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Likelihood and Information Criteria

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Ivan Svetunkov edited this page Jun 16, 2026 · 7 revisions

Likelihood, Information Criteria et al.

This page documents methods for evaluating model fit, comparing models, and assessing forecast accuracy in smooth models.

Note: In Python, loglik, aic, aicc, bic, and bicc are available as properties. Other methods (pointLik, pAIC, pBIC, accuracy) are not yet implemented.

Likelihood Functions

logLik()

Extracts the log-likelihood value from the fitted model.

R Usage

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

# Get log-likelihood
ll <- logLik(model)
print(ll)

# Access the value
as.numeric(ll)

# Degrees of freedom
attr(ll, "df")

Python Usage

from smooth import ADAM

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

# Get log-likelihood
ll = model.loglik

Output

R: Returns an object of class "logLik" containing:

  • The log-likelihood value
  • Degrees of freedom (number of estimated parameters)
  • Number of observations

Python: Returns a float with the log-likelihood value.

pointLik()

Returns a vector of point log-likelihoods for each in-sample observation. Useful for identifying observations that contribute most to the likelihood.

Read the paper about this: DOI 10.1080/01605682.2026.2620516 or on github.

Python: Not yet implemented.

R Usage

library(greybox)  # Required for pointLik functions

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

# Point log-likelihoods
pll <- pointLik(model)

# Sum equals total log-likelihood
sum(pll)
logLik(model)

# Identify problematic observations
which.min(pll)  # Worst fitting observation

# Plot
plot(pll, type="h", main="Point Log-Likelihoods")
abline(h=0, col="red")

Use Cases

  • Identify outliers (observations with very low likelihood)
  • Diagnostic for model specification
  • Used in point information criteria (pAIC, pBIC etc)

Information Criteria

AIC, AICc, BIC, BICc

Standard information criteria for model selection.

R Usage

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

# Individual criteria
AIC(model)   # Akaike Information Criterion
AICc(model)  # Corrected AIC (for small samples)
BIC(model)   # Bayesian Information Criterion
BICc(model)  # Corrected BIC

Python Usage

from smooth import ADAM

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

# Individual criteria
model.aic    # Akaike Information Criterion
model.aicc   # Corrected AIC (for small samples)
model.bic    # Bayesian Information Criterion
model.bicc   # Corrected BIC

Output

R: Returns numeric values for each criterion.

Python: Each property returns a float.

Formulas

Criterion Formula Best For
AIC -2·logLik + 2·k General use
AICc AIC + 2k(k+1)/(n-k-1) Small samples (n/k < 40)
BIC -2·logLik + k·log(n) Large samples, parsimony
BICc BIC + k·(log(n)+1)·k/(n-k-1) Small samples, parsimony

Where k = number of parameters, n = number of observations.

Comparing Models

# Fit different models
model1 <- adam(AirPassengers, "ANN", lags=12)
model2 <- adam(AirPassengers, "AAN", lags=12)
model3 <- adam(AirPassengers, "AAA", lags=12)

# Compare using AICc
AICc(model1)
AICc(model2)
AICc(model3)

# Lower is better
models <- list(model1, model2, model3)
sapply(models, AICc)

Point Information Criteria

Point versions of information criteria based on pointLik(). These help identify observations that contribute most to model complexity.

Python: Not yet implemented.

R Usage

library(greybox)  # Required for point IC functions

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

# Point information criteria
pAIC(model)
pAICc(model)
pBIC(model)
pBICc(model)

These return vectors (one value per observation) rather than single values.

Accuracy

accuracy()

Computes various accuracy measures for the model's forecasts.

Python: Not yet implemented.

R Usage

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

# Compute accuracy measures
accuracy(model)

Measures Returned

See greybox::measured for the full list.

References

See Also

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