Coefficients and Parameters

Coefficients and Parameters

This page documents methods for extracting and analysing model parameters from smooth models.

coef() / coefficients()

Extracts the estimated parameters from the model.

R Usage

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

# Extract all coefficients
coef(model)

# Same as
coefficients(model)

Python Usage

from smooth import ADAM

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

# Extract all coefficients
coefficients = model.coef

Output

R: Returns a named vector containing:

• Smoothing parameters (alpha, beta, gammas, deltas)
• Damping parameter (phi) if applicable
• ARIMA parameters (AR, MA coefficients) if applicable
• Initial states (level, trend, seasonal, ARIMA states)
• Regression coefficients if applicable

Python: Returns an NDArray containing the estimated parameter vector (B).

Example

model <- adam(AirPassengers, "MAM", lags=12)
coef(model)
# Output:
#       alpha        beta       gamma       level      trend   seasonal1 ...
#   0.3521234   0.0012345   0.0001234 126.3456789   0.9987654   0.8765432 ...

confint()

Computes confidence intervals for the estimated parameters.

R Usage

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

# 95% confidence intervals (default)
confint(model)

# 99% confidence intervals
confint(model, level=0.99)

# Bootstrap-based confidence intervals
confint(model, bootstrap=TRUE, nsim=1000)

Parameters

Parameter	Type (R)	Type (Python)	Default	Description
object	adam	-	-	Fitted model
parm	character	-	NULL	Which parameters (NULL = all)
level	numeric	-	0.95	Confidence level
bootstrap	logical	-	FALSE	Use bootstrap for intervals (passed to vcov())
...	-	-	-	Additional parameters passed to vcov() if bootstrap=TRUE

Output

Returns a matrix with columns:

• S.E. - Standard error (if bootstrap=TRUE)
• Lower bound (e.g., "2.5%")
• Upper bound (e.g., "97.5%")

vcov()

Returns the variance-covariance matrix of the estimated parameters.

R Usage

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

# Get covariance matrix (Fisher Information based)
V <- vcov(model)

# Bootstrap-based covariance matrix
V <- vcov(model, bootstrap=TRUE, nsim=1000)

# Heuristic estimation (fast approximation)
V <- vcov(model, heuristics=0.1)

# Standard errors
sqrt(diag(V))

Parameters

Parameter	Type (R)	Type (Python)	Default	Description
object	adam	-	-	Fitted model
bootstrap	logical	-	FALSE	Use bootstrap (calls coefbootstrap())
heuristics	numeric	-	NULL	If set, variance equals abs(coef)*heuristics. Fast approximation
...	-	-	-	Additional parameters passed to coefbootstrap() if bootstrap=TRUE

Output

Returns a square matrix with:

• Rows/columns named by parameters
• Diagonal contains variances
• Off-diagonal contains covariances

coefbootstrap()

Generates bootstrap estimates of model coefficients for inference. Method is defined in greybox and extended for adam.

R Usage

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

# Bootstrap with default settings
bootCoef <- coefbootstrap(model)

# Bootstrap with custom settings
bootCoef <- coefbootstrap(model, nsim=500, method="dsr", parallel=TRUE)

# Examine distribution
hist(bootCoef$coefficients[,"alpha"])

# Bootstrap confidence intervals
apply(bootCoef$coefficients, 2, quantile, probs=c(0.025, 0.975))

# Extract the covariance matrix
bootCoef$vcov

Parameters

Parameter	Type (R)	Type (Python)	Default	Description
object	adam	-	-	Fitted model
nsim	integer	-	1000	Number of bootstrap samples
size	integer	-	floor(0.75*nobs(object))	Size of each bootstrap sample. Used in method="cr"
replace	logical	-	FALSE	Sample with replacement. Needed for method="cr".
prob	numeric vector	-	NULL	Probability weights for sampling. Used in method="cr"
parallel	logical/integer	-	FALSE	Use parallel processing. If integer, specifies number of cores
method	character	-	"cr"	Bootstrap method: "cr" (Case Resampling) or "dsr" (Data Shape Replication)
...	-	-	-	Additional parameters

Bootstrap Methods

Method	Description
"cr"	Case Resampling - resamples observations with varying sample sizes
"dsr"	Data Shape Replication - model free, creates data of the similar shape as the original series

Output

Returns an object of class "bootstrap" containing:

Element	Description
vcov	Covariance matrix of bootstrapped coefficients
coefficients	Matrix of bootstrapped coefficients (rows: samples, columns: parameters)
method	Bootstrap method used ("cr" or "dsr")
nsim	Number of simulations performed
size	Sample size used (NA for some methods)
replace	Whether replacement was used
prob	Probability weights used
parallel	Whether parallel processing was used
model	Model call
timeElapsed	Time elapsed for computation

See Also

• ADAM - Main ADAM function
• Fitted-Values-and-Forecasts - Fitted values, forecasts and simulation
• Refitting-and-Reforecasting - Parameter uncertainty analysis
• Visualisation-and-Output - Plotting forecasts