Dear Bishwajit, can you be more specific? Do you have a predefined model (polynomial, power, exponential) to fit to the data? The data are (x, y) or (x, y --> the 2D plane plus a dependent variable, z)? Could you post an example?

Regards

Popular choices are loess and regression splines, but as Luis says, more information is necessary for people to give more constructive responses.

Points on 2D surface can be referred to their X and Y coordinate and and as piloted  on the graph paper, of course suitably reducing scale. Inspection of plot will show a trend and a function corresponding to that trend may easily be fitted, otherwise a B-Spline curve may be fitted.

try to use simple functions (1s or second order polynomial) then higher order polynomial function. To test if the fit gain is statsistically significant use the F-test

I would avoid higher order polynomials unless you have a theory that predicts say a fourth degree polynomial. A few points changed on one side of a plot can alter the entire in curve.

Thanks everyone for your responses.

To be more elaborate,

I have a set of 2D points (X1, Y1), (X2, Y2)........(Xn, Yn). In case of 3D we can have the Z component value as a Constant (1 or 0). What i need is a algorithm / function / formula which takes these points and gives an output as a curve function.

Is it possible to predict if the set of point can make a curve or a line, whichever appropriate? if so, can we have a way to find if the set of points makes a line more appropriate than a curve or vice verse.  Once we know what the set of points is likely to form, we can then derive the required equation of line / curve etc.

Thanks.

Cubic Spline 3D interpolation is solution of your problem using Trace.

See:http://www.originlab.com/index.aspx?go=Products/Origin/DataAnalysis/Mathematics/InterpolationAndExtrapolation#4

Dear Biswajit,

You can fit the suitable regression line to your data set by least square criteria. You can get the details in the book "Statistics and Data Analysis in Geology" by J. C. Davis. I am attaching one of my publication in "Mathematical Geosciences" for your reference.

It depends on your data. Are they satisfy a functional form or are they just a planar closed curve? How do their scatter plot seem to be?

Biswahit, may you write here a 10 points real data-set of (X,Y). Perhaps I can obtain a non perfect but acceptable fitting equation. (It depends on specific dataset)  emilio

"If few of the points have deviated outside the actual region then it can be ignored."  Before ignoring the outliers it would be worth considering their informational value. 

It can't be done without further information - in the sense that there isn't a unique solution (any number of curves will fit the points exactly). You either have to constrain the solution (e.g., to a linear model) or add further information to get a unique solution.

@Thom,

Please see the link, how is it possible?

http://www.originlab.com/index.aspx?go=Products/Origin/DataAnalysis/Mathematics/InterpolationAndExtrapolation#4

Mohammad - I'm not sure I understand your question. Yes - you can fit a curve to a set of points. The curve you get depends on the information or constraints you use - there is not in general a magic way to go from the points back to the unique function that generated it.

The Lagrange polynomial: http://en.wikipedia.org/wiki/Lagrange_polynomial is your friend. This model has as many parameters as there are data points ("cannot scale"). The whole point about mathematical science (for example, theoretical physics) is to find a model with fewer parameters which explains the data, and hopefully generalizes / predicts unseen data. According to G. Chaitin, as early as the 17th century, Leibniz made remarks on a similar problem (ink dots on paper) and accidentally gave birth to "complexity theory" , a branch of theoretical computer science and of pure mathematics.