Some forecasting methods are very simple and surprisingly effective. Here are four methods that we will use as benchmarks for other forecasting methods.
Average method
Here, the forecasts of all future values are equal to the mean of the historical data. If we let the historical data be denoted by 
, then we can write the forecasts as
![\[ \hat{y}_{T+h|T} = \bar{y} = (y_{1}+\dots+y_{T})/T. \]](http://otexts.com/fpp/wp-content/ql-cache/quicklatex.com-b1e676883925fc0031925993646f5ed8_l3.png "Rendered by QuickLaTeX.com")
The notation 
is a short-hand for the estimate of 
based on the data 
.
Although we have used time series notation here, this method can also be used for cross-sectional data (when we are predicting a value not included in the data set). Then the prediction for values not observed is the average of those values that have been observed. The remaining methods in this section are only applicable to time series data.
meanf(y, h) # y contains the time series # h is the forecast horizon |
Naïve method
This method is only appropriate for time series data. All forecasts are simply set to be the value of the last observation. That is, the forecasts of all future values are set to be 
, where 
is the last observed value. This method works remarkably well for many economic and financial time series.
naive(y, h) rwf(y, h) # Alternative |
Seasonal naïve method
A similar method is useful for highly seasonal data. In this case, we set each forecast to be equal to the last observed value from the same season of the year (e.g., the same month of the previous year). Formally, the forecast for time 
is written as
![\[ y_{T+h-km} \text{~where $m=$ seasonal period, $k=\lfloor (h-1)/m\rfloor+1$,} \]](http://otexts.com/fpp/wp-content/ql-cache/quicklatex.com-8888b462364ee4b837cc1e81c5923fde_l3.png "Rendered by QuickLaTeX.com")
and 
denotes the integer part of 
. That looks more complicated than it really is. For example, with monthly data, the forecast for all future February values is equal to the last observed February value. With quarterly data, the forecast of all future Q2 values is equal to the last observed Q2 value (where Q2 means the second quarter). Similar rules apply for other months and quarters, and for other seasonal periods.
snaive(y, h) |
Drift method
A variation on the naïve method is to allow the forecasts to increase or decrease over time, where the amount of change over time (called the drift) is set to be the average change seen in the historical data. So the forecast for time 
is given by
![\[ y_{T} + \frac{h}{T-1}\sum_{t=2}^n (y_{t}-y_{t-1}) = y_{n} + h \left( \frac{y_{T} -y_{1}}{T-1}\right). \]](http://otexts.com/fpp/wp-content/ql-cache/quicklatex.com-8c0092fc6a785983111cf3d57772b076_l3.png "Rendered by QuickLaTeX.com")
This is equivalent to drawing a line between the first and last observation, and extrapolating it into the future.
rwf(y, h, drift=TRUE) |
Examples
Figure 2.13 shows the first three methods applied to the quarterly beer production data.
Figure 2.13: Forecasts of Australian quarterly beer production.
beer2 <- window(ausbeer,start=1992,end=2006-.1) beerfit1 <- meanf(beer2, h=11) beerfit2 <- naive(beer2, h=11) beerfit3 <- snaive(beer2, h=11) plot(beerfit1, plot.conf=FALSE, main="Forecasts for quarterly beer production") lines(beerfit2$mean,col=2) lines(beerfit3$mean,col=3) legend("topright",lty=1,col=c(4,2,3), legend=c("Mean method","Naive method","Seasonal naive method")) |
In Figure 2.14, the non-seasonal methods were applied to a series of 250 days of the Dow Jones Index.
Figure 2.14: Forecasts based on 250 days of the Dow Jones Index.
dj2 <- window(dj,end=250) plot(dj2,main="Dow Jones Index (daily ending 15 Jul 94)", ylab="",xlab="Day",xlim=c(2,290)) lines(meanf(dj2,h=42)$mean,col=4) lines(rwf(dj2,h=42)$mean,col=2) lines(rwf(dj2,drift=TRUE,h=42)$mean,col=3) legend("topleft",lty=1,col=c(4,2,3), legend=c("Mean method","Naive method","Drift method")) |
Sometimes one of these simple methods will be the best forecasting method available. But in many cases, these methods will serve as benchmarks rather than the method of choice. That is, whatever forecasting methods we develop, they will be compared to these simple methods to ensure that the new method is better than these simple alternatives. If not, the new method is not worth considering.
Proceed to Section 2/4.