2 replies to this topic

[Mathias Devreese]

Dear all,

 

Can anyone help me to interpret the 'CMultStdev' function in the log-additive error model for Phoenix NLME please? Log-additive: C_obs = C_pred + ε . sqrt(1 + C_pred^{2 . (CMultStdev/σ)})²

Is this some sort of a weighing factor for residual variability or how can I interpret this?

 

Thanks,

Mathias

[mittyright]

Dear Mathias,

 

That's just a way to get mixed error model due to the fact that you cannot use more than one epsilon in each observe statement 

 

The best explanation you can find in the User's guide:

 

Another way to specify a mixed error model having a fixed effect, but with the fixed effect signifying the multiplicative sigma, rather than the ratio of multiplicative to additive sigma is Add+Multi. It makes use of a built-in function called "sigma()" that can only be used in this context, and its value is the current estimate of the standard deviation of eps. The formula is:

 

C + eps*sqrt( 1 + ( C*multStdev/sigma() )^2 )

(4) 

where multStdev is the multiplicative sigma.

So when this error model is used, the additive sigma is called stdev, and the multiplicative sigma is called multStdev. Since multiStdev is a fixed effect, its name can be changed as desire.

To justify the above formula, look at the variance. Suppose the additive standard deviation is called sigma1, the multiplicative standard deviation is called sigma2, and suppose the corresponding epsilons eps1 and eps2 are drawn from a unit normal distribution. Then the formula would be:

 

C + eps1*sigma1 + C*eps2*sigma2

(5) 

The variance of this is the sum of the variances from each term, or:

 

sigma1^2 + ( C*sigma2 )^2

(6) 

Now let r be the ratio: r = sigma2/sigma1, then the variance is:

 

sigma1^2 + ( C*r*sigma1 )^2

or

sigma1^2 *( 1 + ( C*r )^2 )

(7) 

which is the variance of:

 

C + eps * sqrt( 1 + ( C*r )^2 )

(8) 

Then replace r with sigma2/sigma1 to obtain:

 

C + eps * sqrt( 1 + ( C*sigma2/sigma1 )^2 )

(9) 

where sigma1 is represented by the sigma() function, and sigma2 is represented by multStdev.

So, by choosing the option Add+Mult, this formula will be used to estimate both stdev (the additive standard deviation) and multStdev (the multiplicative standard deviation).

 

BR,

Mittyright

[Mathias Devreese]

Thanks Mittyright!