ARIMA Models

AutoRegressive Integrated Moving Average - models dependencies between observations.

Model	Description
ARIMA	Classic ARIMA with manual parameter specification
AutoARIMA	Automatic ARIMA with (p,d,q) selection
ARIMAX	ARIMA with exogenous variables

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ARIMA

Classic ARIMA model with manual parameter specification.

Parameters

Parameter	Type	Required	Description
p	INTEGER	Yes	AR (autoregressive) order
d	INTEGER	Yes	Differencing order
q	INTEGER	Yes	MA (moving average) order
seasonal_period	INTEGER	No	For seasonal ARIMA
P	INTEGER	No	Seasonal AR order
D	INTEGER	No	Seasonal differencing
Q	INTEGER	No	Seasonal MA order

Example

-- Non-seasonal ARIMA(1,1,1)
SELECT * FROM ts_forecast_by(
    'sales_data', NULL, date, sales,
    'ARIMA', 28, '1d',
    MAP{'p': '1', 'd': '1', 'q': '1'}
);

-- Seasonal ARIMA(1,1,1)(1,1,1)[7]
SELECT * FROM ts_forecast_by(
    'weekly_data', NULL, date, value,
    'ARIMA', 28, '1d',
    MAP{'p': '1', 'd': '1', 'q': '1', 'P': '1', 'D': '1', 'Q': '1', 'seasonal_period': '7'}
);

Best for: Complex patterns when you know the appropriate order.

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AutoARIMA

Automatic ARIMA with (p,d,q) parameter selection using information criteria.

Parameters

Parameter	Type	Default	Description
seasonal_period	INTEGER	-	Seasonal period (detect with ts_detect_periods_by first)
max_p	INTEGER	5	Maximum AR order
max_d	INTEGER	2	Maximum differencing order
max_q	INTEGER	5	Maximum MA order

Example

SELECT * FROM ts_forecast_by(
    'sales_data', NULL, date, sales,
    'AutoARIMA', 28, '1d',
    MAP{'seasonal_period': '7'}
);

Best for: Complex patterns, long-term forecasts, automatic tuning.

Seasonality Not Auto-Detected

Seasonality is NOT auto-detected. You must detect the period first with ts_detect_periods_by and pass it explicitly via seasonal_period.

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Parameter Selection Guide

Scenario	p	d	q
Stationary data	1-2	0	1-2
Trending data	1-2	1	1-2
Strong trend	1-2	2	1-2
Random walk	0	1	0

When to Use ARIMA vs AutoARIMA

Scenario	Recommended
Known order from domain expertise	ARIMA
Exploratory analysis	AutoARIMA
Production pipeline	AutoARIMA
Performance-critical	ARIMA (faster)
Many time series	AutoARIMA
External predictors	ARIMAX

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ARIMAX

ARIMA with exogenous (external) variables. Incorporates external predictors like temperature, promotions, or economic indicators.

Parameters

Parameter	Type	Default	Description
seasonal_period	INTEGER	-	Seasonal period (detect with ts_detect_periods_by first)
max_p	INTEGER	5	Maximum AR order
max_d	INTEGER	2	Maximum differencing order
max_q	INTEGER	5	Maximum MA order

Example

-- Create future exogenous values
CREATE TABLE future_exog AS
SELECT * FROM (VALUES
    ('2024-01-08'::DATE, 22.0, 1),
    ('2024-01-09'::DATE, 20.0, 0),
    ('2024-01-10'::DATE, 21.0, 1)
) AS t(date, temperature, promotion);

-- Forecast with exogenous variables
SELECT * FROM ts_forecast_exog_by(
    'sales_data', NULL, date, sales,
    ['temperature', 'promotion'],
    'future_exog', date, ['temperature', 'promotion'],
    'ARIMAX', 3,
    MAP{}, '1d'
);

Best for: Complex patterns with external drivers, weather-adjusted demand, promotion planning.

See Exogenous Variables for detailed usage