Seasonal variations or seasonality are the changes in behavior, biological rhythms and physiology etc that occur in response to periodic changes in the environment due to the passage of the seasons. These effects include:

In statistics, many time series exhibit cyclic variation known as seasonality, periodic variation, or periodic fluctuations. This variation can be either regular or semi regular.

Seasonal variation is a component of a time series which is defined as the repetitive and predictable movement around the trend line in one year or less. It is detected by measuring the quantity of interest for small time intervals, such as days, weeks, months or quarters.

Organizations facing seasonal variations, like the motor vehicle industry, are often interested in knowing their performance relative to the normal seasonal variation. The same applies to the ministry of employment which expects unemployment to increase in June because recent graduates are just arriving into the job market and schools have also been given a vacation for the summer. That unemployment increased as predicted is a moot point, the relevant factor is whether the increase is more or less than expected.

Organizations affected by seasonal variation need to identify and measure this seasonality to help with planning for temporary increases or decreases in labor requirements, inventory, training, periodic maintenance, and so forth. Apart from these considerations, the organizations need to know if the variations they have experienced has been more or less would be expected given the usual seasonal variations.

## Detecting seasonality

In this section, techniques for detecting seasonality are discussed. The following graphical techniques can be used to detect seasonality:

• A run sequence plot will often show seasonality
• 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
• The autocorrelation plot can help identify seasonality
• Seasonal Index measures how much the average for a particular period tends to be above (or below) the expected value

The run sequence plot is a recommended first step for analyzing any time series. Although seasonality can sometimes be indicated with 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 (between group patterns) and also the within-group patterns. The box plot shows the seasonal difference (between group patterns) 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.

Both the seasonal subseries plot and the box plot 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 (although the intensity may decrease the further out we go).

Semiregular cyclic variations might be dealt with by spectral density estimation.

## Reasons for studying seasonal variation

There are several main reasons for studying seasonal variation:

• The description of the seasonal effect provides a better understanding of the impact this component has upon a particular series.
• After establishing the seasonal pattern, methods can be implemented to eliminate it from the time-series to study the effect of other components such as cyclical and irregular variations. This elimination of the seasonal effect is referred to as deseasonalizing or seasonal adjustment of data.
• To project the past patterns into the future knowledge of the seasonal variations is a must for the prediction of the future trends.

## Assumptions

A decision maker or analyst can make one of the following assumptions when treating the seasonal component:

• The impact of the seasonal component is constant from year to year.
• The seasonal effect is changing slightly from year to year.
• The impact of the seasonal influence is changing dramatically.

Seasonal adjustment is any method for removing the seasonal component of a time series. The outcome (seasonally adjusted data) are used, for example, when analyzing or reporting non-seasonal trends over durations rather longer than the seasonal period. An appropriate method for seasonal adjustment is chosen on the basis of a particular view taken of the decomposition of time series into components designated with names such as "trend", "cyclic", "seasonal" and "irregular", including how these interact with each other. For example, such components might act additively or multiplicatively. Thus, if a seasonal component acts additively, the adjustment method has two stages:

• estimate the seasonal component of variation in the time series, usually in a form that has a zero mean across series;
• subtract the estimated seasonal component from the original time series, leaving the seasonally adjusted series.

One particular implementation of seasonal adjustment is provided by X-12-ARIMA.

## Seasonal Index

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. There can also be a further 4 index values for quarterly data. The following methods use seasonal indices to measure seasonal variations of a time-series data.

• Method of simple averages
• Ratio to trend method
• Ratio-to-moving average method

## Modeling seasonality

A completely regular cyclic variation in a time series might be dealt with in time series analysis by using a sinusoidal model with one or more sinusoids whose period-lengths may be known or unknown depending on the context. A less completely regular cyclic variation might be dealt with by using a special form of an ARIMA model which can be structured so as to treat cyclic variations semi-explicitly. Such models represent cyclostationary processes.

## Calculation

Now let us try to understand the measurement of seasonal variation by using the Ratio-to-Moving Average method. This technique 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= (1436/4). As the winter-quarter index is 124, we estimate the no. of winter rentals as follows:

359*(124/100)=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.

### Steps

1. Find the centered 12 monthly (or 4 quarterly) moving averages of the original data values in the time-series.

2. Express each original data value of the time-series as a percentage of the corresponding centered moving average values obtained in step(1).In other words, in a multiplicative time-series model, we get(Original data values)/(Trend values) *100 = (T*C*S*I)/(T*C)*100 = (S*I) *100. This implies that the ratio–to-moving average represents the seasonal and irregular components.

3. Arrange these percentages according to months or quarter of given years. Find the averages over all months or quarters of the given years.

4. If the sum of these indices is not 1200(or 400 for quarterly figures), multiply then by a correction factor = 1200/ (sum of monthly indices). Otherwise, the 12 monthly averages will be considered as seasonal indices.

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

Table (1):
Year/Quarters I II III IV
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.

Table (2)
Year Quarter Original Values(Y) 4 Figures Moving Total 4 Figures Moving Average 2 Figures Moving Total 2 Figures Moving Average(T) Ratio-to-Moving Average(%)(Y)/ (T)*100
1996 1 75
2 60
248 62.00
3 54 126.75 63.375 85.21
259 64.75
4 59 130.75 65.375 90.25
264 66.00
1997 1 86 134.25 67.125 128.12
273 68.25
2 65 141.75 70.875 91.71
294 73.50
3 63 148 74 85.13
298 74.50
4 80 150.75 75.375 106.14
305 76.25
1998 1 90 153.25 76.625 117.45
308 77.00
2 72 155.25 77.625 92.75
313 78.25
3 66 159.00 79.50 83.02
323 80.75
4 85 163 81.50 104.29
329 82.25
1999 1 100 166 83 120.48
335 83.75
2 78 169.50 84.75 92.03
343 85.75
3 72
4 93

### Remarks

1. In an additive time-series model, the seasonal component is estimated as S = Y – (T+C+I) Where S is for Seasonal values Y is for actual data values of the time-series T is for trend values C is for cyclical values I is for irregular values.

2. In a multiplicative time-series model, the seasonal component is expressed in terms of ratio and percentage as Seasonal effect = (T*S*C*I)/( T*C*I)*100 = Y/(T*C*I )*100; However in practice the detrending of time-series is done to arrive at S*C*I . This is done by dividing both sides of Y=T*S*C*I by trend values T so that Y/T =S*C*I.

3. The deseasonalized time-series data will have only trend (T) cyclical(C) and irregular (I) components and is expressed as:

• Multiplicative model : Y/S*100 =( T*S*C*I)/S*100 = (T*C*I)*100.
• Additive model: Y – S = (T+S+C+I) – S = T+C+I