RMS error is mostly used method of reliability of fit of a model. Apart from it generally researchers depend on the sum of squares due to error (SSE), R-square and adjusted R-square.  However, Other methods as Ch-square, MAPE and others may be used if one has to compare the observations with model predicted data. Student t-test and correlation coefficients may also serve the same purpose. But, sometimes correlation coefficients in the two cases may be the same, then one has to refer to MAPE.  You can also use Mann–Whitney U test, only if you are ready to loose information and want  a rough guess by converting observations and predictions into ranks.

Sir !  Apart from least squares, global optimization technique (GA ,VFSA, PSO etc...) are best technique to optimize model parameters .However, whether or not this model parameter is actually provides the global minimum cannot be guaranteed, and its acceptability as a solution should be verified by misfit functional.

Thank you Prof. Mohammad Firoz Khan for your suggestions. Thank you Anand, I would try my best to tackle the matter as suggested by you. The line of action is not so clear but horizon of possibilities has definitely expanded.

In the Least square method and regression analysis it is proper to proceed step by step starting from First degree onwards, checking the significance of each equation and the ANOVA due to added terms (increase of the degree of the equation).

Do not confuse GIS RMS with RMSE.  In ArcInfo, the CONTROLPOINTS function reports RMS but uses (n) in the divisor instead of (n-p). 

The choice of the "fitting" procedure will depend upon what are the parameters to be fitted. Can you mathematically model a relation between "the geomorphological parameters like 'rate of erosion' of river islands with changing aggradation-degradation patterns"?  Sometimes dimensional analysis is useful. An example of this I encountered  was the relation between flow parameters and the sizes of solutional patterns on soluble rocks ("scallops"). My attempt is described at .

Thank you Rabindra Nath, Joe and Rane L. Curl for your useful suggestions.. To reply Prof. Rane, we were investigating the plano temporal variability of a relic island surrounded by a highly braided river.Planform measurement was done for the rate of erosion, changing areas of braid bars, channel bearing zones, effective increase in the channel belt areas for different reaches at different times, say the status during 1915, then 1975 and then 2005.The increasing/decreasing trend of braid bar/channel area gives a definite indication of aggradation-degradation pattern which are related to relative variability of accommodation space generation, related again to the interplay of tectonic forcings. Within a channel belt, these are definitely interdependent parameters. The problem was which order of curve matching to choose?

Have you looked at the statistics of the relic island sizes?  There is no direct analogy with "scallops", but the similarity to scallops is in  that they too individually evolve in sizes, while maintaining a log-normal distribution. (If you have a published paper on this island phenomenon, I would appreciate a copy.)

Thank you Rane for showing your interest. I am sending a paper related to the issue. Hoping for interesting comments from you. Warm regards,

Siddhartha

In looking at different methods you need to consider the underlying statistical assumptions they are generally not the same. .E.g., linear regression models (linear in the coefficients) assume a deterministic component plus a random component. There are different possible statistical assumptions for the random component and this affects the choice of statistical analyses that might be used. If you use least squares you can't do any statistical analysis unless you assume the random component has constant variance and is normally distributed with zero mean, i.e., the results of the statistical analyses would not be valid. With respect to RMS, RSME these are basically used to compare whether the model with only a constant term is as good as one with other terms in the regression model. I think you are attempting to oversimplify the question and hence the results will either be mis-leading or not applicable.

Thank you Prof. Donald for your answer and the caution regarding possibilities of oversimplification. Any attempt to generalize multi-parametric events where parameters depending upon the locations keep on changing their relative impact on the overall process is difficult to formulate. When we study landscape evolution in river valleys, the planform measurements of different geomorphic parameters suggest definite correlations. When we keep on changing the 'reach' width; say, 10-12 km or 1-2 km, we find that the impacts of different parameters show significant variation. It is in this context, choosing proper 'order' came into being. Unless things are formulated, future extrapolation is very difficult. We were attempting to find certain 'laws' from the apparent 'randomness' of the events to facilitate effective river management policy. 'Simplifying' the problem is the urgent need. We are yet to understand the threshold beyond which our simplifying bids become victims of 'oversimplification'. Thank you again for the caution.

In the geostatistical literature this is called a "change of support" problem.The problem is that the spatial correlation is affected (this is both a theoretical and a practical problem). As you noticed, the frequency distribution is also affected. With least  squares there is no theoretical basis for incorporating this effect. I suggest you look at  Ecography 25, (2002) and also Scaling of Remote Sensing Data D.A. Quattrochi and M. Goodchild (eds), Lewis Publishers, 

Order of the equation is increased systematically from first order. F-test is done to know its statistical significant. Then the order is increased by one; F-tests are done to know the statistical significance of the second-degree equation and the increase of the order of the equation. This process is continued till the equation and  the increase of the order of the equation become insignificant ( Computed values of Fs become less than the critical values.)