# CLOGLOG - Complementary Log-Log Transform

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

## Syntax

CLOGLOG(X, Lo, Hi, Return)
X
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)).
Lo
is the x-domain lower limit. If missing, Lo is assumed to be 0.
Hi
is the x-domain upper limit. If missing, Hi is assumed to be 1.
Return
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

## Remarks

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,
$$x=\textit{CLogLog}^{-1}(y)=1-e^{-e^y}$$
Where:
• $x_{t}$ 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:
$$\frac{dy}{dx}=\frac{-1}{(b-a)\ln{\frac{b-x}{b-a}}}$$
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 $(-\infty, \infty)$.
8. Unlike Logit and Probit, the complementary log-log transform is asymmetrical.

## Examples

Example 1:

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
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