I wouldn't call myself an expert but it doesn't look like a good fit to me. Whether or not this is good enough for your purposes will depend on what those purposes are. If you're more focused on smaller values of x, this seems to do rather well but I'm usually focused on larger (asymptotic) values, where this doesn't seem to approach the observations.

To be fair, it looks like a good start though. :-)

Thanks a lot, I am a beginner, so I would like to know one more thing from you sir.

As from the graph and also you have mentioned, suppose if i want to consider the higher values of x, then the result is not reliable? Is it feasible/possible to set (perform) it opposite way i.e, fitting focused on higher value of x?

Regards

You did not indicate what the data was nor how the fit was performed. However, my conclusion is the fit is poor, I would have tried either a loess smooth or a nonlinear regression model made up of a logarithmic function.

Books on generalized additive models or nonlinear regression will

help you produce a better fit.

Best, D. Booth

Thank you so much Dr. D. Booth for the answer and suggestion.

In fact, the fitting was done using experimental data in the equation

y = M [1-b/x2 ]

by software. And, I am having doubt that poor fitting is either due to wrong fitting procedure or improper equation (related to properties of concern theory which is usually taken). But experimental data is correct.

Sir, is it good enough to conclude that the experimental data doesn't follow the given equation (if my procedure of fitting is correct)?

With regards

Chitra

The predicted values start by underestimating the true values, then overestimating, and end by again underestimating the true values. Furthermore, it looks like the observed values continue to increase slowly as x exceeds 16000, while the predicted values would suggest that there is a maximum Y of about 53. While the predicted model explains most of the variability (R2=0.95), there are systematic errors. The result is a rough approximation. In some cases this might be useful. However, this is a bad model of system behavior if the intent is to gain a better understanding of how this system works. How you proceed should be determined by the final goal: do you just want a better fit, or are you trying to develop a new paradigm that better explains how your system works.

Thank you so much Dr. Timothy for the nice interpretation and sugestions.

Sir, goal is to explain the system using the model but as it seem to have poor fitting, i'm just searching the possible ways, errors and solutions so as to improve the output result.

With best regards

Chitra

I read off five points of your data roughly and tried a fit. The result is shown in the figure attached. The function used is as follows:

y=a1x^a2/[1+(x/a3)^(a2-a4)] (1)

In the table within the figure, m0, m1, m2, etc. correspond to x, a1, a2, etc. of Eq. (1). I found this function by a logarithmic plot of the data, which showed that, at large values of x, the behavior of y is like y=const∙x^0.3. When x is large, the second term in the denominator of Eq. (1) becomes dominant compared with 1, and y behaves like const∙x^a4 with a4 obtained by least-squares fit as 0.029.

I like your answer Dr.Tatsuo Tabata

I agree with you

Regards!

Thank you so much Dr. Tatsuo for the nice answer.

Sir, I really appreciate your strategy and demonstration.

With the same equation you have given, I perform curve fitting again with my same procedure (same step and software), and got the result attached herewith. I got correct fitting i.e., answer is similar with yours (i can further perform more iterations and enhance more improvement in results shown here). So, I confirmed that my fitting procedure is correct (i.e., also for the previous fitting given in question). Poor fitting results is due to my model (equation not perfect for my experimental result). Now, I can interpret my results obtained from the equation and experimental data as well as further design better model for my experimental data. Thanks a lot again.

With best regards

Chitra

Dear Citra,

Seeing the result of your fit by the function I suggested, I still find systematic deviations of the curve from the data. I cannot guess if the deviations are due to the functional form, the condition to end the process of least-squares fitting, or the not-so-good starting values of parameters. Rather large errors of parameters might suggest that the fitting process did not yet find the true minimum of the chi-square. However, the result seems to be a considerable improvement to your original fitting.

By the way, there was an error in the explanation of my first estimate of the value of a4 from the logarithmic plot. It was not 0.3 but 0.1. The value 0.3 was the starting value I used after some trial runs of the least-squares fit.

Best regards,

Tatsuo

It is not a good fit. You can reduce the associated standard errors by trying a different regression function. If you provide the data, I can find a better regression equation for you.

Dear Tatsuo

Thank you sir, yes i agree still there is some deviation in the new graph, because i still left to do some iteration in my fitting process which improves chi 2 value. But, just by comparing results, i am sure it is correct and will achieve the desired value in completing it.

With best regards

Chitra

Dear Amir Sir

Thank you for the kind response. Well my main purpose is to fit an equation (which is already defined) and study deviation from it rather than to find an equation which exactly fit on it. But, still I think I can make some changes in parameters already defined equation. I am also interested to see your fitting. Please find data in attached file.

With best regards

Chitra

Dear Chitra,

In my previous trial, the x values were smaller than your data by a factor of 10. So, inappropriate starting values for your fitting were probably the cause of systematic deviations. I corrected data and increased the data points by using your data set given above and tried re-fitting. The result (see the figure) indicates that the functional form is not so bad, though the value of a4 obtained is quite smaller than expected.

Regards,

Tatsuo

Dear Tatsuo

Sir, Thank you so much for the detail explanation.

With best regards

Chitra

Dear Respected Colleagues

From your kind responses and suggestions, I think I got the correct fitting with correct equation (which is also appropriate to my model) attached herewith.

Thank you all once again

Best Regards

Chitra

Dear Chitra,

In your last result, the parameters a and b have the same value, and k is quite small. What equation have you used?

Regards,

Tatsuo

Dear Tatsuo

Sir, the equation is y = M(1-a/x-b/x2)+kx

Regards

Chitra

Dear Chitra,

Oh, a minor modification to your original formula worked! From the values of parameters you obtained, I recommend you to try the following simpler equations to see if they also work well:

y=a-b(1/x+1/x^2)+cx

y=a-b(1/x+1/x^2).

Regards,

Tatsuo

Thank you Tatsuo sir

With best regards

Chitra

Enjoyed reading Interaction of prof Tabata with young scientists.

Dear Tatsuo

Sir, I did a mistake in my previous fitting, in which i got same a and b value (you have also guess and mentioned it in previous comment). The corrected fitting is attached herewith. Also, I tried the two equations you have suggested, I got almost similar fitting which are also attached here.

Thank you

With best regards

Chitra

Dear Chitra,

Now, the corrected fit to your equation y=M(1-a/x-b/x^2))+kx seems to give the best result though with minor differences from the simpler equations I wrote.

By the way, what are the quantities for the x and y axes?

Best regards,

Tatsuo

Dear Tatsuo

Sir i'm studying variation of magnetization with applied field.

With best regards

Chitra

Dear Chitra,

Thanks for your kind reply. Then, I suppose this: If you started the measurement of Y from very small values of X, then Y would also have had very small but non-zero values there, and the equation y=M(1-a/x-b/x^2))+kx would fail to be good (y of the equation takes on negative values for small x). However, if you don't mind the behavior of the curve for such values of X, then the equation is good.

Best regards,

Tatsuo

Dear Tatsuo

Sir, your hypothesis is extremely correct. In fact I have chosen the optimum starting value of x mainly based on relevant literature but not directly by looking into the equation. From literature, I got and applied the idea that x must start from some higher values for certain reasons, as is valid one for this specific case. But, from your suggestions, I wish to check the validity of the equation again, so I checked and found it true. In an MS excel sheet, taking the approximate values of M, a, b, k from previous corrected graph and some (arbitrary) lower values of x, the values of y is determined using the equation y=M(1-a/x-b/x^2))+kx . Yes, its true, negative y values at lower values of x. And also, fitting can't be done if the lower values of x are included which is also checked and confirmed by inclusion of low experimental values of x. The excel sheet and fitted curve is attached herewith. It is extremely helpful, because I can also verify and interpret mathematically. Thank you so much.

With best regards

Chitra

Dear Chitra,

If you are interested in using an equation down to x near zero, then my equation y=a1x^a2/[1+(x/a3)^(a2-a4)], which starts from the origin, would be a possibility.

Regards,

Tatsuo

Dear Tatsuo

Sir, thank you for the suggestion. I will try to study on it also.

With best regards

Chitra

Dear Chitra,

your problem is evidently over and I came across it just accidentally. Yet I am sending you another solution which eventualy may come handy. Individual steps are numbered:

1.) Your data set Xi,Yi resembles a hyperbola and so I did a simple transformation first: x = X/1000, y = Y0 - Y, where Y0 was roughly extimated value from the upper end of Yi data.

2.) Obtained new data set x, y was displayed in an EXCEL graph and preliminary fitted by a power function, y = p x^q.

3.) Since the fit was found unsatisfactory, a nonlinear regression function was prepared using the equation Y = Y0 - p x^q.

4.) The overdetemined set of these equations for the paired data x, Y was solved using EXCEL Solver tool. Initial values of parameters were taken from steps 1 and 2; the least squares method (LSQ) was selected for optimisation criterion using EXCEL function SUMXMY2 -> minimum.

5.) Obtained parameters Y0 = 55.41... , p = 19.9......and q = -1.042.... were already used for the Yi approximation and the value of the optimisation criterion 0.235..... gives an estimate of their variance (~ 0.07). If it found unsatisfactory, the procedure may continue by a corrective function which can be additive or multiplicative.

6.) Since the range of all Y or y values is small (all within a decade), the additive correction function is appropriate and the residuals calculated by subtraction of input and output data can be prepared and displayed in another EXCEL graph. Then they are fitted (usually by a polynomial) and the corrective function is obtained this way.

7.) The sum of the main and corrective functions create the optimum approximation, in this case with final variance of about 0.01.

Conclusion remarks: If this procedure fails in the step 2 or 5 the choise of the main approximative function may be wrong. This depends also on the nature of the task, its physical background etc. If the result in the step 7 is not satisfactory don't continue with a second corrective function but try to improve the main function. Anyway, to minimize the number of parameters is worth of effort.

I hope you will find this usefull.

With regards,

Anton

Dear Anton sir, thank you so much for the detail procedures. I'll try your method also.

With best regards

Chitra