I have a variable dataset (say, Y varies between 0 and 1) as a function of distance (X in km). I tried to fit the data with various fitting functions as below:
(a) Y = a - b.X^c , where a,b and c are fit coefficients.
(b) Y = 1 - (X^b)/a , where a and b are fit coefficients.
(c) Y = a + b.EXP(-X/c) , where a,b and c are fit coefficients.
(d) Y = EXP(-(X/a)^b) , where a and b are fit coefficients.
Among the fit functions shown above, I find that (d) fits my data very well but I am not sure if there is any statistical parameter that actually tells how good the fit function is for the dataset. I found that 'goodness of a fit' can be determined by obtaining the ratio of variance of residues to the variance of data set (Y). Any suggestions are welcome to determine the skill level of the fit function in reproducing the actual data set.
try the ROOT MEAN SQUARE ERROR
RMSE=√(1/n ∑_1^n▒(m-o )^2 )
where m and r denote the modelled and observed values
Hello Madhavan
There are two parts to the answer.
1) while selecting a functional form for fitting, ensure that there should be a physical basis for that particular functional relation. Else, you can always find some functional form that would best fit to your data (unless the data is white noise)
2) To see how good your fit is - RMSE is one statistical measure. Other tools such as the 'chi-square test' provide more quantitative estimate. For more details see any good statistics book such as the one by Bendat and Piersol "Random Data - measurement and analysis procedures" (title may be slightly different). I used it quite some time back
Good luck
Dear Madhavan Sir,
I am totally agreed with Moorthy Sir.
I would like to add somethings.
When fitting a nonlinear model to data, your main objective is often to discriminate between different models. Comparing models can answer your question.
There are two main approaches to comparing models
(1) The extra sum-of-squares F tests
The F ratio quantifies the relationship between the relative increase in sum-of-squares and the relative increase in degrees of freedom.
F= [ (SS1-SS2)/(DF1-DF2) ] / (SS2/DF2)
where, SS1 and SS2 are the sum-of-squares for the simpler model and complicated model respectively. DF is the degree of freedom
If the simpler model is correct you expect to get an F ratio near 1.0. If the ratio is much greater than 1.0, there are two possibilities:
• The more complicated model is correct.
• The simpler model is correct, but random scatter led the more complicated model to fit better. The P value tells you how rare this coincidence would be.
The P value answers this question:
If model 1 is really correct, what is the chance that you would randomly obtain data that fits model 2 so much better?
If the P value is low, conclude that model 2 is significantly better than model 1. Otherwise, conclude that there is no compelling evidence supporting model 2, so accept the simpler model (model 1).
(2) The AICc approach
Sir, this is difficult to follow in the theoretical aspect, but we can use with the simple equation.
The fit of any model to a data set can be summarized by an information criterion developed by Akaike, called the AIC.
To compare models, it is only the difference between the two AIC values that matter.
(Delta AIC) = N x ln (SS2/SS1) + 2(Delta DF)
the difference between the two AICc values as the AICc of the simpler model minus the AICc of the more complicated model. When the more complicated model has the lower AICc and so is preferred, the difference of AICc as a positive number. When the simpler model has the lower AICc and so is preferred, the difference of AICc as a negative number.
The equation above helps you get a sense of how AIC works – balancing change in goodness-of-fit vs. the difference in number of parameters. But you don’t have to use that equation. Just look at the individual AIC values, and choose the model with the smallest AIC value. That model is most likely to be correct.
I think the root mean square error (RMSE) method will be appropriate.
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