We resume 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.  
Sometimes we relied on experimental data to corroborate the model and tune it. Other times, we simulated the model to see the predicted outcomes and if it matched up with the observed data. There are some caveats with this form of analysis. It is merely a representation of our understanding based on our assumptions. It is not the truth. The experimental data is closer to the truth than the model. Even the experimental data may be tainted by how we question the nature and not nature itself.  This is what Heisenberg and Covell warn against. A model that is inaccurate may not be reliable in prediction. Even if the model is closer to truth, garbage in may result in garbage out
Any model has a test measure to determine its effectiveness. since the observed and the predicted are both known, a suitable test metric may be chosen. for example the sum of squares of errors or the F-measure may be used to compare and improve systems.
A forward model is a mathematical model that is detailed enough to include the desired level of real world behaviour or features. It is used for simulating realistic experimental data which under the right constraints can be sued to test hypothesis.  While it may be too complicated to fit experimental data, it can be used to generate synthetic data sets for evaluating parameters.
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. 
A forward-inverse modeling is a process to combine data simulation with model fitting so that all parameters can be sufficiently evaluated for robustness, uniqueness and sensitivity. This is very powerful for improving data analysis and understanding the limitations.
A good inverse model should have a good fit and describe the data adequately so that some insights may follow.  The parameters are unique and their values are consistent with the hypothesis and changes to experimental data in response to alterations in the system. 

#codingexercise
Given an array and an integer k, find the maximum for each and every contiguous subarray of size k.
List<int> GetMaxFromSubArraysOfSizeK(List<int> A, int k)
{
    var ret = new List<int>();
    int max = INT_MIN;
    for (int i = 0; i <= A.Count-k; i++)
    {
        max = A[i];

        for (j = 1; j < k; j++)
        {
            if (A[i+j] > max)
               max = A[i+j];
        }
        ret.Add(max);
    }
    return ret;

}