Serial differencing is a technique where values in a time series are subtracted from itself at different time lags or periods. For example, the datapoint f(x) = f(x_t) - f(x_t-n), where n is the period being used.

A period of 1 is equivalent to a derivative with no time normalization: it is simply the change from one point to the next. Single periods are useful for removing constant, linear trends.

Single periods are also useful for transforming data into a stationary series. In this example, the Dow Jones is plotted over ~250 days. The raw data is not stationary, which would make it difficult to use with some techniques.

By calculating the first-difference, we de-trend the data (e.g. remove a constant, linear trend). We can see that the data becomes a stationary series (e.g. the first difference is randomly distributed around zero, and doesn’t seem to exhibit any pattern/behavior). The transformation reveals that the dataset is following a random-walk; the value is the previous value +/- a random amount. This insight allows selection of further tools for analysis.

Figure 15. Dow Jones plotted and made stationary with first-differencing

Larger periods can be used to remove seasonal / cyclic behavior. In this example, a population of lemmings was synthetically generated with a sine wave + constant linear trend + random noise. The sine wave has a period of 30 days.

The first-difference removes the constant trend, leaving just a sine wave. The 30th-difference is then applied to the first-difference to remove the cyclic behavior, leaving a stationary series which is amenable to other analysis.

Figure 16. Lemmings data plotted made stationary with 1st and 30th difference

{
    "serial_diff": {
        "buckets_path": "the_sum",
        "lag": "7"
    }
}

POST /_search
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   "aggs": {
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         "date_histogram": {
            "field": "timestamp",
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         "aggs": {
            "the_sum": {
               "sum": {
                  "field": "lemmings"     
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            "thirtieth_difference": {
               "serial_diff": {                
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