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[Graham Blakey]

I have been undertaking individual curve fits in Phoenix Model.  In the course of the modelling I have used the different options for the residual error component. However, when the power function is used for the residual error I am uncertain how to interpret the resultant standard deviation.  For example if the error were multiplicative an SD of 0.2 equates to a 20%CV.  What is the equivalent for the power model?

[Simon Davis]

Graham,

    I agree it's somewhat complicated so I went back to one of the developers, hope this helps;

 

Here's a power model:

 

                error(CEps = 0.1) # i.e. initial variance of CEps is 0.01

                observe(CObs = C + C ^ (0.75) * CEps)

 

If the power were 0, the model would be additive, because C^0*CEps = 1*CEps = CEps

 

If the power were 1, the model would be multiplicative, because C^1*CEps = C*CEps

 

If the power were 0.5, it would be somewhere between additive and multiplicative.

In other words, when C is large, residuals are also large, but not as large as they would be in a multiplicative model,

and when C is small, residuals also shrink, but not as much as they would in a multiplicative model.

 

All of these three models have a problem, namely that under simulation they can simulate negative values of C.

In an additive model, it happens when CEps is more negative than -C.

In a multiplicative model, it happens when CEps is more negative than -1.

In a power model, it happens when CEps is more negative than -C^(1-power).

If power = 0,      it is -C

If power = 1/2,  it is -sqrt©

If power = 1,      it is -1

If power = 2,      it is -1/C

 

 Part of the problem is the (mis) expectation that there is a constant CV that is independent of the prediction level for different observations.   This is only true for proportional (multiplicative) residual models where both the residual  error in the prediction and well as the prediction itself advance in proportional to the prediction level, so the ratio

(CV=residual error/prediction error )remains constant for all measurements.

 

  For all other residual error models where the residual error is not proportional to the prediction,  the constancy of the CV is not true.  For example, consider a simple additive error model with a residual standard deviation of eps = 10. 

 If he  prediction were 100 at a particular time point, the CV would be 10%   .  But if the prediction were 50, the CV would be 20%.