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Introduction to Time Series

Maths Applications (Year 12) - Time Series Analysis

Bhakti Sharma

Introduction to Time Series

Time Series Data, in simple terms, is used to show data that is collected over time. On a time series graph, you would see time on the x axis, and the variable of interest on the y axis. This shows us the relationship between the variable of interest and a period of time. Time series graphs are very versatile and are used in many different situations, such as for tracking annual rainfall, costs or trends in sales. Here's an example of a time series graph used to illustrate the average rainfall in each season across Summer 2021 to Summer 2023.


There are two main types of time series trends. One of these is the positive upward trend, and the other is the negative downward trend. In the positive upward trend, the data generally seems to go up, in that as time moves on, the y-axis variable seems to increase on average. Of course, there may be troughs and peaks of different sizes, but on average, the plots seem to move up, there is a positive relationship between the two variables. The opposite happens in the negative downward trend. As time goes on, the y-axis variable seems to decrease. Similar to the positive upward trend, we may see peaks and troughs, but on average there is a decrease. The relationship between the two variables is negative. Here are examples of both:



Similar to the trends, there are also two main types of time series patterns. One is the seasonal pattern, in which the graph will have peaks and troughs of similar sizes at regular intervals. The other is the cyclical pattern, in which we see peaks and troughs of different sizes at irregular unpredicted intervals. Here are examples of both:


Time series data can be used to make predictions as well. We must proceed with caution when doing this, however. The first step to take is to decide what regression type is most appropriate for the data. Another thing to remember is that interpolation is generally more reliable than extrapolation, and the further out you extrapolate, the less reliable it is. For example if we are looking at the graph from earlier, and are trying to predict how much rainfall there will be next summer, we could use extrapolation.

By using a regression line, we can see that there is a slight positive upward trend in the data. This means that every year the average rainfall is increasing by a little bit. We can also say that the pattern in the data is seasonal, as there is a consistent trough around Spring/Summer, and a consistent peak around Autumn/Winter each year. Now, going back to our question, what do we predict the average rainfall will be in Summer 2024? Through extrapolation, we could say that it would be around 1mm of rain.

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