# 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 (=1), 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:

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)

where .

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)

Substituting (7) into (6) and taking the likelihood, we obtain:

(8)

where is a constant.

The parameter is the one maximizing defined in (8).

Examples: