# algorithms.statistics.models.family.links

## Module: `algorithms.statistics.models.family.links`

Inheritance diagram for `nipy.algorithms.statistics.models.family.links`:

## Classes

### `CDFLink`

class nipy.algorithms.statistics.models.family.links.CDFLink(dbn=<scipy.stats._continuous_distns.norm_gen object>)

  Bases: `Logit`

  The use the CDF of a scipy.stats distribution as a link function:

  g(x) = dbn.ppf(x)

  __init__(dbn=<scipy.stats._continuous_distns.norm_gen object>)
  clean(p)

    Clip logistic values to range (tol, 1-tol)

    INPUTS:

      p – probabilities

    OUTPUTS: pclip

      pclip – clipped probabilities

  deriv(p)

    Derivative of CDF link

    g(p) = 1/self.dbn.pdf(self.dbn.ppf(p))

    INPUTS:

      x – mean parameters

    OUTPUTS: z

      z – derivative of CDF transform of x

  initialize(Y)
  inverse(z)

    Derivative of CDF link

    g(z) = self.dbn.cdf(z)

    INPUTS:

      z – linear predictors in GLM

    OUTPUTS: p

      p – inverse of CDF link of z

  tol = 1e-10

### `CLogLog`

class nipy.algorithms.statistics.models.family.links.CLogLog

  Bases: `Logit`

  The complementary log-log transform as a link function:

  g(x) = log(-log(x))

  __init__(*args, **kwargs)
  clean(p)

    Clip logistic values to range (tol, 1-tol)

    INPUTS:

      p – probabilities

    OUTPUTS: pclip

      pclip – clipped probabilities

  deriv(p)

    Derivatve of C-Log-Log transform

    g(p) = - 1 / (log(p) * p)

    INPUTS:

      p – mean parameters

    OUTPUTS: z

      z – - 1 / (log(p) * p)

  initialize(Y)
  inverse(z)

    Inverse of C-Log-Log transform

    g(z) = exp(-exp(z))

    INPUTS:

      z – linear predictor scale

    OUTPUTS: p

      p – mean parameters

  tol = 1e-10

### `Link`

class nipy.algorithms.statistics.models.family.links.Link

  Bases: `object`

  A generic link function for one-parameter exponential family, with call, inverse and deriv methods.

  __init__(*args, **kwargs)
  deriv(p)
  initialize(Y)
  inverse(z)

### `Log`

class nipy.algorithms.statistics.models.family.links.Log

  Bases: `Link`

  The log transform as a link function:

  g(x) = log(x)

  __init__(*args, **kwargs)
  clean(x)
  deriv(x)

    Derivative of log transform

    g(x) = 1/x

    INPUTS:

      x – mean parameters

    OUTPUTS: z

      z – derivative of log transform of x

  initialize(Y)
  inverse(z)

    Inverse of log transform

    g(x) = exp(x)

    INPUTS:

      z – linear predictors in GLM

    OUTPUTS: x

      x – exp(z)

  tol = 1e-10

### `Logit`

class nipy.algorithms.statistics.models.family.links.Logit

  Bases: `Link`

  The logit transform as a link function:

  g’(x) = 1 / (x * (1 - x)) g^(-1)(x) = exp(x)/(1 + exp(x))

  __init__(*args, **kwargs)
  clean(p)

    Clip logistic values to range (tol, 1-tol)

    INPUTS:

      p – probabilities

    OUTPUTS: pclip

      pclip – clipped probabilities

  deriv(p)

    Derivative of logit transform

    g(p) = 1 / (p * (1 - p))

    INPUTS:

      p – probabilities

    OUTPUTS: y

      y – derivative of logit transform of p

  initialize(Y)
  inverse(z)

    Inverse logit transform

    h(z) = exp(z)/(1+exp(z))

    INPUTS:

      z – logit transform of p

    OUTPUTS: p

      p – probabilities

  tol = 1e-10

### `Power`

class nipy.algorithms.statistics.models.family.links.Power(power=1.0)

  Bases: `Link`

  The power transform as a link function:

  g(x) = x**power

  __init__(power=1.0)
  deriv(x)

    Derivative of power transform

    g(x) = self.power * x**(self.power - 1)

    INPUTS:

      x – mean parameters

    OUTPUTS: z

      z – derivative of power transform of x

  initialize(Y)
  inverse(z)

    Inverse of power transform

    g(x) = x**(1/self.power)

    INPUTS:

      z – linear predictors in GLM

    OUTPUTS: x

      x – mean parameters