# What is the meaning of results produced by auto.arima() function of R?

I already posted the same question on https://stackoverflow.com/questions/49298634/how-to-interpret-results-of-auto-arima-in-r but some one pointed to ask it on here. I would appreciate any relevant help.

Let I have a time series X and I used fit<-auto.arima() followed by fcast <- forecast(fit, h=100). When I do print(summary(fcast)), I get a result having number of variables (snapshot of an example is attached).

1. What is the meaning of each variable (specially, highlighted in red boxes)? If someone can explain in simple terms, it would be great.
2. What is the meaning of getting -Inf and Inf for MPE and MAPE respectively?
3. What is meaning of Lo 80, Hi 80, Lo 95, and Hi 95? Can I say that it is 80% likely to have actual value equal to Forecast+Lo 80+Hi 80?

1. The first red box gives the estimated variance of the residual noise term, the log-likelihood and various information criteria: the AIC, the small-sample corrected AICc and the BIC.

The second red box contains the Mean Percentage Error (MPE) and the Mean Absolute Percentage Error () on the training set, with some other accuracy measures. You may want to look at ?accuracy.

The third red box contains "the" forecast. The first column contains the forecasted expectation per time period. For the others, see point 3 below.

2. Your percentage errors are almost certainly infinite because you have zero actuals in your training sample, so calculating percentage errors entails a division by zero, which is undefined. In such a case, percentage errors are not helpful. (The MAPE - and also the MPE - has other shortcomings, too.)

3. The "Lo 80" column gives the lower boundary of an 80% . Specifically, it gives the 10% quantile of the predictive density, which is calculated using a normality assumption. The hope is that this 80% PI will contain 80% of future realizations. Note that PIs are notoriously too narrow. "Hi 80" analogously gives the upper boundary of the 80% PI, and the same for "Lo 95" and "Hi 95".

You may be interested in Forecasting: Principles and Practice, a free online forecasting textbook.

• prediction interval and confidence interval are same? – Jitendra Mar 16 '18 at 10:10
• Good question! No, they aren't. CIs pertain to unobservable parameters (e.g., the expectation of the future time series), PIs to observable realizations. See also here. This is frequently confused. – Stephan Kolassa Mar 16 '18 at 10:16
• fcast <- forecast(fit, h=100) forecasts next 100 values based on fit. How can I get forecasted values? Does fcast$fitted gives the forecast? In my case, fcast$fitted produces multiple instances of negative values. – Jitendra Mar 20 '18 at 6:56
• fcast$fitted will give you the in-sample fits. To get the mean point forecasts, use fcast$mean. See ?forecast. If you have nonsensical negative values, that is a separate problem. We have some related questions here. – Stephan Kolassa Mar 20 '18 at 7:10
• fcast$mean gives the mean point forecasts. How can I compute errors in future forecasts? Is it difference between fcast$mean and actual? If possible can you give me an expression that can compute RMSE. For instance, some thing like sqrt(mean((fcast$mean - fcast$x)^2)) – Jitendra Mar 21 '18 at 19:15