**6 March 2007**

**Sensitivity Analysis:**

You have a model fully validated. You are satisfied. Next, you are going to use as an experimental lab to carry out analysis and design improved policies.

Figures are here:

So broadly speaking what is analysis and design?

Analysis means understanding something. Studying something to understand it. IN our context means understanding why and how. By what causal mechanisms and which ways the model behave as it does. Understanding sources of dynamic behavior. This is prerequisite to redesign it. Here main comps of analysis"

One extreme form of analysis is mathematical solution. Dynamical model, if you can solve it, you obtain an equation which is an explicit function of time. The solution is a vector variable =s which are explicit functions of time. Sines, cosines, exponentials can be complicated. But in any case these are time functions. And various parameters of functions freq, exponents come from model. You can quiet precisely say what parameter does what?

Simple birth death population. Solution of a linear pop model is p(t) = p(0) + ... If b = d, you have const pop. The larger difference, larger slopes you obtain. This is an extreme form of precision in math analysis.

This is only true for linear systems. For almost all linear systems, you can obtain math solutions. And for almost all nonlinear systems you can’t obtain a mathematical solution. Well come back to this in what conditions are mathematical solutions possible?

Next level is equilibrium solutions and analysis. We'll talk about this a lot. System stays at these equation points.

Third comp is sensitivity analysis. This is major subfield of analysis. To what extend in what ways (qualitative and quantitative questions) do behavior changes to the changes in initial conditions and structure. This percentage change in dynamic behavior patterns is a major field of analysis.

Policy analysis: Given that you have a certain policy how sensitive is this policy to the parameters, initial conditions of the model. Specific application of sensitivity analysis to policy.

Then there is design. Obtaining improved policy: How can I obtain policy structure that will yield and improvement in system behavior? A dampening of pop growth, stooping cancer growth as a result of medical intervention, higher percentage of students entering in universities as a result of university placement policy.

The limit of design is optimization. Given my objectives this is the best possible policy. This is rarely possible in nonlinear feedback models. There are no easy algorithms. This is a great research area.

Now our topic is sensitivity analysis and policy analysis experimentally.

**Diff****erent**** uses of sensitivity analysis:**

· Model validity testing

· Evaluating importance of parameters: Value of information and importance of parameters. If behavior is sensitivity to a few parameters but insensitive to some other parameters. Then these parameters are import.

· Decide on which parameters you need to spend more effort. These are most sensitivity parameters. Then you know that 10% error in these parameters are import. You need to estimate them better.

· Understand the structure behavior connection. This is mathematical analysis. Mathematical solution constructs time function of behavior in parameters of model. You see the direct links. We can't do this mathematically so let's try to do it with sensitivity analysis. We make series of simulation runs. And decide the role of each parameter. Model behavior connection can be established by doing sensitivity analysis to some extent.

· Evaluating policy alternatives. Sensitivity analysis is applied to some policy. How sensitive is our policy to some set of parameters? If it is very sensitive you don't like it.

High sensitivity may mean:

· weakness in model

· discovery of critical parameters

· need for further research to estimate parameters

· most important parameters for behavior

· weak policies

**Types of sensitivity analysis:**

This is important.

Sensitivity with respect to:

· parameter (or initial) values - i.e. numeric values.

· numerical - a model with behavior like Fig. 4.7. I make some parameter changes. I obtain b). This is numeric change. It is trivial change. Mathematically there will be some numeric sensitivity with any change of the parameter. The question is how much. Whether it exists or not, is not important.

· behavior patterns - This is very different Fig.4.8. b) has a very different type of change. This is oscillation but it is damping. A change from a to b is pattern sensitivity. Can you guess which is more important? Behavior pattern sensitivity is more important than numeric sensitivity.

Next question is: Do you expect behavior change to come from what? From structure changes. Feedback structures that have enough loops, that is control loops, organizations that exist and survive, as a result of negative control loops. These interesting problems have enough negative feedback loops that cause them persist so many years. Even

Alternative structures - you change the structure. You write a new equation. You remove a feedback lop. You modify the structure.

Somewhere between there is form of functions. This may be closer to parameter change or structure change. Fig. 4.5 You modify the level of function. This is parameter change. On the other hand, if you change the form of function, Fig. 4.6., this is more like a structure change. And another change is c). It removes the relation between y and x.

**Some observations:**

Fundamental behavior is created by some dominant feedback loops. We'll talk about this.

Discovery of sensitivity parameters are critical. And of course discovery of structure mostly influential in creating fundamental behavior is important.

S: What if birth rate > death rate or death rate < birth rate in population growth model? This changes the behavior pattern...

When you choose 0 parameter value, you are saying that it doesn't exist at all. This is clear structure change. In your case, a change in parameter value causes a significant behavior change. But this is not structure change. Same structure.

**Tools for sensitivity analysis:**

They are either analytical: What percentage of change in output is caused by what percentage of parameter? For linear systems you can obtain it.

Much more extensive is simulation based experimental sensitivity analysis. How do you do it? Experimental design.

Experimental design is scientific way of organizing experiments.

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