Mathematical Modeling with Multidisciplinary Applications

CHAPTER 8

ESTIMATION OF MODEL PARAMETERS

ROBERT PICHÉ

Tampere University of Technology, Tampere, Finland

8.1 ESTIMATION IS AN INVERSE PROBLEM

Mathematical models are created to help understand real-world data. Estimation is the process of determining the values of parameters of a mathematical model by comparing real-world observations with the corresponding results predicted by the model. Estimation might be done to improve the model’s usability as a tool for prediction or for “what if” scenarios; this is sometimes called model calibration. The estimated values of the parameters are typically of interest too, because they represent important quantities, for example, velocity, population, market volatility.

In other chapters of this book, mathematical modeling procedures are presented for setting up and solving questions of the form: “with given parameters, what observations would result?” In this phase of modeling, we seek to set up a mathematical problem whose solutions exist, are unique, and are continuous with respect to the parameters. In estimation, the question is different: “what do the observations tell me about the parameters?” Estimation is, in this sense, an inverse problem (Figure 8.1).

Figure 8.1 Mathematical model seen as a system with inputs (parameters) and outputs (observations).

This chapter introduces a probability-theory based approach to estimation known as ...