Box Cox transformation¶
We consider a
multivariate stochastic process of dimension
where
and
is an event. We
suppose that the process is
.
We note
the random
variable at the vertex
defined by
.
If the variance of
depends on the vertex
, the Box Cox transformation maps the process
into the process
such that the variance of
is constant (at the first order at least) with
respect to
.
We present here:
the estimation of the Box Cox transformation from a given field of the process
,
the action of the Box Cox transformation on a field generated from
.
We note the Box Cox
transformation which maps the process
into the process
, where
, such that
is independent of
at the first order.
We suppose that
is a positive random variable for
any
. To verify that constraint, it may be needed to
consider the shifted process
.
We illustrate some usual Box Cox transformations
in the
scalar case (
), using the Taylor development of
at the mean point of
.
In the multivariate case, we estimate the Box Cox transformation
component by component and we define the multivariate Box Cox
transformation as the aggregation of the marginal Box Cox
transformations.
Marginal Box Cox transformation¶
The first order Taylor development of around
writes:
which leads to:
and then:
To have constant with respect to
at the first order, we need:
(1)¶
Now, we make some additional hypotheses on the relation between
and
:
If we suppose that
, then (1) leads to the function
and we take
;
If we suppose that
, then (1) leads to the function
and we take
;
More generally, if we suppose that
, then (1) leads to the function
parametrized by the scalar
:
(2)¶
The inverse Box Cox transformation is defined by:
(3)¶
Estimation of the Box Cox transformation¶
The parameter is estimated from a given field of the
process
as follows.
The estimation of
given below is optimized in the case
when
at
each vertex
. If it is not the case, that estimation
can be considered as a proposition, with no guarantee.
The parameters
are then estimated by
the maximum likelihood estimators. We note
and
respectively the cumulative distribution function and the density
probability function of the
distribution.
For all vertices
, we have:
(4)¶
from which we derive the density probability function of
for all vertices
:
(5)¶
Using (5), the likelihood of the values
with respect to the model (4)
writes:
(6)¶
We notice that for each fixed , the likelihood equation
is proportional to the likelihood equation which estimates
. Thus, the maximum likelihood estimator for
for a given
are:
(7)¶
(8)¶
where is a constant.
The parameter is the one maximizing
defined in (8).
Case 1: General linear models¶
In the frame of the general linear model, we consider a functional relation between some input and
output values. Let us consider the following dataset:
.
The general linear model aims at assessing the following prior model :
where:
is a general linear model based upon a functional basis
and a vector of coefficients
,
is a zero-mean stationary Gaussian process whose covariance function reads:
where
is the variance and
is the correlation function that solely depends on the Manhattan distance between input points
and a vector of parameters
.
The optimal parameters of such model are estimated by maximizing a log-likelihood function.
Here we suppose a gaussian prior on . Thus, if we write our various hypotheses,
we get the following log-likelihood function to be optimized:
(9)¶
where is a constant,
(10)¶
Remarks :
The equation (9) applies also if we replace the general linear model by a linear regression model. Indeed a linear model is a specific case of general linear model where the correlation model is a Dirac covariance model.
Note that such estimate might be heavy as we get a double loop optimization. Indeed for each value, we optimize
the parameters of the underlying general linear model. Some practitioners are used to freeze the first general linear model parameters
and then preform a one loop optimization selecting only the best
value.
Case 2: Linear models¶
In the frame of linear models, we consider a functional relation between some input and
output values. Let us consider the following dataset:
.
The general linear model aims at assessing the following prior model :
where:
is a general linear model based upon a functional basis
and a vector of coefficients
,
is a zero-mean stationary white noise process.
The optimal parameters of such model are estimated by maximizing a log-likelihood function.
Here we suppose a gaussian prior on . Thus, if we write our various hypotheses,
we get the following log-likelihood function to be optimized:
(11)¶
where is a constant,
(12)¶
As a remark, the above case is a particular case of (11). Indeed if a linear model is a specific case of general linear model where the correlation model is a Dirac covariance model (White noise model).
In term of costs, a factorization (QR or SVD) is done once for the regression matrix and the parameters defined in (12)
are easily obtained, for each new value, solving the corresponding linear systems.
Sometimes, people perform a grid search for example varying for example
from -3 to 3 using a small step. It allows one
to get both the optimal and assess the robustness of the optimum.