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Thread Subject:
Can PDF be more than one??

Subject: Can PDF be more than one??

From: kumar vishwajeet

Date: 22 Sep, 2010 15:08:05

Message: 1 of 9

Can anyone tell me whteher Probability density function can be more than one. In my opinion, it can be.Because PDf is the derivative of Probability distribution function with respect to change in random variable. If the change in the distribution function is more than the change in the corresponding value of random variable, the PDF will be more than one. Am I correct??
I am asking this question with relevance to Lorenz System. I am trying to plot the pdf of this system. After some time the PDF orient themselves in a particular direction thus streching more in that direction. The value of the peak, meanwhile, goes beyond one reaching upto 6 and then suddenly drops to less than one and keeps on dropping till it becomes almost zero (10^-126).Is this a normal behavior for the PDf of a Lorenz System?/

Thanks

Subject: Can PDF be more than one??

From: Wayne King

Date: 22 Sep, 2010 15:35:21

Message: 2 of 9

"kumar vishwajeet" <kwzeet@gmail.com> wrote in message <i7d64l$36e$1@fred.mathworks.com>...
> Can anyone tell me whteher Probability density function can be more than one. In my opinion, it can be.Because PDf is the derivative of Probability distribution function with respect to change in random variable. If the change in the distribution function is more than the change in the corresponding value of random variable, the PDF will be more than one. Am I correct??
> I am asking this question with relevance to Lorenz System. I am trying to plot the pdf of this system. After some time the PDF orient themselves in a particular direction thus streching more in that direction. The value of the peak, meanwhile, goes beyond one reaching upto 6 and then suddenly drops to less than one and keeps on dropping till it becomes almost zero (10^-126).Is this a normal behavior for the PDf of a Lorenz System?/
>
> Thanks

Hi
Yes, the value of a PDF can exceed one (and they often do). The integral of the pdf over its support cannot exceed one. I cannot speak to what is normal behavior for the PDF of a Lorenz system

Wayne

Subject: Can PDF be more than one??

From: someone

Date: 22 Sep, 2010 15:42:04

Message: 3 of 9

"kumar vishwajeet" <kwzeet@gmail.com> wrote in message <i7d64l$36e$1@fred.mathworks.com>...
> Can anyone tell me whteher Probability density function can be more than one. In my opinion, it can be.Because PDf is the derivative of Probability distribution function with respect to change in random variable. If the change in the distribution function is more than the change in the corresponding value of random variable, the PDF will be more than one. Am I correct??
> I am asking this question with relevance to Lorenz System. I am trying to plot the pdf of this system. After some time the PDF orient themselves in a particular direction thus streching more in that direction. The value of the peak, meanwhile, goes beyond one reaching upto 6 and then suddenly drops to less than one and keeps on dropping till it becomes almost zero (10^-126).Is this a normal behavior for the PDf of a Lorenz System?/
>
> Thanks

The integral of the pdf from -infinity to +infinity must be one. This does not put a limit on the "peak" of the pdf.

Consider a uniform RV centered on 0 from -1/4 to +1/4. The "pdf" is a flat line of height 2 between these two limits.

So, (if I understand your question) yes, the pdf can be more than one.

Subject: Can PDF be more than one??

From: kumar vishwajeet

Date: 22 Sep, 2010 16:08:05

Message: 4 of 9

"someone" <someone@somewhere.net> wrote in message <i7d84c$jrf$1@fred.mathworks.com>...
> "kumar vishwajeet" <kwzeet@gmail.com> wrote in message <i7d64l$36e$1@fred.mathworks.com>...
> > Can anyone tell me whteher Probability density function can be more than one. In my opinion, it can be.Because PDf is the derivative of Probability distribution function with respect to change in random variable. If the change in the distribution function is more than the change in the corresponding value of random variable, the PDF will be more than one. Am I correct??
> > I am asking this question with relevance to Lorenz System. I am trying to plot the pdf of this system. After some time the PDF orient themselves in a particular direction thus streching more in that direction. The value of the peak, meanwhile, goes beyond one reaching upto 6 and then suddenly drops to less than one and keeps on dropping till it becomes almost zero (10^-126).Is this a normal behavior for the PDf of a Lorenz System?/
> >
> > Thanks
>
> The integral of the pdf from -infinity to +infinity must be one. This does not put a limit on the "peak" of the pdf.
>
> Consider a uniform RV centered on 0 from -1/4 to +1/4. The "pdf" is a flat line of height 2 between these two limits.
>
> So, (if I understand your question) yes, the pdf can be more than one.

OK.Thanks. But does anyone have any idea about the behaviour of the PDF of Lorenz System???

Subject: Can PDF be more than one??

From: Roger Stafford

Date: 22 Sep, 2010 19:09:04

Message: 5 of 9

"kumar vishwajeet" <kwzeet@gmail.com> wrote in message <i7d64l$36e$1@fred.mathworks.com>...
> Can anyone tell me whteher Probability density function can be more than one. In my opinion, it can be.Because PDf is the derivative of Probability distribution function with respect to change in random variable. If the change in the distribution function is more than the change in the corresponding value of random variable, the PDF will be more than one. Am I correct??
> I am asking this question with relevance to Lorenz System. I am trying to plot the pdf of this system. After some time the PDF orient themselves in a particular direction thus streching more in that direction. The value of the peak, meanwhile, goes beyond one reaching upto 6 and then suddenly drops to less than one and keeps on dropping till it becomes almost zero (10^-126).Is this a normal behavior for the PDf of a Lorenz System?/
>
> Thanks
- - - - - - - - - - -
  When you say "Lorenz System" do you refer to the Cauchy-Lorentz distribution? With a variable scale parameter, that pdf can certainly exceed one - just have the "scale factor" less than 1/pi. See the site:

 http://en.wikipedia.org/wiki/Cauchy_distribution

  If you refer to the "Lorenz attractor" with its system of three ordinary differential equations which can exhibit chaotic behavior, there doesn't seem to be any probability density function associated with it. It is a deterministic system. Can you be more specific about what you mean by the "Lorenz System".

Roger Stafford

Subject: Can PDF be more than one??

From: kumar vishwajeet

Date: 22 Sep, 2010 20:41:20

Message: 6 of 9

"Roger Stafford" <ellieandrogerxyzzy@mindspring.com.invalid> wrote in message <i7dk8g$hsg$1@fred.mathworks.com>...
> "kumar vishwajeet" <kwzeet@gmail.com> wrote in message <i7d64l$36e$1@fred.mathworks.com>...
> > Can anyone tell me whteher Probability density function can be more than one. In my opinion, it can be.Because PDf is the derivative of Probability distribution function with respect to change in random variable. If the change in the distribution function is more than the change in the corresponding value of random variable, the PDF will be more than one. Am I correct??
> > I am asking this question with relevance to Lorenz System. I am trying to plot the pdf of this system. After some time the PDF orient themselves in a particular direction thus streching more in that direction. The value of the peak, meanwhile, goes beyond one reaching upto 6 and then suddenly drops to less than one and keeps on dropping till it becomes almost zero (10^-126).Is this a normal behavior for the PDf of a Lorenz System?/
> >
> > Thanks
> - - - - - - - - - - -
> When you say "Lorenz System" do you refer to the Cauchy-Lorentz distribution? With a variable scale parameter, that pdf can certainly exceed one - just have the "scale factor" less than 1/pi. See the site:
>
> http://en.wikipedia.org/wiki/Cauchy_distribution
>
> If you refer to the "Lorenz attractor" with its system of three ordinary differential equations which can exhibit chaotic behavior, there doesn't seem to be any probability density function associated with it. It is a deterministic system. Can you be more specific about what you mean by the "Lorenz System".
>
> Roger Stafford

I am talking about the Lorenz attractor. You are correct that the system is considered deterministic. But this attractor behaves in different way based on the initial conditions. In my exapmle, the initial conditions are not deterministic and there is noise associated with the third state as well as to the output of the system. As such the system is modelled as uncertain.I am plotting the PDF for this system.Did you get my question??

Subject: Can PDF be more than one??

From: Roger Stafford

Date: 22 Sep, 2010 22:25:05

Message: 7 of 9

"kumar vishwajeet" <kwzeet@gmail.com> wrote in message <i7dplf$g2o$1@fred.mathworks.com>...
> I am talking about the Lorenz attractor. You are correct that the system is considered deterministic. But this attractor behaves in different way based on the initial conditions. In my exapmle, the initial conditions are not deterministic and there is noise associated with the third state as well as to the output of the system. As such the system is modelled as uncertain.I am plotting the PDF for this system.Did you get my question??
- - - - - - - - -
  The Lorenz oscillator itself is deterministic, so all probabilistic questions of its behavior are dependent entirely on the statistics of initial condition variation and parameter variation, and you haven't given us any information about those. Given that omission, how could you possibly expect a valid answer to your question?

Roger Stafford

Subject: Can PDF be more than one??

From: kumar vishwajeet

Date: 23 Sep, 2010 01:10:12

Message: 8 of 9

"Roger Stafford" <ellieandrogerxyzzy@mindspring.com.invalid> wrote in message <i7dvo1$kat$1@fred.mathworks.com>...
> "kumar vishwajeet" <kwzeet@gmail.com> wrote in message <i7dplf$g2o$1@fred.mathworks.com>...
> > I am talking about the Lorenz attractor. You are correct that the system is considered deterministic. But this attractor behaves in different way based on the initial conditions. In my exapmle, the initial conditions are not deterministic and there is noise associated with the third state as well as to the output of the system. As such the system is modelled as uncertain.I am plotting the PDF for this system.Did you get my question??
> - - - - - - - - -
> The Lorenz oscillator itself is deterministic, so all probabilistic questions of its behavior are dependent entirely on the statistics of initial condition variation and parameter variation, and you haven't given us any information about those. Given that omission, how could you possibly expect a valid answer to your question?
>
> Roger Stafford

Here is the system:-
state eqn:-(continuous)
-------------
x1(dot) = alpha(-x1+x2)
x2(dot) = beta*x1-x2-x1*x3
x3(dot) = -gamma*x3+x1*x2+w(t) where E{ww'} = Q = 1

output:-(discrete)
zk = sqrt(x1(tk)^2+x2(tk)^2+x3(tk)^2)+v; xi(tk) is the state xi at time tk. i=1,2,3

E{vv'} = 1

Initial condition:-
x(t0) ~ 0.9*N([-0.2 -0.2 8]', sqrt(35)*eye(3)) + 0.1*N([0.2 0.2 8]',sqrt(0.35)*eye(3))

Subject: Can PDF be more than one??

From: kumar vishwajeet

Date: 23 Sep, 2010 01:49:21

Message: 9 of 9

"Roger Stafford" <ellieandrogerxyzzy@mindspring.com.invalid> wrote in message <i7dvo1$kat$1@fred.mathworks.com>...
> "kumar vishwajeet" <kwzeet@gmail.com> wrote in message <i7dplf$g2o$1@fred.mathworks.com>...
> > I am talking about the Lorenz attractor. You are correct that the system is considered deterministic. But this attractor behaves in different way based on the initial conditions. In my exapmle, the initial conditions are not deterministic and there is noise associated with the third state as well as to the output of the system. As such the system is modelled as uncertain.I am plotting the PDF for this system.Did you get my question??
> - - - - - - - - -
> The Lorenz oscillator itself is deterministic, so all probabilistic questions of its behavior are dependent entirely on the statistics of initial condition variation and parameter variation, and you haven't given us any information about those. Given that omission, how could you possibly expect a valid answer to your question?
>
> Roger Stafford

The values of alpha, beta and gamma are: 10,8/3 and 28 respectively

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