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 (no intervals by default, matching R)
forecasts = model.predict(h=12)

# With prediction intervals (auto-selects method)
forecasts = model.predict(h=12, interval="prediction", level=0.95)

# Approximate (parametric) intervals
forecasts = model.predict(h=12, interval="approximate", level=0.95)

# Simulation-based intervals
forecasts = model.predict(
    h=12, interval="simulated",
    level=0.95, nsim=1000
)

# Multiple confidence levels at once
forecasts = model.predict(h=12, interval="prediction", level=[0.8, 0.95, 0.99])

# One-sided intervals
forecasts = model.predict(h=12, interval="prediction", level=0.95, side="upper")
forecasts = model.predict(h=12, interval="prediction", level=0.95, side="lower")

# Convenience method for multi-level intervals
intervals = model.predict_intervals(h=12, levels=[0.8, 0.95])

Parameters

Parameter	Type (R)	Type (Python)	Default (R / Python)	Description
object	adam	ADAM	-	Fitted model
h	integer	int	10 / required	Forecast horizon
newdata / X	matrix/data.frame	NDArray	NULL / None	Exogenous variables for forecast period
occurrence	numeric vector	NDArray	NULL / None	Future occurrence probabilities (values in [0,1]) if known
interval	character	str	"none" / "none"	Interval type (see table below)
level	numeric	float or list	0.95 / 0.95	Confidence level(s); see Level parameter below
side	character	str	"both" / "both"	Interval side: "both", "upper", or "lower"; see Side parameter below
cumulative	logical	bool	FALSE / False	Produce cumulative forecasts (useful for inventory control)
nsim	integer	int	NULL / 10000	Number of simulations for simulation-based intervals
scenarios	logical	bool	FALSE / 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	Yes
"approximate"	Analytical formulae for conditional variance	Fast, additive models	Yes	Yes
"simulated"	Monte Carlo simulation	Multiplicative 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	-
"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.

Level parameter

The level parameter specifies the confidence level(s) for prediction intervals. It can be a single value or (in both R and Python) a vector/list of values for producing multiple interval bands simultaneously.

Values above 1 are treated as percentages and automatically divided by 100 (e.g. level=95 becomes level=0.95).

Single level (scalar):

Language	Code	Output columns
R	forecast(model, h=12, level=0.95)	$lower and $upper (single-column matrices)
Python	model.predict(h=12, interval="prediction", level=0.95)	"lower_0.025", "upper_0.975"

Multiple levels (vector/list):

Language	Code	Output columns
R	forecast(model, h=12, level=c(0.8, 0.95, 0.99))	$lower and $upper (3-column matrices)
Python	model.predict(h=12, interval="prediction", level=[0.8, 0.95, 0.99])	"lower_0.1", "lower_0.025", "lower_0.005", "upper_0.9", "upper_0.975", "upper_0.995"

The Python column naming convention uses the quantile value: for side="both" and level=X, the lower column is lower_{(1-X)/2} and the upper column is upper_{(1+X)/2}.

Side parameter

The side parameter controls which bounds of the prediction interval are returned.

Value	Description	Python output columns (level=0.9)
"both"	Both lower and upper bounds (default)	"lower_0.05", "upper_0.95"
"upper"	Upper bound only	"upper_0.9"
"lower"	Lower bound only	"lower_0.1"

When side="upper", the upper quantile equals the level directly (e.g. 90th percentile for level=0.9). When side="lower", the lower quantile equals 1 - level (e.g. 10th percentile for level=0.9). Both side options work with single or multiple levels.

Output

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

Python: predict() returns a pandas DataFrame with a "mean" column and optional interval columns (e.g., "lower_0.025", "upper_0.975" for level=0.95) when interval is not "none". Multiple levels produce multiple column pairs.

Element	Description	R	Python
model	The estimated model object	Yes	-
mean	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	Yes
interval	Whether intervals were produced (logical)	Yes	-
scenarios	Matrix of simulated scenarios (h × nsim)	Yes	Yes

Notes:

• In R, access elements via $ (e.g., fc$mean, fc$lower)
• In both R and Python, scenarios is only populated when interval="simulated" and scenarios=TRUE/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