I need help understanding something.

The box-counting dimension of a set of random data points -- random in the
sense that they are white noise, equally likely to occur anywhere within an
interval -- is theoretically 2. That is computed by adding 1 to the slope
of the log/log plot of the single-variable data points, counted at various
sizes of box measurent..

That works out empirically for me. My measured slope of such a set of data
points is very close to 1.

But what about data points that are distributed not as such white noise, but
as Gaussian-distributed data centered around a mean. These data points
visually do not fill the space as completely, because by definition they
tend to be clustered around the mean. And when I measure the slope of such
a set of points, it is only about .92.

But the literature generally seems to expect a box-counting dimension of 2
for Gaussian data, not just white noise. I don't get it.

Dear Ross Evans,
Are you measuring a time series as:
s[n]->{s[1],s[2], s[3],...,s[n]}
Or
{r[n],s[n]},-> {{r[1],s[1]},{r[2],s[2]},{r[3],s[3]},...,{r[n],s[n]}}
The first will be one dimensional and the second should be two
dimensional.
How are you producing your white noise/ normal variates?
Different randomization method of producing your distribution will
produce different
results.
Ross Evans wrote:

I need help understanding something.

The box-counting dimension of a set of random data points -- random in
the
sense that they are white noise, equally likely to occur anywhere within
an
interval -- is theoretically 2. That is computed by adding 1 to the
slope
of the log/log plot of the single-variable data points, counted at
various
sizes of box measurent..

That works out empirically for me. My measured slope of such a set of
data
points is very close to 1.

But what about data points that are distributed not as such white noise,
but
as Gaussian-distributed data centered around a mean. These data points
visually do not fill the space as completely, because by definition they
tend to be clustered around the mean. And when I measure the slope of
such
a set of points, it is only about .92.

But the literature generally seems to expect a box-counting dimension of
2
for Gaussian data, not just white noise. I don't get it.





--
Respectfully, Roger L. Bagula
tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel:
619-5610814 :
alternative email: rlbtftn@netscape.net
URL : http://home.earthlink.net/~tftn

My white-noise data points were actually one-dimensional, but not a time
series. The white noise points were downloaded from random.org, not created
with a PRNG. For the Gaussian set, I tried several methods: One method
simply added n of these white-noise points and divided by n for each
Gaussian point.. Another method used two of these uniformly random points
at a time and transformed them into Gaussian values using the Box-Muller
algorithm.

Either way, the resulting set of Gaussian values produced slopes in the
range of .92 when run through a box-counting program.

I really do not think my difficulty lies in any artifact of my Gaussian
sets. Rather, there is a fundamental difference between uniform white noise
and Gaussian brown noise. I understand how the former produces a
box-counting slope of 1. The latter does not, because the space is
inherently more sparse in some areas.

With the white noise, as the box size gets smaller, the slope starts at
exactly unity until the boxes are small and numerous enough to produce some
empty boxes. With the Gaussian noise with the same number of data points,
the empty boxes start to occur sooner, with larger box intervals. So the
box-counting slope is lower.

I have not yet tried a cloud of x,y points, but it seems obvious that I
would get a similar result as for a set of points in one dimension.


"Roger Bagula" <tftn@earthlink.net> wrote in message
news:41B72988.4060200@earthlink.net...


Dear Ross Evans,
Are you measuring a time series as:
s[n]->{s[1],s[2], s[3],...,s[n]}
Or
{r[n],s[n]},-> {{r[1],s[1]},{r[2],s[2]},{r[3],s[3]},...,{r[n],s[n]}}
The first will be one dimensional and the second should be two


dimensional.


How are you producing your white noise/ normal variates?
Different randomization method of producing your distribution will
produce different
results.
Ross Evans wrote:



I need help understanding something.

The box-counting dimension of a set of random data points -- random in


the


sense that they are white noise, equally likely to occur anywhere within


an


interval -- is theoretically 2. That is computed by adding 1 to the


slope


of the log/log plot of the single-variable data points, counted at


various


sizes of box measurent..

That works out empirically for me. My measured slope of such a set of


data


points is very close to 1.

But what about data points that are distributed not as such white noise,


but


as Gaussian-distributed data centered around a mean. These data points
visually do not fill the space as completely, because by definition they
tend to be clustered around the mean. And when I measure the slope of


such


a set of points, it is only about .92.

But the literature generally seems to expect a box-counting dimension of


2


for Gaussian data, not just white noise. I don't get it.






--
Respectfully, Roger L. Bagula
tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel:


619-5610814 :


alternative email: rlbtftn@netscape.net
URL : http://home.earthlink.net/~tftn

Dear Ross Evans,
I have posted another way that is used and abused
for finding dimension in sets like this.
The correlation dimension can be applied to such cases as well.
The polar normal noise plotted in two dimensions is much sparser
as you get away from the origin at (0,0) , but is still considered to
be a space filling
curve. In IFS we have space fills that are nonlinear that behave this
way as well.
A finite time snap shot like a picture made by a computer
will give a different answer than the asymptotic limiting value.
Here what you do to determine that. You make a number of pictures
at intervals of time/ numbers of points ( maybe powers of ten).
You do box counting dimension on them all.
You plot them as number of points against the resulting box counting
dimension
and see where the curve tends to for "infinite"
number of points.
It is very difficult to get an IFS for a disk space filling curve that
is "uniform".
Ross Evans wrote:

My white-noise data points were actually one-dimensional, but not a time
series. The white noise points were downloaded from random.org, not
created
with a PRNG. For the Gaussian set, I tried several methods: One method
simply added n of these white-noise points and divided by n for each
Gaussian point.. Another method used two of these uniformly random
points
at a time and transformed them into Gaussian values using the Box-Muller
algorithm.

Either way, the resulting set of Gaussian values produced slopes in the
range of .92 when run through a box-counting program.

I really do not think my difficulty lies in any artifact of my Gaussian
sets. Rather, there is a fundamental difference between uniform
white noise
and Gaussian brown noise. I understand how the former produces a
box-counting slope of 1. The latter does not, because the space is
inherently more sparse in some areas.

With the white noise, as the box size gets smaller, the slope starts at
exactly unity until the boxes are small and numerous enough to
produce some
empty boxes. With the Gaussian noise with the same number of data
points,
the empty boxes start to occur sooner, with larger box intervals. So the
box-counting slope is lower.

I have not yet tried a cloud of x,y points, but it seems obvious that I
would get a similar result as for a set of points in one dimension.


"Roger Bagula" <tftn@earthlink.net> wrote in message
news:41B72988.4060200@earthlink.net...


Dear Ross Evans,
Are you measuring a time series as:
s[n]->{s[1],s[2], s[3],...,s[n]}
Or
{r[n],s[n]},-> {{r[1],s[1]},{r[2],s[2]},{r[3],s[3]},...,{r[n],s[n]}}
The first will be one dimensional and the second should be two


dimensional.


How are you producing your white noise/ normal variates?
Different randomization method of producing your distribution will
produce different
results.
Ross Evans wrote:



I need help understanding something.

The box-counting dimension of a set of random data points -- random in


the


sense that they are white noise, equally likely to occur anywhere
within


an


interval -- is theoretically 2. That is computed by adding 1 to the


slope


of the log/log plot of the single-variable data points, counted at


various


sizes of box measurent..

That works out empirically for me. My measured slope of such a set of


data


points is very close to 1.

But what about data points that are distributed not as such white
noise,


but


as Gaussian-distributed data centered around a mean. These data points
visually do not fill the space as completely, because by definition
they
tend to be clustered around the mean. And when I measure the slope of


such


a set of points, it is only about .92.

But the literature generally seems to expect a box-counting
dimension of


2


for Gaussian data, not just white noise. I don't get it.






--
Respectfully, Roger L. Bagula
tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel:


619-5610814 :


alternative email: rlbtftn@netscape.net
URL : http://home.earthlink.net/~tftn

I need help understanding something.

The box-counting dimension of a set of random data points -- random in the
sense that they are white noise, equally likely to occur anywhere within an
interval -- is theoretically 2. That is computed by adding 1 to the slope
of the log/log plot of the single-variable data points, counted at various
sizes of box measurent..

That works out empirically for me. My measured slope of such a set of data
points is very close to 1.

But what about data points that are distributed not as such white noise, but
as Gaussian-distributed data centered around a mean. These data points
visually do not fill the space as completely, because by definition they
tend to be clustered around the mean. And when I measure the slope of such
a set of points, it is only about .92.

But the literature generally seems to expect a box-counting dimension of 2
for Gaussian data, not just white noise. I don't get it.

Let me say that this is a good quest
and the answer in not obvious
and may well be not all that certain, but
it is what the Einstein theory of Brownian motion tells us.
A French economist actually discovered this diffusion law before
Einstein, but
he is the one who applied it best to physics.
It all ties in with his ideas about curvature of space time as well.
To give you a counter idea: suppose that light got dimmer as the cube
of the distance
instead of as the square; that is the "measure" for space was:
m3(x,y,z)=(x^3+y^3+z^3)^(1/3)
instead of
m2(x,y,z)=(x^2+y^2+z^2)^(1/2)
The result that the diffusion of Brownian motion would
be much slower and be governed by a different set of differential
equations altogether.
The limiting Femat type nth measure is the Hausdorff space measure:
Mhd=Max[x]+Max[y]+Max[z]
Which is a 3d space filling measure.
If we add in an non-Euclidean time in a negative hyperbolic sense:
Mhd4d=Max[x]+Max[y]+Max[z]-c*Max[t]
You see the time probability and the curvature of space is "intrinsic"
in the diffusion measures that Brownian motion is based on.
Ross Evans wrote:

I need help understanding something.

The box-counting dimension of a set of random data points -- random
in the
sense that they are white noise, equally likely to occur anywhere
within an
interval -- is theoretically 2. That is computed by adding 1 to the
slope
of the log/log plot of the single-variable data points, counted at
various
sizes of box measurent..

That works out empirically for me. My measured slope of such a set
of data
points is very close to 1.

But what about data points that are distributed not as such white
noise, but
as Gaussian-distributed data centered around a mean. These data points
visually do not fill the space as completely, because by definition they
tend to be clustered around the mean. And when I measure the slope
of such
a set of points, it is only about .92.

But the literature generally seems to expect a box-counting dimension
of 2
for Gaussian data, not just white noise. I don't get it.

Firstly, I presume you are distributing the points in the plane.
Generically, uniformly random points in n dimensions should have
dimension n.

Secondly, yes indeed a Gaussian cloud of points in n dimensions should
have dimension n.

Thirdly, well the previous statement only holds for some scales. You
must understand that a fractal dimension ONLY applies over a finite
(usually) range of length scales. Suppose your Gaussian cloud of k
points is distrubuted with standard deviation 1:
* on large lengths scales (say >10), the entire cloud will look like
just one point as they are all located at near ly the same pplace, thus
the cloud looks like it has dimension 1
* on medium length scales (say <2 and > k^(1/n)), the cloud of points
looks as if it fills up the space in a fuzz of points and som has
dimension n
* on really small length scales (say <k^(/n)) you will perceive the
points as individuals and the set of points will again have dimension 0.
The right dimension depends upon the length scale at which you want to
interact with the object.

Fourthly, the box counting dimension is a poor methodology which mixes
up the dimension of the middle regime with that of the surrounding
regimes. Thus the box counting dimension (in this example) will be a
little low.

Tony


Ross Evans wrote:

I need help understanding something.

The box-counting dimension of a set of random data points -- random in
the
sense that they are white noise, equally likely to occur anywhere
within an
interval -- is theoretically 2. That is computed by adding 1 to the
slope
of the log/log plot of the single-variable data points, counted at
various
sizes of box measurent..

That works out empirically for me. My measured slope of such a set of
data
points is very close to 1.

But what about data points that are distributed not as such white
noise, but
as Gaussian-distributed data centered around a mean. These data points
visually do not fill the space as completely, because by definition they
tend to be clustered around the mean. And when I measure the slope of
such
a set of points, it is only about .92.

But the literature generally seems to expect a box-counting dimension
of 2
for Gaussian data, not just white noise. I don't get it.

You have not commented on my reply,
Tony

Tony Roberts wrote:
Firstly, I presume you are distributing the points in the plane.
Generically, uniformly random points in n dimensions should have
dimension n.

Secondly, yes indeed a Gaussian cloud of points in n dimensions should
have dimension n.

Thirdly, well the previous statement only holds for some scales. You
must understand that a fractal dimension ONLY applies over a finite
(usually) range of length scales. Suppose your Gaussian cloud of k
points is distrubuted with standard deviation 1:
* on large lengths scales (say >10), the entire cloud will look like
just one point as they are all located at near ly the same pplace, thus
the cloud looks like it has dimension 1
* on medium length scales (say <2 and > k^(1/n)), the cloud of points
looks as if it fills up the space in a fuzz of points and som has
dimension n
* on really small length scales (say <k^(/n)) you will perceive the
points as individuals and the set of points will again have dimension 0.
The right dimension depends upon the length scale at which you want to
interact with the object.

Fourthly, the box counting dimension is a poor methodology which mixes
up the dimension of the middle regime with that of the surrounding
regimes. Thus the box counting dimension (in this example) will be a
little low.

Tony


Ross Evans wrote:

I need help understanding something.

The box-counting dimension of a set of random data points -- random in
the
sense that they are white noise, equally likely to occur anywhere
within an
interval -- is theoretically 2. That is computed by adding 1 to the
slope
of the log/log plot of the single-variable data points, counted at
various
sizes of box measurent..

That works out empirically for me. My measured slope of such a set of
data
points is very close to 1.

But what about data points that are distributed not as such white
noise, but
as Gaussian-distributed data centered around a mean. These data points
visually do not fill the space as completely, because by definition
they
tend to be clustered around the mean. And when I measure the slope of
such
a set of points, it is only about .92.

But the literature generally seems to expect a box-counting dimension
of 2
for Gaussian data, not just white noise. I don't get it.

I don't understand what you mean by choosing the appropriate scale for the
dimension. I thought the way the box-counting method works is to measure
the slope of a near-linear plot of measurements across a wide range of
scales.
Yes indeed, that is why I said "appropriate RANGE of scales".

With a uniformly distributed dust, the box-counting method yields a perfect
slope of 1 until the boxes get small enough to force some boxes to be empty
between points out of pure sparseness. Then the slope rapidly approaches 0
as the size of the boxes is halved, which value it is guaranteed to reach
once the number of filled boxes reaches n.
Exactly my contention: on these small lengths scales the box cunting

One of the intermediate ranges
does not seem to be the "correct" range for the distribution, because the
dimension measured at the intermediate scale -- even for the uniformly
random set -- is less than 1.