condense and summarize the data by fitting it to a “model” that depends on adjustable parameters. Sometimes the model is simply a convenient class of functions, such as polynomials or Gaussians, and the fit supplies the appropriate coefficients. | Chapter 15. Modeling of Data Introduction Given a set of observations one often wants to condense and summarize the data by fitting it to a model that depends on adjustable parameters. Sometimes the model is simply a convenient class of functions such as polynomials or Gaussians and the fit supplies the appropriate coefficients. Other times the model s parameters come from some underlying theory that the data are supposed to satisfy examples are coefficients of rate equations in a complex network of chemical reactions or orbital elements of a binary star. Modeling can also be used as a kind of constrained interpolation where you want to extend a few data points into a continuous function but with some underlying idea of what that function should look like. The basic approach in all cases is usually the same You choose or design a figure-of-merit function merit function for short that measures the agreement between the data and the model with a particular choice of parameters. The merit function is conventionally arranged so that small values represent close agreement. The parameters of the model are then adjusted to achieve a minimum in the merit function yielding best-fit parameters. The adjustment process is thus a problem in minimization in many dimensions. This optimization was the subject of Chapter 10 however there exist special more efficient methods that are specific to modeling and we will discuss these in this chapter. There are important issues that go beyond the mere finding of best-fit parameters. Data are generally not exact. They are subject to measurement errors called noise in the context of signal-processing . Thus typical data never exactly fit the model that is being used even when that model is correct. We need the means to assess whether or not the model is appropriate that is we need to test the goodness-of-fit against some useful statistical standard. We usually also need to know the accuracy with which parameters are determined by the
