Link Functions¶

Standard link functions.

Link functions map the mean μ to the linear predictor η = Xβ.

Standard links: - IdentityLink: g(μ) = μ (Gaussian) - LogLink: g(μ) = log(μ) (Poisson, Gamma, Negative Binomial) - LogitLink: g(μ) = log(μ/(1-μ)) (Binomial) - InverseLink: g(μ) = 1/μ (Gamma canonical) - CLogLogLink: g(μ) = log(-log(1-μ)) (Binomial alternative) - ProbitLink: g(μ) = Φ^{-1}(μ) (Binomial/Beta alternative)

Additional links (Phase 5.5): - SqrtLink: g(μ) = √μ (count data alternative) - PowerLink: g(μ) = μ^p (general power family) - InverseSquareLink: g(μ) = 1/μ² (Inverse Gaussian canonical)

class aurora.distributions.links.IdentityLink[source]¶

Bases: LinkFunction

Identity link function: g(mu) = mu.

The simplest link where the linear predictor equals the response directly. This is the canonical link for the Gaussian family.

Examples

>>> import numpy as np
>>> from aurora.distributions.links import IdentityLink
>>> link = IdentityLink()
>>> mu = np.array([1.0, 2.0, 3.0])
>>> eta = link.link(mu)
>>> np.allclose(eta, mu)
True
>>> mu_back = link.inverse(eta)
>>> np.allclose(mu_back, mu)
True

See also

LogLink: Log link for positive data.
GaussianFamily: Default family for identity link.

link(mu)[source]¶

Apply the link function g(mu) to map mean to linear predictor.

Parameters:: mu (array) – Mean parameter values.
Returns:: Linear predictor eta = g(mu).
Return type:: array

inverse(eta)[source]¶

Apply the inverse link g^{-1}(eta) to map linear predictor to mean.

Parameters:: eta (array) – Linear predictor values.
Returns:: Mean parameter mu = g^{-1}(eta).
Return type:: array

derivative(mu)[source]¶

Compute the derivative dg/dmu of the link function.

Parameters:: mu (array) – Mean parameter values at which to evaluate the derivative.
Returns:: Derivative dg/dmu evaluated at mu.
Return type:: array

class aurora.distributions.links.LogLink[source]¶

Bases: LinkFunction

Log link function: g(mu) = log(mu).

Ensures mu > 0. This is the canonical link for the Poisson family and is commonly used for any strictly positive response.

Notes

The inverse is g^{-1}(eta) = exp(eta). The linear predictor eta is clamped to [-700, 700] to prevent numerical overflow.

Examples

>>> import numpy as np
>>> from aurora.distributions.links import LogLink
>>> link = LogLink()
>>> mu = np.array([1.0, 2.0, 3.0])
>>> eta = link.link(mu)
>>> np.allclose(eta, np.log(mu))
True
>>> mu_back = link.inverse(eta)
>>> np.allclose(mu_back, mu)
True

See also

IdentityLink: Identity link.
PoissonFamily: Default family for log link.
GammaFamily: Often used with log link.

link(mu)[source]¶

Apply the link function g(mu) to map mean to linear predictor.

Parameters:: mu (array) – Mean parameter values.
Returns:: Linear predictor eta = g(mu).
Return type:: array

inverse(eta)[source]¶

Apply the inverse link g^{-1}(eta) to map linear predictor to mean.

Parameters:: eta (array) – Linear predictor values.
Returns:: Mean parameter mu = g^{-1}(eta).
Return type:: array

derivative(mu)[source]¶

Compute the derivative dg/dmu of the link function.

Parameters:: mu (array) – Mean parameter values at which to evaluate the derivative.
Returns:: Derivative dg/dmu evaluated at mu.
Return type:: array

class aurora.distributions.links.LogitLink[source]¶

Bases: LinkFunction

Logit link function: g(mu) = log(mu / (1 - mu)).

Maps probabilities in (0, 1) to the real line. This is the canonical link for the binomial family. The inverse is the logistic sigmoid: g^{-1}(eta) = 1 / (1 + exp(-eta)).

Examples

>>> import numpy as np
>>> from aurora.distributions.links import LogitLink
>>> link = LogitLink()
>>> mu = np.array([0.2, 0.5, 0.8])
>>> eta = link.link(mu)
>>> mu_back = link.inverse(eta)
>>> np.allclose(mu, mu_back)
True

See also

ProbitLink: Normal CDF inverse link.
CLogLogLink: Complementary log-log link.
BinomialFamily: Default family for logit link.
BetaFamily: Often used with logit link.

link(mu)[source]¶

Apply the link function g(mu) to map mean to linear predictor.

Parameters:: mu (array) – Mean parameter values.
Returns:: Linear predictor eta = g(mu).
Return type:: array

inverse(eta)[source]¶

Apply the inverse link g^{-1}(eta) to map linear predictor to mean.

Parameters:: eta (array) – Linear predictor values.
Returns:: Mean parameter mu = g^{-1}(eta).
Return type:: array

derivative(mu)[source]¶

Compute the derivative dg/dmu of the link function.

Parameters:: mu (array) – Mean parameter values at which to evaluate the derivative.
Returns:: Derivative dg/dmu evaluated at mu.
Return type:: array

class aurora.distributions.links.InverseLink[source]¶

Bases: LinkFunction

Inverse link function: g(mu) = 1 / mu.

Ensures mu > 0. This is the canonical link for the Gamma family. The inverse is g^{-1}(eta) = 1 / eta.

Examples

>>> import numpy as np
>>> from aurora.distributions.links import InverseLink
>>> link = InverseLink()
>>> mu = np.array([0.5, 1.0, 2.0])
>>> eta = link.link(mu)
>>> np.allclose(eta, 1.0 / mu)
True
>>> mu_back = link.inverse(eta)
>>> np.allclose(mu_back, mu)
True

See also

InverseSquareLink: Inverse square link for inverse Gaussian.
LogLink: Log link (alternative for Gamma).
GammaFamily: Default family for inverse link.

link(mu)[source]¶

Apply the link function g(mu) to map mean to linear predictor.

Parameters:: mu (array) – Mean parameter values.
Returns:: Linear predictor eta = g(mu).
Return type:: array

inverse(eta)[source]¶

Apply the inverse link g^{-1}(eta) to map linear predictor to mean.

Parameters:: eta (array) – Linear predictor values.
Returns:: Mean parameter mu = g^{-1}(eta).
Return type:: array

derivative(mu)[source]¶

Compute the derivative dg/dmu of the link function.

Parameters:: mu (array) – Mean parameter values at which to evaluate the derivative.
Returns:: Derivative dg/dmu evaluated at mu.
Return type:: array

class aurora.distributions.links.CLogLogLink[source]¶

Bases: LinkFunction

Complementary log-log link function: g(mu) = log(-log(1 - mu)).

Maps probabilities in (0, 1) to the real line. Asymmetric link that approaches 0 more slowly than 1. Commonly used for binomial regression when the probability of success increases slowly at low doses but rapidly at high doses.

The inverse is g^{-1}(eta) = 1 - exp(-exp(eta)).

Examples

>>> import numpy as np
>>> from aurora.distributions.links import CLogLogLink
>>> link = CLogLogLink()
>>> mu = np.array([0.2, 0.5, 0.8])
>>> eta = link.link(mu)
>>> mu_back = link.inverse(eta)
>>> np.allclose(mu, mu_back)
True

See also

LogitLink: Symmetric link for binomial.
ProbitLink: Normal CDF inverse link.
BinomialFamily: Distribution family using these links.

link(mu)[source]¶

Apply the link function g(mu) to map mean to linear predictor.

Parameters:: mu (array) – Mean parameter values.
Returns:: Linear predictor eta = g(mu).
Return type:: array

inverse(eta)[source]¶

Apply the inverse link g^{-1}(eta) to map linear predictor to mean.

Parameters:: eta (array) – Linear predictor values.
Returns:: Mean parameter mu = g^{-1}(eta).
Return type:: array

derivative(mu)[source]¶

Compute the derivative dg/dmu of the link function.

Parameters:: mu (array) – Mean parameter values at which to evaluate the derivative.
Returns:: Derivative dg/dmu evaluated at mu.
Return type:: array

class aurora.distributions.links.ProbitLink[source]¶

Bases: LinkFunction

Probit link g(mu) = Φ^{-1}(mu).

The probit link uses the inverse cumulative distribution function (CDF) of the standard normal distribution. Common alternative to logit for binomial and beta regression models.

Properties: - Lighter tails than logit - Assumes underlying normally distributed latent variable - Results similar to logit in practice for μ ∈ [0.2, 0.8]

Mathematical details: - Link: g(μ) = Φ^{-1}(μ) where Φ is the standard normal CDF - Inverse: μ = Φ(η) - Derivative: dg/dμ = 1/φ(Φ^{-1}(μ)) where φ is the normal PDF

Examples

>>> from aurora.distributions.links import ProbitLink
>>> link = ProbitLink()
>>> import numpy as np
>>> mu = np.array([0.1, 0.5, 0.9])
>>> eta = link.link(mu)  # Transform to linear predictor
>>> mu_back = link.inverse(eta)  # Should equal mu
>>> np.allclose(mu, mu_back)
True

Notes

The probit link is preferred when: - There is a theoretical latent normal process - Lighter tails than logit are desired - Compatibility with other software using probit (e.g., econometrics)

Comparison with logit: - Both are symmetric around 0.5 - Logit has heavier tails (more robust to outliers) - Probit: π × logit(μ) / √3 is a good approximation

References

Bliss, C. I. (1934). “The method of probits.” Science, 79, 38-39.
McCullagh, P., & Nelder, J. A. (1989). Generalized Linear Models.

link(mu)[source]¶: Transform probability to linear predictor: η = Φ^{-1}(μ).

inverse(eta)[source]¶

Transform linear predictor to probability: μ = Φ(η).

No clamping is applied: Φ(η) is evaluable over the whole float64 range (it saturates to 0 or 1 only around |η| ≈ 38). Families are responsible for clipping probabilities away from the boundaries.

derivative(mu)[source]¶

Compute derivative: dg/dμ = 1/φ(Φ^{-1}(μ)).

The derivative is the reciprocal of the normal PDF evaluated at the quantile corresponding to μ.

class aurora.distributions.links.SqrtLink[source]¶

Bases: LinkFunction

Square root link function: g(mu) = sqrt(mu).

Ensures mu > 0. Useful for count data where variance is proportional to the mean. Common alternative to the log link for Poisson-like data.

The inverse is g^{-1}(eta) = eta^2.

Examples

>>> import numpy as np
>>> from aurora.distributions.links import SqrtLink
>>> link = SqrtLink()
>>> mu = np.array([1.0, 4.0, 9.0])
>>> eta = link.link(mu)
>>> np.allclose(eta, np.sqrt(mu))
True
>>> mu_back = link.inverse(eta)
>>> np.allclose(mu_back, mu)
True

See also

LogLink: Log link for count data.
IdentityLink: Identity link.
PoissonFamily: Common family for count data.

link(mu)[source]¶

Apply the link function g(mu) to map mean to linear predictor.

Parameters:: mu (array) – Mean parameter values.
Returns:: Linear predictor eta = g(mu).
Return type:: array

inverse(eta)[source]¶

Apply the inverse link g^{-1}(eta) to map linear predictor to mean.

Parameters:: eta (array) – Linear predictor values.
Returns:: Mean parameter mu = g^{-1}(eta).
Return type:: array

derivative(mu)[source]¶

Compute the derivative dg/dmu of the link function.

Parameters:: mu (array) – Mean parameter values at which to evaluate the derivative.
Returns:: Derivative dg/dmu evaluated at mu.
Return type:: array

class aurora.distributions.links.PowerLink(power=1.0)[source]¶

Bases: LinkFunction

Power link function: g(mu) = mu^power.

General power transformation that generalizes several other link functions as special cases:

power = 1: Identity link
power = 0: Log link (limit as power approaches 0)
power = -1: Inverse link
power = 0.5: Square root link
power = -2: Inverse square link

Parameters:: power (float, default=1.0) – Power parameter for the transformation.

Notes

The Box-Cox transformation is a special case when properly normalized.

For power = 0, this class uses the log link as the limit.

Examples

>>> import numpy as np
>>> from aurora.distributions.links import PowerLink
>>> # Square root link via power
>>> link = PowerLink(power=0.5)
>>> mu = np.array([1.0, 4.0, 9.0])
>>> eta = link.link(mu)
>>> np.allclose(eta, np.sqrt(mu))
True
>>> # Inverse link via power
>>> link = PowerLink(power=-1)
>>> eta = link.link(mu)
>>> np.allclose(eta, 1.0 / mu)
True

See also

IdentityLink: Power link with power=1.
LogLink: Power link with power=0 (limit).
InverseLink: Power link with power=-1.
InverseSquareLink: Power link with power=-2.
TweedieFamily: Uses canonical power link.

__init__(power=1.0)[source]¶

Initialize power link.

Parameters:: power (float) – Power parameter.

link(mu)[source]¶

Apply the link function g(mu) to map mean to linear predictor.

Parameters:: mu (array) – Mean parameter values.
Returns:: Linear predictor eta = g(mu).
Return type:: array

inverse(eta)[source]¶

Apply the inverse link g^{-1}(eta) to map linear predictor to mean.

Parameters:: eta (array) – Linear predictor values.
Returns:: Mean parameter mu = g^{-1}(eta).
Return type:: array

derivative(mu)[source]¶

Compute the derivative dg/dmu of the link function.

Parameters:: mu (array) – Mean parameter values at which to evaluate the derivative.
Returns:: Derivative dg/dmu evaluated at mu.
Return type:: array

class aurora.distributions.links.InverseSquareLink[source]¶

Bases: LinkFunction

Inverse square link function: g(mu) = 1 / mu^2.

Ensures mu > 0. This is the canonical link for the inverse Gaussian distribution. The inverse is g^{-1}(eta) = 1 / sqrt(eta).

Examples

>>> import numpy as np
>>> from aurora.distributions.links import InverseSquareLink
>>> link = InverseSquareLink()
>>> mu = np.array([1.0, 2.0, 3.0])
>>> eta = link.link(mu)
>>> np.allclose(eta, 1.0 / mu**2)
True
>>> mu_back = link.inverse(eta)
>>> np.allclose(mu_back, mu)
True

See also

InverseLink: Inverse link g(mu) = 1/mu.
InverseGaussianFamily: Default family for inverse square link.
PowerLink: General power link (power=-2 is equivalent).

link(mu)[source]¶

Apply the link function g(mu) to map mean to linear predictor.

Parameters:: mu (array) – Mean parameter values.
Returns:: Linear predictor eta = g(mu).
Return type:: array

inverse(eta)[source]¶

Apply the inverse link g^{-1}(eta) to map linear predictor to mean.

Parameters:: eta (array) – Linear predictor values.
Returns:: Mean parameter mu = g^{-1}(eta).
Return type:: array

derivative(mu)[source]¶

Compute the derivative dg/dmu of the link function.

Parameters:: mu (array) – Mean parameter values at which to evaluate the derivative.
Returns:: Derivative dg/dmu evaluated at mu.
Return type:: array

class aurora.distributions.links.common.IdentityLink[source]¶

Bases: LinkFunction

Identity link function: g(mu) = mu.

The simplest link where the linear predictor equals the response directly. This is the canonical link for the Gaussian family.

Examples

>>> import numpy as np
>>> from aurora.distributions.links import IdentityLink
>>> link = IdentityLink()
>>> mu = np.array([1.0, 2.0, 3.0])
>>> eta = link.link(mu)
>>> np.allclose(eta, mu)
True
>>> mu_back = link.inverse(eta)
>>> np.allclose(mu_back, mu)
True

See also

LogLink: Log link for positive data.
GaussianFamily: Default family for identity link.

link(mu)[source]¶

Apply the link function g(mu) to map mean to linear predictor.

Parameters:: mu (array) – Mean parameter values.
Returns:: Linear predictor eta = g(mu).
Return type:: array

inverse(eta)[source]¶

Apply the inverse link g^{-1}(eta) to map linear predictor to mean.

Parameters:: eta (array) – Linear predictor values.
Returns:: Mean parameter mu = g^{-1}(eta).
Return type:: array

derivative(mu)[source]¶

Compute the derivative dg/dmu of the link function.

Parameters:: mu (array) – Mean parameter values at which to evaluate the derivative.
Returns:: Derivative dg/dmu evaluated at mu.
Return type:: array

class aurora.distributions.links.common.LogLink[source]¶

Bases: LinkFunction

Log link function: g(mu) = log(mu).

Ensures mu > 0. This is the canonical link for the Poisson family and is commonly used for any strictly positive response.

Notes

The inverse is g^{-1}(eta) = exp(eta). The linear predictor eta is clamped to [-700, 700] to prevent numerical overflow.

Examples

>>> import numpy as np
>>> from aurora.distributions.links import LogLink
>>> link = LogLink()
>>> mu = np.array([1.0, 2.0, 3.0])
>>> eta = link.link(mu)
>>> np.allclose(eta, np.log(mu))
True
>>> mu_back = link.inverse(eta)
>>> np.allclose(mu_back, mu)
True

See also

IdentityLink: Identity link.
PoissonFamily: Default family for log link.
GammaFamily: Often used with log link.

link(mu)[source]¶

Apply the link function g(mu) to map mean to linear predictor.

Parameters:: mu (array) – Mean parameter values.
Returns:: Linear predictor eta = g(mu).
Return type:: array

inverse(eta)[source]¶

Apply the inverse link g^{-1}(eta) to map linear predictor to mean.

Parameters:: eta (array) – Linear predictor values.
Returns:: Mean parameter mu = g^{-1}(eta).
Return type:: array

derivative(mu)[source]¶

Compute the derivative dg/dmu of the link function.

Parameters:: mu (array) – Mean parameter values at which to evaluate the derivative.
Returns:: Derivative dg/dmu evaluated at mu.
Return type:: array

class aurora.distributions.links.common.LogitLink[source]¶

Bases: LinkFunction

Logit link function: g(mu) = log(mu / (1 - mu)).

Maps probabilities in (0, 1) to the real line. This is the canonical link for the binomial family. The inverse is the logistic sigmoid: g^{-1}(eta) = 1 / (1 + exp(-eta)).

Examples

>>> import numpy as np
>>> from aurora.distributions.links import LogitLink
>>> link = LogitLink()
>>> mu = np.array([0.2, 0.5, 0.8])
>>> eta = link.link(mu)
>>> mu_back = link.inverse(eta)
>>> np.allclose(mu, mu_back)
True

See also

ProbitLink: Normal CDF inverse link.
CLogLogLink: Complementary log-log link.
BinomialFamily: Default family for logit link.
BetaFamily: Often used with logit link.

link(mu)[source]¶

Apply the link function g(mu) to map mean to linear predictor.

Parameters:: mu (array) – Mean parameter values.
Returns:: Linear predictor eta = g(mu).
Return type:: array

inverse(eta)[source]¶

Apply the inverse link g^{-1}(eta) to map linear predictor to mean.

Parameters:: eta (array) – Linear predictor values.
Returns:: Mean parameter mu = g^{-1}(eta).
Return type:: array

derivative(mu)[source]¶

Compute the derivative dg/dmu of the link function.

Parameters:: mu (array) – Mean parameter values at which to evaluate the derivative.
Returns:: Derivative dg/dmu evaluated at mu.
Return type:: array

class aurora.distributions.links.common.ProbitLink[source]¶

Bases: LinkFunction

Probit link g(mu) = Φ^{-1}(mu).

The probit link uses the inverse cumulative distribution function (CDF) of the standard normal distribution. Common alternative to logit for binomial and beta regression models.

Properties: - Lighter tails than logit - Assumes underlying normally distributed latent variable - Results similar to logit in practice for μ ∈ [0.2, 0.8]

Mathematical details: - Link: g(μ) = Φ^{-1}(μ) where Φ is the standard normal CDF - Inverse: μ = Φ(η) - Derivative: dg/dμ = 1/φ(Φ^{-1}(μ)) where φ is the normal PDF

Examples

>>> from aurora.distributions.links import ProbitLink
>>> link = ProbitLink()
>>> import numpy as np
>>> mu = np.array([0.1, 0.5, 0.9])
>>> eta = link.link(mu)  # Transform to linear predictor
>>> mu_back = link.inverse(eta)  # Should equal mu
>>> np.allclose(mu, mu_back)
True

Notes

The probit link is preferred when: - There is a theoretical latent normal process - Lighter tails than logit are desired - Compatibility with other software using probit (e.g., econometrics)

Comparison with logit: - Both are symmetric around 0.5 - Logit has heavier tails (more robust to outliers) - Probit: π × logit(μ) / √3 is a good approximation

References

Bliss, C. I. (1934). “The method of probits.” Science, 79, 38-39.
McCullagh, P., & Nelder, J. A. (1989). Generalized Linear Models.

link(mu)[source]¶: Transform probability to linear predictor: η = Φ^{-1}(μ).

inverse(eta)[source]¶

Transform linear predictor to probability: μ = Φ(η).

derivative(mu)[source]¶

Compute derivative: dg/dμ = 1/φ(Φ^{-1}(μ)).

The derivative is the reciprocal of the normal PDF evaluated at the quantile corresponding to μ.

class aurora.distributions.links.common.InverseLink[source]¶

Bases: LinkFunction

Inverse link function: g(mu) = 1 / mu.

Ensures mu > 0. This is the canonical link for the Gamma family. The inverse is g^{-1}(eta) = 1 / eta.

Examples

>>> import numpy as np
>>> from aurora.distributions.links import InverseLink
>>> link = InverseLink()
>>> mu = np.array([0.5, 1.0, 2.0])
>>> eta = link.link(mu)
>>> np.allclose(eta, 1.0 / mu)
True
>>> mu_back = link.inverse(eta)
>>> np.allclose(mu_back, mu)
True

See also

InverseSquareLink: Inverse square link for inverse Gaussian.
LogLink: Log link (alternative for Gamma).
GammaFamily: Default family for inverse link.

link(mu)[source]¶

Apply the link function g(mu) to map mean to linear predictor.

Parameters:: mu (array) – Mean parameter values.
Returns:: Linear predictor eta = g(mu).
Return type:: array

inverse(eta)[source]¶

Apply the inverse link g^{-1}(eta) to map linear predictor to mean.

Parameters:: eta (array) – Linear predictor values.
Returns:: Mean parameter mu = g^{-1}(eta).
Return type:: array

derivative(mu)[source]¶

Compute the derivative dg/dmu of the link function.

Parameters:: mu (array) – Mean parameter values at which to evaluate the derivative.
Returns:: Derivative dg/dmu evaluated at mu.
Return type:: array

class aurora.distributions.links.common.InverseSquareLink[source]¶

Bases: LinkFunction

Inverse square link function: g(mu) = 1 / mu^2.

Ensures mu > 0. This is the canonical link for the inverse Gaussian distribution. The inverse is g^{-1}(eta) = 1 / sqrt(eta).

Examples

>>> import numpy as np
>>> from aurora.distributions.links import InverseSquareLink
>>> link = InverseSquareLink()
>>> mu = np.array([1.0, 2.0, 3.0])
>>> eta = link.link(mu)
>>> np.allclose(eta, 1.0 / mu**2)
True
>>> mu_back = link.inverse(eta)
>>> np.allclose(mu_back, mu)
True

See also

InverseLink: Inverse link g(mu) = 1/mu.
InverseGaussianFamily: Default family for inverse square link.
PowerLink: General power link (power=-2 is equivalent).

link(mu)[source]¶

Apply the link function g(mu) to map mean to linear predictor.

Parameters:: mu (array) – Mean parameter values.
Returns:: Linear predictor eta = g(mu).
Return type:: array

inverse(eta)[source]¶

Apply the inverse link g^{-1}(eta) to map linear predictor to mean.

Parameters:: eta (array) – Linear predictor values.
Returns:: Mean parameter mu = g^{-1}(eta).
Return type:: array

derivative(mu)[source]¶

Compute the derivative dg/dmu of the link function.

Parameters:: mu (array) – Mean parameter values at which to evaluate the derivative.
Returns:: Derivative dg/dmu evaluated at mu.
Return type:: array

class aurora.distributions.links.common.CLogLogLink[source]¶

Bases: LinkFunction

Complementary log-log link function: g(mu) = log(-log(1 - mu)).

The inverse is g^{-1}(eta) = 1 - exp(-exp(eta)).

Examples

>>> import numpy as np
>>> from aurora.distributions.links import CLogLogLink
>>> link = CLogLogLink()
>>> mu = np.array([0.2, 0.5, 0.8])
>>> eta = link.link(mu)
>>> mu_back = link.inverse(eta)
>>> np.allclose(mu, mu_back)
True

See also

LogitLink: Symmetric link for binomial.
ProbitLink: Normal CDF inverse link.
BinomialFamily: Distribution family using these links.

link(mu)[source]¶

Apply the link function g(mu) to map mean to linear predictor.

Parameters:: mu (array) – Mean parameter values.
Returns:: Linear predictor eta = g(mu).
Return type:: array

inverse(eta)[source]¶

Apply the inverse link g^{-1}(eta) to map linear predictor to mean.

Parameters:: eta (array) – Linear predictor values.
Returns:: Mean parameter mu = g^{-1}(eta).
Return type:: array

derivative(mu)[source]¶

Compute the derivative dg/dmu of the link function.

Parameters:: mu (array) – Mean parameter values at which to evaluate the derivative.
Returns:: Derivative dg/dmu evaluated at mu.
Return type:: array

class aurora.distributions.links.common.SqrtLink[source]¶

Bases: LinkFunction

Square root link function: g(mu) = sqrt(mu).

Ensures mu > 0. Useful for count data where variance is proportional to the mean. Common alternative to the log link for Poisson-like data.

The inverse is g^{-1}(eta) = eta^2.

Examples

>>> import numpy as np
>>> from aurora.distributions.links import SqrtLink
>>> link = SqrtLink()
>>> mu = np.array([1.0, 4.0, 9.0])
>>> eta = link.link(mu)
>>> np.allclose(eta, np.sqrt(mu))
True
>>> mu_back = link.inverse(eta)
>>> np.allclose(mu_back, mu)
True

See also

LogLink: Log link for count data.
IdentityLink: Identity link.
PoissonFamily: Common family for count data.

link(mu)[source]¶

Apply the link function g(mu) to map mean to linear predictor.

Parameters:: mu (array) – Mean parameter values.
Returns:: Linear predictor eta = g(mu).
Return type:: array

inverse(eta)[source]¶

Apply the inverse link g^{-1}(eta) to map linear predictor to mean.

Parameters:: eta (array) – Linear predictor values.
Returns:: Mean parameter mu = g^{-1}(eta).
Return type:: array

derivative(mu)[source]¶

Compute the derivative dg/dmu of the link function.

Parameters:: mu (array) – Mean parameter values at which to evaluate the derivative.
Returns:: Derivative dg/dmu evaluated at mu.
Return type:: array

class aurora.distributions.links.common.PowerLink(power=1.0)[source]¶

Bases: LinkFunction

Power link function: g(mu) = mu^power.

General power transformation that generalizes several other link functions as special cases:

power = 1: Identity link
power = 0: Log link (limit as power approaches 0)
power = -1: Inverse link
power = 0.5: Square root link
power = -2: Inverse square link

Parameters:: power (float, default=1.0) – Power parameter for the transformation.

Notes

The Box-Cox transformation is a special case when properly normalized.

For power = 0, this class uses the log link as the limit.

Examples

>>> import numpy as np
>>> from aurora.distributions.links import PowerLink
>>> # Square root link via power
>>> link = PowerLink(power=0.5)
>>> mu = np.array([1.0, 4.0, 9.0])
>>> eta = link.link(mu)
>>> np.allclose(eta, np.sqrt(mu))
True
>>> # Inverse link via power
>>> link = PowerLink(power=-1)
>>> eta = link.link(mu)
>>> np.allclose(eta, 1.0 / mu)
True

See also

IdentityLink: Power link with power=1.
LogLink: Power link with power=0 (limit).
InverseLink: Power link with power=-1.
InverseSquareLink: Power link with power=-2.
TweedieFamily: Uses canonical power link.

__init__(power=1.0)[source]¶

Initialize power link.

Parameters:: power (float) – Power parameter.

link(mu)[source]¶

Apply the link function g(mu) to map mean to linear predictor.

Parameters:: mu (array) – Mean parameter values.
Returns:: Linear predictor eta = g(mu).
Return type:: array

inverse(eta)[source]¶

Apply the inverse link g^{-1}(eta) to map linear predictor to mean.

Parameters:: eta (array) – Linear predictor values.
Returns:: Mean parameter mu = g^{-1}(eta).
Return type:: array

derivative(mu)[source]¶

Compute the derivative dg/dmu of the link function.

Parameters:: mu (array) – Mean parameter values at which to evaluate the derivative.
Returns:: Derivative dg/dmu evaluated at mu.
Return type:: array

Base Class¶

class aurora.distributions.base.LinkFunction[source]¶

Bases: ABC

Abstract base class for GLM link functions.

A link function g() relates the mean mu to the linear predictor eta:: g(mu) = eta = X^T beta

Subclasses must implement the forward link, its inverse, and its derivative. The link must be monotonic and differentiable.

See also

aurora.distributions.links.LogitLink: Canonical link for binomial/beta
aurora.distributions.links.LogLink: Canonical link for Poisson/Gamma
aurora.distributions.links.IdentityLink: Canonical link for Gaussian

Examples

>>> from aurora.distributions.links import LogitLink
>>> link = LogitLink()
>>> mu = np.array([0.2, 0.5, 0.8])
>>> eta = link.link(mu)      # logit(mu)
>>> link.inverse(eta)        # recovers mu
>>> link.derivative(mu)      # d(mu)/d(eta)

abstractmethod link(mu)[source]¶

Apply the link function g(mu) to map mean to linear predictor.

Parameters:: mu (array) – Mean parameter values.
Returns:: Linear predictor eta = g(mu).
Return type:: array

abstractmethod inverse(eta)[source]¶

Apply the inverse link g^{-1}(eta) to map linear predictor to mean.

Parameters:: eta (array) – Linear predictor values.
Returns:: Mean parameter mu = g^{-1}(eta).
Return type:: array

abstractmethod derivative(mu)[source]¶

Compute the derivative dg/dmu of the link function.

Parameters:: mu (array) – Mean parameter values at which to evaluate the derivative.
Returns:: Derivative dg/dmu evaluated at mu.
Return type:: array