Dear PushKar

I completely agree with Marek Gutowsky about the two meanings that the operation of fit of a series of data can take. Just to complete Marek's answer:

If we have a theory (the model, a known functional relationship) that connects the observable data space to the model parameter space we we are solving an inverse problem. In this first case the objective is to obtain the unknown parameters in our theory (ie the unknown coefficients of a known functional relationship).

In the case in which there is not a theory that connects the data space to the model space we will talk about the simple regression operation whose objective is to carry out an interpolation and / or extrapolation of the same data consistent with the trend of available data and not of a theory.

However, in both cases it should be advisable to have the error on data. The fit will depend on a misfit function (depending on the differences between observed and predicted/regressed data). The availability of the data errors, in fact, allows, on the one hand, to identify which are the data that can be considered outliers, and on the other, to fix a fit threshold: that is the "precision threshold" of our fit.

The "model precision" (functional with minimum mistfit), conceptually, not is the same of "model exactness" (truthfulness of your functional choice)

In the second case (regression of a curve), in the absence of a priori information, or hypotheses, and when we want to regress an experimental data set with a polynomial curve (your first data set), a good practice would be to choose the polynomial curve with the lowest degree within the limits of the error band.

The same principle in my opinion should also be applied in the case of the use of non-polynomial functions (eg power law or logarithm or other functional forms), as in your second second data set; considering the function (predicting your data in the error band) characterized by the lowest number of parameters (for example, as indicated by Marek, a decreasing exponential function characterized by a background parameter and an exponent parameter).

These good practices are in accordance with a general approach that is established by the Occam's principle (pluralitas non est ponenda sine necessitate: it is useless to complicate the model if there is no need).

I can't access the source data, but Excel can itself fit curves to these inside the charts. You just add a trend line to your chart.

This is not generally useful, but in your first chart, an exact 9th degree polynomial can be fit to the data. Excel itself can't do this on a chart, as the limit is 6. For example, if I have the following points (1,.08), (2,.05), (3,.0923), an exact 2nd degree polynomial fits this (coefficients are rounded on the RHS): (x - .08)(x - .05)(x - .0923) = .0362*x^2 - .1385x +.1823.

Another option is this. The data in your first chart does not have a lot of points and looks to the eye to be fairly random. I might consider a Fourier analysis for that data. There is a lot of software that can do that. You could probably find something online in Python that is free. The data set is small enough that you might be able to do it in Wolfram Alpha also.

The data in your second chart looks like it follows a power or exponential distribution. Excel can't fit that because at least one of the values is zero. If I were you, I would probably add a very small quantity to each value that is zero and then Excel should be able to fit either a power or exponential curve to the data right in the chart. You can show the r-squared also. Based upon my experience, you are probably going to find that a power fit is best.

Most likely your graph means something, it is not a collection of random data. If so, then there probably is some theory behind it, and this theory most likely offers one or more functional form which may be applicable here. Your task is then to fit the parameters used in such a theory and see whether their values are sensible.

But it may happen that the theory doesn't exist (yet?) or is simply unknown to you. Then all you can do is to look at the graph and simply guess or assume its functional form. In case of your first graph one would probably try a polynomial, while the second case should be well described by either a rational function or a sum of one or more decaying exponential functions (eventually with appropriate constant background). Another possibility is to treat the first curve as a derivative of few spectral peaks, may be non-symmetric.

Curve fitting is the next step, possible only after you already have a function to be fitted. There are many software packages designed for curve fitting but all they need the function/formula (sometimes even implicit function may be accepted) to be compared with collected data. Unfortunately, I haven't heard about software performing this initial step. That's you job (guessing work).

The two graphs which you have given, I feel there is some kind of noise in the data in second figure and those noises can be avoided. I feel first graph can be approximated by a trigonometric polynomial. The second graph is of the form (x-a)(y-b) = c.

You can give a try to

https://en.wikipedia.org/wiki/Symbolic_regression

and precisely here:

https://en.wikipedia.org/wiki/HeuristicLab

Download from here:

https://dev.heuristiclab.com/trac.fcgi/

Or you can upload data for us to give a try...

you use excel so all you need is to use "add tred line" and you can choose the most good equatin by R sequare wich also be calculated by add tred line, the excel is using the least square method to find the eqauation

just click by right botton mouse on the line ( in graph) and a menue will appear choose "add tred line"

and after that choose the eqaution quadratic , polonomial, e, and choose to make equation appear on graph also choose to find R square,

here I attach your file after finding the best equation by excel

Thank you everyone for the suggestions. I need to find the equation for the graphs because I want to distinguish one graph from another.

Especially for the second case in some of the graphs there is a sharp peak. I would like to know is there any mathematical way to distinguish the graphs consisting the sharp peaks. Or is there any way that I can have a mathematical description that can identify that peak in the graph. Or any other way apart from visual observation that I can characterize the graphs with presence of peak.

I have attached another file which contain 3. What I am looking for is to differentiate the graphs based on presence of peaks from the first one. If possible generate an equation which shows presence of peak.

Maybe I'm wrong but my impression is that your 'peaks' contain a single data point only. If so, then they should be regarded as outliers.

For 'chart2' If we use R package 'infection':

> library(inflection) > plot(xx,yy,type='l') > ede(xx,yy,0) j1 j2 chi EDE 34 192 113 > bb=ese(xx,yy,0) > abline(v=bb[1],col='blue')

we have findpeak.png

If we use R package 'RootsExtremaIflections':

> library(RootsExtremaInflections)

> cc=extremexi(xx,yy,1,length(xx),11,5)

> cc$fextr

[1] 102 102

we end with findpeak2.png.

Actually I have not understood your objective task.

Dear PushKar

I completely agree with Marek Gutowsky about the two meanings that the operation of fit of a series of data can take. Just to complete Marek's answer:

If we have a theory (the model, a known functional relationship) that connects the observable data space to the model parameter space we we are solving an inverse problem. In this first case the objective is to obtain the unknown parameters in our theory (ie the unknown coefficients of a known functional relationship).

In the case in which there is not a theory that connects the data space to the model space we will talk about the simple regression operation whose objective is to carry out an interpolation and / or extrapolation of the same data consistent with the trend of available data and not of a theory.

However, in both cases it should be advisable to have the error on data. The fit will depend on a misfit function (depending on the differences between observed and predicted/regressed data). The availability of the data errors, in fact, allows, on the one hand, to identify which are the data that can be considered outliers, and on the other, to fix a fit threshold: that is the "precision threshold" of our fit.

The "model precision" (functional with minimum mistfit), conceptually, not is the same of "model exactness" (truthfulness of your functional choice)

In the second case (regression of a curve), in the absence of a priori information, or hypotheses, and when we want to regress an experimental data set with a polynomial curve (your first data set), a good practice would be to choose the polynomial curve with the lowest degree within the limits of the error band.

The same principle in my opinion should also be applied in the case of the use of non-polynomial functions (eg power law or logarithm or other functional forms), as in your second second data set; considering the function (predicting your data in the error band) characterized by the lowest number of parameters (for example, as indicated by Marek, a decreasing exponential function characterized by a background parameter and an exponent parameter).

These good practices are in accordance with a general approach that is established by the Occam's principle (pluralitas non est ponenda sine necessitate: it is useless to complicate the model if there is no need).

Hi Pushkar,

When a unique trend cannot be reliably identified, we can fit polynomials of different orders to the data by means of polynomial regression.

For your first data set in Graph.xlsx, the lowest polynomial order to fit all 10 points is 9. Using the 'polyfit' and 'polyval' functions, we can plot the 9th-degree polynomial that best fits the data.

x = [1 2 3 4 5 6 7 8 9 10];

y = [0.001088078 0.000560040 0.000640046 0.000624045 0.000368026 0.000192026 0.0000960061 0.000512033 -0.000192012 0.000752048];

P = polyfit(x, y, 9)

xi = linspace(1, 10);

yi = polyval(P, xi);

plot(xi, yi, x, y, 'o')