---
navigation_title: "Serial differencing"
mapped_pages:
  - https://www.elastic.co/guide/en/elasticsearch/reference/current/search-aggregations-pipeline-serialdiff-aggregation.html
---

# Serial differencing aggregation [search-aggregations-pipeline-serialdiff-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.

:::{image} images/dow.png
:alt: dow
:title: Dow Jones plotted and made stationary with first-differencing
:name: serialdiff_dow
:::

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.

:::{image} images/lemmings.png
:alt: lemmings
:title: Lemmings data plotted made stationary with 1st and 30th difference
:name: serialdiff_lemmings
:::

## Syntax [_syntax_22]

A `serial_diff` aggregation looks like this in isolation:

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

$$$serial-diff-params$$$

| Parameter Name | Description | Required | Default Value |
| --- | --- | --- | --- |
| `buckets_path` | Path to the metric of interest (see [`buckets_path` Syntax](/reference/aggregations/pipeline.md#buckets-path-syntax) for more details | Required |  |
| `lag` | 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, non-zero integer | Optional | `1` |
| `gap_policy` | Determines what should happen when a gap in the data is encountered. | Optional | `insert_zeros` |
| `format` | [DecimalFormat pattern](https://docs.oracle.com/en/java/javase/11/docs/api/java.base/java/text/DecimalFormat.html) for theoutput value. If specified, the formatted value is returned in the aggregation’s`value_as_string` property | Optional | `null` |

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

```console
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](/reference/aggregations/pipeline.md#buckets-path-syntax) for a description of the syntax for `buckets_path`.