Seasonality
Encyclopedia
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.
and then decline after the holidays. So time series of retail sales will typically show increasing sales from September through December and declining sales in January and February.
Seasonality is quite common in economic time series. It is also very common in geophysical and ecological time series. A notable example is the concentration of atmospheric carbon dioxide
: it is at a minimum in September and October, at which point it begins to increase, reaching a peak in April/May, before declining. Another example consists of the famous Milankovitch cycles
.
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
.
One particular implementation of seasonal adjustment is provided by X-12-ARIMA
.
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.
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.
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:
{| class="wikitable" border="0.5"
|-
! 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.
Now the total of seasonal averages is 299.66. Therefore the corresponding correction factor would be 400/299.68 = 1.334. Each seasonal average is multiplied by the correction factor 1.334 to get the adjusted seasonal indices as shown in the above table.
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:
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, many time series
Time series
In statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the...
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
Time series
In statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the...
which is defined as the repetitive and predictable movement around the trend
Linear regression
In statistics, linear regression is an approach to modeling the relationship between a scalar variable y and one or more explanatory variables denoted X. The case of one explanatory variable is called simple regression...
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
Seasonality
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....
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.
Examples
For example, retail sales tend to peak for the Christmas seasonChristmas season
Christmastide is one of the seasons of the liturgical year of most Christian churches. It tends to be defined as the period from Christmas Eve to the evening of 5 January, the day before Epiphany...
and then decline after the holidays. So time series of retail sales will typically show increasing sales from September through December and declining sales in January and February.
Seasonality is quite common in economic time series. It is also very common in geophysical and ecological time series. A notable example is the concentration of atmospheric carbon dioxide
Carbon dioxide
Carbon dioxide is a naturally occurring chemical compound composed of two oxygen atoms covalently bonded to a single carbon atom...
: it is at a minimum in September and October, at which point it begins to increase, reaching a peak in April/May, before declining. Another example consists of the famous Milankovitch cycles
Milankovitch cycles
Milankovitch theory describes the collective effects of changes in the Earth's movements upon its climate, named after Serbian civil engineer and mathematician Milutin Milanković, who worked on it during First World War internment...
.
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 plotSeasonal subseries plotSeasonal subseries plots are a tool for detecting seasonality in a time series. This plot allows one to detect both between-group and within-group patterns. This plot is only useful if the period of the seasonality is already known. In many cases, this will in fact be known. For example, monthly...
is a specialized technique for showing seasonality - Multiple box plotBox plotIn descriptive statistics, a box plot or boxplot is a convenient way of graphically depicting groups of numerical data through their five-number summaries: the smallest observation , lower quartile , median , upper quartile , and largest observation...
s 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
Spectral density estimation
In statistical signal processing, the goal of spectral density estimation is to estimate the spectral density of a random signal from a sequence of time samples of the signal. Intuitively speaking, the spectral density characterizes the frequency content of the signal...
.
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 cyclicalFrequencyFrequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
and irregular variations. This elimination of the seasonal effect is referred to as deseasonalizing or seasonal adjustmentSeasonal adjustmentSeasonal adjustment is a statistical method for removing the seasonal component of a time series that is used when analyzing non-seasonal trends. It is normal to report un-adjusted data for current unemployment rates, as these reflect the actual current situation...
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
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
X-12-ARIMA
X-12-ARIMA is the U.S. Census Bureau's software package for seasonal adjustment. It can be used together with gretl, which provides a graphical user interface for X-12-ARIMA.X-12-ARIMA is the successor to X-11-ARIMA-See also:*AnSWR*ARIMA*CSPro...
.
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 trendTrend estimationTrend estimation is a statistical technique to aid interpretation of data. When a series of measurements of a process are treated as a time series, trend estimation can be used to make and justify statements about tendencies in the data...
method - Ratio-to-moving average method
- Link relatives method
Modeling seasonality
A completely regular cyclic variation in a time series might be dealt with in time series analysis by using a sinusoidal modelSinusoidal model
In statistics, signal processing, and time series analysis, a sinusoidal model to approximate a sequence Yi is:Y_i = C + \alpha\sin + E_i...
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
Arima
The Royal Borough of Arima is the fourth largest town in Trinidad and Tobago. Located east of the capital, Port of Spain, Arima supports the only organised indigenous community in the country, the Santa Rosa Carib Community and is the seat of the Carib Queen...
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:
{| class="wikitable" border="0.5"
|-
! 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 | 78 | 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 | ||||||
Calculation of seasonal index
| Table (3) | ||||
|---|---|---|---|---|
| Years/Quarters | 1 | 2 | 3 | 4 |
| 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 | 91.51 | 69.13 | 63.34 | 75.17 |
| Adjusted Seasonal Average | 122.37 | 92.43 | 84.69 | 100.5 |
Now the total of seasonal averages is 299.66. Therefore the corresponding correction factor would be 400/299.68 = 1.334. Each seasonal average is multiplied by the correction factor 1.334 to get the adjusted seasonal indices as shown in the above table.
Remarks
1. In an additive time-series model, the seasonal component is estimated asS = 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
See also
- OscillationOscillationOscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...
- Periodic functionPeriodic functionIn mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...
- Periodicity (disambiguation)
- PhotoperiodismPhotoperiodismPhotoperiodism is the physiological reaction of organisms to the length of day or night. It occurs in plants and animals.Photoperiodism can also be defined as the developmental responses of plants to the relative lengths of the light and dark periods...
External links
- Seasonality at NIST/SEMATECH e-Handbook of Statistical Methods