CLOGLOG - Complementary Log-Log Transform

1. The complementary log-log link function is commonly used for parameters that lie in the unit interval.
2. The Complementary log-log transformation is defined as follows:
   
   $$y=\textit{CLogLog}(x)=\ln\left( -\ln \left ( 1-x \right) \right) $$ And
   $$x=\textit{CLogLog}^{-1}(y)=1 - e^{-e^{y}} $$
   Where:
   
   • $x_{t}$ is the value of the input time series at time $t$
   • $y_{t}$ is the transformed complementary log-log value at time $t$
   • $\textit{ClogLog}^{-1}(y)$ is the inverse complementary log-log function
   • $\left(x_t +\alpha \right) \gt 0$ for all t values
3. The BOXCOX function accepts a single value or an array of values for X.
4. The shift parameter must be large enough to make all the values of X positive.

Example 1:

Files Examples

Related Links

• Wikipedia - Generalized linear model

References

• John H. Aldrich, Forrest D. Nelson; Linear Probability, Logit, and Probit Models; SAGE Publications, Inc; 1st Edition(Nov 01, 1984), ISBN: 0803921330
• Hamilton, J .D.; Time Series Analysis , Princeton University Press (1994), ISBN 0-691-04289-6
• Tsay, Ruey S.; Analysis of Financial Time Series John Wiley & SONS. (2005), ISBN 0-471-690740