We continue our discussion on modeling. A model articulates how a system behaves quantitatively. Models use numerical methods to examine complex situations and come up with predictions. Most common techniques involved for coming up with a model include statistical techniques, numerical methods, matrix factorization and optimizations.  
An inverse model is a mathematical model that fits experimental data. It aims to provide a best fit to the data. Values for the parameters are obtained from estimation techniques. It generally involves an iterative process to minimize the average difference. The quality of the inverse model is evaluated using well known mathematical techniques as well as intuition. 
The steps for inverse modeling of data include:
1) selecting an appropriate mathematical model using say polynomial or other functions
2) defining an objective function that agrees between the data and the model
3) adjusting model parameters to get a best fit usually by minimizing the objective function
4) evaluating goodness of fit to data by not being perfect due to measurement noise
5) estimating accuracy of best fit parameter values
6) determining whether a much better fit is possible which might be necessary if there is local minima 

There are two ways by which we can select the appropriate model. The first is by observing trend lines which correspond to some well known mathematical formula. The second is on the observation of underlying physical processes which contribute towards the system. These physical interpretations contribute to model parameters. ain order to fit the data points, a model may use least squares of errors. the errors called residuals may be both positive or negative which result in inaccurate measure. Instead the squares of the errors can be minimized to give a better fit.
#codingexercise
Find next greater number using the same digits as the given number. If no other number is possible return the original
Int GetNextGreater(uint n)
{
Var digits = Int.ToDigits(n);
If (digits.IsEmpty()) return 0;
Int I = 0;
Int J = 0;
// find the start for change in digits
For (int i = digits.Count-1;I > 0; I--)
{
If (digits[I] > digits[I-1]) {
       break;
}
If (I == 0) return n;
//find the substitute and sort the digits from position
Int min = I;
For (j = I+1; j < digits.Count; j++)
      If (digits[j] > digits[I-1] && digits[j] < digits[min])
           min = j;
Swap(digits, min, I-1)
returnDigits.GetRange(0,I-1).Union(digts.GetRange(I, digits.Count-I+1).Sort()).ToList().ToInteger();

}

There is an alternative to getting the number as above. It simply rolls the number forward until each number has the other number has the same count of each digits.