Without any additional nformationof the factors - it is simply IMPOSSIBLE.

Thus, if any inference is expected, some additional structure of the processes is to be known.

For instance, if real valued processes X and Y are known as independent stationary gaussian, then their product is stationary too, but not gaussian, and there is a chance to find some procedures detecting e.g. some features of their two correlation functions. Obviously, one cannot separate their sizes.

More precisely, for two independent gaussian variables X and Y with pd-s N(m,v) and N(n,u), respectively, the basic parameters of X*Y we have:

E(X*Y)= E(X)* E(Y)=m*n, E{(X*Y)^2} = E(X^2)* E(Y^2) = (m^2+v)(n^2+u),

hence the variance equals D^2(X*Y) = v*u + u*m^2 + v*n^2.

As we see, some additional knowledge is needed if m and n are to be found.

you can look at the corelation of these random variables

First there is a need to find the correletion between the parameters and how much does they influence on each other as stochastic programming models deals with an unknown parameters. Thus it provides a region within which the impact of parameters on one another can be determined.

Dear Prof. Joachim Domsta thank you very much for your help

I have the correct values of one of the processes. Also I have the real values of the output and one of the processes. therefore, I can find the real values of the other process too. this way I can find their distributions and some statistical parameters. But, these distributions are not determined distributions. Now, with this information, is it possible to find the parameters of the distributions?

Dear Asieh,

please explain what doesit mean "correct values". Are these values withous random perturbation?

Moreover:

Is output equal the product?

What are real values of the process?

I'm from now assuming that you have two-dimensional process (X(t), Y(t)) observed at some instants t1, t2, ,,, tn or the whole trajectory a

Joachim Domsta

I have two variables(for example, t and x). The output (for example y) is the product of these two variables. I have the correct value of t,x,y. But in my experiment all these values have polluted with noise. I can measure the noisy variable t and the noisy output at each point. Therefore, I can also find the noisy value x at each point.

I think t and x do not have direct impact on each other. But maybe some other uncontrollable things influence both t, and x.

Asieh Daneshi

Thank you for the suggestion of indpendence of the two coordinates.

Thus the next two steps are to state qualitatively at least

1. which type of the time dependence of the perturbations is the most acceptable, e.g. either

-- are the perturbations stationary processes? in particular:

-- are they built of independent say additive perturbation created with an observation at each instant separately? OR

-- are they evolving say like a Brownian motion, like some explosion etc.?

2. which would be the type of dependence on time of the processes if they were unperturbed by the noise, e.g. exponential, polynomial, logarithmic or some else and increasing/decreasing etc.

Dear Prof. Joachim Domsta

I really appreciate your help.

perturbations are like Brownian motion, but I think the zero mean Gaussian distribution is not suitable for explaining them, because the perturbed variable is the sum of time intervals, and can't get negative values. Then, assume that the perturbation behaves like Brownian motion but is monotonically rising.

I attached a figure showing one group of the data. The horizontal axis shows one of the parameters, and the vertical axis shows the output. different colors are used to separate the other parameter.

Briefly: And what differentiates the clusters? Best JoaD

Another quick question:Isn't it the case when one color consists of n lines of pairs values taken at instant nr 1, 2, ... k. If this is the case then n would be about 50 and k=6 for the red clouds, and n about 40 and k=3 for blue clouds.

It is VERY important to distiguish such a case from the one of choosing for each instant separately n species and measure them, since then the results between instant are much more independent.

Best, JoaD

Dear Prof. Joachim Domsta

Thank you again for your great support

I think it is better to explain the whole process.

I have two inputs, say x, y and an output, say z. one of the inputs takes three different values, and the other input changes in pseudo-random fashion, meaning that it takes random values around some specific values (that is because there are clusters in the figure).

I know the exact value of the inputs, but I don't know how human subjects have percepted these values. Of course humans' perception has some noise or perturbation.

I have recorded the outputs estimated by humans, and the estimated x values.

the relation between inputs and output is like this:

z=xy

with noise:

z1=(x1+noise(x))(y1+noise(y))

which noise(x) and noise(y) come from the subjects' uncertainty about their estimation.

As I have the estimated output, and the estimated x values, I can find the estimated y values at each point. Then I can find the y1 value from the regression and through this find the noise(y1). Am I right?

what can I do about the other input?