-30 03:00:00 t+1 2,917.69 2,886.00\n",
"4 2014-12-30 04:00:00 t+1 2,946.99 2,963.00"
],
"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>timestamp</th>\n <th>h</th>\n <th>prediction</th>\n <th>actual</th>\n </tr>\n </thead>\n <tbody>\n <tr>\n <th>0</th>\n <td>2014-12-30 00:00:00</td>\n <td>t+1</td>\n <td>3,008.74</td>\n <td>3,023.00</td>\n </tr>\n <tr>\n <th>1</th>\n <td>2014-12-30 01:00:00</td>\n <td>t+1</td>\n <td>2,955.53</td>\n <td>2,935.00</td>\n </tr>\n <tr>\n <th>2</th>\n <td>2014-12-30 02:00:00</td>\n <td>t+1</td>\n <td>2,900.17</td>\n <td>2,899.00</td>\n </tr>\n <tr>\n <th>3</th>\n <td>2014-12-30 03:00:00</td>\n <td>t+1</td>\n <td>2,917.69</td>\n <td>2,886.00</td>\n </tr>\n <tr>\n <th>4</th>\n <td>2014-12-30 04:00:00</td>\n <td>t+1</td>\n <td>2,946.99</td>\n <td>2,963.00</td>\n </tr>\n </tbody>\n</table>\n</div>"
},
"metadata": {},
time series 2
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"execution_count": 15
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}
],
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"source": [
"eval_df = pd.DataFrame(predictions, columns=['t+'+str(t) for t in range(1, HORIZON+1)])\n",
"eval_df['timestamp'] = test.index[0:len(test.index)-HORIZON+1]\n",
"eval_df = pd.melt(eval_df, id_vars='timestamp', value_name='prediction', var_name='h')\n",
"eval_df['actual'] = np.array(np.transpose(test_ts)).ravel()\n",
"eval_df[['prediction', 'actual']] = scaler.inverse_transform(eval_df[['prediction', 'actual']])\n",
"eval_df.head()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Compute the **mean absolute percentage error (MAPE)** over all predictions\n",
"\n",
"$$MAPE = \\frac{1}{n} \\sum_{t=1}^{n}|\\frac{actual_t - predicted_t}{actual_t}|$$"
]
},
{
"cell_type": "code",
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"execution_count": 16,
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"metadata": {},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"h\nt+1 0.01\nt+2 0.01\nt+3 0.02\nName: APE, dtype: float64\n"
]
}
],
time series 1
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"source": [
"if(HORIZON > 1):\n",
" eval_df['APE'] = (eval_df['prediction'] - eval_df['actual']).abs() / eval_df['actual']\n",
" print(eval_df.groupby('h')['APE'].mean())"
]
},
{
"cell_type": "code",
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"execution_count": 17,
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"metadata": {},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"One step forecast MAPE: 0.5570581332313952 %\n"
]
}
],
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"source": [
"print('One step forecast MAPE: ', (mape(eval_df[eval_df['h'] == 't+1']['prediction'], eval_df[eval_df['h'] == 't+1']['actual']))*100, '%')"
]
},
{
"cell_type": "code",
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"execution_count": 18,
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"metadata": {},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"Multi-step forecast MAPE: 1.1460048657704118 %\n"
]
}
],
time series 1
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"source": [
"print('Multi-step forecast MAPE: ', mape(eval_df['prediction'], eval_df['actual'])*100, '%')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Plot the predictions vs the actuals for the first week of the test set"
]
},
{
"cell_type": "code",
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"execution_count": 19,
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"metadata": {},
"outputs": [
{
"output_type": "stream",
"name": "stderr",
"text": [
"No handles with labels found to put in legend.\n"
]
},
{
"output_type": "display_data",
"data": {
"text/plain": "<Figure size 1080x576 with 1 Axes>",
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time series work
4 years ago
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},
"metadata": {
"needs_background": "light"
}
}
],
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"source": [
"if(HORIZON == 1):\n",
" ## Plotting single step forecast\n",
" eval_df.plot(x='timestamp', y=['actual', 'prediction'], style=['r', 'b'], figsize=(15, 8))\n",
"\n",
"else:\n",
" ## Plotting multi step forecast\n",
" plot_df = eval_df[(eval_df.h=='t+1')][['timestamp', 'actual']]\n",
" for t in range(1, HORIZON+1):\n",
" plot_df['t+'+str(t)] = eval_df[(eval_df.h=='t+'+str(t))]['prediction'].values\n",
"\n",
" fig = plt.figure(figsize=(15, 8))\n",
" ax = plt.plot(plot_df['timestamp'], plot_df['actual'], color='red', linewidth=4.0)\n",
" ax = fig.add_subplot(111)\n",
" for t in range(1, HORIZON+1):\n",
" x = plot_df['timestamp'][(t-1):]\n",
" y = plot_df['t+'+str(t)][0:len(x)]\n",
" ax.plot(x, y, color='blue', linewidth=4*math.pow(.9,t), alpha=math.pow(0.8,t))\n",
" \n",
" ax.legend(loc='best')\n",
" \n",
"plt.xlabel('timestamp', fontsize=12)\n",
"plt.ylabel('load', fontsize=12)\n",
"plt.show()"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernel_info": {
"name": "python3"
},
"kernelspec": {
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"name": "python37364bit8d3b438fb5fc4430a93ac2cb74d693a7",
"display_name": "Python 3.7.0 64-bit ('3.7')"
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},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
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"version": "3.7.0"
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},
"nteract": {
"version": "nteract-front-end@1.0.0"
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},
"metadata": {
"interpreter": {
"hash": "70b38d7a306a849643e446cd70466270a13445e5987dfa1344ef2b127438fa4d"
}
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}
},
"nbformat": 4,
"nbformat_minor": 2
}