I want to know how accurate one ARIMA model is for estimating a second time series.

I've built the model:

mod1 <- auto.arima(x)
refit <- Arima(y, model=mod1)
> accuracy(refit)
               ME     RMSE MAE       MPE     MAPE MASE  ACF1
Training set -0.8 1.549193 1.2 -69.33333 77.33333  0.8 -0.05

Now I am not sure how to interpret these results. How good is the fit? Is there an accuracy or some measure like that?


Here are a few points.

  1. Your ARIMA model is not "estimating a second time series", it is filtering it.

  2. The function accuracy gives you multiple measures of accuracy of the model fit: mean error (ME), root mean squared error (RMSE), mean absolute error (MAE), mean percentage error (MPE), mean absolute percentage error (MAPE), mean absolute scaled error (MASE) and the first-order autocorrelation coefficient (ACF1). It is up to you to decide, based on the accuracy measures, whether you consider this a good fit or not. For example, mean percentage error of nearly -70% does not look good to me in general, but that may depend on what your series are and how much predictability you may realistically expect.

  3. It is often a good idea to plot the original series and the fitted values, and also model residuals. You may occasionally learn more from the plot than from the few summarizing measures such as the ones given by the accuracy function.

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Out of all the one simplest to understand is MAPE (Mean absolute percentage error). It considers actual values fed into model and fitted values from the model and calculates absolute difference between the two as a percentage of actual value and finally calculates mean of that.

For example if below are your actual data and results from ARIMA model

ActualData FittedValue AbsolutePercentageError
120        119.5       (abs(120-119)/120)*100 = 0.83%
128        126         (abs(128-126)/128)*100 = 1.56% 

MAPE = (0.83%+1.56%)/2 = 1.195%

Similarly you can do a quick google search to find out how meaning of other criterias. As per my experience MAPE is easiest one to explain to a layman, in case you want to explain model accuracy to a business user who is statistics illiterate. Also, you should forecast for a holdout sample for future and do the similar exercise to see how well it fits for future values Vs. the actuals.

Edit: This is an interesting discussion which accuracy metric to use in what scenario. Why use a certain measure of forecast error (e.g. MAD) as opposed to another (e.g. MSE)?

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I was searching myself how to interpret these indices. Note that the best method remains to plot the predictions over the real data of the same period.

I found these information on various websites including Wikipedia, stackexchange / stackoverflow, statisticshowto and other web places: -You may "Ecosia" some of these phrases to find their source.

  1. ME: Mean Error -- The mean error is an informal term that usually refers to the average of -- all the errors in a set. An “error” in this context is an uncertainty in a measurement, -- or the difference between the measured value and true/correct value

    1. RMSE: Root Mean Squared Error
      2.1 MAE: Mean Absolute Error -- The MAE measures the average magnitude of the errors in a set of forecasts, -- without considering their direction. It measures accuracy for continuous variables. -- The RMSE will always be larger or equal to the MAE; -- the greater difference between them, the greater the variance in the individual errors -- in the sample. If the RMSE=MAE, then all the errors are of the same magnitude -- Both the MAE and RMSE can range from 0 to ∞. -- They are negatively-oriented scores: Lower values are better.

    2. MPE: Mean Percentage Error -- the mean percentage error (MPE) is the computed average of -- percentage errors by which forecasts of a model differ from actual values of the -- quantity being forecast.

    3. MAPE: Mean Absolute Percentage Error -- The MAPE, as a percentage, only makes sense for values where divisions and -- ratios make sense. It doesn't make sense to calculate percentages of temperatures -- MAPEs greater than 100% can occur. -- then this may lead to negative accuracy, which people may have a hard time understanding -- Error close to 0% => Increasing forecast accuracy -- Around 2.2% MAPE implies the model is about 97.8% accurate in predicting the next 15 observations.

    4. MASE: Mean Absolute Scaled Error -- Scale invariance: The mean absolute scaled error is independent of the scale of the data, -- so can be used to compare forecasts across data sets with different scales. -- ok for scales that do not have a meaningful 0, -- penalizes positive and negative forecast errors equally -- Values greater than one indicate that in-sample one-step forecasts from the naïve method perform better than the forecast values under consideration. -- When comparing forecasting methods, the method with the lowest MASE is the preferred method.

    5. ACF1: Autocorrelation of errors at lag 1.' -- it is a measure of how much is the current value influenced by the previous values in a time series. -- Specifically, the autocorrelation function tells you the correlation between points separated by various time lags -- the ACF tells you how correlated points are with each other, -- based on how many time steps they are separated by. That is the gist of autocorrelation, -- it is how correlated past data points are to future data points, for different values of the time separation. -- Typically, you'd expect the autocorrelation function -- to fall towards 0 as points become more separated (i.e. n becomes large in the above notation) -- because its generally harder to forecast further into the future from a given set of data. -- This is not a rule, but is typical. -- ACF(0)=1 (all data are perfectly correlated with themselves), -- ACF(1)=.9 (the correlation between a point and the next point is 0.9), ACF(2)=.4 -- (the correlation between a point and a point two time steps ahead is 0.4)...etc.

      MAPE (???), Correlation and Min-Max Error can be used an RMSE of 100 for a series whose mean is in 1000’s is better than an RMSE of 5 for series in 10’s. So, you can’t really use them to compare the forecasts of two different scaled time series.

.......... All your indicators (ME, RMSE, MAE, MPE, MAPE, MASE, ACF1,...) are aggregations of two types of errors : a bias (you have the wrong model but an accurate fit) + a variance (you have the right model but a inaccurate fit). And there is no statistical method to know if you have a high bias and low variance or a high variance and low bias. So I suggest, you make a plot and make an eye-stimate to select the "best" one, best meaning with the least business consequences if you are wrong.

Generally, all of these values to be as small as possible

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