Time series

Understanding time series data and plots is important for analysing how the response variable changes over time. It involves identifying patterns such as trends, seasonality, structural change, possible outliers, and irregular fluctuations. By applying smoothing techniques, the effects of fluctuations can be minimised. Seasonal indices are also used to remove seasonal component, allowing for clearer visualisation of trends. Deseasonalising the data helps to remove the seasonal effect, providing a more accurate basis for making predictions. Time series analysis is an essential tool in various fields, such as economics, finance, and environmental studies, helping to uncover underlying trends, and forecast future values.

Use this page to revise the following concepts of time series:

• Time Series Data
• Features of Time Series Plots
• Smoothing Techniques
• Deseasonalisation and Seasonal Indices
• Trend Line Forecasting and Predictions

Time Series Data

Time series data is numerical bivariate data where time (e.g., hours, days, months, or years) is always the explanatory variable. The purpose of time series data is to observe how the response variable changes over time. A time series plot is a scatterplot with line segments joining the points in time order, visually representing time series data. Time is always plotted on the horizontal axis.

Features of Time Series Plots

Key features of time series plots include trends, seasonality, structural change, outliers, and irregular fluctuations. By identifying these features in time series plots, the underlying patterns and behaviours of the data over time can be better understood.

Trend

A trend refers to the tendency of data values to increase or decrease over a significant time period. Trends may change over time, and different trends may appear in different time periods within a time series plot. One way to identify trends in a time series plot is by drawing a trend line that smooths out fluctuations while highlighting the overall increasing or decreasing trend.

Increasing trend	Decreasing trend	Different trends within a plot

Seasonality

Seasonality occurs when there is a periodic movement related to a calendar-based period , such as a year, month, or week. Seasonal movements are generally more predictable than other time series features and often arise due to factors like weather change (e.g., higher electricity consumption in winter) or institutional influences (e.g., increased retail sales during the holiday season).

The time series plot reveals consistent peaks and troughs at the same intervals each year, suggesting seasonality. In this case, room occupancy (%) is at its lowest during the June quarter and reaches its highest point in the December quarter.

Structural change

A structural change occurs when the established pattern of a time series plot is suddenly altered.

Outliers

Outliers are present when there are individual values that stand out from the general body of data.

Irregular (random) fluctuations

Irregular fluctuations are random fluctuations in a time series plot that cannot be explained by systematic patterns such as trends, cycles, seasonality, structural change, or outliers. These fluctuations are present in any real-world time series data. There can be multiple sources for these irregular fluctuations, many of which are unknown. These irregular fluctuations cannot be predicted.

​

Smoothing Techniques

When time series data shows significant irregular fluctuations, it can be difficult to identify the underlying trends. In order to reveal the trend, it may be necessary to reduce some of these fluctuations before fitting a trend line. This process is known as smoothing.

There are two basic techniques for smoothing: moving mean and moving median smoothing.

Moving Mean Smoothing

Moving mean smoothing is a technique where individual data points in the time series are replaced with the mean of the data point and its adjacent points. The most straightforward approach is to smooth over a small odd number of data points, such as three or five, although any number of points can be used.

When smoothing data, it is important to decide the number of points to average. Here are some key guidelines:

1. For small data sets, the number of points to average (\(p\)) should be smaller than the total number of points (\(n\)). For example, if \(n = 7\), \(p\) should be about 4.
2. For cyclic or seasonal variations, \(p\) should match the cycle length or the number of seasons. For quarterly data, use \(p = 4\).
3. Common choices for \(p\) include 12 points for monthly sales, 7 for daily sales with daily operation, 5 for stores open only on weekdays, and 4 for quarterly data.
4. Always prefer an odd value for p to avoid symmetry issues. Larger p values smooth the data more but can result in losing more data points.

Three-Moving Mean Smoothing

Three-mean smoothing involves replacing each data value with the mean of itself and the value directly on either side of it. To find the smoothed value of \(y_2\):

\[\frac{Y-1 + y_2 + y_3}{3}\]

The first and last points in the time series are excluded as they do not have values on both sides.

​

Moving Mean Smoothing with Centring

Smoothing with centring is used when applying smoothing with an even number of data points, such as 2- moving or 4-moving mean smoothing. It involves averaging the already smoothed values, however, the final smoothed value may not align with a specific data point from the original dataset but instead lie between data points.

Moving Median Smoothing

Moving median smoothing is a technique where individual data points in the time series are replaced with the median of the data point and its adjacent points. The median point is found by finding the median of the group’s \(x\)-value, and the median of the group’s y-value. It is similar to moving mean smoothing but can be done directly on a graph without calculations.

Median smoothing uses groups of three for three-median smoothing, groups of five for five-median smoothing, and so on.

​

Deseasonalisation and Seasonal Indices

Deseasonalisation/Seasonal adjustment removes the seasonal components from data to reveal underlying trends. This process involves calculating seasonal indices, which show how a particular season (e.g., day, month, or quarter) compares to the average season. For example, the seasonal index for retail sales for the month of December is 1.5 or 150%, indicating that sales tend to be 50% higher than the monthly average.

Seasonal indices are calculated so that their average is 1, meaning the sum of indices equals the number of seasons (e.g., for monthly data: seasonal indices sum to 12; for quarterly data: seasonal indices sum to 4). A seasonal index is calculated using the following formula:

​

​

Seasonally Adjustment of a Time Series

Seasonal indices can be used to either deseasonalise (remove) or reseasonalise (restore) seasonal components in a time series, this process is called seasonally adjusting the data.

• Deseasonalising Data: Divide each actual figure by its seasonal index.
  \[\text{Deaseasonalised figure} = \frac{\text{Actual figure}}{\text{Seasonal index}}\]
• Reasonalising Data: convert a deseasonalised value into an actual data value
  \[\text{Actual figure} = \text{Deseasonalised figure}\ \times\  \text{Seasonal index}\]

​

Interpretation of Seasonal Indices

When an event or occurrence is more frequent during a specific time period, the seasonal index is greater than 1. Conversely, when it is less frequent, the seasonal index is positive but less than 1. A seasonal index is always compared to an average of 1 (or 100%).

For example, the quarterly seasonal indices for sales in the shop are shown in the table below.

A seasonal index of 0.85 (or 85%) for Quarter 2 means that sales in Quarter 2 are typically 15% below the yearly average.

A seasonal index of 1.04 (or 104%) for Quarter 4 means that sales in Quarter 4 are typically 4% above the yearly average.

Trend Line Forecasting and Predictions

If a linear trend is identified in a time series plot, the least squares method is used to model the trend and make predictions about future values. This process is called trend line forecasting. Note that extrapolation (which occurs when predictions are made beyond the observed range of the explanatory variable) can be unreliable.

For time series data with seasonal components, that data is usually deseasonalised before fitting the trend line, and predictions must then be reseasonlised by multiplying by the appropriate seasonal index for proper interpretation:

​

​

Taking it further