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Summary

The webpage provides an introductory guide to time series analysis and forecasting, covering its definition, applications, types, components, decomposition techniques, and characteristics, with a focus on Python implementation and the importance of stationarity.

Abstract

Time series analysis is a statistical method for analyzing sequences of data points collected over time intervals. This guide introduces the concept, emphasizing its utility in forecasting future values, classifying time series data, and inferring causality. It distinguishes between regular and irregular time series, discusses the trend, seasonal, and cyclic components, as well as random fluctuations. The decomposition process is detailed, with both additive and multiplicative models explored, and the Python statsmodel library is utilized for practical implementation. The article also addresses the significance of stationarity in time series data, explaining how to check for it using the Augmented Dickey-Fuller (ADF) test. The content concludes with references to further reading and an invitation to contribute to MLearning.ai as a writer.

Opinions

  • The article advocates for the use of time series analysis in various domains, suggesting it as a valuable tool for decision-making in business and economics.
  • It implies that Python, specifically the statsmodel library, is a preferred tool for time series decomposition and analysis.
  • The author seems to value the clarity and understanding of time series components, emphasizing the importance of decomposing a time series to analyze its underlying patterns.
  • The guide suggests that understanding and modeling the stationarity of a time series is crucial for accurate forecasting.
  • By providing code examples and references to additional resources, the author encourages readers to engage further with the topic and potentially contribute to the field.
  • The mention of the AI service ZAI.chat at the end of the article indicates an endorsement of this tool for those interested in AI and machine learning applications.

Time Series Analysis & Forecasting

A Gentle Introduction to Time Series Analysis and Forecasting

Photo by Aron Visuals on Unsplash

Definition

A time series is a sequence of observations indexed by time. This means that the observations follows a particular order. The subscript t represents the set of allowed times T:

{xt} with t1 … tn

Time series measures how something changes over time.

Applications of Time Series Analysis

There are three main application areas in time series analysis:

  • Time series forecasting: The focus is to predict the future values of a time series.
  • Time series classification: The goal is to predict an action based on past values.
  • Causality: The aim is to drive inference from the relationships among related time series.

We’ll focus on time series forecasting. Some examples of problems where time series forecasting can help are:

  • What is the expected sales volume of food groups in different stores in the next three months?
  • What is the resale value of cars after leasing out for three years?
  • What is the closing price of a stock each day?
  • What are passenger numbers for an international airline route?

Types of Time Series

We distinguish between two types of time series:

  • Regular
  • Irregular

The observations of regular time series have a regular interval of time. Those intervals could be every minute, every hour, or every month. This means that the time difference between each data point is the same. The observations of irregular time series don’t follow a regular interval of time. In the Airplane Passengers Dataset, the observations occur every month.

Airplane Passenger Dataset

Components of Time Series

A time series consists of the following three components:

Components of Time Series

Long-term movement or trend

A trend is the long-term movement in which time series values are developing during a period. The change can be either:

  • Upward (increase in level)
  • Downward (decrease in level)

Short-term movements

There are two types of short-term movements:

  • Seasonal movements are temporal fluctuations that usually occurs at a specific, regular intervals. Those movements are less than a year. It can occur on different time spans, such as daily, weekly, monthly, or yearly variation. Different social conventions, weather seasons, and climatic conditions are reasons for seasonal variations.
  • Cyclic movements are recurrent patterns when data exhibits rises and falls.

Random or irregular fluctuations (Residual)

There is a variability that is irregular. These fluctuations are uncontrollable, unpredictable, and erratic. One examples are Bank holidays which can fall on different calendar days each year. Another examples are promotional campaigns which could depend on different business decisions.

The first three components are also known as signals. The last component, random or irregular fluctuations, is also known as noise.

The Airplane Dataset shows a long-term linear (upward) trend with increasing, seasonal variations. But the seasonal fluctuations increase as the time series increases in size.

Representation of time series components

To remove the component effects, we will perform a decomposition process.

Decomposition a Time Series

In this section, we learn how to visualise time series into its components. The seasonal decomposition is the process of deconstructing a time series. The modeling of the decomposed components can be either additive or multiplicative.

Additive Model

We can reconstruct a time series by adding all three components:

Yt = Tt + St + Rt

We’ll choose an additive model if the seasonal components do not change over time.

Multiplicative Model

We can also reconstruct a time series by multiplying all three components:

Yt = Tt * St * Rt

A multiplicative model is suitable when the seasonal variation fluctuates over time.

There are different implementations to decompose time series. We will use the seasonal_decompose function from the statsmodel library. It is an implementation that uses moving averages and period-adjusted averages. It supports both additive and multiplicative models of decomposition. Here, we should use the multiplicative model.

ts_decomposed = seasonal_decompose(ts,model=’multiplicative’)
ts_decomposed.plot()
pyplot.show()

The result looks as follow:

Seasonal decomposition of a multiplicative model

We can multiply the components to reconstruct the time series.

(ts_decomposed.trend * ts_decomposed.seasonal * ts_decomposed.resid).plot()

It gives the following plot as output.

Reconstructed time series

Different Decomposition Techniques

In the next section, we’ll describe techniques to decompose time series. The general approach is as follows:

  • Detrending: We estimate the trend component and remove it from the time series
  • Deseasonalising: We estimate the seasonality component from the detrended time series.
  • The remaining part of the time series is the residual.

Detrending

There are two different ways:

  • Moving Average
  • Locally estimated scatterplot smoothing (LOESS)
Decomposition Techniques

The moving average is a window moved along the time series in steps. LOESS is a non-parametric method used to fit a smooth curve onto a noisy signal. We will focus on the Moving Average.

At each step, we’ll record the average of all the values in the window. This moving average helps us to estimate the slow change in time series or trend. We can also calculate the seasonal variations as difference between passengers and trend. The seasonal variations are not absolute values, because we use the multiplicative model. The disadvantage is that it is quite noisy. From our example, we can see a trend from February 1949 to August 1949.

Moving Average Calculation

The Pandas DataFrame class offers a rolling function to calculate rolling windows. We will provide the seize of the moving window. We can either provide a fixed number of observations used for each window or a time delta. We will provide 6 months for the first window and 12 months for the second window.

ts['6-month-MA'] = ts['#Passengers'].rolling(window=6).mean()
ts['12-month-MA'] = ts['#Passengers'].rolling(window=12).mean()

After that, we will use Pandas DataFrame plot function to make a plot of the DataFrame.

Moving Average

Characteristics of Time Series

One important characteristic of a time series is stationarity.

Stationarity

The properties of a time series do not change its distribution over time. In other words, the mean and variance remain constant over time. If a time series is stationarity, it means it has no trend and no seasonal variability. Thus, stationary time series processes are easier to analyse and to model. The reasons is that their properties are not dependent on time and will be the same in the future.

Two forms of Stationarity:

  • Strong stationarity: All the parameters of a time series do not change over time.
  • Weak stationarity: The mean and the auto-covariance function do not change over time.

There are four questions to check whether a time series is stationary or not:

  • Does the mean change over time? In other words, is there a trend in the time series?
  • Does the variance change over time? In other words, is the time series heteroscedastic?
  • Does the time series exhibit periodic changes in mean? Or in other words, is there seasonality in the time series?
  • Does the time series have a unit root?

To check whether a time series has a unit root, we will use the Augmented Dickey-Fuller (ADF) test. The null hypothesis is that the AR coefficient in an AR model of the time series is equal to 1. This means that the time series is non-stationary. The alternate hypothesis is that the AR coefficient in the AR model is less than 1.

In Python, we can use the statsmodel library to check for unit roots.

adfuller(ts[‘#Passengers’])

The result of the adfuller test is a tuple. This tuple contains the test-statistic, p-value, and critical values at different confidence levels. Here, we will use the p-value. As the p-value is an easy way to check whether we can reject the null hypothesis or not. If the p-value is less than 0.05, there is a 95 % probability that the series do not have a unit root. Then the time series would be stationary from a unit root perspective. In our case, the p-value is 0.9918802434376408. This means that the time series is non-stationary. In other words, the time series does not have a constant variance over time.

Conclusion

This was an introduction to time series analysis. The code is on GitHub. If you like this article, please clap. If you wish to read similar articles from me, please follow me to receive an email whenever I publish a new article.

References

Atwan, T. (2022) Time Series Analysis with Python Cookbook. 1st ed. Packt Publishing. Available at: https://www.perlego.com/book/3553599/time-series-analysis-with-python-cookbook-pdf (Accessed: 18 November 2023).

Auffarth, B. (2021): Machine Learning for Time-Series with Python. 1st ed. Packt Publishing. Available at: https://www.perlego.com/book/3040281/machine-learning-for-timeseries-with-python-forecast-predict-and-detect-anomalies-with-stateoftheart-machine-learning-methods-pdf (Accessed: 11 November 2023)

Joseph, M. (2022) Modern Time Series Forecasting with Python. 1st ed. Packt Publishing. Available at: https://www.perlego.com/book/3791670/modern-time-series-forecasting-with-python-explore-industryready-time-series-forecasting-using-modern-machine-learning-and-deep-learning-pdf (Accessed: 16 October 2023)

Lazzeri, F. (2020) Machine Learning for Time Series Forecasting with Python. 1st ed. Wiley. Available at: https://www.perlego.com/book/2050082/machine-learning-for-time-series-forecasting-with-python-pdf (Accessed: 12 November 2023).

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