Tutorial

Let us consider chapter 7 of the excellent treatise on the subject of Exponential Smoothing By Hyndman and Athanasopoulos [1]. We will work through all the examples in the chapter as they unfold.

[1] Hyndman, Rob J., and George Athanasopoulos. Forecasting: principles and practice. OTexts, 2014.

Exponential smoothing

First we load some data. We have included the R data in the notebook for expedience.

In [1]:
import os
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from statsmodels.tsa.api import ExponentialSmoothing, SimpleExpSmoothing, Holt

data = [446.6565,  454.4733,  455.663 ,  423.6322,  456.2713,  440.5881, 425.3325,  485.1494,  506.0482,  526.792 ,  514.2689,  494.211 ]
index= pd.DatetimeIndex(start='1996', end='2008', freq='A')
oildata = pd.Series(data, index)
oildata.index = pd.DatetimeIndex(oildata.index, freq=pd.infer_freq(oildata.index))

data = [17.5534,  21.86  ,  23.8866,  26.9293,  26.8885,  28.8314, 30.0751,  30.9535,  30.1857,  31.5797,  32.5776,  33.4774, 39.0216,  41.3864,  41.5966]
index= pd.DatetimeIndex(start='1990', end='2005', freq='A')
air = pd.Series(data, index)
air.index = pd.DatetimeIndex(air.index, freq=pd.infer_freq(air.index))

data = [263.9177,  268.3072,  260.6626,  266.6394,  277.5158,  283.834 , 290.309 ,  292.4742,  300.8307,  309.2867,  318.3311,  329.3724, 338.884 ,  339.2441,  328.6006,  314.2554,  314.4597,  321.4138, 329.7893,  346.3852,  352.2979,  348.3705,  417.5629,  417.1236, 417.7495,  412.2339,  411.9468,  394.6971,  401.4993,  408.2705, 414.2428]
index= pd.DatetimeIndex(start='1970', end='2001', freq='A')
livestock2 = pd.Series(data, index)
livestock2.index = pd.DatetimeIndex(livestock2.index, freq=pd.infer_freq(livestock2.index))

data = [407.9979 ,  403.4608,  413.8249,  428.105 ,  445.3387,  452.9942, 455.7402]
index= pd.DatetimeIndex(start='2001', end='2008', freq='A')
livestock3 = pd.Series(data, index)
livestock3.index = pd.DatetimeIndex(livestock3.index, freq=pd.infer_freq(livestock3.index))

data = [41.7275,  24.0418,  32.3281,  37.3287,  46.2132,  29.3463, 36.4829,  42.9777,  48.9015,  31.1802,  37.7179,  40.4202, 51.2069,  31.8872,  40.9783,  43.7725,  55.5586,  33.8509, 42.0764,  45.6423,  59.7668,  35.1919,  44.3197,  47.9137]
index= pd.DatetimeIndex(start='2005', end='2010-Q4', freq='QS')
aust = pd.Series(data, index)
aust.index = pd.DatetimeIndex(aust.index, freq=pd.infer_freq(aust.index))

Simple Exponential Smoothing

Lets use Simple Exponential Smoothing to forecast the below oil data.

In [2]:
ax=oildata.plot()
ax.set_xlabel("Year")
ax.set_ylabel("Oil (millions of tonnes)")
plt.show()
print("Figure 7.1: Oil production in Saudi Arabia from 1996 to 2007.")
Figure 7.1: Oil production in Saudi Arabia from 1996 to 2007.

Here we run three variants of simple exponential smoothing:

  1. In fit1 we do not use the auto optimization but instead choose to explicitly provide the model with the $\alpha=0.2$ parameter
  2. In fit2 as above we choose an $\alpha=0.6$
  3. In fit3 we allow statsmodels to automatically find an optimized $\alpha$ value for us. This is the recommended approach.
In [3]:
fit1 = SimpleExpSmoothing(oildata).fit(smoothing_level=0.2,optimized=False)
fcast1 = fit1.forecast(3).rename(r'$\alpha=0.2$')
fit2 = SimpleExpSmoothing(oildata).fit(smoothing_level=0.6,optimized=False)
fcast2 = fit2.forecast(3).rename(r'$\alpha=0.6$')
fit3 = SimpleExpSmoothing(oildata).fit()
fcast3 = fit3.forecast(3).rename(r'$\alpha=%s$'%fit3.model.params['smoothing_level'])

ax = oildata.plot(marker='o', color='black', figsize=(12,8))
fcast1.plot(marker='o', ax=ax, color='blue', legend=True)
fit1.fittedvalues.plot(marker='o', ax=ax, color='blue')
fcast2.plot(marker='o', ax=ax, color='red', legend=True)

fit2.fittedvalues.plot(marker='o', ax=ax, color='red')
fcast3.plot(marker='o', ax=ax, color='green', legend=True)
fit3.fittedvalues.plot(marker='o', ax=ax, color='green')
plt.show()

Holt's Method

Lets take a look at another example. This time we use air pollution data and the Holt's Method. We will fit three examples again.

  1. In fit1 we again choose not to use the optimzer and provide explicit values for $\alpha=0.8$ and $\beta=0.2$
  2. In fit2 we do the same as in fit1 but choose to use an exponential model rather than a Holt's additive model.
  3. In fit3 we used a damped versions of the Holt's additive model but allow the dampening parameter $\phi$ to be optimized while fixing the values for $\alpha=0.8$ and $\beta=0.2$
In [4]:
fit1 = Holt(air).fit(smoothing_level=0.8, smoothing_slope=0.2, optimized=False)
fcast1 = fit1.forecast(5).rename("Holt's linear trend")
fit2 = Holt(air, exponential=True).fit(smoothing_level=0.8, smoothing_slope=0.2, optimized=False)
fcast2 = fit2.forecast(5).rename("Exponential trend")
fit3 = Holt(air, damped=True).fit(smoothing_level=0.8, smoothing_slope=0.2)
fcast3 = fit3.forecast(5).rename("Additive damped trend")

ax = air.plot(color="black", marker="o", figsize=(12,8))
fit1.fittedvalues.plot(ax=ax, color='blue')
fcast1.plot(ax=ax, color='blue', marker="o", legend=True)
fit2.fittedvalues.plot(ax=ax, color='red')
fcast2.plot(ax=ax, color='red', marker="o", legend=True)
fit3.fittedvalues.plot(ax=ax, color='green')
fcast3.plot(ax=ax, color='green', marker="o", legend=True)

plt.show()

Seasonally adjusted data

Lets look at some seasonally adjusted livestock data. We fit five Holt's models. The below table allows us to compare results when we use exponential versus additive and damped versus non-damped.

Note: fit4 does not allow the parameter $\phi$ to be otpimized by providing a fixed value of $\phi=0.98$

In [5]:
fit1 = SimpleExpSmoothing(livestock2).fit()
fit2 = Holt(livestock2).fit()
fit3 = Holt(livestock2,exponential=True).fit()
fit4 = Holt(livestock2,damped=True).fit(damping_slope=0.98)
fit5 = Holt(livestock2,exponential=True,damped=True).fit()
params = ['smoothing_level', 'smoothing_slope', 'damping_slope', 'initial_level', 'initial_slope']
results=pd.DataFrame(index=[r"$\alpha$",r"$\beta$",r"$\phi$",r"$l_0$","$b_0$","SSE"] ,columns=['SES', "Holt's","Exponential", "Additive", "Multiplicative"])
results["SES"] =            [fit1.params[p] for p in params] + [fit1.sse]
results["Holt's"] =         [fit2.params[p] for p in params] + [fit2.sse]
results["Exponential"] =    [fit3.params[p] for p in params] + [fit3.sse]
results["Additive"] =       [fit4.params[p] for p in params] + [fit4.sse]
results["Multiplicative"] = [fit5.params[p] for p in params] + [fit5.sse]
results
Out[5]:
SES Holt's Exponential Additive Multiplicative
$\alpha$ 1.000000 0.974306 0.977633 0.978848 0.974911
$\beta$ NaN 0.000000 0.000000 0.000000 0.000000
$\phi$ NaN NaN NaN 0.980000 0.981646
$l_0$ 263.917700 258.882590 260.341582 257.357684 258.951821
$b_0$ NaN 5.010792 1.013780 6.511666 1.019091
SSE 6761.350218 6004.138200 6104.194746 6036.555004 6081.995045

Plots of Seasonally Adjusted Data

The following plots allow us to evaluate the level and slope/trend components of the above table's fits.

In [6]:
for fit in [fit2,fit4]:
    pd.DataFrame(np.c_[fit.level,fit.slope]).rename(
        columns={0:'level',1:'slope'}).plot(subplots=True)
plt.show()
print('Figure 7.4: Level and slope components for Holt’s linear trend method and the additive damped trend method.')
Figure 7.4: Level and slope components for Holt’s linear trend method and the additive damped trend method.

Comparison

Here we plot a comparison Simple Exponential Smoothing and Holt's Methods for various additive, exponential and damped combinations. All of the models parameters will be optimized by statsmodels.

In [7]:
fit1 = SimpleExpSmoothing(livestock2).fit()
fcast1 = fit1.forecast(9).rename("SES")
fit2 = Holt(livestock2).fit()
fcast2 = fit2.forecast(9).rename("Holt's")
fit3 = Holt(livestock2, exponential=True).fit()
fcast3 = fit3.forecast(9).rename("Exponential")
fit4 = Holt(livestock2, damped=True).fit(damping_slope=0.98)
fcast4 = fit4.forecast(9).rename("Additive Damped")
fit5 = Holt(livestock2, exponential=True, damped=True).fit()
fcast5 = fit5.forecast(9).rename("Multiplicative Damped")

ax = livestock2.plot(color="black", marker="o", figsize=(12,8))
livestock3.plot(ax=ax, color="black", marker="o", legend=False)
fcast1.plot(ax=ax, color='red', legend=True)
fcast2.plot(ax=ax, color='green', legend=True)
fcast3.plot(ax=ax, color='blue', legend=True)
fcast4.plot(ax=ax, color='cyan', legend=True)
fcast5.plot(ax=ax, color='magenta', legend=True)
ax.set_ylabel('Livestock, sheep in Asia (millions)')
plt.show()
print('Figure 7.5: Forecasting livestock, sheep in Asia: comparing forecasting performance of non-seasonal methods.')
Figure 7.5: Forecasting livestock, sheep in Asia: comparing forecasting performance of non-seasonal methods.

Holt's Winters Seasonal

Finally we are able to run full Holt's Winters Seasonal Exponential Smoothing including a trend component and a seasonal component. Statsmodels allows for all the combinations including as shown in the examples below:

  1. fit1 additive trend, additive seasonal of period season_length=4 and the use of a Boxcox transformation.
  2. fit2 additive trend, multiplicative seasonal of period season_length=4 and the use of a Boxcox transformation..
  3. fit3 additive damped trend, additive seasonal of period season_length=4 and the use of a boxcox transformation.
  4. fit4 additive damped trend, multiplicative seasonal of period season_length=4 and the use of a boxcox transformation.

The plot shows the results and forecast for fit1 and fit2. The table allows us to compare the results and parameterizations.

In [8]:
fit1 = ExponentialSmoothing(aust, seasonal_periods=4, trend='add', seasonal='add').fit(use_boxcox=True)
fit2 = ExponentialSmoothing(aust, seasonal_periods=4, trend='add', seasonal='mul').fit(use_boxcox=True)
fit3 = ExponentialSmoothing(aust, seasonal_periods=4, trend='add', seasonal='add', damped=True).fit(use_boxcox=True)
fit4 = ExponentialSmoothing(aust, seasonal_periods=4, trend='add', seasonal='mul', damped=True).fit(use_boxcox=True)
results=pd.DataFrame(index=[r"$\alpha$",r"$\beta$",r"$\phi$",r"$\gamma$",r"$l_0$","$b_0$","SSE"])
params = ['smoothing_level', 'smoothing_slope', 'damping_slope', 'smoothing_seasonal', 'initial_level', 'initial_slope']
results["Additive"]       = [fit1.params[p] for p in params] + [fit1.sse]
results["Multiplicative"] = [fit2.params[p] for p in params] + [fit2.sse]
results["Additive Dam"]   = [fit3.params[p] for p in params] + [fit3.sse]
results["Multiplica Dam"] = [fit4.params[p] for p in params] + [fit4.sse]

ax = aust.plot(figsize=(10,6), marker='o', color='black', title="Forecasts from Holt-Winters' multiplicative method" )
ax.set_ylabel("International visitor night in Australia (millions)")
ax.set_xlabel("Year")
fit1.fittedvalues.plot(ax=ax, style='--', color='red')
fit2.fittedvalues.plot(ax=ax, style='--', color='green')

fit1.forecast(8).plot(ax=ax, style='--', marker='o', color='red', legend=True)
fit2.forecast(8).plot(ax=ax, style='--', marker='o', color='green', legend=True)

plt.show()
print("Figure 7.6: Forecasting international visitor nights in Australia using Holt-Winters method with both additive and multiplicative seasonality.")

results
Figure 7.6: Forecasting international visitor nights in Australia using Holt-Winters method with both additive and multiplicative seasonality.
Out[8]:
Additive Multiplicative Additive Dam Multiplica Dam
$\alpha$ 4.546238e-01 3.658627e-01 7.549439e-09 0.000191
$\beta$ 8.173643e-09 1.391288e-15 5.088092e-09 0.000191
$\phi$ NaN NaN 9.440069e-01 0.913506
$\gamma$ 5.244326e-01 2.020959e-09 1.004006e-08 0.000000
$l_0$ 1.421751e+01 1.454894e+01 1.416442e+01 14.534748
$b_0$ 1.307798e-01 1.661327e-01 2.292746e-01 0.444716
SSE 5.001651e+01 4.306936e+01 3.522512e+01 39.843325

The Internals

It is possible to get at the internals of the Exponential Smoothing models.

Here we show some tables that allow you to view side by side the original values $y_t$, the level $l_t$, the trend $b_t$, the season $s_t$ and the fitted values $\hat{y}_t$.

In [9]:
pd.DataFrame(np.c_[aust, fit1.level, fit1.slope, fit1.season, fit1.fittedvalues], 
columns=[r'$y_t$',r'$l_t$',r'$b_t$',r'$s_t$',r'$\hat{y}_t$'],index=aust.index). \
append(fit1.forecast(8).rename(r'$\hat{y}_t$').to_frame())
/usr/lib/python3/dist-packages/pandas/core/frame.py:6211: FutureWarning: Sorting because non-concatenation axis is not aligned. A future version
of pandas will change to not sort by default.

To accept the future behavior, pass 'sort=False'.

To retain the current behavior and silence the warning, pass 'sort=True'.

  sort=sort)
Out[9]:
$\hat{y}_t$ $b_t$ $l_t$ $s_t$ $y_t$
2005-01-01 41.721198 -34.969342 49.317628 -7.593350 41.7275
2005-04-01 24.190175 -35.452837 49.932474 -25.834725 24.0418
2005-07-01 31.460530 -36.532820 51.126192 -19.182998 32.3281
2005-10-01 36.634762 -37.397674 52.210148 -15.209752 37.3287
2006-01-01 45.097771 -38.467250 53.476771 -7.837216 46.2132
2006-04-01 27.191752 -40.276310 55.513858 -27.017377 29.3463
2006-07-01 36.544238 -40.625049 56.224491 -19.713862 36.4829
2006-10-01 41.449478 -42.041777 57.766108 -15.523341 42.9777
2007-01-01 50.934540 -41.543810 57.536750 -7.588664 48.9015
2007-04-01 31.418278 -42.197838 58.151013 -26.874272 31.1802
2007-07-01 38.718329 -42.303501 58.363021 -20.191947 37.7179
2007-10-01 44.140671 -41.085793 57.181904 -14.982804 40.4202
2008-01-01 49.315771 -42.961532 58.853050 -8.619922 51.2069
2008-04-01 32.307043 -43.186569 59.367038 -27.309204 31.8872
2008-07-01 39.207467 -44.827042 61.095666 -20.925631 40.9783
2008-10-01 44.551251 -44.899539 61.462253 -17.320003 43.7725
2009-01-01 54.358174 -46.192274 62.816892 -7.881668 55.5586
2009-04-01 35.153878 -45.980223 62.832217 -28.447937 33.8509
2009-07-01 43.066539 -46.228151 63.082705 -20.550684 42.0764
2009-10-01 45.871154 -46.852571 63.748932 -17.997902 45.6423
2010-01-01 57.166673 -48.770450 65.777658 -7.372176 59.7668
2010-04-01 36.761386 -48.308395 65.650096 -29.816921 35.1919
2010-07-01 44.932524 -48.798713 66.119514 -21.517551 44.3197
2010-10-01 48.399593 -49.269877 66.667502 -18.522359 47.9137
2011-01-01 61.338056 NaN NaN NaN NaN
2011-04-01 37.242889 NaN NaN NaN NaN
2011-07-01 46.842698 NaN NaN NaN NaN
2011-10-01 51.005341 NaN NaN NaN NaN
2012-01-01 64.471008 NaN NaN NaN NaN
2012-04-01 39.777017 NaN NaN NaN NaN
2012-07-01 49.636016 NaN NaN NaN NaN
2012-10-01 53.901644 NaN NaN NaN NaN
In [10]:
pd.DataFrame(np.c_[aust, fit2.level, fit2.slope, fit2.season, fit2.fittedvalues], 
columns=[r'$y_t$',r'$l_t$',r'$b_t$',r'$s_t$',r'$\hat{y}_t$'],index=aust.index). \
append(fit2.forecast(8).rename(r'$\hat{y}_t$').to_frame())
/usr/lib/python3/dist-packages/pandas/core/frame.py:6211: FutureWarning: Sorting because non-concatenation axis is not aligned. A future version
of pandas will change to not sort by default.

To accept the future behavior, pass 'sort=False'.

To retain the current behavior and silence the warning, pass 'sort=True'.

  sort=sort)
Out[10]:
$\hat{y}_t$ $b_t$ $l_t$ $s_t$ $y_t$
2005-01-01 41.861264 -36.533145 51.248219 0.815864 41.7275
2005-04-01 25.838904 -35.869765 50.739864 0.495355 24.0418
2005-07-01 31.659789 -37.286544 52.063706 0.612966 32.3281
2005-10-01 35.189845 -39.171617 54.189974 0.664158 37.3287
2006-01-01 44.929259 -40.308448 55.708718 0.815023 46.2132
2006-04-01 27.933587 -42.089913 57.759021 0.493036 29.3463
2006-07-01 35.824373 -43.102889 59.129954 0.610073 36.4829
2006-10-01 39.768516 -45.646419 61.909774 0.661682 42.9777
2007-01-01 51.174330 -45.124012 61.859336 0.813565 48.9015
2007-04-01 30.814395 -46.411407 63.138251 0.490280 31.1802
2007-07-01 39.009239 -46.389598 63.330573 0.608205 37.7179
2007-10-01 42.486034 -46.174107 63.147120 0.660419 40.4202
2008-01-01 52.174576 -46.762644 63.705097 0.813357 51.2069
2008-04-01 31.677378 -47.838911 64.874194 0.489536 31.8872
2008-07-01 40.035937 -49.242473 66.471124 0.607655 40.9783
2008-10-01 44.515901 -49.578455 67.068821 0.659558 43.7725
2009-01-01 55.343612 -50.605505 68.193125 0.812739 55.5586
2009-04-01 33.773095 -51.518686 69.288237 0.487861 33.8509
2009-07-01 42.644366 -52.028797 69.974313 0.606355 42.0764
2009-10-01 46.778437 -52.314288 70.369433 0.658670 45.6423
2010-01-01 58.009451 -54.097214 72.215285 0.812261 59.7668
2010-04-01 35.648325 -54.503479 72.913514 0.486502 35.1919
2010-07-01 44.784498 -55.167409 73.687042 0.605380 44.3197
2010-10-01 49.174475 -55.393289 74.033812 0.657801 47.9137
2011-01-01 60.967805 NaN NaN NaN NaN
2011-04-01 36.993885 NaN NaN NaN NaN
2011-07-01 46.712642 NaN NaN NaN NaN
2011-10-01 51.482876 NaN NaN NaN NaN
2012-01-01 64.456706 NaN NaN NaN NaN
2012-04-01 39.017298 NaN NaN NaN NaN
2012-07-01 49.292012 NaN NaN NaN NaN
2012-10-01 54.320215 NaN NaN NaN NaN

Finally lets look at the levels, slopes/trends and seasonal components of the models.

In [11]:
states1 = pd.DataFrame(np.c_[fit1.level, fit1.slope, fit1.season], columns=['level','slope','seasonal'], index=aust.index)
states2 = pd.DataFrame(np.c_[fit2.level, fit2.slope, fit2.season], columns=['level','slope','seasonal'], index=aust.index)
fig, [[ax1, ax4],[ax2, ax5], [ax3, ax6]] = plt.subplots(3, 2, figsize=(12,8))
states1[['level']].plot(ax=ax1)
states1[['slope']].plot(ax=ax2)
states1[['seasonal']].plot(ax=ax3)
states2[['level']].plot(ax=ax4)
states2[['slope']].plot(ax=ax5)
states2[['seasonal']].plot(ax=ax6)
plt.show()