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BUILDING AN ADDITIVE MODEL USING TWO DIFFUSION TREES.

Here, I give an example of how a distribution with an additive
structure can be modeled using more than one diffusion tree.  The data
for this example (in file 'adata') consists of 30 cases, each with
eight binary variables.  It was manually constructed so that the first
four variables were unreleated to the last four variables.  With
respect to the first four variables, there are two groups, in which
these variables tend to all be "1", or all be "0".  Similarly, there
are two groups with respect to the last four variables.  The combined
effect is that there are four overall groups, corresponding to
patterns of 00000000, 11110000, 00001111, and 11111111.

We could model this data as a mixture, or using a single Dirichlet
diffusion tree.  Even though this fails to capture the division into
two sets of variables, it does work fairly well.  Here is one way this
could be done with a diffusion tree model:

> dft-spec   alog.dft1 0 8 / 0.2:4:2 - 1
> model-spec alog.dft1 binary
> data-spec  alog.dft1 0 8 2 / adata .
> dft-gen    alog.dft1 fix 2
> mc-spec    alog.dft1 repeat 50 gibbs-latent slice-positions met-terminals\
>                                gibbs-sigmas
> dft-mc     alog.dft1 100

This is similar to what is done in the example of Ex-mixdft-b.doc.
The 'dft-gen' command fixes the diffusion standard deviations to 2
initially.  The subsequent 'gibbs-latent' command will produce values
for the latent variables that fit the data reasonably, starting the
chain off in from a reasonable state.  (Without the 'dft-gen' command,
the latent variables start off rather small.)

The above commands take 82 seconds on the system used (see
Ex-system.doc).  We can look at the tree found using 'dft-display'
with the "-g" option, or using the following command:

> dft-dendrogram alog.dft1 100 alabels | ghostview -

The 'dft-dendrogram' command produces a Postscript picture of the
tree, which is here piped into the ghostview utility (it could instead
by printed).  This command shows the final tree, at iteration 100, and
uses the labels for cases in the file 'alabels', which labels the four
patterns mentioned above with A, B, C, and D .  You should be able to
see that cases with each of the four patterns are mostly grouped
together in subtrees.

To better capture the structure of this data, we can use a model with
two trees, specified as follows:

> dft-spec   alog.dft2 0 8 / 0.1:4:1 - 1 / 0.1:4:1 - 1
> model-spec alog.dft2 binary
> data-spec  alog.dft2 0 8 2 / adata .

The two sets of specifications in 'dft-spec' after the slashes
(identical here) are for two Dirichlet diffusion trees, which are
generated independently in the prior distribution.  The values at the
terminal nodes of these trees are added together to produce the latent
values for the cases, which define the probabilities of the variables
being "0" or "1".  We hope that this will allow one tree to specialize
in modeling the first four variables and the other tree to specialize
in modeling the last four.

Here are the commands used to sample from the posterior distribution:

> dft-gen  alog.dft2 fix 1.4
> mc-spec  alog.dft2 create-latent repeat 50 gibbs-locations sample-latent \
>                                            slice-positions met-terminals \
>                                            gibbs-sigmas
> dft-mc   alog.dft2 100

Here again, the 'dft-gen' command helps start the chain off in a
reasonable state.  The 'create-latent' operation ensures that latent
values exist, which is necessary when there is more than one tree when
the model is non-Gaussian.

These commands take 56 seconds on the system used (see Ex-system.doc).
We can look at the resulting hyperparameters as follows:

> dft-display alog.dft2

DIFFUSION TREE MODEL IN FILE "alog.dft2" WITH INDEX 100

PARAMETERS OF TREE 1

Standard deviation parameters for diffusion process

0.247:     1.380    0.796    7.847    2.466    0.312
0.169    0.554    0.182

Divergence function parameters: - 1.0000 -

PARAMETERS OF TREE 2

Standard deviation parameters for diffusion process

0.147:     0.057    0.350    0.106    0.148    4.766
691.193    4.756    0.956

Divergence function parameters: - 1.0000 -

We see that the first tree has mostly larger diffusion standard
deviations for the first four variables than the last four.  This is
reversed for the second tree, for which the diffusion standard
deviations are small for the first four variables and large for the
last four.  This is consistent with the two trees dividing up the
modeling task.  (In another run, the roles of the two trees might be
reversed.)

We can view the two trees as follows:

> dft-dendrogram alog.dft2 100 1 alabels | ghostview -
> dft-dendrogram alog.dft2 100 2 alabels | ghostview -

This should show that one tree divides the A and B cases from the C
and D cases, whereas the other tree divides the A and C cases from the
B and D cases.  This shows that the two-tree model has discovered the
structure of the data.
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