Science > Physics > Acceleration of Universe As Acceleration of Probable Correlation 5: Increasing Variance
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Science > Physics |
| User: |
"OsherD" |
| Date: |
24 Sep 2005 10:10:46 AM |
| Object: |
Acceleration of Universe As Acceleration of Probable Correlation 5: Increasing Variance |
From Osher Doctorow
If we increase the variance of X and the variance of Y for the linear
correlation coefficient rho very close to 1 (approximately linear
increasing relationship between X and Y with the usual mainstream
statistical assumptions on linear correlation coefficients), then by
inspection of the previous equations we get f(x,y) and f(-x, -y) --> 0
for the bivariate joint Gaussian/normal distribution, and so the
acceleration ofthe universe decreases toward 0 although it remains
positive. Presumably the reverse so to speak (more or less) would
increase the acceleration from near 0, so that we should say a few
words about what f(x, y) and f(-x, -y) increasing would represent.
The answer is simple. For symmetric graph joint bivariate probability
density functions (pdfs) like the bivariate joint Gaussian/normal pdf,
increase in f(x, y) and f(-x, -y) raises the center which is the
maximum of the graph but decreases its spread or variation, so that the
bell shape in our example becomes more highly peaked but less spread
out in the circumference of the base, i.e., a taller and taller bell
which is narrower and narrower in width perpendicular to its "height"
axis. In practical terms, the universe accelerates but for a
relatively short period. This agrees with several recent models and
conclusions of physicists and astrophysicists in arXiv, but they
weren't familiar with PI and didn't use probability-statistics, and
moreover our conclusions are limited to the positive acceleration
rather than to the negative acceleration or deceleration domain under
assumption of bivariate joint Gaussian/normality which is by no means
the only possible probability-statistics distribution or scenario.
The situation gets more complicated when the linear correlation
coefficient isn't near 1, and I'll leave that for homework for now. If
it's near -1 (which is like the 1 case except that X decreases as Y
increases almost linearly), the scenario is similar to the 1 case.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Acceleration of Universe As Acceleration of Probable Correlation 5: Increasing Variance |
24 Sep 2005 01:06:23 PM |
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From Osher Doctorow
Note that x and y are fixed in considering the limits here, so that the
result holds pointwise, that is to say for any fixed x, y.
When rho is very close to 1, the expression x^2 + y^2 - 2rho xy in the
exponential is very close to (x - y)^2 and the quantity f(x, y) is
approximately:
1) f(x, y) = approx 1/[(c1u)(exp(c2/u^2))]
where u = sqrt(1 - rho^2) and c1 = 2pi o1 o2 and c2 is (x - y)^2/2.
Now, to evaluate this consider the limit as x --> 0 of xexp(k/x) which
by rewriting x as 1/(1/x) is the infinity/infinity case of l'Hospital's
rule and by differentiation of numerator and denominator separately
approaches infinity since the 1/x^2 terms of the derivatives cancel and
exp(k/x) --> infinity as x --> 0 and k is a positive constant.
So the denominator of f(x,y) in (1) approaches infinity as u --> 0
which means that as rho^2 --> 1 with fixed x, y, o1, and o2. So
f(x,y) --> 0+.
So the more linearly correlated X and Y are, where one of X, Y is
random time (or arguably randomly increasing with time), the closer the
positive acceleration is to 0 with all else constant. It doesn't
matter whether X and Y increase together or are negatively proportional
like X = -ky where k > 0 as long as they're linearly correlated very
near to a straight line (correlation of 1 or -1).
Going in the opposite direction, as the magnitude of the linear
correlation coefficient rho decreases toward 0, f(x, y) increases and
we get a simlar result to the previous scenario of this part with
f(x,y) increasing in height but with very narrow "width" and "length"
mostly, so the acceleration increases but the increase lasts less and
less time.
A linear correlation coefficient of 0 corresponds to X and Y being
"uncorrelated", which doesn't imply statistical independence but the
latter is a particular case of uncorrelated and statistical
independence does imply uncorrelated.
So arguably the physical universe accelerates because its random
variables becomes uncorrelated with time but so far this is only known
for positive accelerations. Moving from negative to positive or 0
acceleration still remains to be studied unless it's a jump somewhat
like a topological change.
Osher Doctorow
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