Regression of complex-valued functions

 Why are loga40, loga30, loga20, 0 (zero value), and loga0 not independent variables that you control? If so you don't want to estimate those.

If the logaT parameters are not under your control do you still really want to estimate the same set of values of the parameters of the model for each set of your data? And not separate estimates for each curve?

However (putting my head under my shoulder) you can do something like this (Having just noticed Rolf Turner's answer you should absolutely explore that solutions. That would probably the best way to go).

## this seems to run
fit.re <- nls(Re(E) ~ Re(E_2S2P1D(f,c(rep(loga40,5),rep(loga30,5),rep(loga20,5),rep(0,5),rep(loga0,5)),
                                     Eg,Ee,k,h,delta,logbeta,logtau)),
             data = myE, na.action = "na.exclude",  
             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) )

fit.re

## plot fit overlayed observations
png("fitted-Re.png")
plot(f_data, Re(E), type = "p", col = mycol)# log="xy")
lines(f_data, fitted(fit.re), col = "grey80") #, log = "xy"
dev.off()

## this gives an error
## probably due to the 5 curves do not share parameters. See 3rd panel in attached plot-raw-data.png
fit.im <- nls(Im(E) ~ Im(E_2S2P1D(f,c(rep(loga40,5),rep(loga30,5),rep(loga20,5),rep(0,5),rep(loga0,5)),
                                 Eg,Ee,k,h,delta,logbeta,logtau)),
             data = myE, na.action = "na.exclude",
             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) )


## Now try abs()
## which works and gives allmost same estimates as for fit.re
fit.abs <- nls(abs(E) ~ abs(E_2S2P1D(f,c(rep(loga40,5),rep(loga30,5),rep(loga20,5),rep(0,5),rep(loga0,5)),
                                     Eg,Ee,k,h,delta,logbeta,logtau)),
             data = myE, na.action = "na.exclude",
             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))

fit.abs

png("fitted-abs.png")
plot(f_data, abs(E), type = "p", col = mycol)# log="xy")
lines(f_data, predict(fit.abs), col = "grey80") #, log = "xy"
dev.off()










Yours sincerely / Med venlig hilsen


Frede Aakmann T?gersen
Specialist, M.Sc., Ph.D.
Plant Performance & Modeling

Technology & Service Solutions
T +45 9730 5135
M +45 2547 6050
frtog at vestas.com
http://www.vestas.com

Company reg. name: Vestas Wind Systems A/S
This e-mail is subject to our e-mail disclaimer statement.
Please refer to www.vestas.com/legal/notice
If you have received this e-mail in error please contact the sender. 

-----Original Message-----
From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org]
On Behalf Of Andrea Graziani
Sent: 9. februar 2014 23:46
To: r-help at r-project.org
Subject: [R] Regression of complex-valued functions

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(lo
ga0,5)),
                           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

<plot-raw-data.png><fitted-Re.png><fitted-abs.png>