User`s guide

3 Fitting Data

3-4

Parametric Fitting

Parametric fitting involves finding coefficients (parameters) for one or more

models that you fit to data. The data is assumed to be statistical in nature and

is divided into two components: a deterministic component and a random

component.

data = deterministic component + random component

The deterministic component is given by the fit and the random component is

often described as error associated with the data.

data = fit + error

The fit is given by a model that is a function of the independent (predictor)

variable and one or more coefficients. The error represents random variations

in the data that follow a specific probability distribution (usually Gaussian).

The variations can come from many different sources, but are always present

at some level when you are dealing with measured data. Systematic variations

can also exist, but they can be difficult to quantify.

The fitted coefficients often have physical significance. For example, suppose

you have collected data that corresponds to a single decay mode of a radioactive

nuclide, and you want to find the half-life (T

1/2

) of the decay. The law of

radioactive decay states that the activity of a radioactive substance decays

exponentially in time. Therefore, the model to use in the fit is given by

where y

is the number of nuclei at time t = 0, and λ is the decay constant.

Therefore, the data can be described by

Both y

and λ are coefficients determined by the fit. Because T

1/2

= ln(2)/λ, the

fitted value of the decay constant yields the half-life. However, because the

data contains some error, the deterministic component of the equation cannot

completely describe the variability in the data. Therefore, the coefficients and

half-life calculation will have some uncertainty associated with them. If the

uncertainty is acceptable, then you are done fitting the data. If the uncertainty

is not acceptable, then you might have to take steps to reduce the error and

repeat the data collection process.

λt–

data y

λt–

error+=