# Why are exponential smoothing forecasts exactly the same for the following 5 days in hourly sampled 30 days data? R ets forecast package

I would like to use exponential smoothing to forecast for 5 days, but forecasts look all same. I have read the documentation of ets package and tried different Additive, Multiplicative model, but could not fix the problem. My data consists 30 days of hourly measurements and I would like to forecast day from 31 to 35.

Here is my code

library(forecast)

mydatatsfreq <- ts(mydata, frequency = 24)
fit <- ets(mydatatsfreq, model='ZZZ')
summary(fit)


Output of summary

ETS(A,Ad,A)

Call: ets(y = mydatatsfreq, model = "ZZZ")

Smoothing parameters: alpha = 0.9971 beta = 1e-04 gamma = 2e-04 phi = 0.9788

Initial states: l = 6.5994 b = -0.0745 s=-8.5981 -8.3857 -8.2845 -8.4552 -8.5558 -8.6233 -8.662 -6.5815 5.5694 15.1411 20.8226 22.4551 23.014 20.7874 15.5312 7.1746 -3.5179 -8.8709 -8.8073 -8.5763 -8.6457 -8.74 -8.6555 -8.5355

sigma: 1.7493

AIC     AICc      BIC
5593.623 5596.326 5730.958

Training set error measures: ME RMSE MAE MPE MAPE MASE ACF1 Training set 0.00722588 1.749286 0.8136336 NaN Inf 0.6419291 0.05326251


This is the plot of forecasts

Results of auto.arima()

Series: mydatatsfreq
ARIMA(2,0,2)(0,0,2)[24] with non-zero mean

Coefficients:
ar1      ar2      ma1      ma2    sma1    sma2    mean
1.8022  -0.8810  -0.5069  -0.3599  0.4508  0.3917  0.1713
s.e.  0.0190   0.0186   0.0414   0.0397  0.0447  0.0336  0.0056

sigma^2 estimated as 0.002391:  log likelihood=1146.15
AIC=-2276.3   AICc=-2276.1   BIC=-2239.68

• Perhaps because your series has such constant variance. What does auto.arima() give you? – Digio Aug 4 '17 at 11:31
• I have added the output of auto.arima plot is worse than ets it is decreasing than becomes flat. – Reiso Aug 4 '17 at 11:39
• What exactly is the issue? It appears to be a pretty reasonable forecast. Seems like the level is expected to continue, there is no trend and the seasonality looks accounted for. Have you attempted to create a few test set days to see how it would perform? – AnscombesGimlet Aug 4 '17 at 18:39
• the issue is that forecasts should change for each day. For example first 7 days are all different in respect to their max values, but forecasts are all same. I have changed frequency value to 24*7 from 24 to catch weekly seasonality. With this frequency, forecasts are different for all 5 days, but now fitted forecasts are not that good as they are with frequency=24 and error also increased for fitted forecasts. – Reiso Aug 5 '17 at 2:39

ETS is not very useful in my opinion for data such as you describe due to the POSSIBLE presence of anomalies/level shifts/time trends etc. , changing error variability , the need for deterministic seasonal pulses rather than seasonal ARIMA , etc.. There are often hourly effects , day-of-the-week effects and other latent variables to be exploited. Often the hour effect depends on which day-of-the-week . Please post your data and I will try and help you using potentially more powerful procedures than what you are using/trying.

EDITED AFTER RECEIPT OF DATA:

You have 24 values per day for 30 days where non-zero values arise for only 10 of the 24 hours . An ARIMA approach (your approach) is flawed because of the fact that so many values are 0.0 thus creating the impression of strong autocorrelation for short term lags. This is why your forecasts are the SAME. You really have 10 NON-ZERO observations per day for 30 days ( 300 observations) and wish to predict say for the next 5 days (50 values).

Using AUTOBOX automatically (my tool of choice) a reasonable model is here . An unusual value was detected at the 26th day hour 2 of 10 via Intervention Detection procedures. A significant reduction in the variance of the errors was detected using the Tsay procedure at observation 61 (first reading for the 7th week) thus Weighted Least Squares was employed . A significant upwards shift (level/step ) was detected at week 14 period 4 (134th point of 300) . Finally an AR(1) component was identified and used.

Here is the ACF of the model residuals suggesting sufficiency . Here is the plot of the forecasts for the next 5 weeks (50 values) . The actual.fit and forecast graph is here and here with forecast limits

In summary 8 seasonal pulses (deterministic effects for hour of the day modulo 10) were found to be significant and used.

In terms of software , if you have a simple problem , simple tools will suffice. Your problem in my opinion required a fairly comprehensive approach as simple tools failed to characterize the data. Simple tools rarely (never !) deal with complex data.

NEW EDIT:

The model is driven by fundamentally deterministic structure i.e. 1) a constant 2) 8 hourly dummies 3) a level shift 4 ) a pulse indicator PLUS an AR(1) factor of .6 which results in an assymtotic forecast. The hourly dummy variables are dominant resulting in an approximate constant expectation. The forecasts are slightly different at the third decimal position.

• thanks for your answer. I have uploaded the data to the following link. [link]( mega.nz/#!t4QAhKpC!msD1scrhdqIZe_qcWENaOUCkW59lO6h9tJgG19xufRs) . Please let me know, if you cannot access the data. Your help will be really appreciated. – Reiso Aug 7 '17 at 1:23
• Thanks for the edit and your effort. As I understood you removed zero values from the data and applied ARIMA model. But same forecasts are still there. You forecasted for 5 following days and their forecasts look exactly the same. Is that problem something we cannot overcome? I think it is happening because of the nature of data. Additional information, when I make frequency = 24*5 I don't get same forecasts, but now ets cannot fit data well as it can while freq=24. (Freq=24 same forecasts, Freq=24*5 different forecast but not good fit.) – Reiso Aug 8 '17 at 6:34
• I can't agree with your conclusion that "Your problem in my opinion required a fairly comprehensive approach" since your solution (looking at the plots) doesn't seem to differ very much from the ETS solution... – Tim Aug 8 '17 at 10:22
• There are a number of subtle differences suggesting that the OP can glean understanding from the data. Time series analysis is often a lot more than forecasting. The identified pulse suggests an unspecified exogenous factor . The change in level and error variance suggests unspecified exogenous factors. Having said that this particular data set is so strongly driven by seasonal dummies that your comment is true in terms of the forecast function BUT not so in terms of "information/signal extraction" BUT that is just my opinion. I would like to hear what the OP has to opine in this regard. – IrishStat Aug 8 '17 at 11:11
• Rereading the post .... the OP is concerned that the max values are changing on a daily basis....suggests to me that in the absence of the expected value changing the forecast variance might be the culprit. The identification of the error variance changing at period 60 (of 300) suggests that the maximum expected value might have changed over time . – IrishStat Aug 8 '17 at 11:30