Nonlinear Estimation Introductory Overview - General Purpose

Nonlinear Estimation

In the most general terms, Nonlinear Estimation computes the relationship between a set of independent variables and a dependent variable. For example, we may want to compute the relationship between the dose of a drug and its effectiveness, the relationship between training and subsequent performance on a task, the relationship between the price of a house and the time it takes to sell it, etc. You may recognize research issues in these examples that are commonly addressed by such techniques as multiple regression (see Multiple Regression) or analysis of variance (see, ANOVA/MANOVA). In fact, you may think of nonlinear estimation as a generalization of those methods. Specifically, multiple regression (and ANOVA) assumes that the relationship between the independent variable(s) and the dependent variable is linear in nature. Nonlinear estimation leaves it up to you to specify the nature of the relationship; for example, you can specify the dependent variable to be a logarithmic function of the independent variable(s), an exponential function, a function of some complex ratio of independent measures, etc. (However, if all variables of interest are categorical in nature, or can be converted into categorical variables, you can also consider Correspondence Analysis.)

When allowing for any type of relationship between the independent variables and the dependent variable, two issues raise their heads. First, what types of relationships "make sense", that is, are interpretable in a meaningful manner? Note that the simple linear relationship is very convenient in that it allows us to make such straightforward interpretations as "the more of x (e.g., the higher the price of a house), the more there is of y (the longer it takes to sell it); and given a particular increase in x, a proportional increase in y can be expected." Nonlinear relationships cannot usually be interpreted and verbalized in such a simple manner. The second issue that needs to be addressed is how to exactly compute the relationship, that is, how to arrive at results that allow us to say whether or not there is a nonlinear relationship as predicted.

The following links are to topics that discuss the nonlinear regression problem in a somewhat more formal manner, that is, introduce the common terminology that will enable you to examine the nature of these techniques more closely, and how they are used to address important questions in various research domains (medicine, social sciences, physics, chemistry, pharmacology, engineering, etc.).