Seasonality

In time series data, seasonality refers to the trends that occur at specific regular intervals less than a year, such as weekly, monthly, or quarterly. Seasonality may be caused by various factors, such as weather, vacation, and holidays and consists of periodic, repetitive, and generally regular and predictable patterns in the levels of a time series.
Seasonal fluctuations in a time series can be contrasted with cyclical patterns. The latter occur when the data exhibits rises and falls that are not of a fixed period. Such non-seasonal fluctuations are usually due to economic conditions and are often related to the "business cycle"; their period usually extends beyond a single year, and the fluctuations are usually of at least two years.
Organisations facing seasonal variations, such as ice-cream vendors, are often interested in knowing their performance relative to the normal seasonal variation. Seasonal variations in the labour market can be attributed to the entrance of school leavers into the job market as they aim to contribute to the workforce upon the completion of their schooling. These regular changes are of less interest to those who study employment data than the variations that occur due to the underlying state of the economy; their focus is on how unemployment in the workforce has changed, despite the impact of the regular seasonal variations.

It is necessary for organisations to identify and measure seasonal variations within their market to help them plan for the future. This can prepare them for the temporary increases or decreases in labour requirements and inventory as demand for their product or service fluctuates over certain periods. This may require training, periodic maintenance, and so forth that can be organized in advance. Apart from these considerations, the organisations need to know if variation they have experienced has been more or less than the expected amount, beyond what the usual seasonal variations account for.

Motivation

There are several main reasons for studying seasonal variation:

Detection

The following graphical techniques can be used to detect seasonality:

A run sequence plot will often show seasonality
A seasonal plot will show the data from each season overlapped
A seasonal subseries plot is a specialized technique for showing seasonality
Multiple box plots can be used as an alternative to the seasonal subseries plot to detect seasonality
An autocorrelation plot and a spectral plot can help identify seasonality.

A really good way to find periodicity, including seasonality, in any regular series of data is to remove any overall trend first and then to inspect time periodicity.
The run sequence plot is a recommended first step for analyzing any time series. Although seasonality can sometimes be indicated by this plot, seasonality is shown more clearly by the seasonal subseries plot or the box plot. The seasonal subseries plot does an excellent job of showing both the seasonal differences and also the within-group patterns. The box plot shows the seasonal difference quite well, but it does not show within group patterns. However, for large data sets, the box plot is usually easier to read than the seasonal subseries plot.
The seasonal plot, seasonal subseries plot, and the box plot all assume that the seasonal periods are known. In most cases, the analyst will in fact, know this. For example, for monthly data, the period is 12 since there are 12 months in a year. However, if the period is not known, the autocorrelation plot can help. If there is significant seasonality, the autocorrelation plot should show spikes at lags equal to the period. For example, for monthly data, if there is a seasonality effect, we would expect to see significant peaks at lag 12, 24, 36, and so on.
An autocorrelation plot can be used to identify seasonality, as it calculates the difference between a Y value and a lagged value of Y. The result gives some points where the two values are close together, but other points where there is a large discrepancy. These points indicate a level of seasonality in the data.
Semiregular cyclic variations might be dealt with by spectral density estimation.

Calculation

Seasonal variation is measured in terms of an index, called a seasonal index. It is an average that can be used to compare an actual observation relative to what it would be if there were no seasonal variation. An index value is attached to each period of the time series within a year. This implies that if monthly data are considered there are 12 separate seasonal indices, one for each month. The following methods use seasonal indices to measure seasonal variations of a time-series data.

Method of simple averages

The measurement of seasonal variation by using the ratio-to-moving-average method provides an index to measure the degree of the seasonal variation in a time series. The index is based on a mean of 100, with the degree of seasonality measured by variations away from the base. For example, if we observe the hotel rentals in a winter resort, we find that the winter quarter index is 124. The value 124 indicates that 124 percent of the average quarterly rental occur in winter. If the hotel management records 1436 rentals for the whole of last year, then the average quarterly rental would be 359=. As the winter-quarter index is 124, we estimate the number of winter rentals as follows:
359*=445;
Here, 359 is the average quarterly rental. 124 is the winter-quarter index. 445 the seasonalized winter-quarter rental.
This method is also called the percentage moving average method. In this method, the original data values in the time-series are expressed as percentages of moving averages. The steps and the tabulations are given below.

Ratio to trend method

Find the centered 12 monthly moving averages of the original data values in the time-series.
:
Express each original data value of the time-series as a percentage of the corresponding centered moving average values obtained in step. In other words, in a multiplicative time-series model, we get / × 100 = / × 100 = × 100.

This implies that the ratio-to-moving average represents the seasonal and irregular components.

:
:
Arrange these percentages according to months or quarter of given years. Find the averages over all months or quarters of the given years.
:
If the sum of these indices is not 1200, multiply then by a correction factor = 1200 /. Otherwise, the 12 monthly averages will be considered as seasonal indices.
Ratio-to-moving-average method

Let us calculate the seasonal index by the ratio-to-moving-average method from the following data:

Year/Quarters	1	2	3	4
1996	75	60	54	59
1997	86	65	63	80
1998	90	72	66	85
1999	100	78	72	93

Now calculations for 4 quarterly moving averages and ratio-to-moving-averages are shown in the below table.

Year	Quarter	Original Values	4 Figures Moving Total	4 Figures Moving Average	2 Figures Moving Total	2 Figures Moving Average	Ratio-to-Moving-Average/ *100
1996	1	75			—	—	—
1996	1	75	—	—	—	—	—
1996	2	60	—	—	—	—	—
1996	2	60	248	62.00	—	—	—
1996	3	54	248	62.00	126.75	63.375	85.21
1996	3	54	259	64.75	126.75	63.375	85.21
1996	4	59	259	64.75	130.75	65.375	90.25
1996	4	59	264	66.00	130.75	65.375	90.25
1997	1	86	264	66.00	134.25	67.125	128.12
1997	1	86	273	68.25	134.25	67.125	128.12
1997	2	65	273	68.25	141.75	70.875	91.71
1997	2	65	294	73.50	141.75	70.875	91.71
1997	3	63	294	73.50	148.00	74.00	85.13
1997	3	63	298	74.50	148.00	74.00	85.13
1997	4	80	298	74.50	150.75	75.375	106.14
1997	4	80	305	76.25	150.75	75.375	106.14
1998	1	90	305	76.25	153.25	76.625	117.45
1998	1	90	308	77.00	153.25	76.625	117.45
1998	2	72	308	77.00	155.25	77.625	92.75
1998	2	72	313	78.25	155.25	77.625	92.75
1998	3	66	313	78.25	159.00	79.50	83.02
1998	3	66	323	80.75	159.00	79.50	83.02
1998	4	85	323	80.75	163.00	81.50	104.29
1998	4	85	329	82.25	163.00	81.50	104.29
1999	1	100	329	82.25	166.00	83.00	120.48
1999	1	100	335	83.75	166.00	83.00	120.48
1999	2	78	335	83.75	169.50	84.75	92.03
1999	2	78	343	85.75	169.50	84.75	92.03
1999	3	72	343	85.75	—	—	—
1999	3	72	—	—	—	—	—
1999	4	93	—	—	—	—	—
1999	4	93			—	—	—

Years/Quarters	1	2	3	4	Total
1996	—	—	85.21	90.25
1997	128.12	91.71	85.13	106.14
1998	117.45	92.75	83.02	104.29
1999	120.48	92.04	—	—
Total	366.05	276.49	253.36	300.68
Seasonal Average	122.01	92.16	84.45	100.23	398.85
Adjusted Seasonal Average	122.36	92.43	84.69	100.52	400

Now the total of seasonal averages is 398.85. Therefore, the corresponding correction factor would be 400/398.85 = 1.00288. Each seasonal average is multiplied by the correction factor 1.00288 to get the adjusted seasonal indices as shown in the above table.