380 lines
16 KiB
Plaintext
380 lines
16 KiB
Plaintext
[[search-aggregations-pipeline-movavg-aggregation]]
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=== Moving Average Aggregation
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Given an ordered series of data, the Moving Average aggregation will slide a window across the data and emit the average
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value of that window. For example, given the data `[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]`, we can calculate a simple moving
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average with windows size of `5` as follows:
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- (1 + 2 + 3 + 4 + 5) / 5 = 3
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- (2 + 3 + 4 + 5 + 6) / 5 = 4
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- (3 + 4 + 5 + 6 + 7) / 5 = 5
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- etc
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Moving averages are a simple method to smooth sequential data. Moving averages are typically applied to time-based data,
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such as stock prices or server metrics. The smoothing can be used to eliminate high frequency fluctuations or random noise,
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which allows the lower frequency trends to be more easily visualized, such as seasonality.
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==== Syntax
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A `moving_avg` aggregation looks like this in isolation:
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[source,js]
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--------------------------------------------------
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{
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"moving_avg": {
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"buckets_path": "the_sum",
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"model": "holt",
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"window": 5,
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"gap_policy": "insert_zero",
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"settings": {
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"alpha": 0.8
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}
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}
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}
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--------------------------------------------------
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.`moving_avg` Parameters
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|===
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|Parameter Name |Description |Required |Default Value
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|`buckets_path` |Path to the metric of interest (see <<bucket-path-syntax, `buckets_path` Syntax>> for more details |Required |
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|`model` |The moving average weighting model that we wish to use |Optional |`simple`
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|`gap_policy` |Determines what should happen when a gap in the data is encountered. |Optional |`insert_zero`
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|`window` |The size of window to "slide" across the histogram. |Optional |`5`
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|`settings` |Model-specific settings, contents which differ depending on the model specified. |Optional |
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|===
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`moving_avg` aggregations must be embedded inside of a `histogram` or `date_histogram` aggregation. They can be
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embedded like any other metric aggregation:
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[source,js]
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--------------------------------------------------
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{
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"my_date_histo":{ <1>
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"date_histogram":{
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"field":"timestamp",
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"interval":"day"
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},
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"aggs":{
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"the_sum":{
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"sum":{ "field": "lemmings" } <2>
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},
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"the_movavg":{
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"moving_avg":{ "buckets_path": "the_sum" } <3>
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}
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}
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}
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}
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--------------------------------------------------
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<1> A `date_histogram` named "my_date_histo" is constructed on the "timestamp" field, with one-day intervals
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<2> A `sum` metric is used to calculate the sum of a field. This could be any metric (sum, min, max, etc)
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<3> Finally, we specify a `moving_avg` aggregation which uses "the_sum" metric as its input.
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Moving averages are built by first specifying a `histogram` or `date_histogram` over a field. You can then optionally
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add normal metrics, such as a `sum`, inside of that histogram. Finally, the `moving_avg` is embedded inside the histogram.
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The `buckets_path` parameter is then used to "point" at one of the sibling metrics inside of the histogram (see
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<<bucket-path-syntax>> for a description of the syntax for `buckets_path`.
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==== Models
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The `moving_avg` aggregation includes four different moving average "models". The main difference is how the values in the
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window are weighted. As data-points become "older" in the window, they may be weighted differently. This will
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affect the final average for that window.
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Models are specified using the `model` parameter. Some models may have optional configurations which are specified inside
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the `settings` parameter.
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===== Simple
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The `simple` model calculates the sum of all values in the window, then divides by the size of the window. It is effectively
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a simple arithmetic mean of the window. The simple model does not perform any time-dependent weighting, which means
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the values from a `simple` moving average tend to "lag" behind the real data.
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[source,js]
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--------------------------------------------------
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{
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"the_movavg":{
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"moving_avg":{
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"buckets_path": "the_sum",
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"model" : "simple"
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}
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}
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}
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--------------------------------------------------
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A `simple` model has no special settings to configure
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The window size can change the behavior of the moving average. For example, a small window (`"window": 10`) will closely
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track the data and only smooth out small scale fluctuations:
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[[movavg_10window]]
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.Moving average with window of size 10
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image::images/pipeline_movavg/movavg_10window.png[]
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In contrast, a `simple` moving average with larger window (`"window": 100`) will smooth out all higher-frequency fluctuations,
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leaving only low-frequency, long term trends. It also tends to "lag" behind the actual data by a substantial amount:
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[[movavg_100window]]
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.Moving average with window of size 100
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image::images/pipeline_movavg/movavg_100window.png[]
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==== Linear
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The `linear` model assigns a linear weighting to points in the series, such that "older" datapoints (e.g. those at
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the beginning of the window) contribute a linearly less amount to the total average. The linear weighting helps reduce
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the "lag" behind the data's mean, since older points have less influence.
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[source,js]
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--------------------------------------------------
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{
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"the_movavg":{
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"moving_avg":{
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"buckets_path": "the_sum",
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"model" : "linear"
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}
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}
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--------------------------------------------------
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A `linear` model has no special settings to configure
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Like the `simple` model, window size can change the behavior of the moving average. For example, a small window (`"window": 10`)
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will closely track the data and only smooth out small scale fluctuations:
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[[linear_10window]]
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.Linear moving average with window of size 10
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image::images/pipeline_movavg/linear_10window.png[]
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In contrast, a `linear` moving average with larger window (`"window": 100`) will smooth out all higher-frequency fluctuations,
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leaving only low-frequency, long term trends. It also tends to "lag" behind the actual data by a substantial amount,
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although typically less than the `simple` model:
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[[linear_100window]]
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.Linear moving average with window of size 100
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image::images/pipeline_movavg/linear_100window.png[]
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==== EWMA (Exponentially Weighted)
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The `ewma` model (aka "single-exponential") is similar to the `linear` model, except older data-points become exponentially less important,
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rather than linearly less important. The speed at which the importance decays can be controlled with an `alpha`
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setting. Small values make the weight decay slowly, which provides greater smoothing and takes into account a larger
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portion of the window. Larger valuers make the weight decay quickly, which reduces the impact of older values on the
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moving average. This tends to make the moving average track the data more closely but with less smoothing.
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The default value of `alpha` is `0.5`, and the setting accepts any float from 0-1 inclusive.
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[source,js]
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--------------------------------------------------
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{
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"the_movavg":{
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"moving_avg":{
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"buckets_path": "the_sum",
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"model" : "ewma",
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"settings" : {
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"alpha" : 0.5
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}
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}
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}
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--------------------------------------------------
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[[single_0.2alpha]]
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.EWMA with window of size 10, alpha = 0.2
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image::images/pipeline_movavg/single_0.2alpha.png[]
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[[single_0.7alpha]]
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.EWMA with window of size 10, alpha = 0.7
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image::images/pipeline_movavg/single_0.7alpha.png[]
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==== Holt-Linear
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The `holt` model (aka "double exponential") incorporates a second exponential term which
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tracks the data's trend. Single exponential does not perform well when the data has an underlying linear trend. The
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double exponential model calculates two values internally: a "level" and a "trend".
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The level calculation is similar to `ewma`, and is an exponentially weighted view of the data. The difference is
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that the previously smoothed value is used instead of the raw value, which allows it to stay close to the original series.
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The trend calculation looks at the difference between the current and last value (e.g. the slope, or trend, of the
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smoothed data). The trend value is also exponentially weighted.
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Values are produced by multiplying the level and trend components.
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The default value of `alpha` and `beta` is `0.5`, and the settings accept any float from 0-1 inclusive.
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[source,js]
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--------------------------------------------------
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{
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"the_movavg":{
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"moving_avg":{
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"buckets_path": "the_sum",
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"model" : "holt",
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"settings" : {
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"alpha" : 0.5,
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"beta" : 0.5
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}
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}
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}
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--------------------------------------------------
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In practice, the `alpha` value behaves very similarly in `holt` as `ewma`: small values produce more smoothing
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and more lag, while larger values produce closer tracking and less lag. The value of `beta` is often difficult
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to see. Small values emphasize long-term trends (such as a constant linear trend in the whole series), while larger
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values emphasize short-term trends. This will become more apparently when you are predicting values.
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[[double_0.2beta]]
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.Holt-Linear moving average with window of size 100, alpha = 0.5, beta = 0.2
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image::images/pipeline_movavg/double_0.2beta.png[]
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[[double_0.7beta]]
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.Holt-Linear moving average with window of size 100, alpha = 0.5, beta = 0.7
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image::images/pipeline_movavg/double_0.7beta.png[]
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==== Holt-Winters
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The `holt_winters` model (aka "triple exponential") incorporates a third exponential term which
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tracks the seasonal aspect of your data. This aggregation therefore smooths based on three components: "level", "trend"
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and "seasonality".
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The level and trend calculation is identical to `holt` The seasonal calculation looks at the difference between
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the current point, and the point one period earlier.
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Holt-Winters requires a little more handholding than the other moving averages. You need to specify the "periodicity"
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of your data: e.g. if your data has cyclic trends every 7 days, you would set `period: 7`. Similarly if there was
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a monthly trend, you would set it to `30`. There is currently no periodicity detection, although that is planned
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for future enhancements.
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There are two varieties of Holt-Winters: additive and multiplicative.
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===== "Cold Start"
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Unfortunately, due to the nature of Holt-Winters, it requires two periods of data to "bootstrap" the algorithm. This
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means that your `window` must always be *at least* twice the size of your period. An exception will be thrown if it
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isn't. It also means that Holt-Winters will not emit a value for the first `2 * period` buckets; the current algorithm
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does not backcast.
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[[holt_winters_cold_start]]
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.Holt-Winters showing a "cold" start where no values are emitted
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image::images/pipeline_movavg/triple_untruncated.png[]
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Because the "cold start" obscures what the moving average looks like, the rest of the Holt-Winters images are truncated
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to not show the "cold start". Just be aware this will always be present at the beginning of your moving averages!
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===== Additive Holt-Winters
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Additive seasonality is the default; it can also be specified by setting `"type": "add"`. This variety is preferred
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when the seasonal affect is additive to your data. E.g. you could simply subtract the seasonal effect to "de-seasonalize"
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your data into a flat trend.
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The default value of `alpha`, `beta` and `gamma` is `0.5`, and the settings accept any float from 0-1 inclusive.
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The default value of `period` is `1`.
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[source,js]
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--------------------------------------------------
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{
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"the_movavg":{
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"moving_avg":{
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"buckets_path": "the_sum",
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"model" : "holt_winters",
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"settings" : {
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"type" : "add",
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"alpha" : 0.5,
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"beta" : 0.5,
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"gamma" : 0.5,
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"period" : 7
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}
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}
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}
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--------------------------------------------------
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[[holt_winters_add]]
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.Holt-Winters moving average with window of size 120, alpha = 0.5, beta = 0.7, gamma = 0.3, period = 30
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image::images/pipeline_movavg/triple.png[]
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===== Multiplicative Holt-Winters
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Multiplicative is specified by setting `"type": "mult"`. This variety is preferred when the seasonal affect is
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multiplied against your data. E.g. if the seasonal affect is x5 the data, rather than simply adding to it.
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The default value of `alpha`, `beta` and `gamma` is `0.5`, and the settings accept any float from 0-1 inclusive.
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The default value of `period` is `1`.
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[WARNING]
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======
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Multiplicative Holt-Winters works by dividing each data point by the seasonal value. This is problematic if any of
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your data is zero, or if there are gaps in the data (since this results in a divid-by-zero). To combat this, the
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`mult` Holt-Winters pads all values by a very small amount (1*10^-10^) so that all values are non-zero. This affects
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the result, but only minimally. If your data is non-zero, or you prefer to see `NaN` when zero's are encountered,
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you can disable this behavior with `pad: false`
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======
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[source,js]
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--------------------------------------------------
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{
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"the_movavg":{
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"moving_avg":{
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"buckets_path": "the_sum",
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"model" : "holt_winters",
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"settings" : {
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"type" : "mult",
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"alpha" : 0.5,
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"beta" : 0.5,
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"gamma" : 0.5,
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"period" : 7,
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"pad" : true
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}
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}
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}
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--------------------------------------------------
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==== Prediction
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All the moving average model support a "prediction" mode, which will attempt to extrapolate into the future given the
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current smoothed, moving average. Depending on the model and parameter, these predictions may or may not be accurate.
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Predictions are enabled by adding a `predict` parameter to any moving average aggregation, specifying the nubmer of
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predictions you would like appended to the end of the series. These predictions will be spaced out at the same interval
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as your buckets:
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[source,js]
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--------------------------------------------------
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{
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"the_movavg":{
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"moving_avg":{
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"buckets_path": "the_sum",
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"model" : "simple",
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"predict" 10
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}
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}
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--------------------------------------------------
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The `simple`, `linear` and `ewma` models all produce "flat" predictions: they essentially converge on the mean
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of the last value in the series, producing a flat:
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[[simple_prediction]]
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.Simple moving average with window of size 10, predict = 50
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image::images/pipeline_movavg/simple_prediction.png[]
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In contrast, the `holt` model can extrapolate based on local or global constant trends. If we set a high `beta`
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value, we can extrapolate based on local constant trends (in this case the predictions head down, because the data at the end
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of the series was heading in a downward direction):
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[[double_prediction_local]]
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.Holt-Linear moving average with window of size 100, predict = 20, alpha = 0.5, beta = 0.8
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image::images/pipeline_movavg/double_prediction_local.png[]
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In contrast, if we choose a small `beta`, the predictions are based on the global constant trend. In this series, the
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global trend is slightly positive, so the prediction makes a sharp u-turn and begins a positive slope:
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[[double_prediction_global]]
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.Double Exponential moving average with window of size 100, predict = 20, alpha = 0.5, beta = 0.1
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image::images/pipeline_movavg/double_prediction_global.png[]
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The `holt_winters` model has the potential to deliver the best predictions, since it also incorporates seasonal
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fluctuations into the model:
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[[holt_winters_prediction_global]]
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.Holt-Winters moving average with window of size 120, predict = 25, alpha = 0.8, beta = 0.2, gamma = 0.7, period = 30
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image::images/pipeline_movavg/triple_prediction.png[]
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