## Multivariate transformations: confirmatory
analysis

The purpose of this page is to show how it is
possible to validate a certain set of transformation parameters which have been
found. In this page we use the same dataset which had been used in page mult_fsmtra.html

We had found that $\lambda = (0.5, 0, 0.5, 0, 0)$ was a reasonable set of transformation parameters and that this transformation
was supported by all the data except the outliers.

In this page we use the multivariate fan plot of Figure below to confirm these values of λ.

% Mussels data.
load('mussels.mat');
Y=mussels{:,:};
plotslrt=struct;
plotslrt.ylim=[-6.2 6.2];
[out]=FSMfan(Y,[0.5 0 0.5 0 0],'laAround',[-1 -0.5 0 1/3 0.5 1],'init',58,'plotslrt',plotslrt);
% Compare this plot with Figure 4.24 p. 182 of ARC (2004)

This Figure shows the evolution of the
score statistics during 30 forward searches. The panels confirm
the transformation we have found, but also show what other
transformation is acceptable for each variable. The transformation
for $y_1$ is not very tightly defined, with the statistics for 0.5
and 1/3 close to zero throughout the search and the log
transformation within the 99% band. For $y_2$ the statistic for
the log transformation is closest to zero throughout, although 1/3
is also acceptable. For $y_3$ the value of 1/3 is a little better
than is 0.5: the other four values are unacceptable. The
transformation for $y_4$ is unambiguously the log. The statistics
for the other five values of the parameter increase steadily in
magnitude throughout the search. Interestingly, there is no
evidence of an effect of the outliers. The
effect of the
outliers is however evident in the last panel, that for
transformation of $y_5$. The statistic for $\lambda_5=0$ is close
to zero until the end of the search when the outliers enter. The
one third transformation gives a statistic which is stable and
close to zero at the end of the search. However it has negative
values earlier on, which are significant at the 5% level.

In Atkinson Riani and Cerioli (2004) it is possible to find a detailed comparison of the data
before and after the transformation.