Time Series Analysis, trend, moving averages?

Question

I'm home studying for a statistics exam (school level). Part of the syllabus is "seasonal variation, trend, short term and random variation. However, it's not in my text book! Any basic formulae for "moving averages to estimate seasonal effect, to deseasonalise series and make short term forecasts" would be appreciated.

Answer 1

We assume that time series data is made up of the following:
trend + seasonal variation + short term variation + random variation.
Trend is the long-term pattern.
Seasonal variation is a periodic cycle of predictable changes to the data. For instance, retail stores typically have a sharp jump in sales during December (and a slump in January when everyone is trying to pay off their credit cards). Each element of the cycle is called a season; for monthly sales data the seasons would be the months.
Short-term variation is caused by specific events that are not on a repeated cycle. Data corresponding to identified short-term variations should generally be excluded from analysis. An example of this might be a low business volume on a particular day due to power failure, or a boycott.
Random variation is pretty much what the name suggests; it's the day-to-day (or minute-to-minute or whatever) variation caused by random factors such as people deciding where to eat for lunch.

In analysing time series data, we are generally interested in the trend and seasonal variations. There's not much we can do to analyse the other two. To determine seasonal variations we use moving averages.

Moving averages are very simple. You define an average based over a certain number of points, say n. This number is generally chosen with some idea of what it's likely to be (i.e. by cheating). For instance, if you have monthly sales data, you probably want to choose n = 12 to correspond to one year's worth of data. If the expected cycle is daily, weekly, or monthly, you choose an appropriate number of points to make that the range covered. For instance, if you have a temperature reading every 4 hours and you want to eliminate the daily variation between day and night, you'd take a 6 point moving average to cover 24 hours of data.

The moving average itself is defined at the centre of the n points. If n is odd, this will correspond to one of the data points; if n is even it will be half way between two data points. Don't be alarmed by this. Note that there are fewer points in the moving average than in the original data series, as you need to be about n/2 points in before you can get n data points to work with.

To get deseasonalised data, replace the original data with a moving average designed to cover one cycle at a time, as mentioned above. You can then use the moving average data to construct trend lines to find the long-term trend. The average of each season minus the trend line, across all years, gives you the seasonal variation for that season. For instance, if we have Q1-Q4 data for 2001-2006, construct 4-point moving averages to get deseasonalised data and work out the trend line. Then take the average of (Q1 - trend line) for 2001 through 2006 to get the average seasonal variation for Q1, and do the same for Q2 to Q4. To predict sales for Q1 2007, you need to take the trend line and add the average seasonal variation for Q1.

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Answer 2

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