Solved – getting flat time series forecasts from most of the techniques

arimaexponential-smoothingforecastingrtime series

I have a simple example time series:

Data:

 Jan   Feb   Mar   Apr   May   Jun   Jul   Aug   Sep   Oct   Nov   Dec
2000 200.1 199.5 199.4 198.9 199.0 200.2 198.6 200.0 200.3 201.2 201.6 201.5
2001 201.5 203.5 204.9 207.1 210.5 210.5 209.8 208.8 209.5 213.2 213.7 215.1
2002 218.7 219.8 220.5 223.8 222.8 223.8 221.7 222.3 220.8 219.4 220.1 220.6
2003 218.9 217.8 217.7 215.0 215.3 215.9 216.7 216.7 217.7 218.7 222.9 224.9
2004 222.2 220.7 220.0 218.7 217.0 215.9 215.8 214.1 212.3 213.9 214.6 213.6
2005 212.1 211.4 213.1 212.9 213.3 211.5 212.3 213.0 211.0 210.7 210.1 211.4
2006 210.0 209.7 208.8 208.8 208.8 210.6 211.9 212.8 212.5 214.8 215.3 217.5
2007 218.8 220.7 222.2 226.7 228.4 233.2 235.7 237.1 240.6 243.8 245.3 246.0
2008 246.3 247.7 247.6 247.8 249.4 249.0 249.9 250.5 251.5 249.0 247.6 248.8
2009 250.4 250.7 253.0 253.7 255.0 256.2 256.0 257.4 260.4 260.0 261.3 260.4
2010 261.6 260.8 259.8 259.0 258.9 257.4 257.7 257.9 257.4 257.3 257.6 258.9
2011 257.8 257.7 257.2 257.5 256.8 257.5 257.0 257.6 257.3 257.5 259.6 261.1
2012 262.9 263.3 262.8 261.8 262.2 262.7

I then ran 4 different time series forecast models: Holt Winters smoothing, TBATS Smoothing, ARIMA, and AR Neural Nets with the following functions in R, using the "forecast" package: HoltWinters, tbats, auto.arima, nnetar

I forecasted 36 periods (3 years) ahead. The following results:

My question is why does the HoltWinters seem to be the only meaningful forecast. I have enough data that getting flat lines for all the other forecasts seems odd. Like something is breaking or I am not understanding something. Especially since TBATS is a generalized form of Holt Winters. And just looking at the series ARIMA should output something more than a straight average? Right? The (1,1,1) even implies its taking a difference into account. Also none of the models seem to fail and return a null model. Very curious why I am seeing these results and how to interpret.

Any help or explanation is much appreciated!

demand is a ts object by the way.
Below is my code:

> hw_test = HoltWinters(demand)
> hw_forecast = forecast(hw_test, h=36)
> plot(hw_forecast)
> arima_test = auto.arima(demand)
> arima_forecast = forecast(arima_test, h=36)
> plot(arima_forecast)
> tbats_test = tbats(demand)
> tbats_forecast = forecast(tbats_test, h=36)
> plot(tbats_forecast)
> nn_test = nnetar(demand)
> nn_forecast = forecast(nn_test, h=36)
> plot(nn_forecast)

Best Answer

Au contraire , you have a fairly complex time series which can't be (easily) solved with simple tools. Your data set suggests a change in the error variance at period 93 . This is confirmed visually visually (if you look real close ) as the series is highly autocorrelated lag 1. Analysis suggests a change in the model error variance at on around period 93 ..

There are three time trends in addition to a first order differencing and an ma(1) . . The residual plot suggests near randomness except for a possible level shift in the time range 78-96 which is confirmed by the residual acf culminating in a very reasonable upwards trending forecast . The interesting thing to me there are no anomalies(pulses) that need remedying. I used AUTOBOX ( my tool of choice which i have helped to develop ) .

To find out what is wrong with the models you have tried , look at the residuals to assess thoroughness or the effectiveness of the equation to separate signal and noise.

The actual/fit and forecast is here . Note that the AUTOBOX forecast is quite similar to your HW forecast without the false seasonal component (model injected).

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Solved – Forecasting beyond one season using Holt-Winters’ exponential smoothing

I am not very familiar with Holt-Winters, however I have this excellent book by @Rob Hyndman. The package forecast (which is based on the book) of statistical package R gives the following result on your data:

> hw<-read.table("~/R/stackoverflow/hw.txt")
> tt<-ts(hw[,3],start=c(1999,1),freq=12)

> aa<-forecast(tt)
> plot(aa)
> summary(aa)

Forecast method: ETS(M,N,A)

Model Information:
ETS(M,N,A) 

Call:
 ets(y = object) 

  Smoothing parameters:
    alpha = 0.1701 
    gamma = 1e-04 

  Initial states:
    l = 870.4847 
    s = -278.0815 -143.6584 151.959 -135.595 514.2527 236.9216
           -32.7679 128.8337 115.0829 47.5922 -234.4105 -370.1288

  sigma:  0.1122

     AIC     AICc      BIC 
1892.756 1896.346 1933.115 

In-sample error measures:
         ME        RMSE         MAE         MPE        MAPE        MASE 
 18.1543007 121.8594668  70.7086492   0.8480306   7.0006920   0.2893504

Here is the graph of the forecast together with the confidence intervals: enter image description here

Note that the function forecast picks automatically the best exponential smoothing model from 30 models which are classified by the type of trend model, seasonal part model and the additivity or multiplicity of error.

The best model found in your data is with multiplicative error, no trend and additive seasonality, which is less complicated model than you are trying to fit. The way function forecast works is however that the more complicated model was considered and rejected in favor the final model.

If you provide the exact formulas it would be possible to fit the precise model to see whether the problem you described is really property of the model.

Solved – How to adjust for a temporary 12-month level shift in time series

In the absence of the knowledge of the event , what you are looking for is a procedure to simultaneously identify and refine an arima model AND also automatically identify and include 2 level/step shift indicators (possibly collapsing into 1).... reflecting the temporary effect via Intervention Detection procedures http://docplayer.net/12080848-Outliers-level-shifts-and-variance-changes-in-time-series.html . If you post your actual data in a column oriented csv file I will try to help you further.

Alternatively if you are aware of the timing and length of the intervention you can construct an X variable of the form ...0,0,0,0,0,...,1,1,1,1,...0,0,0,0,0, detailing the known beginning and termination points and then try to identify the arima portion of this armaX model.

EDITED AFTER RECEIPT OF DATA:

The data that you posted is different from the graph you posted.

Here is a graph of the data you posted which is the data I analyzed.

Your data suggest the need for a differencing factor of order 1 ....thus a level shift detection requires 2 pulses. When you difference a step/level you get a pulse ... thus a model that has differencing requires pulses to reflect the abrupt upwards effect and the abrupt downwards effect. A partial picture of the model is here .. .272 up and .241 down suggesting a different return to the baseline.

I submitted the 151 monthly numbers to my favorite time series program and it automatically developed a use model .Here is the Actual/Fit and Forecast graph and less cluttered here .

The equation is here detailing four seasonal pulses covering Feb, Sept and Nov suggesting non-seasonal activity for the other 9 months and 4 additional pulses .

Note that the differencing operator is distributed across all series in the equation. Also note that {1-B}level = pulse thus {1-B]pulse = {1-B}{1-B}level . The AUTOBOX equation shows {1-B}pulse which if you wish can be restated as {1-B}{1-B}level .

Restated a pulse in a non-stationary can be interpreted as an intercept change. Visually one can confirm the identified Pulses as points of change for the model-implied intercept.

A significant change in error variance (downwards) was found at or about time period 60.

The model statistics are here and here

The forecasts are detailed here .

EDITED TO ANSWER THE OP'S COMMENT

Adjusting the 12 observations and then identifying an ARIMA model is a sound approach. The only problem is there are 4 seasonal factors ( seasonal pulses ) and 3 pulses that need to be adjusted for before identifying the first difference model (0,1,0)(0,0,0) with a constant while dealing with a non-constant error variance. Your resultant ACF of the errors should look something like this ...suggesting sufficiency.

By the way .. why did you post data that was different from your graph ????

Best Answer

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