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Coefficients and Parameters

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Ivan Svetunkov edited this page May 21, 2026 · 11 revisions

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

# Matching parameter names (coef is a bare NDArray)
model.coef_names    # e.g. ['alpha', 'beta', 'gamma', 'level', 'trend', 'seasonal_1', ...]

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). The matching labels (in the same order) are available via model.coef_names, which mirror R's naming (alpha, beta, gamma/gamma1, phi, level, trend, seasonal_1…, ARIMA phi1[1]/theta1[1], regressor names, constant).

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)

Python Usage

from smooth import ADAM

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

# 95% confidence intervals (default) — returns a pandas DataFrame
model.confint()

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

# Only selected parameters
model.confint(parm=["alpha", "beta"])

Note: The Python confint() reproduces R's interval construction, including the clamping of bounds to the admissible region for ETS smoothing parameters (alpha, beta, gamma, phi) and ARIMA AR/MA coefficients. Bootstrap intervals (bootstrap=TRUE) are R-only.

Parameters

Parameter Type (R) Type (Python) Default Description
object / model adam ADAM (self) - Fitted model
parm character list / str / None NULL / None Which parameters (NULL/None = all)
level numeric float 0.95 Confidence level
bootstrap logical (R only) FALSE Use bootstrap for intervals (passed to vcov())
step_size - float / None - / None Finite-difference step for the Fisher Information
... - - - Additional parameters passed to vcov() if bootstrap=TRUE (R)

Output

  • R: a matrix with columns S.E. (if bootstrap=TRUE), lower bound (e.g. 2.5%) and upper bound (e.g. 97.5%).
  • Python: a pandas.DataFrame indexed by the parameter names (model.coef_names) with columns ["S.E.", "<lo>%", "<hi>%"] (e.g. 2.5%, 97.5%).

Python occurrence models: confint() is not yet available for OM / OMG (the cumulative-logistic occurrence distribution is unsupported in the Fisher Information path) — see OM.

vcov()

Returns the variance-covariance matrix of the estimated parameters. It is obtained by inverting the observed Fisher Information matrix (the negative Hessian of the log-likelihood at the optimum).

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))

Python Usage

import numpy as np
from smooth import ADAM

model = ADAM(model="AAN")
model.fit(y)

# Covariance matrix (Fisher Information based) — pandas DataFrame
V = model.vcov()

# Heuristic estimation (fast diagonal approximation)
V = model.vcov(heuristics=0.1)

# Standard errors
np.sqrt(np.diag(V))

Note: When the model was fitted with fi=True, the cached Fisher Information is reused; otherwise it is computed on demand. Bootstrap covariance (bootstrap=True) is not implemented in Python and raises NotImplementedError.

Parameters

Parameter Type (R) Type (Python) Default Description
object / model adam ADAM (self) - Fitted model
bootstrap logical bool (raises if True) FALSE / False Use bootstrap (R: calls coefbootstrap())
heuristics numeric float / None NULL / None If set, variance equals abs(coef)*heuristics. Fast approximation
step_size - float / None - / None Finite-difference step for the Fisher Information
... - - - Additional parameters passed to coefbootstrap() if bootstrap=TRUE (R)

Output

Returns a square matrix (R) / pandas.DataFrame (Python) with:

  • Rows/columns named by parameters (Python: model.coef_names)
  • Diagonal contains variances
  • Off-diagonal contains covariances

Python occurrence models: vcov() is not yet available for OM / OMG — see OM.

coefbootstrap()

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

Python: Not yet implemented.

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

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