Researchers in different engineering fields mostly come to me for fitting their obtained data in laboratory, by using some mathematical methods. Their most complain to me is that these types of methods, for instance a simple linear regression or least square, do not express the treatment of their data as well as they appear in real in the nature.
There is a sure way to express the relation between two groups of data. Try the following:
1. Let A and B be sets of data. Identify the features for a datum in $A\cup B$.
2. Give a probe function $\phi$d (mapping from a piece of data to a real number) that can be used to extract feature values for the elements of $A\cup B$. Let
\[
\Phi = \{\phi_1, …, \phi_n\}
\]
and let $\Phi(x) = (\phi_1(x),…., \phi_n(x)\}
be a feature vector that describes datum x.
3. Define the descriptive similarity distance D(A,B) between A and B to be
\[
D(A,B) = inf{d(\Phi(a),\Phi(b)): a\in A, b\in B\}
\]
where $d(\Phi(a),\Phi(b))$ is the Manhattan distance between the pair of feature vectors.
You may perform the Pearson correlation analysis between the groups of data, for dependency analysis between the data sets.
Dear Marzieh (and other participants),
I agree with Joydip that the more popular measure of association between paired data is the Pearson correlation and related techniques as regression. However, I should remark that Pearson correlation has been designed for real random variables. I.e. the way of meassuring differences is by subtraction (absolute scale) and, in principle, they are able to attain both possitive and negative values. When Pearson correlation is ussed to measure association of variables not fullfilling these conditions, you can obtain extrange and confussing results. This phenomenon was detected by K. Pearson in 1896 and has been called "spurious correlation". See for instance the book by J. Aitchison (1986) [reprinted in 2003 (the Blackburn Press).
Although, I do not know which kind of data you are dealing with, I guess that sometimes they are compositional (expressed as percentages, concentrations, proportions). In these cases the P-correlation results are meaningless or spurious. Less dramatic, but still inadequate, is to correlate positive variables which scale is thought relarive (ratio scale). This cases are easily transformed into absolute scale by taking log's. I would recommend to check whether the variables you are dealing with are compositional, positive, directional... It may be the origin of bad results in any P-correlation analysis.
On the other hand, Pearson correlation only capture "linear" relations between real variables. In some cases, non-linear relation can be captured using other correlation coefficients like Sperman correlation or Kendall's-tau.
Least squares curve fitting can cope with complicated functional dependences. But, at least, two question should be addressed:
--How the residuals are going to be meassured? In a standard multiple regression, residuals are measured in the scale and along the values of the response variable. If the scale of the response is not absolute (real response variable), you are going again into problems. A tranformation of such a variable is needed (for compositional data, log-ratio transformations; for positive-relative scale data, log transformation).
--Are both (real) variables random (or affected by measurement errors)? If yes, ordinary regression may be not a suitable approach. Regression with orthogonal residuals (e.g. principal component analysis), regression on the principal axis and other methods may be useful.
Hope this comment is useful.
I am totally agree to Juan Jose Egozcue about the Pearson correlation limitations.
Similarity measures and clustering methods may be some time useful with respect to purpose of data analysis.
There is a sure way to express the relation between two groups of data. Try the following:
1. Let A and B be sets of data. Identify the features for a datum in $A\cup B$.
2. Give a probe function $\phi$d (mapping from a piece of data to a real number) that can be used to extract feature values for the elements of $A\cup B$. Let
\[
\Phi = \{\phi_1, …, \phi_n\}
\]
and let $\Phi(x) = (\phi_1(x),…., \phi_n(x)\}
be a feature vector that describes datum x.
3. Define the descriptive similarity distance D(A,B) between A and B to be
\[
D(A,B) = inf{d(\Phi(a),\Phi(b)): a\in A, b\in B\}
\]
where $d(\Phi(a),\Phi(b))$ is the Manhattan distance between the pair of feature vectors.
A detailed introduction to similarity distance (mentioned in my previous post) can her
found in a new book on visual patterns (see the attached cover for this book, now in press, which will be available early in 20140. See, also, my articles on near sets downloadable from my RG page.
Dear @Marzieh, in addition to what @James and @Juan mentioned, you can use the so-called least squares methods that enables you to find the polynomial of degree n that has the least distance from the data, or you can find approximation of exponential behavior or other type. Splines works also well, however, in this case you should have information about derivatives.
Dear @Prof. Dr. Rababah, many thanks for your response. Actually, the aim of my question was to find out any new way other than the existing commonly used methods such as least square method and regression as you mentioned above. Since in some cases as I explained above these mentioned methods are not satisfactory enough. So I appreciate any new suggestion for curve fitting which works better rather than the common existing ways.
Another way to express the relationship between two groups of data is to consider a bipartite graph showing the connections between the data. See, for example,
Y. Zhao, Bipartite graph tiling:
http://www2.gsu.edu/~matyxz/publications/bip_til.pdf
This paper has some interesting propositions and proofs.
A good overview and introduction to bipartite graphs is given by
J. Dompierre, Bipartite graphs, Laurtentian Univ., 2011:
http://www.cs.laurentian.ca/jdompierre/html/MATH2056E_W2011/cours/s9.2_graph_bipartite.pdf
For a way to use multicolour Ramsey numbers in bipartite graphs, see
F.R.K. Chung, On multicolor Ramsey numbers for complete bipartite graphs,
J. of Combinatorial Theory (B) 18, 1975, 164-169:
http://www.math.ucsd.edu/~ronspubs/75_05_multicolor.pdf
In general you can try to find, if there exist, correspondences or functions or linear transformations between the two sets. It is not easy, since you must know many of the hard corps of Maths, but, at least you get rid of regression and 'least norms analysis'. Finally, if you want to find just a measurable 'object', you can use non parametric methods (For example for an inflection point see my profile and R package inflection).
@Demetris Christopoulos:
One approach to finding correspondences between to groups of data is to consider isometries. Let X, Y be two metric spaces. Then define a bijective map f on X to Y that preserves distances so that
\[
d_Y(f(x), f(y)) = d_X(x,y),
\]
where x,y are members of $X$, f(x), f(y) are members of Y, and d is a distance function. If X can be transformed into Y, then X and Y are isometric. A nice example of an isometry in the attached image from
http://en.wikipedia.org/wiki/Isometry
Dear Professor Dr. James F. Peters
Thanks for your interesting responses. About your last comment, respectfully I have a question. The idea to find an isometric mapping between to sets is nice. My question is about the type of relation we will find between two sets using this isometry. In other words, if I have understood well, in case that we find such this isometry, we will find a structural relation between our assumed sets rather than a mathematical formula. Am I right?
Dear Marzieh,
Yes, you are correct! Instead of a formula, try doing the following:
1. identify two or more sets of interest.
2. identify probe functions that can be used to extract feature values from the points in the sets from step 1.
3. Let X. Y be sets of interest. Start with X and define a metric topology on X endowed with two or more proximity relations. In other words, endow X with a relator containing proximity relations. Do the same with Y. Now X and Y have have structures induced by the relator.
4. Define a simiilarity distance function D(X,Y) that can be used to compare the similarity distance between subsets in X and Y.
5. Assume that set A in X and set B in Y are sufficiently near. Then define a descriptive isometry f on X into Y, i.e., f: X --> Y.
After step 5, it will be possible to investigate isometries between subsets in the set of interprets identified in step 1. The basic approach is explained in chapters 1, 4 and 12 in my book published this year:
J.F. Peters, Topology of Digital Images, springer, 2014
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