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Generalized Linear Models (GLM)

GLM models for binary classification, count data, and other non-normal response distributions.

Penalized IRLS

All GLM expressions (logistic, poisson, negative_binomial, tweedie, probit, cloglog) accept a lambda_ kwarg (default 0.0) that adds an L2 (ridge) penalty on the coefficients inside the IRLS update. Set lambda_ > 0 to stabilize estimation under collinearity or quasi-separation.

The sklearn-style logistic_regression exposes the same penalty via C = 1 / lambda_ and an explicit penalty choice.

logistic

Logistic regression for binary classification.

ps.logistic(
    y: Union[pl.Expr, str],  # Binary (0/1)
    *x: Union[pl.Expr, str],
    lambda_: float = 0.0,        # L2 (Ridge) regularization strength
    with_intercept: bool = True,
) -> pl.Expr

Returns: See GLM Output

Example: 

df.group_by("group").agg(ps.logistic("success", "x1", "x2").alias("model"))

logistic_regression

Sklearn-style logistic regression. Distinct from logistic: uses inverse-strength regularization C = 1 / lambda_ and an explicit penalty choice. Returns the same GLM Output schema.

ps.logistic_regression(
    y: Union[pl.Expr, str],      # Binary (0/1)
    *x: Union[pl.Expr, str],
    penalty: str = "l2",          # "l2" or "none"
    C: float = 1.0,               # Inverse of regularization strength
    threshold: float = 0.5,
    max_iter: int = 100,
    tol: float = 1e-8,
    with_intercept: bool = True,
) -> pl.Expr

Returns: See GLM Output

Example: 

df.group_by("group").agg(
    ps.logistic_regression("success", "x1", "x2", penalty="l2", C=0.5).alias("model")
)

poisson

Poisson regression for count data.

ps.poisson(
    y: Union[pl.Expr, str],  # Non-negative counts
    *x: Union[pl.Expr, str],
    lambda_: float = 0.0,        # L2 (Ridge) regularization strength
    with_intercept: bool = True,
) -> pl.Expr

Returns: See GLM Output

negative_binomial

Negative Binomial regression for overdispersed count data.

ps.negative_binomial(
    y: Union[pl.Expr, str],
    *x: Union[pl.Expr, str],
    theta: float | None = None,  # Dispersion; None = estimate
    lambda_: float = 0.0,        # L2 (Ridge) regularization strength
    with_intercept: bool = True,
) -> pl.Expr

Returns: See GLM Output

tweedie

Tweedie GLM for flexible variance structures.

ps.tweedie(
    y: Union[pl.Expr, str],
    *x: Union[pl.Expr, str],
    var_power: float = 1.5,      # 0=Gaussian, 1=Poisson, 2=Gamma, 3=InvGaussian
    lambda_: float = 0.0,        # L2 (Ridge) regularization strength
    with_intercept: bool = True,
) -> pl.Expr

Returns: See GLM Output

Variance Power Interpretation: | var_power | Distribution | |-----------|--------------| | 0 | Gaussian (Normal) | | 1 | Poisson | | (1, 2) | Compound Poisson-Gamma | | 2 | Gamma | | 3 | Inverse Gaussian |

probit

Probit regression for binary classification.

ps.probit(
    y: Union[pl.Expr, str],  # Binary (0/1)
    *x: Union[pl.Expr, str],
    lambda_: float = 0.0,        # L2 (Ridge) regularization strength
    with_intercept: bool = True,
) -> pl.Expr

Returns: See GLM Output

cloglog

Complementary log-log regression for binary classification.

ps.cloglog(
    y: Union[pl.Expr, str],  # Binary (0/1)
    *x: Union[pl.Expr, str],
    lambda_: float = 0.0,        # L2 (Ridge) regularization strength
    with_intercept: bool = True,
) -> pl.Expr

Returns: See GLM Output

Regularization

All GLM models support L2 (Ridge) regularization via the lambda_ parameter:

# Unregularized logistic regression
ps.logistic("y", "x1", "x2")

# Ridge-regularized logistic regression
ps.logistic("y", "x1", "x2", lambda_=1.0)

Regularization helps with: - Preventing overfitting - Stabilizing estimation when predictors are correlated - Handling quasi-separation in binary response models

See Also

  • Linear Models - Standard linear regression
  • ALM - Augmented Linear Model with 24+ distributions
  • Diagnostics - Quasi-separation detection