<s>
In	O
statistics	O
,	O
data	B-General_Concept
transformation	I-General_Concept
is	O
the	O
application	O
of	O
a	O
deterministic	O
mathematical	O
function	O
to	O
each	O
point	O
in	O
a	O
data	O
set	O
—	O
that	O
is	O
,	O
each	O
data	O
point	O
zi	O
is	O
replaced	O
with	O
the	O
transformed	O
value	O
yi	O
=	O
f(zi )	O
,	O
where	O
f	O
is	O
a	O
function	O
.	O
</s>
<s>
Transforms	O
are	O
usually	O
applied	O
so	O
that	O
the	O
data	O
appear	O
to	O
more	O
closely	O
meet	O
the	O
assumptions	O
of	O
a	O
statistical	O
inference	O
procedure	O
that	O
is	O
to	O
be	O
applied	O
,	O
or	O
to	O
improve	O
the	O
interpretability	O
or	O
appearance	O
of	O
graphs	B-Application
.	O
</s>
<s>
The	O
transformation	B-Algorithm
is	O
usually	O
applied	O
to	O
a	O
collection	O
of	O
comparable	O
measurements	O
.	O
</s>
<s>
Guidance	O
for	O
how	O
data	O
should	O
be	O
transformed	O
,	O
or	O
whether	O
a	O
transformation	B-Algorithm
should	O
be	O
applied	O
at	O
all	O
,	O
should	O
come	O
from	O
the	O
particular	O
statistical	O
analysis	O
to	O
be	O
performed	O
.	O
</s>
<s>
For	O
example	O
,	O
a	O
simple	O
way	O
to	O
construct	O
an	O
approximate	O
95%	O
confidence	O
interval	O
for	O
the	O
population	O
mean	O
is	O
to	O
take	O
the	O
sample	O
mean	O
plus	O
or	O
minus	O
two	O
standard	B-General_Concept
error	I-General_Concept
units	O
.	O
</s>
<s>
However	O
,	O
if	O
the	O
population	O
is	O
substantially	O
skewed	B-General_Concept
and	O
the	O
sample	O
size	O
is	O
at	O
most	O
moderate	O
,	O
the	O
approximation	O
provided	O
by	O
the	O
central	O
limit	O
theorem	O
can	O
be	O
poor	O
,	O
and	O
the	O
resulting	O
confidence	O
interval	O
will	O
likely	O
have	O
the	O
wrong	O
coverage	O
probability	O
.	O
</s>
<s>
If	O
desired	O
,	O
the	O
confidence	O
interval	O
can	O
then	O
be	O
transformed	O
back	O
to	O
the	O
original	O
scale	O
using	O
the	O
inverse	O
of	O
the	O
transformation	B-Algorithm
that	O
was	O
applied	O
to	O
the	O
data	O
.	O
</s>
<s>
For	O
example	O
,	O
suppose	O
we	O
have	O
a	O
scatterplot	B-Application
in	O
which	O
the	O
points	O
are	O
the	O
countries	O
of	O
the	O
world	O
,	O
and	O
the	O
data	O
values	O
being	O
plotted	O
are	O
the	O
land	O
area	O
and	O
population	O
of	O
each	O
country	O
.	O
</s>
<s>
Another	O
reason	O
for	O
applying	O
data	B-General_Concept
transformation	I-General_Concept
is	O
to	O
improve	O
interpretability	O
,	O
even	O
if	O
no	O
formal	O
statistical	O
analysis	O
or	O
visualization	O
is	O
to	O
be	O
performed	O
.	O
</s>
<s>
Data	B-General_Concept
transformation	I-General_Concept
may	O
be	O
used	O
as	O
a	O
remedial	O
measure	O
to	O
make	O
data	O
suitable	O
for	O
modeling	O
with	O
linear	B-General_Concept
regression	I-General_Concept
if	O
the	O
original	O
data	O
violates	O
one	O
or	O
more	O
assumptions	O
of	O
linear	B-General_Concept
regression	I-General_Concept
.	O
</s>
<s>
For	O
example	O
,	O
the	O
simplest	O
linear	B-General_Concept
regression	I-General_Concept
models	I-General_Concept
assume	O
a	O
linear	O
relationship	O
between	O
the	O
expected	O
value	O
of	O
Y	O
(	O
the	O
response	O
variable	O
to	O
be	O
predicted	O
)	O
and	O
each	O
independent	O
variable	O
(	O
when	O
the	O
other	O
independent	O
variables	O
are	O
held	O
fixed	O
)	O
.	O
</s>
<s>
For	O
example	O
,	O
addition	O
of	O
quadratic	O
functions	O
of	O
the	O
original	O
independent	O
variables	O
may	O
lead	O
to	O
a	O
linear	O
relationship	O
with	O
expected	O
value	O
of	O
Y	O
,	O
resulting	O
in	O
a	O
polynomial	O
regression	O
model	O
,	O
a	O
special	O
case	O
of	O
linear	B-General_Concept
regression	I-General_Concept
.	O
</s>
<s>
Another	O
assumption	O
of	O
linear	B-General_Concept
regression	I-General_Concept
is	O
homoscedasticity	B-General_Concept
,	O
that	O
is	O
the	O
variance	O
of	O
errors	O
must	O
be	O
the	O
same	O
regardless	O
of	O
the	O
values	O
of	O
predictors	O
.	O
</s>
<s>
if	O
the	O
data	O
is	O
heteroscedastic	B-General_Concept
)	O
,	O
it	O
may	O
be	O
possible	O
to	O
find	O
a	O
transformation	B-Algorithm
of	O
Y	O
alone	O
,	O
or	O
transformations	O
of	O
both	O
X	O
(	O
the	O
predictor	O
variables	O
)	O
and	O
Y	O
,	O
such	O
that	O
the	O
homoscedasticity	B-General_Concept
assumption	O
(	O
in	O
addition	O
to	O
the	O
linearity	O
assumption	O
)	O
holds	O
true	O
on	O
the	O
transformed	O
variables	O
and	O
linear	B-General_Concept
regression	I-General_Concept
may	O
therefore	O
be	O
applied	O
on	O
these	O
.	O
</s>
<s>
Yet	O
another	O
application	O
of	O
data	B-General_Concept
transformation	I-General_Concept
is	O
to	O
address	O
the	O
problem	O
of	O
lack	O
of	O
normality	O
in	O
error	O
terms	O
.	O
</s>
<s>
Univariate	O
normality	O
is	O
not	O
needed	O
for	O
least	B-Algorithm
squares	I-Algorithm
estimates	O
of	O
the	O
regression	O
parameters	O
to	O
be	O
meaningful	O
(	O
see	O
Gauss	O
–	O
Markov	O
theorem	O
)	O
.	O
</s>
<s>
For	O
illustrative	O
purposes	O
,	O
if	O
base-10	O
logarithm	O
were	O
used	O
instead	O
of	O
natural	O
logarithm	O
in	O
the	O
above	O
transformation	B-Algorithm
and	O
the	O
same	O
symbols	O
(	O
a	O
and	O
b	O
)	O
are	O
used	O
to	O
denote	O
the	O
regression	B-General_Concept
coefficients	I-General_Concept
,	O
then	O
a	O
unit	O
increase	O
in	O
X	O
would	O
lead	O
to	O
a	O
times	O
increase	O
in	O
Y	O
on	O
an	O
average	O
.	O
</s>
<s>
Generalized	O
linear	O
models	O
(	O
GLMs	O
)	O
provide	O
a	O
flexible	O
generalization	O
of	O
ordinary	O
linear	B-General_Concept
regression	I-General_Concept
that	O
allows	O
for	O
response	O
variables	O
that	O
have	O
error	O
distribution	O
models	O
other	O
than	O
a	O
normal	O
distribution	O
.	O
</s>
<s>
The	O
logarithm	O
transformation	B-Algorithm
and	O
square	O
root	O
transformation	B-Algorithm
are	O
commonly	O
used	O
for	O
positive	O
data	O
,	O
and	O
the	O
multiplicative	O
inverse	O
transformation	B-Algorithm
(	O
reciprocal	O
transformation	B-Algorithm
)	O
can	O
be	O
used	O
for	O
non-zero	O
data	O
.	O
</s>
<s>
The	O
power	B-General_Concept
transformation	I-General_Concept
is	O
a	O
family	O
of	O
transformations	O
parameterized	O
by	O
a	O
non-negative	O
value	O
λ	O
that	O
includes	O
the	O
logarithm	O
,	O
square	O
root	O
,	O
and	O
multiplicative	O
inverse	O
transformations	O
as	O
special	O
cases	O
.	O
</s>
<s>
To	O
approach	O
data	B-General_Concept
transformation	I-General_Concept
systematically	O
,	O
it	O
is	O
possible	O
to	O
use	O
statistical	O
estimation	O
techniques	O
to	O
estimate	O
the	O
parameter	O
λ	O
in	O
the	O
power	B-General_Concept
transformation	I-General_Concept
,	O
thereby	O
identifying	O
the	O
transformation	B-Algorithm
that	O
is	O
approximately	O
the	O
most	O
appropriate	O
in	O
a	O
given	O
setting	O
.	O
</s>
<s>
Since	O
the	O
power	B-General_Concept
transformation	I-General_Concept
family	O
also	O
includes	O
the	O
identity	O
transformation	B-Algorithm
,	O
this	O
approach	O
can	O
also	O
indicate	O
whether	O
it	O
would	O
be	O
best	O
to	O
analyze	O
the	O
data	O
without	O
a	O
transformation	B-Algorithm
.	O
</s>
<s>
In	O
regression	O
analysis	O
,	O
this	O
approach	O
is	O
known	O
as	O
the	O
Box	O
–	O
Cox	O
transformation	B-Algorithm
.	O
</s>
<s>
The	O
reciprocal	O
transformation	B-Algorithm
,	O
some	O
power	B-General_Concept
transformations	I-General_Concept
such	O
as	O
the	O
Yeo	O
–	O
Johnson	O
transformation	B-Algorithm
,	O
and	O
certain	O
other	O
transformations	O
such	O
as	O
applying	O
the	O
inverse	O
hyperbolic	O
sine	O
,	O
can	O
be	O
meaningfully	O
applied	O
to	O
data	O
that	O
include	O
both	O
positive	O
and	O
negative	O
values	O
(	O
the	O
power	B-General_Concept
transformation	I-General_Concept
is	O
invertible	O
over	O
all	O
real	O
numbers	O
if	O
λ	O
is	O
an	O
odd	O
integer	O
)	O
.	O
</s>
<s>
However	O
,	O
when	O
both	O
negative	O
and	O
positive	O
values	O
are	O
observed	O
,	O
it	O
is	O
sometimes	O
common	O
to	O
begin	O
by	O
adding	O
a	O
constant	O
to	O
all	O
values	O
,	O
producing	O
a	O
set	O
of	O
non-negative	O
data	O
to	O
which	O
any	O
power	B-General_Concept
transformation	I-General_Concept
can	O
be	O
applied	O
.	O
</s>
<s>
A	O
common	O
situation	O
where	O
a	O
data	B-General_Concept
transformation	I-General_Concept
is	O
applied	O
is	O
when	O
a	O
value	O
of	O
interest	O
ranges	O
over	O
several	O
orders	O
of	O
magnitude	O
.	O
</s>
<s>
Power	B-General_Concept
transforms	I-General_Concept
,	O
and	O
in	O
particular	O
the	O
logarithm	O
,	O
can	O
often	O
be	O
used	O
to	O
induce	O
symmetry	O
in	O
such	O
data	O
.	O
</s>
<s>
If	O
values	O
are	O
naturally	O
restricted	O
to	O
be	O
in	O
the	O
range	O
0	O
to	O
1	O
,	O
not	O
including	O
the	O
end-points	O
,	O
then	O
a	O
logit	O
transformation	B-Algorithm
may	O
be	O
appropriate	O
:	O
this	O
yields	O
values	O
in	O
the	O
range	O
( ∞	O
,	O
∞	O
)	O
.	O
</s>
<s>
However	O
,	O
if	O
symmetry	O
or	O
normality	O
are	O
desired	O
,	O
they	O
can	O
often	O
be	O
induced	O
through	O
one	O
of	O
the	O
power	B-General_Concept
transformations	I-General_Concept
.	O
</s>
<s>
Nevertheless	O
,	O
usage	O
of	O
Gaussian	O
statistics	O
is	O
perfectly	O
possible	O
by	O
applying	O
data	B-General_Concept
transformation	I-General_Concept
.	O
</s>
<s>
To	O
assess	O
whether	O
normality	O
has	O
been	O
achieved	O
after	O
transformation	B-Algorithm
,	O
any	O
of	O
the	O
standard	O
normality	B-General_Concept
tests	I-General_Concept
may	O
be	O
used	O
.	O
</s>
<s>
A	O
graphical	O
approach	O
is	O
usually	O
more	O
informative	O
than	O
a	O
formal	O
statistical	O
test	O
and	O
hence	O
a	O
normal	B-Application
quantile	I-Application
plot	I-Application
is	O
commonly	O
used	O
to	O
assess	O
the	O
fit	O
of	O
a	O
data	O
set	O
to	O
a	O
normal	O
population	O
.	O
</s>
<s>
Alternatively	O
,	O
rules	O
of	O
thumb	O
based	O
on	O
the	O
sample	O
skewness	B-General_Concept
and	O
kurtosis	B-Error_Name
have	O
also	O
been	O
proposed	O
.	O
</s>
<s>
A	O
variance-stabilizing	B-General_Concept
transformation	I-General_Concept
aims	O
to	O
remove	O
a	O
variance-on-mean	O
relationship	O
,	O
so	O
that	O
the	O
variance	O
becomes	O
constant	O
relative	O
to	O
the	O
mean	O
.	O
</s>
<s>
Examples	O
of	O
variance-stabilizing	B-General_Concept
transformations	I-General_Concept
are	O
the	O
Fisher	B-Algorithm
transformation	I-Algorithm
for	O
the	O
sample	O
correlation	O
coefficient	O
,	O
the	O
square	O
root	O
transformation	B-Algorithm
or	O
Anscombe	O
transform	O
for	O
Poisson	O
data	O
(	O
count	O
data	O
)	O
,	O
the	O
Box	O
–	O
Cox	O
transformation	B-Algorithm
for	O
regression	O
analysis	O
,	O
and	O
the	O
arcsine	O
square	O
root	O
transformation	B-Algorithm
or	O
angular	O
transformation	B-Algorithm
for	O
proportions	O
(	O
binomial	O
data	O
)	O
.	O
</s>
<s>
While	O
commonly	O
used	O
for	O
statistical	O
analysis	O
of	O
proportional	O
data	O
,	O
the	O
arcsine	O
square	O
root	O
transformation	B-Algorithm
is	O
not	O
recommended	O
because	O
logistic	O
regression	O
or	O
a	O
logit	O
transformation	B-Algorithm
are	O
more	O
appropriate	O
for	O
binomial	O
or	O
non-binomial	O
proportions	O
,	O
respectively	O
,	O
especially	O
due	O
to	O
decreased	O
type-II	O
error	O
.	O
</s>
<s>
It	O
is	O
also	O
possible	O
to	O
modify	O
some	O
attributes	O
of	O
a	O
multivariate	O
distribution	O
using	O
an	O
appropriately	O
constructed	O
transformation	B-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
when	O
working	O
with	O
time	O
series	O
and	O
other	O
types	O
of	O
sequential	O
data	O
,	O
it	O
is	O
common	O
to	O
difference	B-Algorithm
the	O
data	O
to	O
improve	O
stationarity	B-Algorithm
.	O
</s>
<s>
If	O
data	O
generated	O
by	O
a	O
random	O
vector	O
X	O
are	O
observed	O
as	O
vectors	O
Xi	O
of	O
observations	O
with	O
covariance	O
matrix	O
Σ	O
,	O
a	O
linear	B-Architecture
transformation	I-Architecture
can	O
be	O
used	O
to	O
decorrelate	B-Algorithm
the	O
data	O
.	O
</s>
<s>
Then	O
the	O
transformed	O
vector	O
Yi	O
=	O
A−1X''i	O
has	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
as	O
its	O
covariance	O
matrix	O
.	O
</s>
