# CLOGLOG - Complementary Log-Log Transform

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

## Syntax

CLOGLOG(X, Return_type)
X
the real number for which we compute the transformation.
Return_type
is a number that determines the type of return value: 1 (or missing)= C-Log-Log , 2= Inverse C-Log-Log.
RETURN_TYPE NUMBER RETURNED
1 or omitted C-Log-Log Transform
2 Inverse of C-LOg-Log transform

## Remarks

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.

## Examples

Example 1:

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A B C D
January 10, 2008 0.66 0.64 0.66
January 11, 2008 0.02 -3.99 0.02
January 12, 2008 0.54 0.18 0.54
January 13, 2008 0.21 -1.34 0.21
January 14, 2008 0.73 1.02 0.73
January 15, 2008 0.37 -0.52 0.37
January 16, 2008 1.00 6.25 1.00
January 17, 2008 0.42 -0.32 0.42
January 18, 2008 0.99 5.27 0.99
January 19, 2008 0.04 -3.22 0.04
January 20, 2008 0.23 -1.20 0.23
January 21, 2008 0.31 -0.79 0.31
January 22, 2008 0.69 0.82 0.69
January 23, 2008 0.37 -0.54 0.37
January 24, 2008 0.78 1.28 0.78
January 25, 2008 0.30 -0.86 0.30
January 26, 2008 0.97 3.45 0.97
January 27, 2008 0.91 2.29 0.91
January 28, 2008 0.92 2.40 0.92
January 29, 2008 0.88 1.97 0.88
January 30, 2008 0.14 -1.78 0.14
January 31, 2008 0.06 -2.81 0.06
February 1, 2008 0.19 -1.42 0.19
February 2, 2008 0.61 0.46 0.61