Moving Average

Moving averages and means to smoothen noisy (time series) data, unearthing the underlying trends. The process is also known as filtering the data.

Moving averages (MA) are a series of averages estimated on a pre-defined number of consecutive data. For example, MA₅ contains the set of all the averages of 5 successive members of the original series. The following relationship represents how a centred or two-sided moving average is estimated.

MA₅ = (X_t-2 + X_t-1 + X_t + X_t+1 + X_t+2)/5

‘t’ represents the time at which the moving average is estimated. t-1 is the observation just before t, and t+1 means one observation immediately after t. To illustrate the concept, we develop the following table representing consecutive 10 points (electric signals) in a dataset.

Date	Signal
1	72.5052
2	70.6720
3	62.4502
4	57.4714
5	55.3151
6	58.0904
7	62.6202
8	63.2485
9	60.5846
10	56.3154

The centred MA starts from point 3 (the midpoint of 1, 2, 3, 4, 5). The value at 3 is the mean of the first five points (72.5052 + 70.6720 + 62.4502 + 57.4714 + 55.3151) /5 = 63.68278.

Date	Signal	Average
1	72.5052
2	70.6720
3	62.4502	63.68278
4	57.4714
5	55.3151
6	58.0904
7	62.6202
8	63.2485
9	60.5846
10	56.3154

This process is continued – MA on point 4 is the mean of points 2, 3, 4, 5 and 6, etc.

Date	Signal	MA
1	72.5052
2	70.6720
3	62.4502	63.68278 [1-5]
4	57.4714	60.79982 [2-6]
5	55.3151	59.18946 [3-7]
6	58.0904	59.34912 [4-8]
7	62.6202	59.97176 [5-9]
8	63.2485	60.17182 [6-10]
9	60.5846
10	56.3154

The one-sided moving average is different. It estimates MA at the end of the interval.
MA₅ = (X_t-4 + X_t-3 + X_t-2 + X_t-1 + X_t)/5

Date	Signal	MA
1	72.5052
2	70.6720
3	62.4502
4	57.4714
5	55.3151	63.68278 [1-5]
6	58.0904	60.79982 [2-6]
7	62.6202	59.18946 [3-7]
8	63.2485	59.34912 [4-8]
9	60.5846	59.97176 [5-9]
10	56.3154	60.17182 [6-10]

We will look at the impact of moving averages in the next post.