the moddeling that we use in this question is grey box modelling 2. An experimen

Question

the moddeling that we use in this question is grey box modelling

2. An experiment is conducted to measure the effect of water intake on urine sodium level. Every day the water intake of the subject is recorded and sodium level on their urine are measured in the end of the day. If we want to construct a mathematical model describing this experiment, what kind of modelling would you choose? Explain your reasoning. What is your chosen method to create the model? Explain each step of the modelling process a. b. c. Once you obtain your general model equation, how would you estimate the parameters of the model? Assume that the order of the model is 2.

Explanation / Answer

Part a)

As it is given that we will use the grey box modelling method .

Let us understand when we use grey box model :

Grey box modelling is used when  peculiarities of what is going on inside the system are not entirely known . Hence , on basis of experimental data this grey box model is used .

So ,in our case we know the experimental data but we do not know much about what is hapeening insight or we can say at molecular level hence we used grey box modelling.

Part b)

Grey Box modelling is being used . In this model :

~ we need to find parameters or variable paramters relation.

~For a particular structure it is arbitrarily assumed that the data consists of sets of feed vectors f, product vectors p, and operating condition vectors c.

~ c will contain values extracted from f, as well as other values

~model can be converted to a function of the form:

m(f,p,q)

here ,the vector function m gives the errors between the data p, and the model predictions. The vector q gives some variable parameters that are the model's unknown parts.

~parameters q vary with the operating conditions c in a manner to be determined.

This relation can be specified as q = Ac where A is a matrix of unknown coefficients, and cas in linear regression includes a constant term and possibly transformed values of the original operating conditions to obtain non-linear relations between the original operating conditions and q. It is then a matter of selecting which terms in A are non-zero and assigning their values. The model completion becomes an optimisation problem to determine the non-zero values in A that minimizes the error terms m(f,p,Ac) over the data.

Part c)

Once a selection of non-zero values is made, the remaining coefficients in A can be determined by minimizing m(f,p,Ac) over the data with respect to the nonzero values in A, typically by non-linear least squares. Selection of the nonzero terms can be done by optimization methods such as simulated annealing and evolutionary algorithms. Also the non-linear least squares can provide accuracy estimates for the elements of A that can be used to determine if they are significantly different from zero, thus providing a method of term selection.

It is sometimes possible to calculate values of q for each data set, directly or by non-linear least squares. Then the more efficient linear regression can be used to predict q using cthus selecting the non-zero values in A and estimating their values. Once the non-zero values are located non-linear least squares can be used on the original model m(f,p,Ac) to refine these values

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