Re: [R] Regression of complex-valued functions

Rolf Turner Tue, 11 Feb 2014 00:44:52 -0800

I have not the mental energy to go through your somewhat complicatedexample, but I suspect that your problem is simply the following: Thefunction nls() is trying to minimize a sum of squares, and that does notmake sense in the context of complex observations. That is, nls() istrying to minimize a sum of terms of the form


        (f(x_i,theta) - z_i)^2

where f(x_i,theta) and z_i are complex quantities (and theta is thevector of parameters with respect to which minimization is "trying to bedone". But as I said, this makes no sense. Not that(f(x_i,theta) - z_i)^2 is a complex quantity and *not* a positive realquantity.


You need to minimize a sum of terms of the form

        |f(x_i,theta) - z_i|^2

where |w|^2 equal to w times the complex conjugate of w. You get thisin R via the Mod() function, i.e. |w|^2 is (Mod(w))^2.

Thus it seems to me that nls() cannot handle what you want handled. Youneed to code up the function to be minimized yourself, using Mod() andthen use optim() or nlm() to minimize it that function with respect to"theta".


HTH

cheers,

Rolf Turner


On 10/02/14 11:45, Andrea Graziani wrote:

Hi everyone,

I previously posted this question but my message was not well written and did 
not contain any code so I will try to do a better job this time.

The goal is to perform a non-linear regression on complex-valued data.
I will first give a short description of the data and then describe the 
complex-valued non-linear function.

***The Data
Obtained from mechanical tests performed at 5 loading frequencies 0.1, 0.25, 1, 
0.4 and 12 Hz, and at 5 temeperatures.
The independent variable used in the regression is:
f_data <- rep(c(0.1,0.25,1,4,12),5)

The measured values of the response variable are:
E <- c(289.7411+ 225.0708i , 386.4417+ 303.5021i ,  671.5132+ 521.1253i , 
1210.8638+ 846.6253i , 1860.9623+1155.4139i ,
862.8984+ 636.2637i , 1159.0436+ 814.5609i , 1919.0369+1186.5679i , 
3060.7207+1573.6088i  , 4318.1781+1868.4761i  ,
2760.7782+1418.5450i , 3306.3013+1612.2712i , 4746.6958+1923.8662i , 
6468.5769+2148.9502i , 8072.2642+2198.5344i  ,
6757.7680+2061.3110i , 7591.9787+2123.9168i , 9522.9471+2261.8489i , 
11255.0952+2166.6411i , 12601.3970+2120.7178i ,
11913.6543+2016.0828i , 12906.8294+2030.0610i , 14343.7693+1893.4877i , 
15942.7703+1788.0910i , 16943.2261+1665.9847i)

To visualize the data:
plot(f_data,Re(E),log="xy")
plot(f_data,Im(E),log="xy")
plot(E)

***Non-linear regression function:
Obtained from an analytical model

E_2S2P1D <- function(f,logaT,Eg,Ee,k,h,delta,logbeta,logtau)
            Ee+(Eg-Ee)*(
            1+delta*(2i*pi*10^(logtau)*f*10^logaT)^-k +
                    (2i*pi*10^(logtau)*f*10^logaT)^-h +
                    (2i*pi*10^(logtau)*f*10^logaT*10^(logbeta))^-1
                      )^-1

E_2S2P1D is a complex-valued function (note the imaginary unit "i") where:
"f" is the real-valued independent variable (i.e. the frequency),
"logaT","Eg","Ee","k","h","delta","logbeta","logtau" are real-valued parameters.

For example:

E_2S2P1D(1,0,27000,200,0.16,0.47,1.97,2.2,0.02)

[1] 9544.759+2204.974i

E_2S2P1D(1,-1,27000,200,0.16,0.47,1.97,2.2,0.02)

[1] 6283.748+2088.473i

In order to find the parameters of the regression function I use “nls".

Executing

fit <- nls(E ~ 
E_2S2P1D(f_data,c(rep(loga40,5),rep(loga30,5),rep(loga20,5),rep(0,5),rep(loga0,5)),
                             Eg,Ee,k,h,delta,logbeta,logtau),
           start=list(loga40=-3.76,loga30=-2.63,loga20=-1.39,loga0=1.68,
                      
Eg=27000,Ee=200,k=0.16,h=0.47,delta=1.97,logbeta=2.2,logtau=-0.02) )

results in a lot of warnings (my translation into english):
"In numericDeriv(form[[3L]], names(ind), env) :
imaginary parts removed during the conversion”

I’ve the same error trying:
y <- 
E_2S2P1D(f_data,c(rep(-3.76,5),rep(-2.63,5),rep(-1.39,5),rep(0,5),rep(1.68,5)),27000,200,0.16,0.47,1.97,2.2,0.02)
plot(E,y)

If I got it right R is not able to handle regression problems on complex-valued 
functions, correct?
If yes, is there some workaround?
For example using MSExcel I write the Real and Imaginary parts separately, and 
than I minimize

error = sum [ ( Re(E) - Re(E_E_2S2P1D) ) / Re(E) )^2 ] + sum [ ( Im(E) - 
Im(E_E_2S2P1D) ) / Im(E) )^2 ]

using the MSExcel solver function.

thank you for your help
Andrea

PS
Thanks to David Winsemius for his suggestions

______________________________________________
R-help@r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.


______________________________________________
R-help@r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.

Re: [R] Regression of complex-valued functions

Reply via email to