I think the question is not phrased correctly. Whether two parameters in a model are dependent of each other is something that the user of the model must know - since they are PARAMETERS. Please let us know if you really meant one variable, or solution, being dependent of one or more parameters in the model, and how to analyze that situation.

Consider a simple example

if

y=ax^2       1

thentaking differentials

dy=2axdx     2

divide equation 2 by equation 1 gives

dy/y=2dx/x

so if x changes by 1 per cent, y changes by 2 per cent

ie if x changes ----> y changes a lot more, in fact changes twice as much.

Hence this method will allow  you to compare percentage change of one variable with respect to another.

Try Monte Carlo simulation method

You have to use well known sensitivity analysis methods.

https://en.wikipedia.org/wiki/Sensitivity_analysis

the desired sensitivity information is given by the derivative of y with respect to x.

large values of the derivative indicate strong sensitivities, small values indicate low sensitivities.

if your model is such that the derivative exists, you can calculate it (depending on the complexitiy of the model)

a) by pen and paper

b) by numerical approximation

c) by symbolic calculation

Maybe you need to check the availability of the Implicit Function Theorem

Philipp Kügler's answer ist the right one. You may add automatic differentiation, which is implemented in several software toolsfor any language; that translates a given routine into another one which computes derivatives of output variables with respect to input variables.

Refs.: See Andreas Griewank, HU Berlin; Andrea Walther, U Paderborn; Christian Bischof, RWTH Aachen.

The first step should follow the answer by Michael Patriksson, then other answers are applicable as a function of the first-step outcome.

First of all take the bounds of the variable X, then assume X is distributed normally ( according to the nature of the problem say). Now consider the Partial derivative of Y is equals some functions ( as per your model)  of the distributions of X. Then proceed as traditional method. Thanks

Michael Patriksson's is a correct answer to what I think is not really Alaa Hlafta's real question.    Typically one would have a model involving some unknown parameters and believes correctly describes a real-world situation if one did know the parameter values correctly.  One may then wish to determine these values from empirical observation.  If the observations are exact, then this is an Inverse Problem --- possible (by the Implicit Function Theorem) if a certain derivative is invertible.  More often, the observation is inexact, with errors due to measurement quantification or noise or ...  and the theory of such Inverse Problems becomes more complicated, especially if the unknown parameters are functions rather than numbers.  

Alaa -

It sounded to me like you were thinking that when you want to find a local min-max for y with respect to x, you take the partial of y with respect to x and set it equal to zero, so that you can find where y is instantaneously not changing with x, but that what you wanted to know now, instead, was where is y changing the most with respect to x, or at least how it changes, correct? If so, then the answer is that partial derivative. Think of a slope b in y = bx, for a tangential line at the point of interest in the x-y plane. The partial derivative you found is b. For a local min-max indicator for y, b is zero, but you might want to find the maximum absolute magnitude for b, not the lowest absolute magnitude for b (which can only reach the lowest such value, zero, at a y min/max). So you need to take the partial of y with respect to x; use that for b, the slope of a tangential line in the x-y plane, but instead of setting it equal to zero, to see if there is a solution, you just want to see the range of such possible b values, and perhaps in particular, which is the largest one. If this is a partial derivative, b might depend upon a lot of things, and have an operations research type solution for maximum sensitivity values.

Cheers - Jim

Hi,

If you have access to Engineering Equation Solver (EES) it can help you. EES directly calculates partial derivative based sensitivity analysis of input parameters. The command is named as Uncertainty Propagation in the software.

cheers

Alaa H Lafta May I know what you did at the end?

Best!

This UncertaInty Propagation is just what we have been talking about. In the most common, most interesting cases the input parameter is a vector (i.e., several components, often dependent) or perhaps even a function for nonlinear problems and one considers partial derivatives (or functional derivatives) to get the propagation by the chain rule. This combined derivative becomes a matrix (or linear operator) and what you are seeking is given by the inverse, providing this inverse exists when evaluated where needed.

Thomas I. Seidman is there any reference you could share? Thank you! Best regards

I think there might be several approaches that might be followed to answer this question.

1) Scaling the differential equation

You can try to scale de equation such that the behaviour of the solution will be influenced by a single number given by the ratio of your parameters.

e.g.

Given u=f(x,t) and a heat equation of the form a u_t = b u_xx, defining a set of reference quantities for both the dependent and independent variables (u -> [u], x->[x], t->[x]/[u]) the equation can be rewritten in non-dimensional form (a[u][x]/b)u_hat_t_hat = u_hat_x_hat2. The behaviour of the non-dimensional solution will depend on (a[u][x]/b).

2) Considering the two parameters as random variables

In this case, you can estimate the linear correlation matrix between the two parameters. The differential equation is considered as a black box, where the input is given by the two parameters and the output by the solution. To this end, one can use a set of numerical techniques provided by python or matlab.

3) Sensitivity analysis

You can carry out a sensitivity analysis of your operator by either using a variational or an adjoint-based approach. In this way, you will know how the solution is changing with the parameters. Then, you can use this information to find some correlation between the parameters.

4) Continuation

Another interesting approach would be to fix a set of initial conditions and a trivial solution and then changing the parameters in order to maximize a pre-defined functional(e.g. the max value of the solution) and then performing a continuation in the parameter space to find the combination of parameters which progressively optimize your objective.

You may want to check also this RG thread:

https://www.researchgate.net/post/How_to_run_a_sensitivity_analysis_of_a_chamical_process_with_monte_carlo_method