This example explains how to calculate the central moving average of a time series using the SPMF open-source data mining library.
How to run this example?
What is the calculation of the central moving average for time series?
Calculating the central moving average is a simple but popular way of smoothing a time series to remove noise. It takes as parameter a window size w (a number of data point), which must be an odd number greater than 1. Then, for a time series, it replaces each data point X by the average of X with the (w-1)/2 data points appearing before, and the (w-1)/2 data points appearing after.
What is the input of this algorithm?
The input is one or more time series. A time series is a sequence of floating-point decimal numbers (double values). A time-series can also have a name (a string).
Time series are used in many applications. An example of time series is the price of a stock on the stock market over time. Another example is a sequence of temperature readings collected using sensors.
For this example, consider the following time series:
Name Data points ECG1 3,2,8,9,8,9,8,7,6,7,5,4,2,7,9,8,5
This example time series database is provided in the file contextMovingAverage.txt of the SPMF distribution.
In SPMF, to read a time-series file, it is necessary to indicate the "separator", which is the character used to separate data points in the input file. In this example, the "separator" is the comma ',' symbol.
To calculate the central moving average, it is necessary to provide a window size w, which is a number of data points. In this example, this parameter will be set to 3 data points. Thus, the central moving average will be calculated for each of the above time series using a window size of 3 data points.
What is the output?
The output is the moving average of the time series received as input. The moving average is calculated by replacing each data point X in a time series by the average of X with the (w-1)/2 data points appearing before, and the (w-1)/2 data points appearing after.
For example, in the above example, if the window size is set to 3 data points, the result is:
Name Data points ECG1_CEMAVG 2.5,4.333333333333333,6.333333333333333,8.333333333333334,8.666666666666666,8.333333333333334,8.0,7.0,6.666666666666667,6.0,5.333333333333333,3.6666666666666665,4.333333333333333,6.0,8.0,7.333333333333333,6.5
To see the result visually, it is possible to use the SPMF time series viewer, described in another example of this documentation. Here is the original time series and the central moving average for window = 3.
It is possible to see that the time series are less noisy.We can increase the values of the window parameter to obtain a yet more smooth time series. For example, if we set window = 7:
Input file format
The input file format is defined as follows. It is a text file. The text file contains one or more time series. Each time series is represented by two lines in the input file. The first line contains the string "@NAME=" followed by the name of the time series. The second line is a list of data points, where data points are floating-point decimal numbers separated by a separator character (here the ',' symbol).
For example, the input file of the previous example, named contextMovingAverage.txt is defined as follows:
Consider the first two lines. It indicates that the first time series name is "ECG2" and that it consits of the data points: 3,2,8,9,8,9,8,7,6,7,5,4,2,7,9,8, and 5. Then, three other time series are provided in the same file, which follows the same format.
But note that it is possible to have more than one time series per file. For example, this is another input file called contextSax.txt, which contains 4 time series.
Output file format
The output file format is the same as the input format. For example, there is the result for window = 3:
It is sometimes said that the central moving average of a time series should not include the first (w-1)/2 or last (w-1)/2 points because for these points there is not enough previous or suceeding points for calculating the central average with w points. In this implementation, a design decision is that the moving average should contain as many data points as the original time series. To calculate the central moving average of the first(w-1)/2 points, it was decided to ignore non existing previous points for the calculation, and for the last (w-1)/2 points, it was decided to ignore non existing succeeding points for the calculation.
Note that in SPMF, other variations of the moving average called the "cumulative moving average" and "prior moving average" are also offered.
Where can I get more information about the central moving average?
The central moving average is a basic operation for analyzing time series. It is described in many websites and books.
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