The classical method of time series is done by decomposing a time series into trend, seasonal, cyclic and irregular components.

## What is decomposing?

The meaning of the word “decomposing” in Mathematics is breaking something into parts, that together are the same as the original.

For example, we can decompose 664 like this;

**664 –> 600 + 60 +4**

This method can be used for forecasting as well as to get a basic understanding of the time series.

## Introduction

Decomposition methods assume that Y_{t}, the actual time series value at period t, is a function of four components namely;

**Trend Component (T)****Seasonal Component (S)****Cyclic Component (C)****Irregular or Error Component (I)**

There are two main techniques to analyze a time series using the Decomposition Method. They are Additive Method and Multiplicative Method.

Before jumping into those, let’s get a better understanding about the Components of a Time Series.

## Components of a Time Series

### Trend $(T)$

Trend is the long term movement of a time series. In other words, trend is the general tendency of time series to increase, decrease or stagnate.

Tendency of a time series is always not the same throughout the time period. There is a possibility of short-term up trends, down trends or even sideline movements. But the overall long-term tendency may be upward, downward or stable.

Here the interpretation of “long-term” depends on the variable of interest. If its sales, long-term can be interpreted as 5 years. If its a highly volatile stock, long term may be interpreted as 1 year or less.

Trend can be linear or non linear.

Trend is usually resulted by long term effects of factors such as socio economic, political and demographical characteristics.

### Seasonal Variation (S)

Short term variations in time series as a result of seasonal factors are defined as Seasonal Variation (S) in a time series. Typically, seasonal variations occur within a specific period of time (one year or shorter).

For an example, ice-cream sales of a shop may increase each year during summer, or sales of a toys store may increase every year during christmas.

Time Series Data can also exhibit seasonal patterns of less than year in duration. For an example, daily traffic volume shows within the day “seasonal” behavior, with peak levels occurring during rush hours. Weekly sales in a fashion store shows “weekly” seasonal behavior where highest sales occur during weekend.

### Cyclic Variation (C)

Long term oscillations occurring in a time series are called Cyclic Variations. Term can be 5 years, 10 years or even more. These variations are mostly observed in economic and business data.

Most economic and business time series are influenced by wave like changes of prosperity and depreciation. There is periodic up and down movements and these movements occur in a cycle.

Cyclic Variation is extremely difficult to forecast. Hence, it is often combined with long-term trend effect.

### Irregular Variation (I)

Irregular Variations are fluctuations in time series that are short in duration, unpredictable in nature and follow no regularity in the occurrence pattern.

The irregular component is what is left after trend, seasonal variation and cyclic variation of a time series are estimated and removed. Hence, they are also known as ‘residual variations’.

Irregular variations can be either due to random effects or non-random effects such as earthquakes, tsunami or war.

Trend and Cyclic Variations are long term movements while seasonal and irregular variations are short term movements.

## Mathematical Models in Classical Analysis

In this section, two models most commonly used in decomposition method will be discussed.

### Additive Model

**Y _{t} = T + S + C + I**

In the additive model, it is assumed that the effects of four components are additive in nature.

- Y
_{t}– value of the original time series - T – Trend Value
- S – Seasonal Value
- C – Cyclical Value
- I – Irregular Value

In this model S,C and I are absolute quantities. This means that when you take the long term trend, and you calculate the S,C,I values you can add or subtract those values from the long term trend. And most importantly, the gradient of the time series plot remains constant.

Additive model assumes that the four components of the time series are independent of each other.

### Multiplicative Model

In a multiplicative model, a time series is expressed as;

**Y _{t} = T x S x C x I**

Here, the decomposition of the time series is done based on the assumption that the effects of four components of a time series are not necessarily independent of each other.

In this model S,C and I are not absolute quantities. They are relative variations.

That is, S,C, and I values respond proportionally to increases and decreases in the trend. Hence, the gradient of a multiplicative model is not constant.

## Conclusion

To decide whether the model is additive or multiplicative, time series plot can be used.

If the size of the seasonal effect is directly proportional to the mean, then the multiplicative model is used.

If the size of the seasonal effect does not change with the mean, then an additive model is used.