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_{tn}), 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 firstdifference, we detrend 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 randomwalk; 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 firstdifference removes the constant trend, leaving just a sine wave. The 30thdifference is then applied to the firstdifference 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" } }
Table 14. serial_diff
Parameters
Parameter Name  Description  Required  Default Value 


Path to the metric of interest (see 
Required 


The historical bucket to subtract from the current value. E.g. a lag of 7 will subtract the current value from the value 7 buckets ago. Must be a positive, nonzero integer 
Optional 


Determines what should happen when a gap in the data is encountered. 
Optional 


Format to apply to the output value of this aggregation 
Optional 

serial_diff
aggregations must be embedded inside of a histogram
or date_histogram
aggregation:
POST /_search { "size": 0, "aggs": { "my_date_histo": { "date_histogram": { "field": "timestamp", "interval": "day" }, "aggs": { "the_sum": { "sum": { "field": "lemmings" } }, "thirtieth_difference": { "serial_diff": { "buckets_path": "the_sum", "lag" : 30 } } } } } }
A 

A 

Finally, we specify a 
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
.