PREDICT: Statistical Forecasting Algorithm

The Predict is a forecasting feature that generates future values by analyzing historical data and identifying patterns such as trends, seasonality, and historical relationships. It applies statistical forecasting techniques to estimate future values for a specified forecast period and supports configurable forecasting options to meet different planning requirements. The behavior of the forecast depends on the forecasting algorithm, the forecasting model it uses, and the model order selected.

A forecasting algorithm is a mathematical or statistical method that analyzes historical time-series data to identify patterns such as level, trend, seasonality, and relationships between observations, and uses those patterns to estimate future values.

A statistical model is the mathematical representation of a historical time series created by a forecasting algorithm. It captures characteristics, such as level, trend, seasonality, and relationships between observations, and is used to generate forecast values

A model order is a set of parameters that defines the structure and complexity of a statistical model. It tells the algorithm how many components or parameters are included in the model. Below table contains list of Forecast Algorithm, Statistical model and Model order available in Fabric plan

Forecast Algorithm	Statistical Model	Model Order
Trend Decomposition with MSTL
Exponential Smoothing	Auto ETS Simple Exponential Smoothing Holt (Double Exponential) Holt (Double Exponential), Damped Trend Holt Winters, Additive (A, A, A) Holt Winters, Additive, Damped (A, Ad, A) Holt Winters, Multiplicative (M, M, M) Holt Winters, Multiplicative, Damped (M, Ad, M) Holt Winters, Multiplicative, Seasonal (M, N, M)
ARIMA	Auto ARIMA Non - Seasonal (ARIMA) Seasonal (SARIMA)	Non - Seasonal (ARIMA) ARIMA / (1,1,1) Random Walk / (0,1,0) AR(1) / (1,0,0) IMA(1,1) / (0,1,1) Differenced AR(1) / (1,1,0) Seasonal (SARIMA) Airline Model (0,1,1,12) General SARIMA (1,1,1,12) Seasonal AR (1,0,0,12) Quarterly (1,0,1,4)

1. Trend Decomposition with MSTL - Multiple Seasonal Trend decomposition using LOESS Trend Decomposition with MSTL is a time-series decomposition technique that separates a time series into trend, multiple seasonal components, and a remainder using LOESS smoothing. The decomposed components can then be used to improve forecasting accuracy for data with multiple seasonal patterns. It is particularly useful for analyzing data with more than one seasonal pattern and can improve forecasting by isolating the underlying structure of the time series. Use MSTL when your data has:

   · More than one seasonal pattern.

   · Long-term trends.

   · Complex recurring cycles.

2. Exponential Smoothing Exponential Smoothing is a statistical forecasting algorithm that predicts future values by calculating weighted averages of historical observations, assigning greater weight to more recent data and progressively smaller weights to older data. It is commonly used for time-series forecasting and can be extended to model trends and seasonal patterns through methods such as Holt's Linear Trend and Holt-Winters. Use Exponential Smoothing when your data has:

   · A relatively stable level.

   · No significant trend.

   · No repeating seasonal pattern.

• Simple Exponential Smoothing: Simple Exponential Smoothing predicts future values by computing a weighted average of past observations, giving the greatest weight to the most recent data, and is best for time series with no trend and no seasonality.

• Holt (Double Exponential): Holt's (Double Exponential) Smoothing is a forecasting method that models and forecasts future values by estimating both the current level and the underlying trend of a time series, making it suitable for data with a trend but no seasonality.

• Holt (Double Exponential), Damped Trend: Holt's (Double Exponential) Smoothing with Damped Trend is a forecasting method that models and forecasts future values by estimating the current level and trend while gradually reducing the influence of the trend over time, making it suitable for non-seasonal data where growth or decline is expected to slow.

• Holt Winters, Additive (A, A, A): Holt-Winters Additive (A, A, A) is a forecasting method that predicts future values by combining an additive error, an additive trend, and an additive seasonal component, making it suitable for time series with a trend and constant-sized seasonal fluctuations. Seasonal fluctuations remain approximately constant in magnitude regardless of the level of the series.

• Holt Winters, Additive, Damped (A, Ad, A): Holt-Winters Additive Damped (A, Ad, A) is an ETS forecasting method that predicts future values using an additive error, a trend that gradually weakens over time, and a constant additive seasonal pattern, making it suitable for data with trend and constant-sized seasonality where long-term growth is expected to slow.

• Holt Winters, Multiplicative (M, M, M): Holt-Winters Multiplicative (M, M, M) is an ETS forecasting method that predicts future values using multiplicative error, multiplicative trend, and multiplicative seasonal components, making it ideal for time series where both growth and seasonal fluctuations increase proportionally with the level of the data. Seasonal fluctuations change proportionally with the level of the series.

• Holt Winters, Multiplicative, Damped (M, Ad, M): Holt-Winters Multiplicative Damped (M, Ad, M) is an ETS forecasting method that predicts future values using multiplicative errors, a gradually weakening trend, and multiplicative seasonality, making it ideal for data where seasonal effects grow with the level of the series but long-term growth is expected to slow down.

• Holt Winters, Multiplicative, Seasonal (M, N, M): Holt-Winters Multiplicative Seasonal (M, N, M) is an ETS forecasting method that predicts future values using multiplicative errors, no trend, and multiplicative seasonality, making it suitable for stable time series with seasonal effects that change proportionally with the level of the data.

• Auto ETS: Automatically evaluates multiple ETS statistical models and selects the model that best fits the historical time series using statistical selection criteria.

After running forecast, you can optimize it by choosing the following parameters:

• Alpha (α) – Level Smoothing Parameter: Alpha controls how quickly the model updates the current level (average) based on the newest observation. The Parameter range of Alpha is (0 < α < 1).

• Beta (β) – Trend Smoothing Parameter: Beta controls how quickly the model updates the trend. The Parameter range of Beta is (0 < β < 1).

• Gamma (γ) – Seasonal Smoothing Parameter: Gamma controls how quickly the seasonal pattern is updated. The Parameter range of Gamma is (0 < γ < 1).

• Phi (φ) – Damping Parameter: Phi controls how much the trend is reduced (damped) into the future. The Parameter range of Phi is (0 < φ < 1).

1. ARIMA (AutoRegressive Integrated Moving Average) ARIMA is a forecasting algorithm that predicts future values by combining information from past observations (AutoRegressive), differencing the data to achieve stationarity, and past forecast errors (Moving Average), making it well suited for non-seasonal time series forecasting. Use ARIMA when your data:

   · Has no seasonal pattern.

   · May have a trend that can be removed through differencing.

   · Shows autocorrelation, where past values help predict future values.

   · Contains sufficient historical observations to estimate the model.

   · Requires short- to medium-term forecasting based on historical patterns.

• Non Seasonal ARIMA: Non-Seasonal ARIMA is the standard ARIMA forecasting method that predicts future values using past observations, differencing, and past forecast errors, and is designed for time series that have trends but no repeating seasonal patterns.

You can also choose the model order:

• ARIMA / (1,1,1): ARIMA(1,1,1) is a non-seasonal forecasting model that first differences the data once to remove the trend, then predicts future values using one previous observation and one previous forecast error.

• Random Walk / (0,1,0): Random Walk (ARIMA(0,1,0)) is the simplest ARIMA model, which differences the data once and predicts the next value as the current observed value, assuming future changes are random.

• AR(1) / (1,0,0): AR(1), or ARIMA(1,0,0), is a forecasting model that predicts the next value using only the immediately previous observation, assuming the data is already stationary and has no trend or seasonality.

• IMA(1,1) / (0,1,1): IMA(1,1), or ARIMA(0,1,1), is a forecasting model that removes the trend by differencing the data once and predicts future values by adjusting for the most recent forecast error, without using autoregressive terms.

• Differenced AR(1) / (1,1,0): Differenced AR(1), or ARIMA(1,1,0), is a forecasting model that first removes the trend by differencing the data once and then predicts future values by modeling the relationship between consecutive changes using one autoregressive term.

Model Order	Description
ARIMA (1,1,1)	Uses both recent observations and recent forecast errors to forecast non-seasonal data with a trend.
Random Walk (0,1,0)	Assumes the future value will be similar to the latest observed value.
AR(1) (1,0,0)	Uses the most recent observation to predict the next value.
IMA(1,1) (0,1,1)	Uses recent forecast errors to improve predictions after removing the trend.
Differenced AR(1) (1,1,0)	Uses recent changes in the data to forecast future values after removing the trend.

• Seasonal ARIMA: Seasonal ARIMA (SARIMA) is a forecasting method that extends ARIMA by modeling both non-seasonal patterns (trend and short-term relationships) and repeating seasonal patterns, making it ideal for time series with regular cycles such as monthly, quarterly, or weekly data.

You can choose the model order:

• Airline Model (0,1,1,12): The Airline Model (0,1,1,12) is a seasonal forecasting model that removes both trend and yearly seasonality, then uses recent and seasonal forecast errors to predict future values, making it especially effective for monthly data with a repeating annual pattern.

• General SARIMA (1,1,1,12): The General SARIMA (1,1,1,12) is a seasonal forecasting model that removes both trend and yearly seasonality, then predicts future values using the non-seasonal autoregressive and moving average components together with seasonal autoregressive and seasonal moving average components.

• Seasonal AR (1,0,0,12): Seasonal AR(1,0,0,12) is a seasonal autoregressive forecasting model that predicts a value using the observation from one seasonal cycle earlier (such as the same month last year), with no seasonal differencing and no seasonal moving average component.

• Quarterly (1,0,1,4): Quarterly (1,0,1,4) is the seasonal component of a SARIMA model that uses one seasonal autoregressive term and one seasonal moving average term with a seasonal period of four quarters, making it suitable for quarterly data with a stable yearly seasonal pattern.

  Model Order	Description
  Airline Model (0,1,1,12)	Models yearly seasonality and trend using recent forecast errors.
  General SARIMA (1,1,1,12)	Models trend, recent observations, and yearly seasonal patterns.
  Seasonal AR (1,0,0,12)	Predicts future values using observations from the same season in previous years.
  Quarterly AR (1,0,1,4)	Models recurring quarterly patterns using previous quarters and seasonal forecast errors.

• Auto ARIMA: Automatically evaluates multiple ARIMA model orders and selects the best-fitting ARIMA model using statistical model selection criteria.

Refer the below table for a summary or when you have a question as Which algorithm to choose?"

Data Pattern	Recommended Algorithm
Stable data with no trend or seasonality	Simple Exponential Smoothing
Trend only	Holt's Linear Trend
Trend gradually weakens	Holt's Damped Trend
Trend with constant seasonal variation	Holt-Winters Additive
Trend with seasonal variation proportional to the level	Holt-Winters Multiplicative
Multiple seasonal patterns	MSTL
Non-seasonal data with autocorrelation	ARIMA
Seasonal data	SARIMA
Unsure which ETS model to use	Auto ETS
Unsure which ARIMA model to use	Auto ARIMA