ETS models

The ETS models are a family of time series models with an underlying state space model consisting of a level component, a trend component (T), a seasonal component (S), and an error term (E).

This notebook shows how they can be used with statsmodels. For a more thorough treatment we refer to [1], chapter 8 (free online resource), on which the implementation in statsmodels and the examples used in this notebook are based.

statsmodels implements all combinations of: - additive and multiplicative error model - additive and multiplicative trend, possibly dampened - additive and multiplicative seasonality

However, not all of these methods are stable. Refer to [1] and references therein for more info about model stability.

[1] Hyndman, Rob J., and George Athanasopoulos. Forecasting: principles and practice, 3rd edition, OTexts, 2019. https://www.otexts.org/fpp3/7

[1]:
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
%matplotlib inline
from statsmodels.tsa.exponential_smoothing.ets import ETSModel
[2]:
plt.rcParams['figure.figsize'] = (12, 8)

Simple exponential smoothing

The simplest of the ETS models is also known as simple exponential smoothing. In ETS terms, it corresponds to the (A, N, N) model, that is, a model with additive errors, no trend, and no seasonality. The state space formulation of Holt’s method is:

\begin{align} y_{t} &= y_{t-1} + e_t\\ l_{t} &= l_{t-1} + \alpha e_t\\ \end{align}

This state space formulation can be turned into a different formulation, a forecast and a smoothing equation (as can be done with all ETS models):

\begin{align} \hat{y}_{t|t-1} &= l_{t-1}\\ l_{t} &= \alpha y_{t-1} + (1 - \alpha) l_{t-1} \end{align}

Here, \(\hat{y}_{t|t-1}\) is the forecast/expectation of \(y_t\) given the information of the previous step. In the simple exponential smoothing model, the forecast corresponds to the previous level. The second equation (smoothing equation) calculates the next level as weighted average of the previous level and the previous observation.

[3]:
oildata = [
    111.0091, 130.8284, 141.2871, 154.2278,
    162.7409, 192.1665, 240.7997, 304.2174,
    384.0046, 429.6622, 359.3169, 437.2519,
    468.4008, 424.4353, 487.9794, 509.8284,
    506.3473, 340.1842, 240.2589, 219.0328,
    172.0747, 252.5901, 221.0711, 276.5188,
    271.1480, 342.6186, 428.3558, 442.3946,
    432.7851, 437.2497, 437.2092, 445.3641,
    453.1950, 454.4096, 422.3789, 456.0371,
    440.3866, 425.1944, 486.2052, 500.4291,
    521.2759, 508.9476, 488.8889, 509.8706,
    456.7229, 473.8166, 525.9509, 549.8338,
    542.3405
]
oil = pd.Series(oildata, index=pd.date_range('1965', '2013', freq='AS'))
oil.plot()
plt.ylabel("Annual oil production in Saudi Arabia (Mt)");
../../../_images/examples_notebooks_generated_ets_4_0.png

The plot above shows annual oil production in Saudi Arabia in million tonnes. The data are taken from the R package fpp2 (companion package to prior version [1]). Below you can see how to fit a simple exponential smoothing model using statsmodels’s ETS implementation to this data. Additionally, the fit using forecast in R is shown as comparison.

[4]:
model = ETSModel(oil, error='add', trend='add', damped_trend=True)
fit = model.fit(maxiter=10000)
oil.plot(label='data')
fit.fittedvalues.plot(label='statsmodels fit')
plt.ylabel("Annual oil production in Saudi Arabia (Mt)");

# obtained from R
params_R = [0.99989969, 0.11888177503085334, 0.80000197, 36.46466837, 34.72584983]
yhat = model.smooth(params_R).fittedvalues
yhat.plot(label='R fit', linestyle='--')

plt.legend();
../../../_images/examples_notebooks_generated_ets_6_0.png

By default the initial states are considered to be fitting parameters and are estimated by maximizing log-likelihood. Additionally it is possible to only use a heuristic for the initial values. In this case this leads to better agreement with the R implementation.

[5]:
model_heuristic = ETSModel(oil, error='add', trend='add', damped_trend=True,
                          initialization_method='heuristic')
fit_heuristic = model_heuristic.fit()
oil.plot(label='data')
fit.fittedvalues.plot(label='estimated')
fit_heuristic.fittedvalues.plot(label='heuristic', linestyle='--')
plt.ylabel("Annual oil production in Saudi Arabia (Mt)");

# obtained from R
params = [0.99989969, 0.11888177503085334, 0.80000197, 36.46466837, 34.72584983]
yhat = model.smooth(params).fittedvalues
yhat.plot(label='with R params', linestyle=':')

plt.legend();
../../../_images/examples_notebooks_generated_ets_8_0.png

The fitted parameters and some other measures are shown using fit.summary(). Here we can see that the log-likelihood of the model using fitted initial states is a bit lower than the one using a heuristic for the initial states. Additionally, we see that \(\beta\) (smoothing_trend) is at the boundary of the default parameter bounds, and therefore it’s not possible to estimate confidence intervals for \(\beta\).

[6]:
print(fit.summary())
                                 ETS Results
==============================================================================
Dep. Variable:                      y   No. Observations:                   49
Model:                      ETS(AAdN)   Log Likelihood                -258.108
Date:                Thu, 06 Aug 2020   AIC                            528.216
Time:                        13:04:47   BIC                            539.567
Sample:                    01-01-1965   HQIC                           532.523
                         - 01-01-2013   Scale                         2202.048
Covariance Type:               approx
===================================================================================
                      coef    std err          z      P>|z|      [0.025      0.975]
-----------------------------------------------------------------------------------
smoothing_level     0.9999      0.136      7.326      0.000       0.732       1.267
smoothing_trend  9.999e-05        nan        nan        nan         nan         nan
damping_trend       0.8798      0.059     14.901      0.000       0.764       0.996
initial_level      54.2331     55.750      0.973      0.331     -55.034     163.500
initial_trend      52.9975     34.392      1.541      0.123     -14.409     120.404
===================================================================================
Ljung-Box (Q):                        1.42   Jarque-Bera (JB):                20.73
Prob(Q):                              0.49   Prob(JB):                         0.00
Heteroskedasticity (H):               0.52   Skew:                            -1.02
Prob(H) (two-sided):                  0.20   Kurtosis:                         5.45
===================================================================================

Warnings:
[1] Covariance matrix calculated using numerical (complex-step) differentiation.
[7]:
print(fit_heuristic.summary())
                                 ETS Results
==============================================================================
Dep. Variable:                      y   No. Observations:                   49
Model:                      ETS(AAdN)   Log Likelihood                -258.881
Date:                Thu, 06 Aug 2020   AIC                            525.761
Time:                        13:04:47   BIC                            533.328
Sample:                    01-01-1965   HQIC                           528.632
                         - 01-01-2013   Scale                         2272.587
Covariance Type:               approx
===================================================================================
                      coef    std err          z      P>|z|      [0.025      0.975]
-----------------------------------------------------------------------------------
smoothing_level     0.9999      0.097     10.289      0.000       0.809       1.190
smoothing_trend     0.1183        nan        nan        nan         nan         nan
damping_trend       0.8000        nan        nan        nan         nan         nan
==============================================
              initialization method: heuristic
----------------------------------------------
initial_level                          33.6309
initial_trend                          34.8115
===================================================================================
Ljung-Box (Q):                        0.27   Jarque-Bera (JB):                19.93
Prob(Q):                              0.88   Prob(JB):                         0.00
Heteroskedasticity (H):               0.50   Skew:                            -0.95
Prob(H) (two-sided):                  0.18   Kurtosis:                         5.48
===================================================================================

Warnings:
[1] Covariance matrix calculated using numerical (complex-step) differentiation.

Holt-Winters’ seasonal method

The exponential smoothing method can be modified to incorporate a trend and a seasonal component. In the additive Holt-Winters’ method, the seasonal component is added to the rest. This model corresponds to the ETS(A, A, A) model, and has the following state space formulation:

\begin{align} y_t &= l_{t-1} + b_{t-1} + s_{t-m} + e_t\\ l_{t} &= l_{t-1} + b_{t-1} + \alpha e_t\\ b_{t} &= b_{t-1} + \beta e_t\\ s_{t} &= s_{t-m} + \gamma e_t \end{align}
[8]:
austourists_data = [
    30.05251300, 19.14849600, 25.31769200, 27.59143700,
    32.07645600, 23.48796100, 28.47594000, 35.12375300,
    36.83848500, 25.00701700, 30.72223000, 28.69375900,
    36.64098600, 23.82460900, 29.31168300, 31.77030900,
    35.17787700, 19.77524400, 29.60175000, 34.53884200,
    41.27359900, 26.65586200, 28.27985900, 35.19115300,
    42.20566386, 24.64917133, 32.66733514, 37.25735401,
    45.24246027, 29.35048127, 36.34420728, 41.78208136,
    49.27659843, 31.27540139, 37.85062549, 38.83704413,
    51.23690034, 31.83855162, 41.32342126, 42.79900337,
    55.70835836, 33.40714492, 42.31663797, 45.15712257,
    59.57607996, 34.83733016, 44.84168072, 46.97124960,
    60.01903094, 38.37117851, 46.97586413, 50.73379646,
    61.64687319, 39.29956937, 52.67120908, 54.33231689,
    66.83435838, 40.87118847, 51.82853579, 57.49190993,
    65.25146985, 43.06120822, 54.76075713, 59.83447494,
    73.25702747, 47.69662373, 61.09776802, 66.05576122,
]
index = pd.date_range("1999-03-01", "2015-12-01", freq="3MS")
austourists = pd.Series(austourists_data, index=index)
austourists.plot()
plt.ylabel('Australian Tourists');
../../../_images/examples_notebooks_generated_ets_13_0.png
[9]:
# fit in statsmodels
model = ETSModel(austourists, error="add", trend="add", seasonal="add",
                damped_trend=True, seasonal_periods=4)
fit = model.fit()

# fit with R params
params_R = [
    0.35445427, 0.03200749, 0.39993387, 0.97999997, 24.01278357,
    0.97770147, 1.76951063, -0.50735902, -6.61171798, 5.34956637
]
fit_R = model.smooth(params_R)

austourists.plot(label='data')
plt.ylabel('Australian Tourists')

fit.fittedvalues.plot(label='statsmodels fit')
fit_R.fittedvalues.plot(label='R fit', linestyle='--')
plt.legend();
../../../_images/examples_notebooks_generated_ets_14_0.png
[10]:
print(fit.summary())
                                 ETS Results
==============================================================================
Dep. Variable:                      y   No. Observations:                   68
Model:                      ETS(AAdA)   Log Likelihood                -152.627
Date:                Thu, 06 Aug 2020   AIC                            327.254
Time:                        13:04:48   BIC                            351.668
Sample:                    03-01-1999   HQIC                           336.927
                         - 12-01-2015   Scale                            5.213
Covariance Type:               approx
======================================================================================
                         coef    std err          z      P>|z|      [0.025      0.975]
--------------------------------------------------------------------------------------
smoothing_level        0.3399      0.111      3.070      0.002       0.123       0.557
smoothing_trend        0.0259      0.008      3.158      0.002       0.010       0.042
smoothing_seasonal     0.4011      0.080      5.041      0.000       0.245       0.557
damping_trend          0.9800        nan        nan        nan         nan         nan
initial_level         29.4484        nan        nan        nan         nan         nan
initial_trend          0.6148      0.392      1.568      0.117      -0.154       1.383
initial_seasonal.0    -3.4422        nan        nan        nan         nan         nan
initial_seasonal.1    -5.9552        nan        nan        nan         nan         nan
initial_seasonal.2   -11.4853        nan        nan        nan         nan         nan
initial_seasonal.3          0        nan        nan        nan         nan         nan
===================================================================================
Ljung-Box (Q):                        5.76   Jarque-Bera (JB):                 7.68
Prob(Q):                              0.67   Prob(JB):                         0.02
Heteroskedasticity (H):               0.46   Skew:                            -0.63
Prob(H) (two-sided):                  0.07   Kurtosis:                         4.05
===================================================================================

Warnings:
[1] Covariance matrix calculated using numerical (complex-step) differentiation.

Predictions

The ETS model can also be used for predicting. There are several different methods available: - forecast: makes out of sample predictions - predict: in sample and out of sample predictions - simulate: runs simulations of the statespace model - get_prediction: in sample and out of sample predictions, as well as prediction intervals

We can use them on our previously fitted model to predict from 2014 to 2020.

[11]:
pred = fit.get_prediction(start='2014', end='2020')
[12]:
df = pred.summary_frame(alpha=0.05)
df
[12]:
mean pi_lower pi_upper
2014-03-01 67.611065 63.136093 72.086036
2014-06-01 42.814605 38.339633 47.289577
2014-09-01 54.106484 49.631513 58.581456
2014-12-01 57.928423 53.453451 62.403394
2015-03-01 68.422007 63.947035 72.896978
2015-06-01 47.277848 42.802876 51.752820
2015-09-01 58.954826 54.479855 63.429798
2015-12-01 63.982357 59.507385 68.457329
2016-03-01 75.905428 71.430457 80.380400
2016-06-01 51.418089 46.653967 56.182211
2016-09-01 63.703365 58.629433 68.777297
2016-12-01 67.978251 62.575872 73.380630
2017-03-01 78.315920 71.735791 84.896049
2017-06-01 53.780371 46.883306 60.677435
2017-09-01 66.018401 58.788184 73.248618
2017-12-01 70.246986 62.668762 77.825211
2018-03-01 80.539281 71.892652 89.185909
2018-06-01 55.959264 46.968012 64.950517
2018-09-01 68.153717 58.805069 77.502364
2018-12-01 72.339596 62.621785 82.057406
2019-03-01 82.590038 71.864546 93.315530
2019-06-01 57.969006 46.875619 69.062394
2019-09-01 70.123264 58.651859 81.594669
2019-12-01 74.269752 62.410975 86.128529
2020-03-01 84.481591 71.656118 97.307064

In this case the prediction intervals were calculated using an analytical formula. This is not available for all models. For these other models, prediction intervals are calculated by performing multiple simulations (1000 by default) and using the percentiles of the simulation results. This is done internally by the get_prediction method.

We can also manually run simulations, e.g. to plot them. Since the data ranges until end of 2015, we have to simulate from the first quarter of 2016 to the first quarter of 2020, which means 17 steps.

[13]:
simulated = fit.simulate(anchor="end", nsimulations=17, repetitions=100)
[14]:
for i in range(simulated.shape[1]):
    simulated.iloc[:,i].plot(label='_', color='gray', alpha=0.1)
df["mean"].plot(label='mean prediction')
df["pi_lower"].plot(linestyle='--', color='tab:blue', label='95% interval')
df["pi_upper"].plot(linestyle='--', color='tab:blue', label='_')
pred.endog.plot(label='data')
plt.legend()
[14]:
<matplotlib.legend.Legend at 0x7f58f617a2d0>
../../../_images/examples_notebooks_generated_ets_21_1.png

In this case, we chose “end” as simulation anchor, which means that the first simulated value will be the first out of sample value. It is also possible to choose other anchor inside the sample.

[ ]: