Temporal Structure - Part 3 - Decompose Time Series Data

Time series decomposition involves thinking of a series as a combination of level, trend, seasonality, and noise components. 

After completing this tutorial, you will know:

• The time series decomposition method of analysis and how it can help with forecasting.
• How to automatically decompose time series data in Python.
• How to decompose additive and multiplicative time series problems and plot the results.

A. Time Series Components

A useful abstraction for selecting forecasting methods is to break a time series down into systematic and unsystematic components.

• Systematic: Components of the time series that have consistency or recurrence and can be described and modeled.
• Non-Systematic: Components of the time series that cannot be directly modeled.

A given time series is thought to consist of three systematic components including level, trend, seasonality, and one non-systematic component called noise. These components are defined as follows:

• Level: The average value in the series.
• Trend: The increasing or decreasing value in the series.
• Seasonality: The repeating short-term cycle in the series.
• Noise: The random variation in the series.

B. Combining Time Series Components

A series is thought to be an aggregate or combination of these four components. All series have a level and noise. The trend and seasonality components are optional. It is helpful to think of the components as combining either additively or multiplicatively.

1. Additive Model

An additive model suggests that the components are added together as follows:

y(t) = Level + Trend + Seasonality + Noise

2. Multiplicative Model

A multiplicative model suggests that the components are multiplied together as follows:

y(t) = Level × Trend × Seasonality × Noise

C. Decomposition as a Tool

Decomposition is primarily used for time series analysis, and as an analysis tool it can be used to inform forecasting models on your problem. It provides a structured way of thinking about a time series forecasting problem.

Each of these components are something you may need to think about and address during data preparation, model selection, and model tuning.

You may address it explicitly in terms of modeling the trend and subtracting it from your data, or implicitly by providing enough history for an algorithm to model a trend if it may exist.

Real-world problems are messy and noisy. There may be additive and multiplicative components. There may be an increasing trend followed by a decreasing trend. There may be non-repeating cycles mixed in with the repeating seasonality components. Nevertheless, these abstract models provide a simple framework that you can use to analyze your data and explore ways to think about and forecast your problem.

D. Automatic Time Series Decomposition

There are methods to automatically decompose a time series. The Statsmodels library provides an implementation of the naive, or classical, decomposition method in a function called seasonal decompose(). It requires that you specify whether the model is additive or multiplicative.

For example, the snippet below shows how to decompose a series into trend, seasonal, and residual components assuming an additive model. The result object provides access to the trend and seasonal series as arrays. It also provides access to the residuals, which are the time series after the trend, and seasonal components are removed.

from statsmodels.tsa.seasonal import seasonal_decompose

series = ...

result = seasonal_decompose(series, model='additive')

print(result.trend)

print(result.seasonal)

print(result.resid)

print(result.observed)

These four time series can be plotted directly from the result object by calling the plot() function.

from statsmodels.tsa.seasonal import seasonal_decompose

from matplotlib import pyplot

series = ...

result = seasonal_decompose(series, model='additive')

result.plot()

pyplot.show()

1. Additive Decomposition

We can create a time series comprised of a linearly increasing trend from 1 to 99 and some random noise and decompose it as an additive model.

# additive decompose a contrived additive time series

from random import randrange

from matplotlib import pyplot

from statsmodels.tsa.seasonal import seasonal_decompose

series = [i+randrange(10) for i in range(1,100)]

result = seasonal_decompose(series, model='additive', freq=1)

result.plot()

pyplot.show()

-----Result-----

Line plots of a decomposed additive time series

2. Multiplicative Decomposition

We can contrive a quadratic time series as a square of the time step from 1 to 99, and then decompose it assuming a multiplicative model.

# multiplicative decompose a contrived multiplicative time series

from matplotlib import pyplot

from statsmodels.tsa.seasonal import seasonal_decompose

series = [i**2.0 for i in range(1,100)]

result = seasonal_decompose(series, model='multiplicative', freq=1)

result.plot()

pyplot.show()

-----Result-----

Line plots of a decomposed multiplicative time series

Exponential changes can be made linear by data transforms. In this case, a quadratic trend can be made linear by taking the square root. An exponential growth in seasonality may be made linear by taking the natural logarithm.

Line plots of a decomposed multiplicative time series

We can see that the trend and seasonality information extracted from the series does seem reasonable. The residuals are also interesting, showing periods of high variability in the early and later years of the series.