In this chapter we will introduce two case studies:

1. multi-variate linear regression (Section 9.1), and

2. state estimation in an electric power system (Section 9.2).

Both problems will turn out to be unconstrained optimization problems of the special class of least-squares data fitting problems [84, chapter 13].

Multi-variate linear regression

Some of this section is based on [103] and further details can be found there. The development assumes a background in probability. See, for example, [31, 103].

Motivation

In many applications, we have a hypothesized functional relationship between variables. That is, we believe that there are some dependent variables that vary according to some function of some independent variables. The simplest relationship that we can imagine is a linear or affine relationship between the variables.

For example, we may be trying to estimate the circuit parameters of a black-box circuit by measuring the relationship between currents and voltages at the terminals of the circuit. We will have to try several values of current and voltage to characterize the circuit parameters. As in the circuit case study of Section 4.1, we could either:

• apply vectors of current injections and measure voltages, interpreting the currents as the independent variables and the voltages as the dependent variables, or

• apply vectors of voltages and measure currents, interpreting the voltages as the independent variables and the currents as the dependent variables.