Serial differencing aggregation

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(xt) - f(xt-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.

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.

A serial_diff aggregation looks like this in isolation:

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

serial_diff aggregations must be embedded inside of a histogram or date_histogram aggregation:

 POST /_search {
   "size": 0,
   "aggs": {
      "my_date_histo": {                  1
         "date_histogram": {
            "field": "timestamp",
            "calendar_interval": "day"
         },
         "aggs": {
            "the_sum": {
               "sum": {
                  "field": "lemmings"     2
               }
            },
            "thirtieth_difference": {
               "serial_diff": {                3
                  "buckets_path": "the_sum",
                  "lag" : 30
               }
            }
         }
      }
   }
}

1. A date_histogram named "my_date_histo" is constructed on the "timestamp" field, with one-day intervals
2. A sum metric is used to calculate the sum of a field. This could be any metric (sum, min, max, etc)
3. Finally, we specify a serial_diff aggregation which uses "the_sum" metric as its input.

Serial differences are built by first specifying a histogram or date_histogram over a field. You can then optionally add normal metrics, such as a sum, inside of that histogram. Finally, the serial_diff is embedded inside the histogram. The buckets_path parameter is then used to "point" at one of the sibling metrics inside of the histogram (see buckets_path Syntax for a description of the syntax for buckets_path.