we love the normal form!
bcz of that,in most cases we try to make them act like normal.its not classifiers point of view,but its feature extraction view!
Which transformation?
The main criterion in choosing a transformation is: what works with the
data? As examples above indicate, it is important to consider as well two
questions.
What makes physical (biological, economic, whatever) sense, for example
in terms of limiting behaviour as values get very small or very large?
This question often leads to the use of logarithms.
Can we keep dimensions and units simple and convenient? If possible, we
prefer measurement scales that are easy to think about. The cube root of
a volume and the square root of an area both have the dimensions of
length, so far from complicating matters, such transformations may
simplify them. Reciprocals usually have simple units, as mentioned
earlier. Often, however, somewhat complicated units are a sacrifice that
has to be made.
When use what?
The most useful transformations in introductory data analysis are the reciprocal, logarithm, cube root, square root, and square. In what follows, even when it is not emphasised, it is supposed that transformations are used only over ranges on which they yield (finite) real numbers as results.
**Reciprocal**
The reciprocal, x to 1/x, with its sibling the negative reciprocal, x to
-1/x, is a very strong transformation with a drastic effect on
distribution shape. It can not be applied to zero values. Although it
can be applied to negative values, it is not useful unless all values are
positive. The reciprocal of a ratio may often be interpreted as easily as
the ratio itself: e.g.
population density (people per unit area) becomes area per person;
persons per doctor becomes doctors per person;
rates of erosion become time to erode a unit depth.
(In practice, we might want to multiply or divide the results of taking
the reciprocal by some constant, such as 1000 or 10000, to get numbers
that are easy to manage, but that itself has no effect on skewness or
linearity.)
The reciprocal reverses order among values of the same sign: largest
becomes smallest, etc. The negative reciprocal preserves order among
values of the same sign.
**Logarithm**
The logarithm, x to log base 10 of x, or x to log base e of x (ln x), or
x to log base 2 of x, is a strong transformation with a major effect on
distribution shape. It is commonly used for reducing right skewness and
is often appropriate for measured variables. 855
It can not be applied to
zero or negative values. One unit on a logarithmic scale means a
multiplication by the base of logarithms being used. Exponential growth
or decline
y = a exp(bx)
is made linear by
ln y = ln a + bx
so that the response variable y should be logged. (Here exp() means
raising to the power e, approximately 2.71828, that is the base of
natural logarithms.)
An aside on this exponential growth or decline equation: put x = 0, and
y = a exp(0) = a,
so that a is the amount or count when x = 0. If a and b > 0, then y grows
at a faster and faster rate (e.g. compound interest or unchecked
population growth), whereas if a > 0 and b < 0, y declines at a slower
and slower rate (e.g. radioactive decay).
Power functions y = ax^b are made linear by log y = log a + b log x so
that both variables y and x should be logged.
An aside on such power functions: put x = 0, and for b > 0,
y = ax^b = 0,
so the power function for positive b goes through the origin, which often
makes physical or biological or economic sense. Think: does zero for x
imply zero for y? This kind of power function is a shape that fits many
data sets rather well.
Consider ratios y = p / q where p and q are both positive in practice.
Examples are
males / females;
dependants / workers;
downstream length / downvalley length.
Then y is somewhere between 0 and infinity, or in the last case, between
1 and infinity. If p = q, then y = 1. Such definitions often lead to
skewed data, because there is a clear lower limit and no clear upper
limit. 945