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.
Loading data
First we load some data. We have included the R data in the notebook for expedience.
[1]:
import os import numpy as np import pandas as pd import matplotlib.pyplot as plt from statsmodels.tsa.api import ExponentialSmoothing, SimpleExpSmoothing, Holt %matplotlib inline 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.
[2]:
ax = oildata.plot() ax.set_xlabel("Year") ax.set_ylabel("Oil (millions of tonnes)") 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 Figure 7.1: Oil production in Saudi Arabia from 1996 to 2007. 2 we do not use the auto optimization but instead choose to explicitly provide the model with the \(\alpha=0.2\) parameter 2. In Figure 7.1: Oil production in Saudi Arabia from 1996 to 2007. 3 as above we choose an \(\alpha=0.6\) 3. In Figure 7.1: Oil production in Saudi Arabia from 1996 to 2007. 4 we allow statsmodels to automatically find an optimized \(\alpha\) value for us. This is the recommended approach.
[3]:
fit1 = SimpleExpSmoothing(oildata, initialization_method="heuristic").fit( smoothing_level=0.2, optimized=False ) fcast1 = fit1.forecast(3).rename(r"$\alpha=0.2$") fit2 = SimpleExpSmoothing(oildata, initialization_method="heuristic").fit( smoothing_level=0.6, optimized=False ) fcast2 = fit2.forecast(3).rename(r"$\alpha=0.6$") fit3 = SimpleExpSmoothing(oildata, initialization_method="estimated").fit() fcast3 = fit3.forecast(3).rename(r"$\alpha=%s$" % fit3.model.params["smoothing_level"]) plt.figure(figsize=(12, 8)) plt.plot(oildata, marker="o", color="black") plt.plot(fit1.fittedvalues, marker="o", color="blue") (line1,) = plt.plot(fcast1, marker="o", color="blue") plt.plot(fit2.fittedvalues, marker="o", color="red") (line2,) = plt.plot(fcast2, marker="o", color="red") plt.plot(fit3.fittedvalues, marker="o", color="green") (line3,) = plt.plot(fcast3, marker="o", color="green") plt.legend([line1, line2, line3], [fcast1.name, fcast2.name, fcast3.name])
[3]:
<matplotlib.legend.Legend at 0x7f0627052860>
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 Figure 7.1: Oil production in Saudi Arabia from 1996 to 2007. 2 we again choose not to use the optimizer and provide explicit values for \(\alpha=0.8\) and \(\beta=0.2\) 2. In Figure 7.1: Oil production in Saudi Arabia from 1996 to 2007. 3 we do the same as in Figure 7.1: Oil production in Saudi Arabia from 1996 to 2007. 2 but choose to use an exponential model rather than a Holt’s additive model. 3. In Figure 7.1: Oil production in Saudi Arabia from 1996 to 2007. 4 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 \(\beta=0.2\)
[4]:
import os import numpy as np import pandas as pd import matplotlib.pyplot as plt from statsmodels.tsa.api import ExponentialSmoothing, SimpleExpSmoothing, Holt %matplotlib inline 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) 0
[4]:
import os import numpy as np import pandas as pd import matplotlib.pyplot as plt from statsmodels.tsa.api import ExponentialSmoothing, SimpleExpSmoothing, Holt %matplotlib inline 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) 2
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: Figure 7.1: Oil production in Saudi Arabia from 1996 to 2007. 9 does not allow the parameter \(\phi\) to be optimized by providing a fixed value of \(\phi=0.98\)
import os import numpy as np import pandas as pd import matplotlib.pyplot as plt from statsmodels.tsa.api import ExponentialSmoothing, SimpleExpSmoothing, Holt %matplotlib inline 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) 3
import os import numpy as np import pandas as pd import matplotlib.pyplot as plt from statsmodels.tsa.api import ExponentialSmoothing, SimpleExpSmoothing, Holt %matplotlib inline 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) 4
import os import numpy as np import pandas as pd import matplotlib.pyplot as plt from statsmodels.tsa.api import ExponentialSmoothing, SimpleExpSmoothing, Holt %matplotlib inline 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) 3
SESHolt'sExponentialAdditiveMultiplicative$\alpha$1.0000000.9743089.776329e-010.9788520.974891$\beta$NaN0.0000004.016578e-120.0000000.000000$\phi$NaNNaNNaN0.9800000.981637$l_0$263.917688258.8826002.603440e+02257.357526258.940454$b_0$NaN5.0107831.013780e+006.6447411.038159SSE6761.3502356004.1382006.104195e+036036.5550056081.995166
Plots of Seasonally Adjusted Data
The following plots allow us to evaluate the level and slope/trend components of the above table’s fits.
import os import numpy as np import pandas as pd import matplotlib.pyplot as plt from statsmodels.tsa.api import ExponentialSmoothing, SimpleExpSmoothing, Holt %matplotlib inline 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) 6
import os import numpy as np import pandas as pd import matplotlib.pyplot as plt from statsmodels.tsa.api import ExponentialSmoothing, SimpleExpSmoothing, Holt %matplotlib inline 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) 7
import os import numpy as np import pandas as pd import matplotlib.pyplot as plt from statsmodels.tsa.api import ExponentialSmoothing, SimpleExpSmoothing, Holt %matplotlib inline 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) 8
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.
import os import numpy as np import pandas as pd import matplotlib.pyplot as plt from statsmodels.tsa.api import ExponentialSmoothing, SimpleExpSmoothing, Holt %matplotlib inline 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) 9
[2]: 0
[2]: 1
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. Figure 7.1: Oil production in Saudi Arabia from 1996 to 2007. 2 additive trend, additive seasonal of period [3]: 1 and the use of a Box-Cox transformation. 1. Figure 7.1: Oil production in Saudi Arabia from 1996 to 2007. 3 additive trend, multiplicative seasonal of period [3]: 1 and the use of a Box-Cox transformation.. 1. Figure 7.1: Oil production in Saudi Arabia from 1996 to 2007. 4 additive damped trend, additive seasonal of period [3]: 1 and the use of a Box-Cox transformation. 1. Figure 7.1: Oil production in Saudi Arabia from 1996 to 2007. 9 additive damped trend, multiplicative seasonal of period [3]: 1 and the use of a Box-Cox transformation.
The plot shows the results and forecast for Figure 7.1: Oil production in Saudi Arabia from 1996 to 2007. 2 and Figure 7.1: Oil production in Saudi Arabia from 1996 to 2007. 3. The table allows us to compare the results and parameterizations.
[2]: 2
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AdditiveMultiplicativeAdditive DamMultiplica Dam$\alpha$1.490116e-081.490116e-081.490116e-081.490116e-08$\beta$1.409865e-080.000000e+006.490761e-095.042250e-09$\phi$NaNNaN9.430416e-019.536043e-01$\gamma$7.544975e-161.889367e-157.434023e-161.273086e-16$l_0$1.119348e+011.106376e+011.084022e+019.899296e+00$b_0$1.205396e-011.198957e-012.456749e-011.975447e-01SSE4.402746e+013.611262e+013.527620e+013.062033e+01
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 \(\hat{y}_t\). Note that these values only have meaningful values in the space of your original data if the fit is performed without a Box-Cox transformation.
[2]: 6
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ax = oildata.plot() ax.set_xlabel("Year") ax.set_ylabel("Oil (millions of tonnes)") print("Figure 7.1: Oil production in Saudi Arabia from 1996 to 2007.") 0
[2]: 8
$\hat{y}_t$$b_t$$l_t$$s_t$$y_t$2005-01-0144.5841280.59782234.29758010.28654841.72752005-04-0124.9381890.59782234.895402-9.95721324.04182005-07-0133.0057650.59782235.493224-2.48745832.32812005-10-0137.0311070.59782236.0910460.94006237.32872006-01-0146.9754150.59782236.68886810.28654846.21322006-04-0127.3294770.59782237.286690-9.95721329.34632006-07-0135.3970530.59782237.884512-2.48745836.48292006-10-0139.4223950.59782238.4823340.94006242.97772007-01-0149.3667030.59782239.08015610.28654848.90152007-04-0129.7207650.59782239.677978-9.95721331.18022007-07-0137.7883410.59782240.275800-2.48745837.71792007-10-0141.8136830.59782240.8736220.94006240.42022008-01-0151.7579910.59782241.47144410.28654851.20692008-04-0132.1120530.59782242.069266-9.95721331.88722008-07-0140.1796290.59782242.667088-2.48745840.97832008-10-0144.2049710.59782243.2649100.94006243.77252009-01-0154.1492790.59782243.86273210.28654855.55862009-04-0134.5033410.59782244.460554-9.95721333.85092009-07-0142.5709170.59782245.058376-2.48745842.07642009-10-0146.5962590.59782245.6561970.94006245.64232010-01-0156.5405670.59782246.25402010.28654859.76682010-04-0136.8946280.59782246.851841-9.95721335.19192010-07-0144.9622050.59782247.449663-2.48745844.31972010-10-0148.9875470.59782248.0474850.94006247.91372011-01-0158.931855NaNNaNNaNNaN2011-04-0139.285916NaNNaNNaNNaN2011-07-0147.353493NaNNaNNaNNaN2011-10-0151.378835NaNNaNNaNNaN2012-01-0161.323143NaNNaNNaNNaN2012-04-0141.677204NaNNaNNaNNaN2012-07-0149.744781NaNNaNNaNNaN2012-10-0153.770123NaNNaNNaNNaN
ax = oildata.plot() ax.set_xlabel("Year") ax.set_ylabel("Oil (millions of tonnes)") print("Figure 7.1: Oil production in Saudi Arabia from 1996 to 2007.") 2
ax = oildata.plot() ax.set_xlabel("Year") ax.set_ylabel("Oil (millions of tonnes)") print("Figure 7.1: Oil production in Saudi Arabia from 1996 to 2007.") 3
ax = oildata.plot() ax.set_xlabel("Year") ax.set_ylabel("Oil (millions of tonnes)") print("Figure 7.1: Oil production in Saudi Arabia from 1996 to 2007.") 4
ax = oildata.plot() ax.set_xlabel("Year") ax.set_ylabel("Oil (millions of tonnes)") print("Figure 7.1: Oil production in Saudi Arabia from 1996 to 2007.") 2
$\hat{y}_t$$b_t$$l_t$$s_t$$y_t$2005-01-0143.0053880.62093435.0160021.22816441.72752005-04-0126.3529490.62093435.6369360.73948424.04182005-07-0133.2847280.62093436.2578700.91800032.32812005-10-0136.7195080.62093436.8788040.99568137.32872006-01-0146.0558230.62093437.4997381.22816446.21322006-04-0128.1896330.62093438.1206720.73948429.34632006-07-0135.5647990.62093438.7416060.91800036.48292006-10-0139.1925160.62093439.3625400.99568142.97772007-01-0149.1062580.62093439.9834741.22816448.90152007-04-0130.0263160.62093440.6044080.73948431.18022007-07-0137.8448690.62093441.2253420.91800037.71792007-10-0141.6655240.62093441.8462770.99568140.42022008-01-0152.1566930.62093442.4672111.22816451.20692008-04-0131.8630000.62093443.0881450.73948431.88722008-07-0140.1249390.62093443.7090790.91800040.97832008-10-0144.1385310.62093444.3300130.99568143.77252009-01-0155.2071280.62093444.9509471.22816455.55862009-04-0133.6996830.62093445.5718810.73948433.85092009-07-0142.4050090.62093446.1928150.91800042.07642009-10-0146.6115390.62093446.8137490.99568145.64232010-01-0158.2575630.62093447.4346831.22816459.76682010-04-0135.5363670.62093448.0556170.73948435.19192010-07-0144.6850790.62093448.6765510.91800044.31972010-10-0149.0845470.62093449.2974850.99568147.91372011-01-0161.307998NaNNaNNaNNaN2011-04-0137.373051NaNNaNNaNNaN2011-07-0146.965149NaNNaNNaNNaN2011-10-0151.557555NaNNaNNaNNaN2012-01-0164.358434NaNNaNNaNNaN2012-04-0139.209734NaNNaNNaNNaN2012-07-0149.245220NaNNaNNaNNaN2012-10-0154.030563NaNNaNNaNNaN
Finally lets look at the levels, slopes/trends and seasonal components of the models.
ax = oildata.plot() ax.set_xlabel("Year") ax.set_ylabel("Oil (millions of tonnes)") print("Figure 7.1: Oil production in Saudi Arabia from 1996 to 2007.") 6
ax = oildata.plot() ax.set_xlabel("Year") ax.set_ylabel("Oil (millions of tonnes)") print("Figure 7.1: Oil production in Saudi Arabia from 1996 to 2007.") 7
By using a state space formulation, we can perform simulations of future values. The mathematical details are described in Hyndman and Athanasopoulos [2] and in the documentation of fit1 = SimpleExpSmoothing(oildata, initialization_method="heuristic").fit( smoothing_level=0.2, optimized=False ) fcast1 = fit1.forecast(3).rename(r"$\alpha=0.2$") fit2 = SimpleExpSmoothing(oildata, initialization_method="heuristic").fit( smoothing_level=0.6, optimized=False ) fcast2 = fit2.forecast(3).rename(r"$\alpha=0.6$") fit3 = SimpleExpSmoothing(oildata, initialization_method="estimated").fit() fcast3 = fit3.forecast(3).rename(r"$\alpha=%s$" % fit3.model.params["smoothing_level"]) plt.figure(figsize=(12, 8)) plt.plot(oildata, marker="o", color="black") plt.plot(fit1.fittedvalues, marker="o", color="blue") (line1,) = plt.plot(fcast1, marker="o", color="blue") plt.plot(fit2.fittedvalues, marker="o", color="red") (line2,) = plt.plot(fcast2, marker="o", color="red") plt.plot(fit3.fittedvalues, marker="o", color="green") (line3,) = plt.plot(fcast3, marker="o", color="green") plt.legend([line1, line2, line3], [fcast1.name, fcast2.name, fcast3.name]) 0.
Similar to the example in [2], we use the model with additive trend, multiplicative seasonality, and multiplicative error. We simulate up to 8 steps into the future, and perform 1000 simulations. As can be seen in the below figure, the simulations match the forecast values quite well.
[2] Hyndman, Rob J., and George Athanasopoulos. Forecasting: principles and practice, 2nd edition. OTexts, 2018.
ax = oildata.plot() ax.set_xlabel("Year") ax.set_ylabel("Oil (millions of tonnes)") print("Figure 7.1: Oil production in Saudi Arabia from 1996 to 2007.") 8
ax = oildata.plot() ax.set_xlabel("Year") ax.set_ylabel("Oil (millions of tonnes)") print("Figure 7.1: Oil production in Saudi Arabia from 1996 to 2007.") 9
Simulations can also be started at different points in time, and there are multiple options for choosing the random noise.