# SARIMA

SARIMA is the abbreviation for Seasonal Autoregressive Integrated Moving Average, a time series analysis in the fields of statistics and econometrics. In Kloudfuse, we implement the SARIMA algorithm as the **agile** option for anomaly detection.

## Parameters

Without accounting for seasonality, we utilize three parameters:

- **p**: Number of historical points considered for auto-regression (AR)
- **q**: Number of historical points considered for moving averages (MA).
- **d**: Number of times to apply differencing. Specifies that calculations should be made on the differences between consecutive points, rather than the raw points.

When considering seasonality, we add these additional parameters:

- **sp**: Seasonally-adjusted number of historical points considered for auto-regression (AR)
- **sq**: Seasonally-adjusted number of historical points considered for moving averages (MA).
- **sd**: Seasonally-adjusted number of times to apply differencing.
- **sm**: Number of discrete timestamps in a period.

## How it works

To make predictions, we maximize **p** and **q** historical points. This means that we use **\$max(p,q)+d\$** historical points to make a prediction.

The model produces a predicted value per timestamp; the upper and lower bands sit `bound` standard deviations around the prediction, and values that escape the band are anomalous.

## In Dashboards

To use the `sarima` operator in a dashboard, apply the following function:

```none
sarima( 
  ${promql}, 
  2, 1, 2, 0, 0, 0, 0, 
  ${bound}, 
  ${band} 
)
```

|     |     |
| --- | --- |
| **1** | `${promql}`: PromQL query to evaluate |
| **2** | `${bound}`: Number of standard deviations (stdv): 1, 2, or 3 |
| **3** | `${band}`: 4 = lower band, 5 = upper band, 6 = both upper and lower bands |

For the operator reference — syntax, parameters, and a validated example — see [sarima](https://docs.kloudfuse.com/platform/latest/query-languages/promql/algorithm/#sarima) in the PromQL documentation.

## Limitations

- If the evaluated metrics do not exhibit true seasonality, SARIMA may create incorrect (invalid) alerts, or mask valid alerting conditions.

- SARIMA fits a model **per input series**; cost grows with the number of series. Aggregate before applying the function rather than banding every raw series.

- Training needs \$max(p,q)+d\$ historical points — and \$(max(sp,sq)+sd)sm\$ when seasonal terms are set. The service fetches this history beyond the visible chart range; series with insufficient history are skipped silently rather than reported as errors.

- High seasonal orders multiply both the history requirement and the fitting time.

## Next steps

For an in-depth discussion of the SARIMA functions, see these external resources:

- [Seasonal AutoRegressive Integrated Moving Average with eXogenous regressors model](https://www.statsmodels.org/0.9.0/generated/statsmodels.tsa.statespace.sarimax.SARIMAX.html) in StatsModels
- [Autoregressive integrated moving average](https://en.wikipedia.org/wiki/Autoregressive_integrated_moving_average.html) in Wikipedia