# Relating parameters to a measured variable

I have an ordinary differential equation based model for a system which depends on 16 parameters (all continuous and positive).

I have 10000 random sets of parameters where each set has 12 elements. Within each set, the value of the parameters are randomly sampled between 0.1 to 10x of a predefined value.

For example if the predefined set is [1, 2, 3, 4] (showing only 4 elements for simplicity), then each random set will have values from a uniform distribution in the ranges [0.01, 10], [0.02, 20], [0.03, 30] and [0.04, 40], respectively. Assuming, for simplicity, that any value lower then the predefined value is "low" and vice versa, then each set will have a random combination of high and low values of different parameters.

Now. I used these randomly sampled sets for simulating the ODE and calculate a metric — Q. With each parameter set, the ODE simulation returns a distinct value of Q. Finally, I get a distribution of Q for different parameter sets.

In the scatter plot shown below, I have retained the region of interest. The y-axis denotes the value of Q. The points highlighted in red denote "high" values of Q and those in black denote "low" values. I know the parameter sets that generate the red dots and those that generate black dots. I want to see if there are specific patterns of parameter values that generate red dots that are distinct from those that generate black dots. I am assuming that this problem can be addressed using clustering.

I am not asking for software or tools that can be used for clustering. I wish to know if clustering can be at all used for this kind of analysis and if so, how should I proceed? How do I make a comparison between the "red sets" and the "black sets"?

I agree that this question is a little broad but I am fine with answers that point me to techniques/methods that are used to solve these kind of problems. I do not have much experience in this area and any clue about how to proceed would be helpful.

Edit:

Would binomial logistic regression be appropriate in this case?

• What kind of space do the parameters belong to? How many parameters are there? – shadowtalker Jul 17 '15 at 19:45
• @ssdecontrol The parameters are continuous positive numbers. There are, in total 16 parameters. – WYSIWYG Jul 18 '15 at 4:14
• Is each axis in the plot a parameter? – shadowtalker Jul 18 '15 at 4:15
• @ssdecontrol no. The plot depicts two metrics (on each axis) that are dependent on these parameters. I want to see what parameter patterns produce the subset of the results marked in red. I have a one to one map between parameter sets and the metrics. I just have to find out how different are the parameters that produce the different sets of output (And also find a pattern). – WYSIWYG Jul 18 '15 at 4:24
• How complicated are those metrics? What kinds of functions are they? – shadowtalker Jul 18 '15 at 11:35