Choose model for time series forecasting with R I try to forecast my web visitors on the web site for 10 future days using time series.
I have used a auto.arima() model My time series is daily. 
How can I choose between ets() model and auto.arima() model concidering that:
accuracy(auto.arima)

                   ME     RMSE      MAE        MPE    MAPE MASE      ACF1
Training set 26.84448 877.9984 697.2491 -0.7432541 9.72338  NaN -0.126274

and 
accuracy(ets)

                   ME     RMSE      MAE      MPE     MAPE MASE       ACF1
Training set 10.38619 1039.264 833.8032 -1.01747 11.62925  NaN 0.02922446

 A: You should use forecast and not predict to forecast your web visitors.
Here an example based on simulated data (I have no access to your data).
You may adapt this example to your data. Hope this may be of help.
Time series with daily data
Data simulation
data <- rnorm(3650, m=10, sd=2)

Use ts() to create time series
series <- ts(data, frequency=365, start=c(1919, 1))

attributes(series)
class(series)
length(series)

Plot the data
plot(series, col ="aquamarine")

The accuracy of forecasts can only be determined by considering how well a model performs on new data that were not used when fitting the model.
The size of the test set is typically about 20% of the total sample 
Training set
Use data from 1919 to 1926 for forecasting
sr = window(series, start=c(1919,1), end=c(1926,365))

Test set
Use remaining data from 1927 to 1928 to test accuracy
ser = window(series, start=c(1927,1), end=c(1928, 365))

Herewith enclosed is an exercise I've done on the basis of the online book of Hyndman and Athanasopoulos: "Forecasting: principles and practice", a very good reading.You may find the book here: https://www.otexts.org/fpp
Since this is a very simply simulation, there is no real trend in it, so the two models produce similar forecasting of a linera trend. Your data in all likelihood will have a seasonal component.

Assign the data to a vector of convenience
trainData <- sr     ### Training set (8 years, daily data)
testData <- ser     ### Test set    (2 years, daily data)


Auto-arima without stepwise (more precise)
library(fpp)

The default value in auto.arima() is test="kpss". 
A KPSS test has a null hypothesis of stationarity
In general, all the defaults are set to the values that give the best forecasts on average.
Caution, takes a while to compute
arimaMod <- auto.arima(trainData, stepwise=FALSE, approximation=FALSE)
arimaMod.Fr <-forecast(arimaMod,h=730)      ### h is the length of the forecast, in this case 2 years = 365*2 

Plot of the prediction and the test set
plot(arimaMod.Fr, plot.conf=FALSE)
lines(testData, col="red")
legend("topleft",lty=1,bty = "n",col=c("red","blue"),c("testData","ARIMAPred"))


Only the prediction
Plot the test set
plot(ser, main="Daily data", ylab="", xlab="Months", col = "red")

Plot the forecast on the training set
lines(arimaMod.Fr$mean, col="blue", lwd=3)


Accuracy
Accuracy for forecasting of sr (forecasted data) on ser (original data used as test set)
The best model had the lowest error (particularly the MAPE, Mean absolute percentage error)
accuracy(arimaMod.Fr,testData)


Test residues of arima
tsdisplay(residuals(arimaMod))


ETS
fit.ets <- ets(trainData)

fr.ets <- forecast(fit.ets,h=730)

plot(forecast(fit.ets,h=730))

Compare accuracy of ETS vs ARIMA
There is an overlap between the ETS and the ARIMA forecasting,
So I've plotted the ETS forecasting with a wider line
plot(testData)

lines(fr.ets$mean, col="red", lwd=5)
    lines(arimaMod.Fr$mean, col="green")
legend("topleft",lty=1,cex=1,y.intersp=0.6,bty = "n",col=c("black","red","green"),c("data","ETS","ARIMA"))

Accuracy test
accuracy(fr.ets,testData)
accuracy(arimaMod.Fr,testData)

A: The best model is the model with the the lowest error (particularly the MAPE, Mean absolute percentage error).
However, the accuracy of forecasts can only be determined by considering how well a model performs on new data that were not used when fitting the model.
The size of the test set is typically about 20% of the total sample. 
So, you need to split the series in a training set and a test set.
You may consult this tutorial, which is based on the online book of Hyndman and Athanasopoulos "Forecasting: principles and practice", a very good reading. The book explains with details what MAPE and the other measures of error are.
You may find the book here: https://www.otexts.org/fpp
The tutorial is here: https://rpubs.com/RatherBit/90267
I hope this is helpful.
