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.date_range(start='1996', end='2008', freq='A')
oildata = pd.Series(data, 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.date_range(start='1990', end='2005', freq='A')
air = pd.Series(data, 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.date_range(start='1970', end='2001', freq='A')
livestock2 = pd.Series(data, index)

data = [407.9979 ,  403.4608,  413.8249,  428.105 ,  445.3387,  452.9942, 455.7402]
index= pd.date_range(start='2001', end='2008', freq='A')
livestock3 = pd.Series(data, 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.date_range(start='2005', end='2010-Q4', freq='QS-OCT')
aust = pd.Series(data, 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 optimizer 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 optimized 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 260.341513 257.357600 258.951854
$b_0$ NaN 5.010792 1.013780 6.644539 1.038144
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 Box-Cox transformation.
  2. fit2 additive trend, multiplicative seasonal of period season_length=4 and the use of a Box-Cox transformation..
  3. fit3 additive damped trend, additive seasonal of period season_length=4 and the use of a Box-Cox transformation.
  4. fit4 additive damped trend, multiplicative seasonal of period season_length=4 and the use of a Box-Cox 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).rename('Holt-Winters (add-add-seasonal)').plot(ax=ax, style='--', marker='o', color='red', legend=True)
fit2.forecast(8).rename('Holt-Winters (add-mul-seasonal)').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
/usr/lib/python3/dist-packages/statsmodels/tsa/holtwinters.py:712: ConvergenceWarning: Optimization failed to converge. Check mle_retvals.
  ConvergenceWarning)
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
$\alpha$ 4.546459e-01 3.659223e-01 7.771267e-15 0.000189
$\beta$ 7.794133e-09 7.316443e-19 5.975782e-20 0.000189
$\phi$ NaN NaN 9.446193e-01 0.913789
$\gamma$ 5.243837e-01 1.060914e-12 4.955475e-14 0.000000
$l_0$ 1.421754e+01 1.454811e+01 1.417767e+01 14.534881
$b_0$ 1.307698e-01 1.661090e-01 2.399716e-01 0.484079
SSE 5.001687e+01 4.307029e+01 3.530344e+01 39.666447

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]:
df = 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)
df.append(fit1.forecast(8).rename(r'$\hat{y}_t$').to_frame(), sort=True)
Out[9]:
$\hat{y}_t$ $b_t$ $l_t$ $s_t$ $y_t$
2005-01-01 41.721301 -34.969418 49.317727 -7.593396 41.7275
2005-04-01 24.190405 -35.452731 49.932372 -25.834541 24.0418
2005-07-01 31.460402 -36.532791 51.126130 -19.182969 32.3281
2005-10-01 36.634633 -37.397668 52.210116 -15.209764 37.3287
2006-01-01 45.097596 -38.467294 53.476795 -7.837304 46.2132
2006-04-01 27.191816 -40.276313 55.513851 -27.017281 29.3463
2006-07-01 36.544210 -40.625014 56.224439 -19.713825 36.4829
2006-10-01 41.449390 -42.041777 57.766084 -15.523320 42.9777
2007-01-01 50.934403 -41.543765 57.536686 -7.588718 48.9015
2007-04-01 31.418131 -42.197821 58.150971 -26.874297 31.1802
2007-07-01 38.718290 -42.303417 58.362916 -20.191886 37.7179
2007-10-01 44.140504 -41.085655 57.181734 -14.982807 40.4202
2008-01-01 49.315669 -42.961440 58.852914 -8.619795 51.2069
2008-04-01 32.306929 -43.186468 59.366898 -27.309123 31.8872
2008-07-01 39.207416 -44.826955 61.095541 -20.925483 40.9783
2008-10-01 44.551342 -44.899323 61.462001 -17.319727 43.7725
2009-01-01 54.357955 -46.192151 62.816712 -7.881574 55.5586
2009-04-01 35.153811 -45.980030 62.831978 -28.447763 33.8509
2009-07-01 43.066343 -46.227986 63.082485 -20.550579 42.0764
2009-10-01 45.871156 -46.852335 63.748644 -17.997619 45.6423
2010-01-01 57.166477 -48.770316 65.777461 -7.372023 59.7668
2010-04-01 36.761409 -48.308126 65.649780 -29.816638 35.1919
2010-07-01 44.932417 -48.798443 66.119177 -21.517279 44.3197
2010-10-01 48.399507 -49.269581 66.667135 -18.522045 47.9137
2011-01-01 61.337759 NaN NaN NaN NaN
2011-04-01 37.242799 NaN NaN NaN NaN
2011-07-01 46.842530 NaN NaN NaN NaN
2011-10-01 51.005064 NaN NaN NaN NaN
2012-01-01 64.470463 NaN NaN NaN NaN
2012-04-01 39.776728 NaN NaN NaN NaN
2012-07-01 49.635627 NaN NaN NaN NaN
2012-10-01 53.901136 NaN NaN NaN NaN
In [10]:
df = 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)
df.append(fit2.forecast(8).rename(r'$\hat{y}_t$').to_frame(), sort=True)
Out[10]:
$\hat{y}_t$ $b_t$ $l_t$ $s_t$ $y_t$
2005-01-01 41.860019 -36.529901 51.244121 0.815914 41.7275
2005-04-01 25.838329 -35.866598 50.735921 0.495384 24.0418
2005-07-01 31.659094 -37.283662 52.060074 0.613001 32.3281
2005-10-01 35.189675 -39.168718 54.186389 0.664203 37.3287
2006-01-01 44.928949 -40.305567 55.705171 0.815073 46.2132
2006-04-01 27.933400 -42.087113 57.755569 0.493065 29.3463
2006-07-01 35.824218 -43.100038 59.126477 0.610108 36.4829
2006-10-01 39.768767 -45.643430 61.906160 0.661726 42.9777
2007-01-01 51.174397 -45.120736 61.855424 0.813614 48.9015
2007-04-01 30.814215 -46.408183 63.134340 0.490309 31.1802
2007-07-01 39.009005 -46.386264 63.326558 0.608240 37.7179
2007-10-01 42.486130 -46.170456 63.142772 0.660464 40.4202
2008-01-01 52.174114 -46.759046 63.700745 0.813407 51.2069
2008-04-01 31.677045 -47.835416 64.869947 0.489565 31.8872
2008-07-01 40.035566 -49.239072 66.466994 0.607690 40.9783
2008-10-01 44.516066 -49.574721 67.064385 0.659603 43.7725
2009-01-01 55.343283 -50.601802 68.188673 0.812789 55.5586
2009-04-01 33.772845 -51.515006 69.283811 0.487890 33.8509
2009-07-01 42.644046 -52.025095 69.969874 0.606390 42.0764
2009-10-01 46.778562 -52.310277 70.364685 0.658714 45.6423
2010-01-01 58.009042 -54.093333 72.210619 0.812312 59.7668
2010-04-01 35.648092 -54.499545 72.908816 0.486531 35.1919
2010-07-01 44.784147 -55.163485 73.682353 0.605416 44.3197
2010-10-01 49.174608 -55.389037 74.028798 0.657846 47.9137
2011-01-01 60.967374 NaN NaN NaN NaN
2011-04-01 36.993488 NaN NaN NaN NaN
2011-07-01 46.712003 NaN NaN NaN NaN
2011-10-01 51.482604 NaN NaN NaN NaN
2012-01-01 64.455898 NaN NaN NaN NaN
2012-04-01 39.016675 NaN NaN NaN NaN
2012-07-01 49.291081 NaN NaN NaN NaN
2012-10-01 54.319649 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()