Re: (statistics)how to make date more like Laplacian distribution?

Hello Walala,

I don't think anyone has accused you of lying. But I see the problem as you
have stated it is that theory predicts your data should be Laplacian, and
for some reason you think your sample data is inconsistent with this notion.
It probably is consistent but at first blush it appears otherwise.

Now for a simple example. Let's say you have a fair die. Each time you toss
it, the probabily of each outcome is 1/6. So theory says we have a uniform
distribution. Now toss the die 100 times and count how many times each value
shows up. Now theory says on average each value shows up 1/6 of the time.
However in any given sample (set of measurements), you can't expect this
perfect a distribution every time. Why you ask? That is because each trial
is independent of the prior ones. And in this situation I only tossed it 100
times and 100 is not a multiple of six, so not all bins can have the same
number of counts. Since the counts must be integers and 100/6 = 16.6666666,
you can see the one problem.

Now to illustrate another problem, we will use simple coin tosses as this
has a smaller sample space than the dice problem.   In a simple coin toss
experiment where on average heads or tails shows up 50% of time, the results
of the next toss can't depend on the result of the last toss. Otherwise we
could accurately predict the coin toss result for every throw and we know
that is not true.

But let's look at the results of a coin toss trial and try to fit a
distribution to it.  If you toss a single coin 4 times or just simply toss 4
coins once, we know there are 2^4=16 possible outcomes. And if you enumerate
all 16 sets you will notice that only 6 out of the 16 will have heads half
of the time and tails half of the time. Wow more than half of the time, the
sample distribution doesn't fit the theoretical distribution. So if I go and
toss 4 coins and get 3 heads and 1 tail, can I proclaim my coins are unfair?
Clearly I can't, because probability says that will happen 25% of the time.

What all I done so far is show how a sample statistic (extracted from one
set of data) is not very good at predicting a population statistic
(extracted from all possible sets of data). And when you are trying to
determine a distribution you are in effect trying to find all of the
population statistics.

Now for your problem. The 1st thing is the theory correct in saying the
distribution is Laplacian? That can be answered by going through the details
of the derivation. I don't have your papers, so I can't help you here. Now
you can test this hypothesis statistically by one of several goodness of fit
tests. A common one is a chi square test.  But these test work well with
large amounts of data. But since you are reducing pictures and the world has
a lot pictures, you shouldn't be lacking for data. Another way is to develop
confidence intervals for the Laplacian distribution, for example 90%. Now if
you repeat the experiment many times, will your measurements fall inside or
outside with too high a frequency. I.e., If my 3 heads and 1 tail
combination showed up 50% of the time, instead of the expected 25%,  in a
large number of trials, I'd be suspicious that my distribution isn't
uniform.

Now for the question you posed about how to make your data more Laplacian
like. First you will have to decide what distribution it does have. Then you
can find a transformation that turns it into a Laplacian and you will have
to account for the Jacobian here. But if you know the distribution there
isn't much point is transforming it into another distribution.

To find the distribution when you don't have a theoretical way to get there
is by the process of resampling. Ie., you extract the distribution by
processing subsets of a large chunk of sample data. Some good resampling
methods that are used to find the distributions are known collectively as
bootstrapping methods. These really hit the stats scene during the 1970s. A
simple example of resampling can be found in the "jackknife method."

I hope this helps clarify some of the issues. I think it is time to study
some stats.

Clay



"walala" <[EMAIL PROTECTED]> wrote in message
news:[EMAIL PROTECTED]
>
> "Clay S. Turner" <[EMAIL PROTECTED]> wrote in message
> news:[EMAIL PROTECTED]
> > Hello Walala,
> >
> > How big is your sample size? If you grab only100 samples, you wouldn't
> > expect it to exactly fit the generating distribution. Can you grab
> 1,000,000
> > samples? If still it doesn't fit, then suspect your generating dist. is
> > different from a Laplacian. You can look into bootstrapping methods to
get
> > an estimate of the dist.
> >
> > Clay
> >
> >
> >
> >
> > "walala" <[EMAIL PROTECTED]> wrote in message
> > news:[EMAIL PROTECTED]
> > > Dear all,
> > >
> > > If I already know a prior that my data distribution should follow the
> > shape
> > > of Laplacian distribution... the data obtained from measurement is of
> > course
> > > a little off(not very symmtrical), how can I make the measured data
more
> > > Laplacian distribution like(make it at least a little more
symmtrical)?
> > >
> > > Can anybody give me an example or detailed explanation? I am kind of
> > afraid
> > > of statistics... :=)
> > >
> > > Thanks a lot,
> > >
> > > -Walala
> > >
> > >
> >
> >
>
> Dear Clay,
>
> Why people think I want to lie upon seeing my question? Oh, it's my
problem
> that I did not clearly present the background...
>
> Here is the story: in deblocking of block DCT coded JPEG images, it was
> known that the DCTed coefficients are Laplacian distributed... But now I
am
> looking at low bit rate JPEG images, so there are someking of artifacts...
> in order to reconstruct the original images... many algorithms have been
> devised... one possibility is to make the image coefficients more Laplcian
> like...
>
> So that came my question: how to make data more Laplican like...Please
give
> me some detailed explanation as I am not veteran in statistics...
>
> Thanks a lot,
>
> -Walala
>
> (I am just a poor student, not lieing government agency, issurance
company,
> weapon dealer, lawyers, and politicians... so please help me!)
>
>