Introduction to Time Series (Part II)

Wait wait wait, have you read the first part? If not here is the link :)

Now that we have had a look at the different components of a Time Series do you think it’s always straightforward to figure out which components are present in a given dataset? (Well, sometimes it might be sometimes it might not be.)

// Presence and Absence of the components

Trend and Seasonality can be absent or present, if they are present then they can either be additive or multiplicative in nature. So we have the following 9 possibilities:

1. No Trend

• No seasonality
• Additive seasonality
• Multiplicative seasonality

2. Additive Trend

• No seasonality
• Additive seasonality
• Multiplicative seasonality

3. Multiplicative Trend

• No seasonality
• Additive seasonality
• Multiplicative seasonality

I know too much theory without graphs is confusing. Let’s have a look at all 9 possibilities -

No Trend with different scenarios of seasonality

We can pass a straight line in between and see all the observations are more or less around that line which means overall there is no increasing or decreasing pattern hence we can say there is no trend present in all of the above 3 graphs.

For checking the seasonality part, we can easily say that the leftmost graph shows no seasonality i.e. it is an example of ‘No Trend with No Seasonality’. The middle graph shows there is a repetitive pattern after a certain interval of time which means seasonality is present and since the pattern is constant we can say that seasonality is additive in nature. Similarly, there is seasonality present in the right graph but this time the width & height are changing with each season which implies this is an example of ‘No Trend with Multiplicative Seasonality’.

Additive Trend with different scenarios of seasonality

In this case, we can pass a linearly increasing line (or decreasing line) in between and see all the observations are more or less around that line, and because of this linearity, we say that an additive trend is present in all of the above 3 graphs.

For checking the seasonality part, the leftmost graph shows no seasonality i.e. it is an example of an ‘Additive Trend with No Seasonality’. The middle graph shows seasonality, now to identify whether the seasonality is additive or multiplicative we try passing a line touching all the crests and another line touching all the troughs. If the first line is linearly increasing or decreasing, and the second line roughly straight then we say seasonality is additive in nature. If both lines show a non-linear increasing/decreasing pattern then we say seasonality is multiplicative in nature which can be seen in the right graph.

Multiplicative Trend with different scenarios of seasonality

In this case, we can pass a non-linearly increasing line (or decreasing line) in between and see all the observations are more or less around that line, and because of this non-linearity, we say that a multiplicative trend is present in all of the above 3 graphs.

For checking the seasonality part, the leftmost graph shows no seasonality i.e. it is an example of a ‘Multiplicative Trend with No Seasonality’. To identify whether the seasonality is additive or multiplicative we use the same method of passing the two lines. If both lines are non-linearly increasing or decreasing and in the same direction then we say that seasonality is additive in nature as seen in the middle graph. If both lines are in the opposite direction then we say that seasonality is multiplicative in nature which can be seen in the right graph.

Woosh that was too much in one go, so is there any way to check which components are present rather than just observing from the plot? Yess!!!

// Decomposition of a Time Series
In simple words, a dataset might not necessarily contain only one of the four time series components, it might be a combination of more than one component and when such scenarios occur it might not be easily interpretable from the plot which components are present in the data. To deal with such situations we can use the decomposition method i.e. simply break it down into its components.

To decompose a series we must assume that some type of relationship exists among the four components contained in it. The combinations are called models and there are two types:

1. Additive
The additive forecasting model is one of the most widely used models. True to its name, at any time point t, the time series is the addition of all the components present and they are independent, there is no interaction between them.

Suppose Yₜ is our observed time series values, Tₜ is the trend component, Sₜ is the seasonal component, and Rₜ is the random/irregular component then the additive model is defined as:

Yₜ = Tₜ + Sₜ + Rₜ

2. Multiplicative
In this forecasting model, at any time point t, the time series is the product of all the components present which means that they are not independent, there is some interaction between them.

With the same notation stated above the multiplicative model is defined as:

Yₜ = Tₜ * Sₜ * Rₜ

// Now you might think about how to identify when to use which model?

• We use the additive model when the seasonal variations are roughly constant over time i.e. they are independent of time.
• We use the multiplicative model when the seasonal variations are either increasing or decreasing over time.

Will you now be able to identify correctly the nature of the trend and seasonality? Try out this link and find out.

Thanks for reading !!