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# FM Time Series Data

## 4.5 Analysis of De-seasonalised Data

### Linear Regression of Ordinary Time Series Data

• As with any other type of bivariate data, it is often useful to apply linear regression to time series data in order to predict values for which we have no data.
• For time series data, time is always the explanatory variable.
• Unprocessed data with seasonality is generally poorly modelled by a linear fit.

Note: if you cannot remember how to construct and interpret a linear fit, revise notes for 3.1 Least Squares Linear Regression and 3.2 Modelling Linear Associations.

### Re-seasonalising Data

Read More »4.5 Analysis of De-seasonalised Data

## 4.4 Introduction to Seasonal Indices

### Seasonal Indices

• Seasonal indices provide a method to de-seasonalise data.
• The seasonal index of a season/month/period/etc. compares the average value of a particular season to the average of all seasons in a cycle.
• A seasonal index of 1 indicates the average value of the season is exactly equal to the average value of the entire cycle.
• A seasonal index greater than 1 indicates the average value of the season is greater than that of the entire cycle (e.g. a seasonal index of 1.2 indicates the season’s average is 20% higher than the cycle’s average).
Read More »4.4 Introduction to Seasonal Indices

## 4.3 Numerical Smoothing using the Moving Median Method

Note: if you can’t remember the basics of numerical smoothing, revise notes for 4.2 Numerical Smoothing using the Moving Mean Method.

### Moving Median Smoothing

• The moving median smoothing method involves taking the median of each group.
• This method is particularly effective when the exact values of data points are unknown (e.g. if the data is shown in a time series plot without the raw dataset).

### Smooth a Single Point using an Odd Moving Median

Read More »4.3 Numerical Smoothing using the Moving Median Method

## 4.2 Numerical Smoothing using the Moving Mean Method

### The Idea of Numerical Smoothing

• Time series plots are often ‘noisy’, with many random fluctuations that make it difficult to analyse the long-term pattern of the data.
• Numerical smoothing provides a method of lessening the impact of those random fluctuations so that the pattern is easier to discern.
• The two methods for numerical smoothing used in further maths are moving mean and moving median smoothing.
• In both methods, a new set of values are created by taking a group of data points, finding the mean or median, then moving to the next group (by replacing the first data point in the group with the next data point not yet included).
Read More »4.2 Numerical Smoothing using the Moving Mean Method

## 4.1 Introduction to Time Series Plots

### Time Series Plots

• Time series plots are a specific type of graph, where the explanatory variable is time.
• They are used to analyse how a system changes over time.
• When describing a time series plot, the trend, seasonality, irregular fluctuations, structural change and outliers are all important aspects.
• Time series plots can either be shown as a graph with data points connected by lines, or as a dot plot.
Read More »4.1 Introduction to Time Series Plots