Time Series Decomposition Models

Time Series Decomposition Models

Time Series Decomposition is the process of separating a time series into its different components to understand the underlying pattern of the data.

A time series generally consists of four major components:

Trend (T) – Long-term upward or downward movement.

Seasonal Variation (S) – Regular and recurring variations within a specific period.

Cyclical Variation (C) – Long-term fluctuations around the trend.

Irregular Variation (I) – Random or unpredictable movements.

The two important models used for time series decomposition are:

1. Additive Model
2. Multiplicative Model

Additive Model

In the Additive Model, the different components of a time series are added together to obtain the observed value.

Formula

Y=T+S+C+I

Where:

Y = Observed value

T = Trend component

S = Seasonal component

C = Cyclical component

I = Irregular component

In simple words

Observed Value = Trend + Seasonal + Cyclical + Irregular

In this model, the effect of seasonal and other variations is assumed to remain approximately constant in absolute terms.

Example

Suppose the components of a company’s sales are:

Then:

Y=T+S+C+I

Y=100+20+10−5

Y=125​

Therefore, the observed sales value is 125 units.

When is the Additive Model used?

The Additive Model is appropriate when the seasonal fluctuations remain approximately constant over time, regardless of the level of the time series.

For example:

Since the seasonal effect is approximately constant, the Additive Model is suitable.

Multiplicative Model

In the Multiplicative Model, the different components are multiplied together to obtain the observed value.

Formula

Y=T×S×C×I

Where:

Y = Observed value

T = Trend component

S = Seasonal component

C = Cyclical component

I = Irregular component

In simple words

Observed Value = Trend × Seasonal × Cyclical × Irregular

In this model, variations are generally expressed in terms of ratios, percentages, or indices.

Example

Suppose:

Then:

Y=T×S×C×I

Y=100×1.20×1.10×0.95

Y=125.40​

Therefore, the observed value is 125.40 units.

When is the Multiplicative Model used?

The Multiplicative Model is appropriate when the size of seasonal fluctuations changes according to the level of the time series.

For example:

Here, the seasonal effect increases as the trend increases. Therefore, the Multiplicative Model is more appropriate.

Difference between Additive and Multiplicative Models

Numerical Example: Additive Model

Suppose the following values are given:

Trend = 500

Seasonal variation = +50

Cyclical variation = +20

Irregular variation = −10

Using the Additive Model:

Y=T+S+C+I

Y=500+50+20−10

Y=560​

Therefore, the observed value is 560 units.

Numerical Example: Multiplicative Model

Suppose:

Trend = 500

Seasonal Index = 1.10

Cyclical Index = 1.05

Irregular Index = 0.98

Using the Multiplicative Model:

Y=T×S×C×I

Y=500×1.10×1.05×0.98

Y=565.95​

Therefore, the observed value is 565.95 units.

How to Identify the Appropriate Model?

Additive Model

Use the Additive Model when the seasonal variation is expressed as an absolute value and remains relatively constant.

Example:

Trend = 100
Seasonal effect = +20

Y=100+20=120

Multiplicative Model

Use the Multiplicative Model when seasonal variation is expressed as a percentage, ratio, or index and changes with the level of the series.

Example:

Trend = 100
Seasonal Index = 1.20

Y=100×1.20=120

Key Points to Remember

Additive Model: Y=T+S+C+I​

Multiplicative Model: Y=T×S×C×I​

Additive Model → Constant seasonal effect

Multiplicative Model → Proportional or changing seasonal effect

Thus, the choice between the two models depends mainly on whether the seasonal variation is constant in absolute terms or changes proportionally with the level of the time series.

FAQ: Time Series Decomposition Models

What is Time Series Decomposition?

Time Series Decomposition is the process of separating a time series into its major components: Trend, Seasonal, Cyclical, and Irregular variations.

What are the two main decomposition models?

The two main models are:

Additive Model

Multiplicative Model

What is the formula of the Additive Model?

Y=T+S+C+I

Where Y is the observed value and T, S, C, and I represent Trend, Seasonal, Cyclical, and Irregular components respectively.

What is the formula of the Multiplicative Model?

Y=T×S×C×I

When is the Additive Model appropriate?

The Additive Model is appropriate when seasonal fluctuations remain approximately constant in absolute magnitude over time.

When is the Multiplicative Model appropriate?

The Multiplicative Model is appropriate when seasonal fluctuations vary according to the level of the time series.

What is the main difference between the two models?

The Additive Model combines components through addition, whereas the Multiplicative Model combines them through multiplication.

How is the seasonal component represented in the Additive Model?

It is represented as an absolute amount, such as +20 or −15.

How is the seasonal component represented in the Multiplicative Model?

It is generally represented as a ratio or index, such as 1.20 or 0.90.

What does a seasonal index of 1.20 mean?

A seasonal index of 1.20 indicates that the value is approximately 20% above the trend level, assuming other components are neutral.

What does a seasonal index of 0.80 mean?

A seasonal index of 0.80 indicates that the value is approximately 20% below the trend level, assuming other components are neutral.

What are the four components of a time series?

The four components are:

Trend

Seasonal

Cyclical

Irregular

Which model is commonly used when seasonal variation increases as the series level increases?

The Multiplicative Model.

Can the Additive Model have negative seasonal variations?

Yes. For example, the seasonal component may be +20 in one period and −15 in another period.

Why is decomposition useful?

Decomposition helps researchers and analysts understand the different sources of variation in a time series and helps in forecasting, planning, and decision-making.

MCQs: Time Series Decomposition Models

What is the main purpose of time series decomposition?

A. To remove all data
B. Separate a time series into its components
C. Calculate only the mean
D. To calculate only the median

Answer: B. To separate a time series into its components

Which of the following is an important component of a time series?

A. Trend
B. Seasonal variation
C. Irregular variation
D. All of the above

Answer: D. All of the above

What is the formula for the Additive Model?

A. Y=T×S×C×I
B. Y=T−S−C−I
C. Y=T+S+C+I
D. Y=T/S/C/I

Answer: C. Y=T+S+C+I

What is the formula for the Multiplicative Model?

A. Y=T+S+C+I
B. Y=T×S×C×I
C. Y=T−S+C−I
D. Y=T/S+C/I

Answer: B. Y=T×S×C×I

In the Additive Model, the components are:

A. Multiplied
B. Divided
C. Added
D. Squared

Answer: C. Added

In the Multiplicative Model, the components are:

A. Added
B. Multiplied
C. Subtracted
D. Averaged

Answer: B. Multiplied

Which model is suitable when seasonal variations are approximately constant in absolute terms?

A. Multiplicative Model
B. Additive Model
C. Regression Model
D. Moving Average Model

Answer: B. Additive Model

Which model is suitable when seasonal variations change proportionally with the level of the series?

A. Additive Model
B. Multiplicative Model
C. Semi-Average Model
D. Graphical Model

Answer: B. Multiplicative Model

In an Additive Model, seasonal variation is generally measured in:

A. Absolute units
B. Ratios only
C. Percentages only
D. Index numbers only

Answer: A. Absolute units

In a Multiplicative Model, seasonal variation is commonly expressed as:

A. Absolute difference
B. Ratio or index
C. Arithmetic mean
D. Standard deviation

Answer: B. Ratio or index

If T=100, S=20, C=10, and I=−5, what is Y under the Additive Model?

A. 115
B. 120
C. 125
D. 135

Answer: C. 125

Calculation:

Y=100+20+10−5=125

If T=100, S=1.20, C=1.10, and I=0.95, what is Y?

A. 110.50
B. 115.40
C. 125.40
D. 130.50

Answer: C. 125.40

A seasonal index of 1.20 indicates approximately:

A. 20% below trend
B. 20% above trend
C. 120% below trend
D. No seasonal effect

Answer: B. 20% above trend

A seasonal index of 0.80 indicates approximately:

A. 20% above trend
B. 80% above trend
C. 20% below trend
D. No variation

Answer: C. 20% below trend

Which component represents the long-term general movement of a time series?

A. Seasonal
B. Trend
C. Irregular
D. Random

Answer: B. Trend

Which component represents regular variations occurring within a year, month, quarter, etc.?

A. Trend
B. Cyclical
C. Seasonal
D. Irregular

Answer: C. Seasonal

Which component represents unpredictable changes caused by random events?

A. Trend
B. Seasonal
C. Cyclical
D. Irregular

Answer: D. Irregular

Which model is more appropriate when seasonal variation increases as sales increase?

A. Additive Model
B. Multiplicative Model
C. Semi-Average Model
D. Moving Median Model

Answer: B. Multiplicative Model

Which of the following is NOT a component of the classical time series decomposition?

A. Trend
B. Seasonal
C. Cyclical
D. Correlation

Answer: D. Correlation

The Additive Model can be represented as:

A. Y=T+S+C+I
B. Y=T×S×C×I
C. Y=T/S/C/I
D. Y=TS

Answer: A. Y=T+S+C+I

Some More: 

• POP- Introduction to Programming Using ‘C’
• DS – Data structure Using C
• OOP – Object Oriented Programming 
• Java Programming
• DBMS – Database Management System
• RDBMS – Relational Database Management System
• https://defineinfoloop.blogspot.com/?m=1