Classical Decomposition of a time series

A time series is a data set that has a time component. Yes, it is just what you think about, in the optimal case you have one value in a fixed time interval. It is a pretty actual topic to create massive datasets on behalf of sensors in an Internet of Things scenario, and you usually get too much values (>100 measurements per second), so to create a useful time series requires some preprocessing (filter or average). Missing values can be a problem, also it is usually not difficult to find a good estimate.

You can therefore build a time line which could look like this:

Here I used R's default dataset "AirPassengers" which reflects the monthly international airline passenger numbers in the years 1949 to 1960.

Natural questions are: 

• Can you see which of the years were good, which of them were bad?
• Can you split up the time series into more homogenious components?
• Can you predict the values for future months? And how good are these predictions?

Time series analysis seems not too complex, however in reality often there are combinations of models required to make good predictions. In this post I want to show you how standard approches work. In reality, especially in retail, the timelines are usually not that easy to handle, as it is widely influenced by customer reviews.

The central idea is, that a time series $Y = (Y_t)$ is a combination of a three independent sub - time series:
- A trend component T is a long-term tendence in the data, it does not have to be linear.
- A seasonal component S is a pattern that reoccurs regularily after a fixed period (like every summer, every january or every day at 10:30).
- A random component I, also called irregular or noise.

We want to try to find these three time series in the upper mentioned example. First we have to decide on the type of decomposition, we can choose from additive and multiplicative.
In an additive model we add the 3 sub time series up to get the original time series: $$Y_t = T_t + S_t + I_t$$ You should use it when the seasonal variance does not change that much.

In a multiplicative model we multiply the 3 sub time series: $$Y_t= T_t * S_t * I_t$$ Use it when you see the peeks growing with time, like in the earlier mentioned example of airplane passengers. Here we should go for a multiplicative model.

Tip: A multiplicative model often can be changed into an additive model using the log function.

How would we get the values for the trend, seasonal and random sub time series? We will go step by step, just to motivate, here the result calculated by R with function decompose:

Here the corresponding coding in R: $$plot(decompose(ts(AirPassengers, frequency = 12, start =1949), type = "mult"))$$

To go on:

1. Here is how to determine the trend component
2. Here is how to determine the seasonal and random component
3. Here is a summary on the classical decomposition of time series

Seasonality and Random Determination

In this post we saw how the three components trend, seasonality and random of a time series. How to extract the trend was shown there, now we focus on how the seasonal component and the random component is determined.

Assume we have a detrended time series (we take here the AirPassengers time series and remove the trend). We assume a seasonality of a fixed period. In reality the assumption to have a fixed seasonality is too strict, as the period could shorten or change its structure over time. But under this assumption the determination of the seasonality is easy: To get the seasonal value of January, we take all values of January and build the average. This is the pattern we use for all periods.

The last step is to determine the random component $I$, we get it by simply removing the trend $T$ and seasonal component $S$ from the original time series $Y$, in an additive model this would be $I_t = Y_t - T_t - S_t$ and in a multiplicative model $I_t = Y_t / (T_t * S_t)$.
In our example, this is how the random components looks like:

What can we get out of it?
The random component shows the noise in the data, the values that do not fit the model. It helps to get a feeling how well the data is explained by the assumption to have a trend and a seasonality. The classical decomposition also could help to find outliers, which will show up with a high peek.

For completeness here again the whole picture holding all the steps discussed:

 

Classical Decomposition - Summary

The classical decomposition of a time series can help to get an overview on the tendencies (trend component), periodic patterns (seasonal component) and quality of the model (random component). In addition it helps to identify outliers in a time series.

To forecast a time series it is often useful to have a decomposition and to forecast each of the components in the decomposition seperately. A seasonal component would just be repeated constantly (naive forecast), meanwhile you could use exponential smoothing methods to forecast the trend and random component.

On the other hand the classical decomposition shows some disadvantages: We saw in this post that the trend and therefore also the random component cannot be determined at the beginning and at the end of a time series. Also we saw in that post that it relies on the assumption that we have a stable period with a pretty constant pattern. In reality this is often not the case: e.g. 100 years ago the energy consumption was high in winter was high due to heaters, now in summer it is equally high due to air condition.

To overcome these bounderies other decomposition methods have been developed, see for instance the Seasonal and Trend Determination using LOESS (1990). I will describe it in a new post.

I hope you liked and got a picture on the classical decomposition, I really enjoyed building up this example and encourage you to comment and extend.