CLOGLOG - Complementary Log-Log Transform

Computes the complementary log-log transformation, including its inverse.


CLOGLOG(X, Lo, Hi, Return)

is the real value(s) for which we compute the transformation: a single value or a one-dimensional array of cells (e.g., rows or columns)).
is the x-domain lower limit. If missing, Lo is assumed to be 0.
is the x-domain upper limit. If missing, Hi is assumed to be 1.
is a number that determines the type of return value: 1 (or missing) = C-Log-Log, 2 = Inverse C-Log-Log.
Return Description
1 or omitted C-Log-Log Transform.
2 The inverse of the C-Log-Log Transform.


  1. The X value(s) must be between Lo and Hi (exclusive).
  2. The complementary log-log link function is commonly used for parameters that lie in the unit interval.
  3. The standard complementary log-log transformation is defined as follows:
    $$y=\textit{CLogLog}(x)=ln{(-ln{(1-x)})}$$ And,
    • $x_{t}$ is the input value of the input time series at time $t$. X must be between 0 and 1, exclusive.
    • $y_{t}$ is the transformed complementary log-log value at time $t$.
    • $\textit{CLogLog}^{-1}(y)$ is the inverse complementary log-log function.
  4. To support a generic interval (a, b), we perform the following mapping:
    $$z=\frac{x - a}{b - a}$$ So, the revised transform function is expressed as follows:
    $$y=\ln{(-\ln{(\frac{b-x}{b-a})})}$$ The inverse transform is expressed as follows:
    $$-e^y=\ln{(\frac{b-x}{b-a})}\Rightarrow x=b-(b-a)e^{-e^y}$$
  5. The transform first-order derivative is calculated as follows:
  6. The transform first-order derivative (i.e., $\frac{dy}{dx}$ is positive for all x-values and goes to infinity as $x$ approaches the interval endpoints.
  7. In essence, the complementary log-log function converts a bounded x-range from (Lo, Hi) to $(-\infty, \infty)$.
    This figure demonstrates the x-domain mapping from the interval (-2, 2) to a full real-value domain using the complementary log-log function.
  8. Unlike Logit and Probit, the complementary log-log transform is asymmetrical.

Files Examples

Related Links


  • John H. Aldrich, Forrest D. Nelson; Linear Probability, Logit, and Probit Models; SAGE Publications, Inc; 1st Edition(Nov 01, 1984), ISBN: 0803921330.
  • Hastie, T. J.; Tibshirani, R. J. (1990). Generalized Additive Models. Chapman & Hall/CRC. ISBN 978-0-412-34390-2.


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