1) Such cases shouldn't be too rare. It should even be rathre frequent in any case where a function is non-linear in higher dimensions and you pass through a valley on a respective path. Just one common example: engine speed as function of consumption at same required load. Mind the change of dependent vs. independent variable relative to common display habits. But it depends on an experimental setup what can be considered as independent and dependent.

2) The ambiguity is only present if you disregard a basic functional dependency that re-creates a connection between the two values. Linear correlations will fail, of course, but you might describe the curve with an additional parametric variable?

Cheers,

Petra

Dear Emilio

From statistical point of view, there is few data. Given data are non-linear, you can not use the linear methods. But you could try find a transformation to carry the data to a linear behavior. For example if you use something like Q'=Q-2.76 and Y'=(Y-4.62)^2, you can get some of linearity with a correlation of 0.99. But I repeat there are too few data.

Thanks Petra and Henry for your replies. Both of you agree that linear methods do not help in this case.  

About Petra´s points, your engine example is very clear and proper; your words about "connection between the two values" are well founded. In my case, the experimental setup was to use Q=3-2*x+4*x2 and  Y=5-x-2*x2 for the ten decil points (0; 0.1, 0.2 ... 0.9, 1.0) for X. In this case X links Q and Y, and they are not parametric, just polynomic expressions.

About Henry´s answer, we may generate thousands of points with this setup and results do not change too much. Transforming data to save linearity model is possible but quite risky as a research method, in my opinion.

Thanks again for your reaction, emilio

ok I now understand. The data come from known parametric equations, where the parameter is x. I think you have is a deterministic model. i.e, you know always how is the behavior of data. I do not find uncertainty to use statistical methods. What do you want to explain from data?

Henry, indeed I used the computer as a lab, but the generated data is non parametric because it does not contain parameters, but only coefficients and exponents on variable x; it is made of polynomic expressions. They are linked through X, the receptor cummulative population. Q is related to Y, Y to Q, and both to X. None of them is an independent variable that makes the other dependent; they are simply related because in the moment I declare the table is the dataset, we may graph X vs Q, Q vs. Y, Y vs. Q, etc. With Q,Y data we may graph them and observe their curved retalionship at once, with some polysemic interval. These cases only admits nonparametric analysis. You may design many examples made of more complex shapes and declare them as your data.

In nature we observe winds with positive velocities and diferent orientation, with their own temperature, humidity, pressure, sunlight, etc. When they meet they create an unsteady situation that must be solved someway. Many times they dissipate, other times they resolve as rain, as a thunder, as a tornado, as snow, etc.. This is a polysemic situation for us, not for nature that finds something to grasp itself and discharge its energy getting a new more quiet balance. Those events contain a natural random component with several options and non-linear behaviour, that we hardly understand.

I have strong doubts about linear models and correlations to solve  simpler cases. So I seek some help and new ideas about it. This motivated my question. OK, my regards.emilio

Dear Emilio, by elimination of variables your relation is found to be:

1/4*x^2+x*y+y^2-7*x-12*y+155/4 = 0

with x=Q and y=Y as abbreviations, which seems to be a parabola curve.

See attachment graph.

Hi Demetris. Nice to read you again and thanks for your time and applied math method to the setup. Graph looks like a rotated parabolic curve and once that the limiting values of [o

Depending on the nature of the underlying measurements I would be very cautious to extend/extrapolate the given data, especially so at the very short curved end. It might imply hypotheses for practically senseless regions.

I agree Petra. That is the reason why it were added the limits of the cummulative receptor population fraction as 0

Btw, your original curve (section) could be interpreted as a function, at least if seen from the perspective of an exchange of dependent vs. independent variable. With Demetris's extrapolation this interpretation is no more possible.

Perhaps, Petra. But the result is that X, Q and Y are mutually dependent. What Demetris made was a mathematical excercise without thinking about the limits present in dataset, he only used equations to get a new one. It helps because it matches dataset inside the region of its values. The responsibility of analyst is also to understand the limits that make plaussible the model, Demetris only showed the isomorphism between dataset and mathematics and graphed his results for a wider range of Q,Y values, but we may ask him about the point. Your comment is relevant Petra, because you called the attention to this important point that shows the dangers of uncontroled extrapolation and the need to be precise about the limits that dataset contains and must be followed by any model that seeks to represent data relationships. emilio