Chapter 6

Question 1

What are the for types of minima?

Answer 1

• Global Minimum
• Strict Global Minimum
• Local Minimum
• Strict Local Minimum

Question 2

What are the three types of cost functions?

Answer 2

• Linear
• Quadratic
• Non-Linear

Question 3

What are the four types of constraints?

Answer 3

• Unconstrained
• Non-Linear Equality Constr.
• Non-Linear Inequality. Constr.
• Box Constraints

Question 4

What is the feasible set in optimization?

Answer 4

The region where possible solutions may be found

Question 5

What is the Newton Raphson Algorithm?

Answer 5

An iterative algorithm in optimization that solves the problem of non-linear equation resulting from the minimum gradient condition

Question 6

What are limitations of the Newton Raphson Algorithm?

Answer 6

• Assumes convexity of cost function
• Strongly dependent on starting values

Question 7

What are possible remedies to limitations of Newton Raphson?

Answer 7

Step size control and BFGS Hessian approximation

Question 8

What is output sensitivity?

Answer 8

The measure for the influence of a parameter on the model output

Question 9

What are the Pros and Cons of Sensitivity Equations?

Answer 9

Pros:
- Exact, no mistakes
- Relatively fast

Cons:
- All analytical derivatives of mode have to be computed
- No differentiable functions
- Not flexible with change in model structure
- Decrease in computational time might be negligible for complex, non-linear system

Question 10

What are the pros and cons of finite differences?

Answer 10

Pros:
- Flexible
- Easy to implement since no analytic derivatives need to be calculated

Cons:
- Inexact due to round off errors
- Big computational burden in case of many parameters
- Bad choice of perturbation error can deteriorate accuracy

Question 11

What are the basic steps for implementing maximum likelihood methods?(5)

Answer 11

1. Choose starting value
2. Compute system response
3. Compute residuals and residual covariance matrix
4. Compute parameter update with one of the non-linear optimization algorithms
5. Iterate until convergance

Question 12

What are three methods for computing parameter updates for maximum likelihood methods?

Answer 12

Output error method
Filter error method
Equation error method

Question 13

What two things are used to maximize the maximum likelihood function?

Answer 13

Parameters and Residual Covariance Matrix

Question 14

What is the iterative two step approach for maximizing the maximum likelihood function?

Answer 14

Optimize with parameters fixed

Optimize with residual covariance matrix fixed

Question 15

What are the two approaches for computing output sensivtivities?

Answer 15

• Finite difference approximations
• Solutions to Sensitivity Equations

Question 16

How are solutions to non-linear model equations computed?

Answer 16

Multi-step integration methods

Question 17

What are the assumptions of the output error method?

Answer 17

• No process noise
• Only additive, independent gaussian measurement noise
• inputs are measurable without error, exogenous i.e. independent of system outputs ((no feedback controller!)) and sufficiently and adequately varied to excite the various modes of the dynamical system

Question 18

What kind of noise does the output error method consider?

Answer 18

additive, white, and Gaussian measurement noise

Question 19

What can be problematic for output error methods>

Answer 19

correlated parameters

data with significant process noise

Question 20

What are the assumptions for the equation error method?

Answer 20

• inputs measurable without error
• all states measurable without error
• All state derivatives measurable with additive measurement noise

Question 21

What are the properties of system for equation error method?

Answer 21

It is NOT necessary to integrate any state equations; the problem becomes static!