Databases Reference
In-Depth Information
1.2
Uniform
Gaussian
Laplacian
Gamma
1.0
0.8
0.6
0.4
0.2
0
−6
−4
−2
0
2
4
6
F I GU R E 8 . 7
Uniform, Gaussian, Laplacian, and gamma distributions.
given by
4
√
3
√
8
exp
−
√
3
|
x
|
f
X
(
x
)
=
(86)
πσ
|
|
2
σ
x
The shapes of these four distributions, assuming a mean of zero and a variance of one, are
shown in Figure
8.7
.
One way of obtaining the estimate of the distribution of a particular source is to divide the
range of outputs into “bins” or intervals
I
k
. We can then find the number of values
n
k
that
fall into each interval. A plot of
n
k
n
T
, where
n
T
is the total number of source outputs being
considered, should give us some idea of what the input distribution looks like. Be aware that
this is a rather crude method and can at times be misleading. For example, if we were not
careful in our selection of the source output, we might end up modeling some local peculiarities
of the source. If the bins are too large, we might effectively filter out some important properties
of the source. If the bin sizes are too small, we may miss out on some of the gross behavior of
the source.
Once we have decided on some candidate distributions, we can select between them using
a number of sophisticated tests. These tests are beyond the scope of this topic but are described
in [
115
].
Many of the sources that we deal with when we design lossy compression schemes have
a great deal of structure in the form of sample-to-sample dependencies. The probability
models described here capture none of these dependencies. Fortunately, we have a lot of
models that can capture most of this structure. We describe some of these models in the next
section.
