You need to define what is "vector quantity". Is it the probabilistic variable which represents the vector length? If yes, then this is a reverse problem to what we solved in paper "teration-free Vector Orientation using Maximum Relative Entropy with Observational Priors" back in MaxEnt 2011. One more thing: it is also very important to define how your "vector quantity" relates with "vector components", i.e. what type of the constraint you want to enforce in your variational problem.

There are two options: either soft constraint on the vector length like in our mentioned paper or the constraint on probability cumulative distribution which is used in signal processing (see my paper's "The Kalman Filter Revisited Using Maximum Relative Entropy" section on nonlinear filter ).

So these two definitions are required before suggesting a solution for this problem.

Thanks Mr. Renaldas for your answer. The vector quantity in question here is the transverse velocity vector (t in the expression above is the modulus of this vectorial quantity). I would like to find out the corresponding PDF of each component knowing the law of the modulus. Yes I think that I should be more precise in asking the question. Could you please be of help? Thanks!

Yes of course (I assume that PDF = probability distribution function). Let us consider three dimensions and the Cartesian coordinates. Here for the distance r from the centre of rotation symmetry (since we are dealing with 'isotropy') one has r = (x^2+y^2+z^2)^{1/2}. Your PDF being isotropic, it means that it is a function of r (of course the dependence on r may be very complicated). Along the x direction one has y=z=0, along the y direction x=z=0 and along the z direction.

Let your PDF be f(r). Since it is normalized to unity, one has

\int dx dy dz f(r) = 1.

Now you could define

g(x) = \int dy dz f(r)

Since \int dx g(x) = 1, and since like f(r) the function g(x) is non-negative, g(x) is a probability distribution function. The function f(r) may be viewed as a conditional distribution function for x (for given values of y and z), and the function g(x) the total distribution function for x. Perhaps you have such total distribution function in mind. If so, note that similar to g(x) one can define h(y) = \int dx dz f(r), and k(z) = \int dx dy f(r). In the case where the planar areas where the points (x,y), (x,z) and (y,z) are defined are identical, naturally g(s) =h(s)=k(s), where s a real quantity.

If the region of integration (that is the three-dimensional region where x, y and z are defined) is spherical, one may define the function

p(r) = 4\pi r^2 f(r)

which is the probability distribution function for distance r, defined over r \ge 0. The function p(r) is non-negative and its integral over [0,R] (the radius of the integration region) is equal to 1.

Dear Abdallah, I wrote the above comment without being aware of the earlier comment and response (your response) on this page. My above comment remains applicable if you replace everywhere in my comment r with t and x, y and z with t_x, t_y, and t_z, where (t_x,t_y,t_z) are the Cartesian coordinates of the vector whose amplitude is t.

Behnam, pdf of r does not necessarily mean the equation r = (x^2+y^2+z^2)^{1/2} exists between four probabilistic variables x, y, z and r. In real life a person might mean that this equation exists between the means of probabilistic variables, which is a different thing and thus this problem should have a different constraint than you describe. One has to be very explicit what the constraint means. In your case r^2=x^2+y^2+z^2 (if all four are probabilistic variables, but not their means, in your head probably means \int dx dy dz dr (r^2-x^2-y^2-z^2) f(r,x,y,z)=0, and this says the following : "the sum of variances/uncertainty of vector components is equal to the variance/uncertainty of the vector length for pdf of f(x,y,z,r)". But whether this assumption is right for Abdallah real life problem we do not know, because we do not know his context. In spite of the constraint over the means which has not yet been discussed by us, there is third possibility: maybe the transformation r^2=x^2+y^2+z^2 exists not between the probabilistic variables, but between their cumulative distribution functions like in signal processing. As I mentioned look at my paper on Kalman filter the section called "Nonlinear filter", or feel free to look into the "Handbook of Formulas and tables for signal processing" section "34.7 Function of Two Random Variables", especially Examples 2 and 3 with Z=X^2+Y^2.

Dear Renaldas, my above answers are based on the limited information provided by the question. Very generally, however, if by |r| we are meaning the Cartesian norm of the vector r with Cartesian coordinates (x,y,z), then |r| = (x^2+y^2+z^2)^{1/2}. Now, for x, y and z being independent stochastic variables (conform the statement of the problem that refers to a vector-valued stochastic variable), |r| = (x^2+y^2+z^2)^{1/2} is NOT an independent stochastic variable. Therefore, denoting the distribution function under discussion as f(x,y,z,|r|) is utterly meaningless (the function in question is over-specified, since to any combination {x,y,z} corresponds a unique value for |r|). Besides being meaningless, the latter function is not what one calls 'probability distribution function'. Perhaps the question is about something different than what I have understood, but if I have understood the question correctly, then I am afraid that I strongly disagree with you.

Since you talk about a constraint, a relevant question is the following: given the function g(x,y,z) and given the statistical distributions of, say x and y (these variables may have identical distribution functions -- for instance, both may be normally distributed with identical average and variance), what is the statistical distribution of z under the condition g(x,y,z) = a, where a is a constant, or some stochastic variable of specified distribution? The equality g(x,y,z) = a constrains the variation of z for independent variations of x and y.

I think it is the responsibility of the person posing the question to clarify the problem. As outsiders to the question, it does not make much sense that we argue about issues that are likely to prove to be irrelevant to the question at hand.

Dear Renaldas&Behnam. Please here is again my question with more details. Given the probability density function of the modulus of a given vectorial quantity (here is a the transverse velocity of a particle embedded in a turbulent flow). The modulus is just a Cartesian norm as mentioned by Behnam. Since my turbulence is homogeneous and isotropic, the PDFs of each component would be equal and they are all symmetric with zero mean (they take values in all the axis of reals), with a large tail as obtained by numerical calculations. I would like here to find out whether theoretical evidences are possible in order to get such a distribution based on the fact that the PDF of the Euclidean norm of this vectorial quantity is known. May I add more information to the question? Thanks!

Behnam, I agree that context was missing. I also understand that we can use marginalization like you suggested.

Abdallah, now when we know that a vector is a function of time this problem becomes even more interesting. Could you tell us what you are measuring: either x,y and z components or the modulus of the vector? Also why do you need pdfs of x, y and z? Are your measurements discrete values, is discretization interval the same for all time series?

Dear Abdallah, but what you are now asking is completely different from the main question of this page. Your present question is whether one can come up with a mathematical model that reproduces your numerically-calculated distribution function. This is actually an inverse problem, which I do not believe can have a unique solution. Many apparently different random precesses generate similar distribution functions. But I think this is a side issue. You have actually obtained the distribution function at issue through a numerical calculation, that is you already know your model. This leads me to think that I actually do not understand your question.

Incidentally, since you are referring to your distribution function as having a "large tail" (is this by the way the same distribution function that on the logarithmic scale does not have a linearly-decaying tail? -- we had a discussion about that on a separate page), does it actually have, e.g., a finite variance? In any case, you may wish to look into the subject matter of 'Heavy-tailed distribution', and in particular 'Lévy flight'. You may begin with Wikipedia for orientation.

Renaldas&Behnam the reason why I am looking to the PDF of each component is to make a prediction with the multifractal (MF) model and accurately reproduce the numerically observed PDFs for different increments (scales). In the MF formalism, we need the PDF of the LARGE SCALE transverse velocity fluctuation for each component ( = the quantity that I am looking to now). The PDF mentioned above is the PDF of the LARGE SCALE transverse velocity fluctuation for the MODULUS. Once this PDF is known, I could then, using the MF model, fit my numerical results. I don't know if I am clear now? I have put in attach there a numerical result of one component obtained for a small increment in which we clearly notice the intermittent character of turbulence in small scales.

In resumé, knowing that the PDF of the LARGE SCALE transverse velocity MODULUS is given above, is it possible to get the PDF of the LARGE SCALE transverse velocity for each component? If this could be done, it is easy to reproduce the shape observed in this plot for any small scale. Thank you!

Here is the solution (in the following, x, y and z represent respectively v_x, v_y and v_z). You have already calculated P(z), displayed in the above pdf file. Let the full distribution function be Q(r), where r = (x^2+y^2+z^2)^{1/2}. With reference to my earlier comments on the page:

P(z) = \int dx dy Q(r).

Now describe the two-dimensional vector whose Cartesian coordinates are x and y in the cylindrical coordinates u and \phi, where u varies over [0,\infty) and \phi over [0,2\pi). One has r = (u^2+z^2)^{1/2} and thus

P(z) = 2\pi \int_{0}^{\infty} du u Q((u^2+z^2)^{1/2}).

Let w = (u^2+z^2)^{1/2}. One has dw/du = u/w, so that

P(z) = 2\pi \int_{|z|}^{\infty} dw w Q(w).

Let now z>0. Differentiating both sides of the above equation with respect to z, one obtains: dP(z)/dz = -2\pi z Q(z). For z

Thanks Mr. Behnam for this interesting discussion, I have got exactly what I was looking for!

best

a

You are welcome Abdallah.

Dear Abdallah, thank you for your message and the attached pdf file. To begin with, the function P(z) = 4π1/2 (1 + z2) exp(-z2), in your Eq. (8), is not properly normalized; integrating it with respect to z over (-∞,∞) yields 6π. This is not strange, since integrating the function Q(t) in your Eq. (6) over [0,∞) one obtains 1. In reality, one must have 4π t2 Q(t) integrated over [0,∞) equal to 1 (see my above note on this page as well as your Eq. (2) in the pdf file). It follows that the function Q(t) in Eq. (6) is incorrect. The correct Q(t) must be Q(t) = exp(-t2)/π3/2, which results in P(z) = exp(-z2)/π1/2, as expected.

Thank you Dr. Behnam for your detailed answer. It looks good now!

You are welcome Abdallah.