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)")
[3]:
Text(0, 0.5, 'Annual oil production in Saudi Arabia (Mt)')
../../../_images/examples_notebooks_generated_ets_4_1.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)
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()
RUNNING THE L-BFGS-B CODE

           * * *

Machine precision = 2.220D-16
 N =            2     M =           10

At X0         0 variables are exactly at the bounds

At iterate    0    f=  6.27365D+00    |proj g|=  8.99900D-01

At iterate    1    f=  5.31675D+00    |proj g|=  6.49880D-04

At iterate    2    f=  5.30939D+00    |proj g|=  5.55378D-04

At iterate    3    f=  5.29115D+00    |proj g|=  5.80869D-05

At iterate    4    f=  5.29096D+00    |proj g|=  1.95399D-06

           * * *

Tit   = total number of iterations
Tnf   = total number of function evaluations
Tnint = total number of segments explored during Cauchy searches
Skip  = number of BFGS updates skipped
Nact  = number of active bounds at final generalized Cauchy point
Projg = norm of the final projected gradient
F     = final function value

           * * *

   N    Tit     Tnf  Tnint  Skip  Nact     Projg        F
    2      4     13      5     0     1   1.954D-06   5.291D+00
  F =   5.2909564634183583

CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_<=_PGTOL
[4]:
<matplotlib.legend.Legend at 0x7fb1b5c03250>
../../../_images/examples_notebooks_generated_ets_6_2.png

By default the initial states are considered to be fitting parameters and are estimated by maximizing log-likelihood. It is possible to only use a heuristic for the initial values:

[5]:
model_heuristic = ETSModel(oil, 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()
RUNNING THE L-BFGS-B CODE

           * * *

Machine precision = 2.220D-16
 N =            1     M =           10

At X0         0 variables are exactly at the bounds

At iterate    0    f=  6.27365D+00    |proj g|=  8.99900D-01

At iterate    1    f=  5.31675D+00    |proj g|=  0.00000D+00

           * * *

Tit   = total number of iterations
Tnf   = total number of function evaluations
Tnint = total number of segments explored during Cauchy searches
Skip  = number of BFGS updates skipped
Nact  = number of active bounds at final generalized Cauchy point
Projg = norm of the final projected gradient
F     = final function value

           * * *

   N    Tit     Tnf  Tnint  Skip  Nact     Projg        F
    1      1      2      1     0     1   0.000D+00   5.317D+00
  F =   5.3167544390512402

CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_<=_PGTOL
[5]:
<matplotlib.legend.Legend at 0x7fb1a8bb4650>
../../../_images/examples_notebooks_generated_ets_8_2.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 fractionally lower than the one using a heuristic for the initial states.

[6]:
print(fit.summary())
                                 ETS Results
==============================================================================
Dep. Variable:                      y   No. Observations:                   49
Model:                       ETS(ANN)   Log Likelihood                -259.257
Date:                Tue, 14 Feb 2023   AIC                            524.514
Time:                        22:28:59   BIC                            530.189
Sample:                    01-01-1965   HQIC                           526.667
                         - 01-01-2013   Scale                         2307.767
Covariance Type:               approx
===================================================================================
                      coef    std err          z      P>|z|      [0.025      0.975]
-----------------------------------------------------------------------------------
smoothing_level     0.9999      0.132      7.551      0.000       0.740       1.259
initial_level     110.7864     48.110      2.303      0.021      16.492     205.081
===================================================================================
Ljung-Box (Q):                        1.87   Jarque-Bera (JB):                20.78
Prob(Q):                              0.39   Prob(JB):                         0.00
Heteroskedasticity (H):               0.49   Skew:                            -1.04
Prob(H) (two-sided):                  0.16   Kurtosis:                         5.42
===================================================================================

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(ANN)   Log Likelihood                -260.521
Date:                Tue, 14 Feb 2023   AIC                            525.042
Time:                        22:28:59   BIC                            528.826
Sample:                    01-01-1965   HQIC                           526.477
                         - 01-01-2013   Scale                         2429.964
Covariance Type:               approx
===================================================================================
                      coef    std err          z      P>|z|      [0.025      0.975]
-----------------------------------------------------------------------------------
smoothing_level     0.9999      0.132      7.559      0.000       0.741       1.259
==============================================
              initialization method: heuristic
----------------------------------------------
initial_level                          33.6309
===================================================================================
Ljung-Box (Q):                        1.85   Jarque-Bera (JB):                18.42
Prob(Q):                              0.40   Prob(JB):                         0.00
Heteroskedasticity (H):               0.44   Skew:                            -1.02
Prob(H) (two-sided):                  0.11   Kurtosis:                         5.21
===================================================================================

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")
[8]:
Text(0, 0.5, 'Australian Tourists')
../../../_images/examples_notebooks_generated_ets_13_1.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()
RUNNING THE L-BFGS-B CODE

           * * *

Machine precision = 2.220D-16
 N =            9     M =           10

At X0         1 variables are exactly at the bounds

At iterate    0    f=  3.40132D+00    |proj g|=  9.88789D-01

At iterate    1    f=  2.58255D+00    |proj g|=  9.99244D-01

At iterate    2    f=  2.49918D+00    |proj g|=  2.90033D-01

At iterate    3    f=  2.48198D+00    |proj g|=  2.44942D-01

At iterate    4    f=  2.43118D+00    |proj g|=  7.29711D-02

At iterate    5    f=  2.42924D+00    |proj g|=  7.03570D-02

At iterate    6    f=  2.42851D+00    |proj g|=  4.66400D-02

At iterate    7    f=  2.42794D+00    |proj g|=  2.92420D-02

At iterate    8    f=  2.42784D+00    |proj g|=  2.53309D-02

At iterate    9    f=  2.42721D+00    |proj g|=  1.89540D-02

At iterate   10    f=  2.42622D+00    |proj g|=  3.18642D-02

At iterate   11    f=  2.42512D+00    |proj g|=  3.53009D-02

At iterate   12    f=  2.42383D+00    |proj g|=  3.65632D-02

At iterate   13    f=  2.42196D+00    |proj g|=  4.83549D-02

At iterate   14    f=  2.41828D+00    |proj g|=  5.99626D-02

At iterate   15    f=  2.41131D+00    |proj g|=  6.89752D-02

At iterate   16    f=  2.40200D+00    |proj g|=  7.65524D-02

At iterate   17    f=  2.39385D+00    |proj g|=  8.77262D-02

At iterate   18    f=  2.37916D+00    |proj g|=  1.52280D-01

At iterate   19    f=  2.35438D+00    |proj g|=  2.40331D-01

At iterate   20    f=  2.33829D+00    |proj g|=  2.52950D-01

At iterate   21    f=  2.33771D+00    |proj g|=  2.54712D-01

At iterate   22    f=  2.33762D+00    |proj g|=  2.49728D-01

At iterate   23    f=  2.33762D+00    |proj g|=  2.49785D-01
  Positive dir derivative in projection
  Using the backtracking step

At iterate   24    f=  2.32725D+00    |proj g|=  2.22964D-01

At iterate   25    f=  2.29782D+00    |proj g|=  1.16416D-01

At iterate   26    f=  2.29781D+00    |proj g|=  1.19525D-01

At iterate   27    f=  2.29781D+00    |proj g|=  1.21800D-01

At iterate   28    f=  2.29779D+00    |proj g|=  1.25588D-01

At iterate   29    f=  2.29098D+00    |proj g|=  8.12570D-02

At iterate   30    f=  2.27580D+00    |proj g|=  1.86477D-01

At iterate   31    f=  2.27428D+00    |proj g|=  9.08980D-02

At iterate   32    f=  2.27301D+00    |proj g|=  6.75787D-02

At iterate   33    f=  2.27170D+00    |proj g|=  1.33350D-02

At iterate   34    f=  2.27161D+00    |proj g|=  7.76454D-03

At iterate   35    f=  2.27151D+00    |proj g|=  7.06417D-03

At iterate   36    f=  2.27126D+00    |proj g|=  1.35173D-02

At iterate   37    f=  2.27079D+00    |proj g|=  3.13304D-02

At iterate   38    f=  2.27040D+00    |proj g|=  3.68805D-02

At iterate   39    f=  2.27011D+00    |proj g|=  2.89762D-02

At iterate   40    f=  2.26990D+00    |proj g|=  8.62204D-03

At iterate   41    f=  2.26982D+00    |proj g|=  1.13697D-02

At iterate   42    f=  2.26972D+00    |proj g|=  1.99753D-02

At iterate   43    f=  2.26953D+00    |proj g|=  3.51412D-02

At iterate   44    f=  2.26918D+00    |proj g|=  4.74065D-02

At iterate   45    f=  2.26860D+00    |proj g|=  4.83962D-02

At iterate   46    f=  2.26699D+00    |proj g|=  5.08225D-02

At iterate   47    f=  2.26415D+00    |proj g|=  1.36901D-01

At iterate   48    f=  2.25997D+00    |proj g|=  8.06429D-02

At iterate   49    f=  2.25582D+00    |proj g|=  3.97484D-02

At iterate   50    f=  2.25468D+00    |proj g|=  5.97888D-02

At iterate   51    f=  2.25315D+00    |proj g|=  6.74137D-02

At iterate   52    f=  2.24917D+00    |proj g|=  3.69321D-02

At iterate   53    f=  2.24770D+00    |proj g|=  2.18101D-02

At iterate   54    f=  2.24663D+00    |proj g|=  1.87953D-02

At iterate   55    f=  2.24634D+00    |proj g|=  1.01867D-02

At iterate   56    f=  2.24597D+00    |proj g|=  1.05881D-02

At iterate   57    f=  2.24544D+00    |proj g|=  9.89684D-03

At iterate   58    f=  2.24525D+00    |proj g|=  3.97566D-03

At iterate   59    f=  2.24522D+00    |proj g|=  2.84928D-03

At iterate   60    f=  2.24521D+00    |proj g|=  5.90927D-03

At iterate   61    f=  2.24520D+00    |proj g|=  4.01243D-03

At iterate   62    f=  2.24518D+00    |proj g|=  2.27796D-03

At iterate   63    f=  2.24516D+00    |proj g|=  2.96954D-03

At iterate   64    f=  2.24507D+00    |proj g|=  1.21805D-02

At iterate   65    f=  2.24495D+00    |proj g|=  1.14386D-02

At iterate   66    f=  2.24470D+00    |proj g|=  6.46780D-03

At iterate   67    f=  2.24459D+00    |proj g|=  3.54023D-03

At iterate   68    f=  2.24454D+00    |proj g|=  3.16773D-03

At iterate   69    f=  2.24453D+00    |proj g|=  7.06564D-03

At iterate   70    f=  2.24452D+00    |proj g|=  2.94338D-03

At iterate   71    f=  2.24451D+00    |proj g|=  1.06000D-03

At iterate   72    f=  2.24451D+00    |proj g|=  1.21227D-03

At iterate   73    f=  2.24451D+00    |proj g|=  5.10170D-04

At iterate   74    f=  2.24451D+00    |proj g|=  8.05578D-05

At iterate   75    f=  2.24451D+00    |proj g|=  6.59917D-05

           * * *

Tit   = total number of iterations
Tnf   = total number of function evaluations
Tnint = total number of segments explored during Cauchy searches
Skip  = number of BFGS updates skipped
Nact  = number of active bounds at final generalized Cauchy point
Projg = norm of the final projected gradient
F     = final function value

           * * *

   N    Tit     Tnf  Tnint  Skip  Nact     Projg        F
    9     75     97     83     0     1   6.599D-05   2.245D+00
  F =   2.2445131888487211

CONVERGENCE: REL_REDUCTION_OF_F_<=_FACTR*EPSMCH
[9]:
<matplotlib.legend.Legend at 0x7fb1a8ce2110>
../../../_images/examples_notebooks_generated_ets_14_2.png
[10]:
print(fit.summary())
                                 ETS Results
==============================================================================
Dep. Variable:                      y   No. Observations:                   68
Model:                      ETS(AAdA)   Log Likelihood                -152.627
Date:                Tue, 14 Feb 2023   AIC                            327.254
Time:                        22:28:59   BIC                            351.668
Sample:                    03-01-1999   HQIC                           336.928
                         - 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      3.157      0.002       0.010       0.042
smoothing_seasonal     0.4009      0.080      5.040      0.000       0.245       0.557
damping_trend          0.9800        nan        nan        nan         nan         nan
initial_level         29.4460        nan        nan        nan         nan         nan
initial_trend          0.6146      0.392      1.567      0.117      -0.154       1.383
initial_seasonal.0    -3.4299        nan        nan        nan         nan         nan
initial_seasonal.1    -5.9571        nan        nan        nan         nan         nan
initial_seasonal.2   -11.4949        nan        nan        nan         nan         nan
initial_seasonal.3          0        nan        nan        nan         nan         nan
===================================================================================
Ljung-Box (Q):                        5.76   Jarque-Bera (JB):                 7.71
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 67.611002 63.136027 72.085977
2014-06-01 42.815041 38.340066 47.290016
2014-09-01 54.106430 49.631455 58.581405
2014-12-01 57.928005 53.453030 62.402980
2015-03-01 68.422270 63.947295 72.897245
2015-06-01 47.278272 42.803298 51.753247
2015-09-01 58.954912 54.479937 63.429887
2015-12-01 63.982081 59.507106 68.457056
2016-03-01 75.905175 71.430200 80.380150
2016-06-01 51.418321 46.654138 56.182504
2016-09-01 63.703334 58.629288 68.777381
2016-12-01 67.977966 62.575424 73.380508
2017-03-01 78.315606 71.735535 84.895677
2017-06-01 53.780544 46.883487 60.677601
2017-09-01 66.018313 58.788056 73.248569
2017-12-01 70.246645 62.668337 77.824953
2018-03-01 80.538911 71.892377 89.185446
2018-06-01 55.959383 46.968180 64.950586
2018-09-01 68.153575 58.804935 77.502215
2018-12-01 72.339202 62.621359 82.057045
2019-03-01 82.589617 71.864246 93.314989
2019-06-01 57.969075 46.875769 69.062381
2019-09-01 70.123073 58.651712 81.594434
2019-12-01 74.269310 62.410541 86.128079
2020-03-01 84.481123 71.655796 97.306451

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 0x7fb1a6c70350>
../../../_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.