SARIMA Model for longterm trend limitation

Question

Unfortunately i do not find anything regarding longterm forecast with SARIMA. What is the problem with longerm forecasts for exmaple 140 data points are used to forecasts another 140 data points?

What is the limitation?

Reagrds,

Max

Answer 1

With any time series model - not just SARIMA - the longer the forecast, the harder it might prove for the time series model to predict accurately.

The challenge with time series modelling is to capture all relevant seasonality trends and repeating patterns. This is why using 10 years of weather data to predict that of next year is feasible - whereas using only 1 year of training data might lead to issues.

As an example, here is a graph of the maximum recorded air temperature in Dublin, Ireland over an extended period of time.

Decomposing the series reveals an upward trend along with clear seasonality patterns:

When a SARIMA model was used to forecast the test data (185 periods forward) - over 70% of the forecasts deviated from the actual by less than 10 percent:

>>> print(results.summary())
                                 Statespace Model Results                                 
==========================================================================================
Dep. Variable:                              maxtp   No. Observations:                  740
Model:             SARIMAX(1, 0, 0)x(2, 1, 0, 12)   Log Likelihood                 468.218
Date:                            Thu, 14 Mar 2019   AIC                           -926.436
Time:                                    16:54:47   BIC                           -903.485
Sample:                                12-01-1941   HQIC                          -917.580
                                     - 07-01-2003                                         
Covariance Type:                              opg                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
intercept      0.0006      0.005      0.126      0.900      -0.009       0.011
ar.L1          0.1728      0.032      5.469      0.000       0.111       0.235
ar.S.L12      -0.6074      0.023    -26.858      0.000      -0.652      -0.563
ar.S.L24      -0.3256      0.023    -14.108      0.000      -0.371      -0.280
sigma2         0.0161      0.000     39.691      0.000       0.015       0.017
===================================================================================
Ljung-Box (Q):                      129.08   Jarque-Bera (JB):              2081.35
Prob(Q):                              0.00   Prob(JB):                         0.00
Heteroskedasticity (H):               0.76   Skew:                            -0.97
Prob(H) (two-sided):                  0.04   Kurtosis:                        11.05
===================================================================================

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).

>>> predictions=results.predict(741, 925, typ='levels')
>>> predictions=np.exp(predictions)
>>> test=np.exp(test)
>>> mse=(predictions-test)/test
>>> mse=abs(mse)
>>> below10=mse[mse < 0.10].count()
>>> all=mse.count()
>>> accuracy=below10/all
>>> accuracy
0.7081081081081081

Therefore, forecasting 140 data points may well be feasible, but ideally your training data would be significantly greater than 140 data points. Depending on the type of data under analysis, too few data points in the training set risks that the model will not adequately capture the appropriate trend and fluctuations influenced by seasonality.

Answer 2

"What is the problem with longerm forecasts for exmaple 140 data points are used to forecasts another 140 data points?"

The confidence limits may broaden due to the form of the model . If there are "many outliers" i.e. empirically identified pulses allowing for them in the future can have a major effect in the widening of the prediction limits via bootstrapping.

See How to set (p,d,q) and (P,D,Q) for SARIMA time series model for an example of how temperature can be efficiently modelled using pseudo-causals ( seasonal dummies) identified from the data suggesting month of the year effects along with anomalies and a level shift rather than arima effects as the better choice.

Unwarranted arima differencing yields unecessarilily wide limits .