15.4.3 Model Diagnosis Using Dependency Values

We can also assess the quality of a fit model using the Dependency value. A value close to 1 indicates that the function may be (but is not necessarily) overparameterized and that the parameter may be redundant. You should include restrictions to make sure that the fitting model is meaningful. In this example, where sample data is fitted by the ExpDecay1 function:

Model Diagnosis Using Dependency Values-1.jpg

You can see that the Dependency values for parameters x0 and A1 are large, which suggests that these two variables are highly dependent. In other words, one parameter varies with the other, without changing the R^2 value. This means that the result may not valid. We can see from the graph that the X value starts from about 500. To compensate for this, we can perform the fitting by fixing x0 as 500 to generate a reasonable dependency result:

Model Diagnosis Using Dependency Values-2.jpg

Moreover, dependent parameters may also lead to some meaningless results. For example, if we fit the same data by ExpDec1 model, the result may be as follows:

Model Diagnosis Using Dependency Values-3.jpg

You can see that the parameter value A is too large and may not make sense. In this case, we should use bounds or constraints to limit the parameter values so that they fall within a reasonable range.