Fitted Values and Forecasts

Fitted Values and Forecasts

This page documents methods for extracting fitted values and generating forecasts from smooth models.

predict()/forecast()

Generates point forecasts and optionally prediction intervals for h steps ahead.

In R, the standard method is forecast(), although predict() is also available, but with a limited functionality and focusing on the in-sample prediction.

In Python, use predict() method.

R Usage

model <- adam(AirPassengers, "MMM", lags=12)

# Out-of-sample prediction
pred <- forecast(model, h=12, interval="prediction")

# In-sample confidence intervals
pred <- predict(model, interval="confidence")

Python Usage

from smooth import ADAM

# Fit model
model = ADAM(model="ZZZ", lags=12)
model.fit(y)

# Basic point forecast
forecasts = model.predict(h=12)

# With prediction intervals (parametric)
forecasts = model.predict(
    h=12,
    calculate_intervals=True,
    interval_method='parametric',
    level=0.95
)

# Simulation-based intervals
forecasts = model.predict(
    h=12,
    calculate_intervals=True,
    interval_method='simulation',
    level=[0.8, 0.95],
    nsim=1000
)

# One-sided intervals
forecasts = model.predict(
    h=12,
    calculate_intervals=True,
    interval_method='parametric',
    level=0.95,
    side='upper'
)

# Access prediction intervals after calling predict()
intervals = model.predict_intervals(h=12, levels=[0.8, 0.95])

Parameters

Parameter	Type (R)	Type (Python)	Default	Description
object	adam	ADAM	-	Fitted model
h	integer	int	10	Forecast horizon
newdata	matrix/data.frame	-	NULL	New explanatory variables for forecasting
X	-	NDArray	None	Exogenous variables for forecast period
occurrence	numeric vector	-	NULL	Future occurrence probabilities (values in [0,1]) if known
calculate_intervals	-	bool	True	Whether to calculate prediction intervals
interval	character	-	"none"	Interval type (see table below)
interval_method	-	str	"parametric"	Method: "parametric", "simulation", or "bootstrap"
level	numeric	float/list	0.95	Confidence level(s)
side	character	str	"both"	Interval side: "both", "upper", "lower"
cumulative	logical	-	FALSE	Produce cumulative forecasts (useful for inventory control)
nsim	integer	int	NULL	Number of simulations.
scenarios	logical	-	FALSE	Return simulated scenarios (only for interval="simulated")
...	-	-	-	Other parameters passed to underlying functions

Interval Types

Type	Description	Best For	R	Python
"none"	No intervals	Point forecasts only	Yes	Yes
"prediction"	Analytical for additive, simulated for others	General use (recommended)	Yes	-
"parametric" / "approximate"	Analytical formulae for conditional variance	Fast, additive models	Yes	Yes
"semiparametric"	Based on multi-step forecast errors and assumed distribution	When residuals are not i.i.d.	Yes	-
"nonparametric"	Quantile regression (Taylor & Bunn, 1999)	Distribution-free	Yes	-
"empirical"	Empirical quantiles of multi-step forecast errors	Distribution and model free	Yes	-
"simulated" / "simulation"	Monte Carlo simulation	Multiplicative models	Yes	Yes
"bootstrap"	Bootstrap-based intervals	Parameter uncertainty	-	Yes
"confidence"	Confidence intervals for point forecast via reforecast()	Uncertainty in the forecast line	Yes	-
"complete"	Accounts for parameter uncertainty via reforecast()	Full uncertainty quantification	Yes	-

Note: "semiparametric", "nonparametric" and "empirical" use the rmultistep() method to extract multistep in-sample forecast errors. See more in Residuals-and-Errors.

Output

R: Returns an object of class "adam.forecast" / "smooth.forecast".

Python: predict() returns a NumPy array of point forecasts. Prediction intervals are stored in model.forecast_results and can be accessed via predict_intervals().

Element	Description	R	Python
model	The estimated model object	Yes	-
method	Name of the model (e.g., "ETS(M,M,M)")	Yes	-
mean / forecast	Point forecasts (conditional mean)	Yes	Yes
lower	Lower prediction bounds (matrix with columns per level)	Yes	Yes
upper	Upper prediction bounds (matrix with columns per level)	Yes	Yes
level	Confidence level(s) used	Yes	Yes
side	Which side of intervals was requested	-	Yes
interval	Whether intervals were produced (logical)	Yes	-
scenarios	Matrix of simulated scenarios (h × nsim)	Yes	-

Notes:

• In R, access elements via $ (e.g., fc$mean, fc$lower)
• In Python, predict() returns point forecasts directly; use predict_intervals() to get a dictionary with forecast, lower, upper keys
• In R, scenarios is only populated when interval="simulated" and scenarios=TRUE

fitted()

Extracts fitted (in-sample predicted) values from the model.

R Usage

model <- adam(AirPassengers, "MMM", lags=12)

# Get fitted values
fitted(model)

Python Usage

from smooth import ADAM

model = ADAM(model="MMM", lags=12)
model.fit(y)

# Get fitted values
fitted_values = model.fitted

Output

R: Returns a time series with fitted values for each in-sample observation.

Python: Returns an NDArray with fitted values for each in-sample observation.

actuals()

Extracts the actual (observed) values used in model estimation.

R Usage

model <- adam(AirPassengers, "MMM", lags=12, h=12, holdout=TRUE)

# Get actuals
y <- actuals(model)

Python Usage

from smooth import ADAM

model = ADAM(model="MMM", lags=12)
model.fit(y)

# Get actuals (the data used for fitting)
actuals = model.actuals

Output

R: Returns a time series with the actual in-sample observations.

Python: Returns an NDArray with the actual in-sample observations.

See Also

• ADAM - Main ADAM function
• Coefficients-and-Parameters - Extracting model parameters
• Simulation-Functions - simulate() and sim.* functions
• Refitting-and-Reforecasting - Parameter uncertainty analysis
• Visualisation-and-Output - Plotting forecasts