Re: [R] Complicated nls formula giving singular gradient message
I don't Think that viewing lack of convergence by some R routine as a uuseful tool for diagnosing model or data inadequacy is a very useful approach. It is far better to fit the model. Then standard techniques can be employed to investigate these matters. For the model considered here there are 5 parameters and 96 observations. So a priori no reason to suspect that the data are insufficient. So where lies the problem? Fitting the model and using the very accurate Hessian approximation provided by AD Model Builder provides some immediate clues. The eigenvalues of the Hessian are 3.943982727e-08104.6301825150.7527476203.044988959736.68735 so the condition number is about 1.e+13. With such a badly scaled problem it is difficult to fit with finite difference approximations to the derivatives. The approximate std devs of the parameter estimates are index name value std dev 1 NS 1.1254e-02 7.1128e-03 2 LogKi -8.8933e+00 8.2411e-02 3 LogKi -5.2005e+00 9.2179e-02 4 LogKi -7.2677e+00 7.7047e-02 5 BMax 2.1226e+05 5.1699e+03 so there is no initial indication that the parameter estimates are badly determined. __ 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] Complicated nls formula giving singular gradient message
Though my topic is slightly old already, I feel that it is necessary to post an update on my situation. I ended being able to estimate the parameters for this problem without having to worry as much about initial parameter estimates using AD Model Builder. It calculates the exact gradient using automatic differentiation so it's able to avoid the singular gradient problem nls can give. I also used the R package PBSadmb, which allowed me to run AD Model Builder and retrieve the results from within R. Then I could do what I liked with the results: generate graphs, more analysis, etc. Thanks to everyone who helped, Jared On Wed, Dec 15, 2010 at 8:47 AM, dave fournier da...@otter-rsch.com wrote: Jared Blashka wrote: Hi, Can you write a little note to the R list saying something like Re: SOLVED[R] Complicated nls formula giving singular gradient message I was able to estimate the parameters for this problem using AD Mode Builder which calculates the exact gradient for you using automatic differentiation and is thus able to avoid the singular gradient problem I encountered in nls. That way other R users who might be able to take advantage of the software will hear about it. Cheers, Dave Dave, That's exactly what I was looking for! Thanks for all your help! Jared On Tue, Dec 14, 2010 at 7:13 AM, dave fournier da...@otter-rsch.commailto: da...@otter-rsch.com wrote: Jared Blashka wrote: The source code for that is in jared.tpl I changed from least squares to a concentrated likelihood so that you could get estimated std devs via the delta method. they are in jared.std I rescaled the parameters so that the condition number of the Hessian is close to 1. You can see the eigenvalues of the Hessian in jared.eva. Your data are in jared.dat and the initial parameter values are in jared.pin. The parameter estimates with their estiamted std devs are: index name value std dev 1 NS 1.1254e-02 7.1128e-03 2 LogKi -8.8933e+00 8.2411e-02 3 LogKi -5.2005e+00 9.2179e-02 4 LogKi -7.2677e+00 7.7047e-02 5 BMax 2.1226e+05 5.1699e+03 ~How does it look? Cheers, Dave Dave - AD Model Builder looks like a great tool that I can use, but I'm curious if it can also perform global parameter estimations across multiple data sets. In regards to the example I have provided, I have two similar data sets that also need to be analyzed, but the values for NS and BMax between the three data sets should be the same. Each data set has a unique LogKi value however. In R, I accomplished this by merging the three data sets and adding an additional field for each data point that identified which set it was originally from. Then in the regression formula I specified the LogKi term as a vector: LogKi[dset]. The results of the regression gave me one value each for NS and BMax, but three LogKi values. I haven't had much time to look through the AD Model Builder documentation yet, but are you aware if such an analysis method is possible? Here's one such example of a data set all -structure(list(X = c(-13, -11, -10, -9.5, -9, -8.5, -8, -7.5, -7, -6.5, -6, -5, -13, -11, -10, -9.5, -9, -8.5, -8, -7.5, -7, -6.5, -6, -5, -13, -11, -10, -9.5, -9, -8.5, -8, -7.5, -7, -6.5, -6, -5, -13, -11, -10, -9.5, -9, -8.5, -8, -7.5, -7, -6.5, -6, -5, -13, -11, -10, -9.5, -9, -8.5, -8, -7.5, -7, -6.5, -6, -5, -13, -11, -10, -9.5, -9, -8.5, -8, -7.5, -7, -6.5, -6, -5, -13, -11, -10, -9.5, -9, -8.5, -8, -7.5, -7, -6.5, -6, -5, -13, -11, -10, -9.5, -9, -8.5, -8, -7.5, -7, -6.5, -6, -5), Y = c(3146L, 3321L, 2773L, 2415L, 2183L, 1091L, 514L, 191L, 109L, 65L, 54L, 50L, 3288L, 3243L, 2826L, 2532L, 2060L, 896L, 517L, 275L, 164L, 106L, 202L, 53L, 3146L, 3502L, 2658L, 3038L, 3351L, 3238L, 2935L, 3212L, 3004L, 3088L, 2809L, 1535L, 3288L, 2914L, 2875L, 2489L, 3104L, 2771L, 2861L, 3309L, 2997L, 2361L, 2687L, 1215L, 3224L, 3131L, 3126L, 2894L, 2495L, 2935L, 2516L, 2994L, 3074L, 3008L, 2780L, 1454L, 3146L, 2612L, 2852L, 2774L, 2663L, 3097L, 2591L, 2295L, 1271L, 1142L, 646L, 68L, 3288L, 2606L, 2838L, 1320L, 2890L, 2583L, 2251L, 2155L, 1164L, 695L, 394L, 71L, 3224L, 2896L, 2660L, 2804L, 2762L, 2525L, 2615L, 1904L, 1364L, 682L, 334L, 64L), dset = c(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3)), .Names = c(X
Re: [R] Complicated nls formula giving singular gradient message
Jared: You realize, of course, that just because you get estimates of the parameters from the software is no guarantee that the estimates mean anything? Nor does it mean that they mean nothing, I hasten to add. If, as one might suspect, the model is overparameterized, the estimates may be so imprecise that they are effectively useless -- but the fitted values may nevertheless (__Especially__ if overarameterized) fit your data very well. The model just won't fit future data. In other words, you may have a well-fitting, scientifically meaningless model. Cheers, Bert On Mon, Dec 20, 2010 at 10:26 AM, Jared Blashka evilamaran...@gmail.com wrote: Though my topic is slightly old already, I feel that it is necessary to post an update on my situation. I ended being able to estimate the parameters for this problem without having to worry as much about initial parameter estimates using AD Model Builder. It calculates the exact gradient using automatic differentiation so it's able to avoid the singular gradient problem nls can give. I also used the R package PBSadmb, which allowed me to run AD Model Builder and retrieve the results from within R. Then I could do what I liked with the results: generate graphs, more analysis, etc. Thanks to everyone who helped, Jared On Wed, Dec 15, 2010 at 8:47 AM, dave fournier da...@otter-rsch.com wrote: Jared Blashka wrote: Hi, Can you write a little note to the R list saying something like Re: SOLVED [R] Complicated nls formula giving singular gradient message I was able to estimate the parameters for this problem using AD Mode Builder which calculates the exact gradient for you using automatic differentiation and is thus able to avoid the singular gradient problem I encountered in nls. That way other R users who might be able to take advantage of the software will hear about it. Cheers, Dave Dave, That's exactly what I was looking for! Thanks for all your help! Jared On Tue, Dec 14, 2010 at 7:13 AM, dave fournier da...@otter-rsch.commailto: da...@otter-rsch.com wrote: Jared Blashka wrote: The source code for that is in jared.tpl I changed from least squares to a concentrated likelihood so that you could get estimated std devs via the delta method. they are in jared.std I rescaled the parameters so that the condition number of the Hessian is close to 1. You can see the eigenvalues of the Hessian in jared.eva. Your data are in jared.dat and the initial parameter values are in jared.pin. The parameter estimates with their estiamted std devs are: index name value std dev 1 NS 1.1254e-02 7.1128e-03 2 LogKi -8.8933e+00 8.2411e-02 3 LogKi -5.2005e+00 9.2179e-02 4 LogKi -7.2677e+00 7.7047e-02 5 BMax 2.1226e+05 5.1699e+03 ~ How does it look? Cheers, Dave Dave - AD Model Builder looks like a great tool that I can use, but I'm curious if it can also perform global parameter estimations across multiple data sets. In regards to the example I have provided, I have two similar data sets that also need to be analyzed, but the values for NS and BMax between the three data sets should be the same. Each data set has a unique LogKi value however. In R, I accomplished this by merging the three data sets and adding an additional field for each data point that identified which set it was originally from. Then in the regression formula I specified the LogKi term as a vector: LogKi[dset]. The results of the regression gave me one value each for NS and BMax, but three LogKi values. I haven't had much time to look through the AD Model Builder documentation yet, but are you aware if such an analysis method is possible? Here's one such example of a data set all -structure(list(X = c(-13, -11, -10, -9.5, -9, -8.5, -8, -7.5, -7, -6.5, -6, -5, -13, -11, -10, -9.5, -9, -8.5, -8, -7.5, -7, -6.5, -6, -5, -13, -11, -10, -9.5, -9, -8.5, -8, -7.5, -7, -6.5, -6, -5, -13, -11, -10, -9.5, -9, -8.5, -8, -7.5, -7, -6.5, -6, -5, -13, -11, -10, -9.5, -9, -8.5, -8, -7.5, -7, -6.5, -6, -5, -13, -11, -10, -9.5, -9, -8.5, -8, -7.5, -7, -6.5, -6, -5, -13, -11, -10, -9.5, -9, -8.5, -8, -7.5, -7, -6.5, -6, -5, -13, -11, -10, -9.5, -9, -8.5, -8, -7.5, -7, -6.5, -6, -5), Y = c(3146L, 3321L, 2773L, 2415L, 2183L, 1091L, 514L, 191L, 109L, 65L, 54L, 50L, 3288L, 3243L, 2826L, 2532L, 2060L, 896L, 517L, 275L, 164L, 106L, 202L, 53L, 3146L, 3502L, 2658L, 3038L, 3351L, 3238L, 2935L, 3212L, 3004L, 3088L, 2809L, 1535L, 3288L, 2914L, 2875L, 2489L, 3104L, 2771L, 2861L, 3309L, 2997L, 2361L, 2687L, 1215L, 3224L, 3131L, 3126L, 2894L, 2495L, 2935L, 2516L
[R] Complicated nls formula giving singular gradient message
I'm attempting to calculate a regression in R that I normally use Prism for, because the formula isn't pretty by any means. Prism presents the formula (which is in the Prism equation library as Heterologous competition with depletion, if anyone is curious) in these segments: KdCPM = KdnM*SpAct*Vol*1000 R=NS+1 S=(1+10^(X-LogKi))*KdCPM+Hot a=-1*R b=R*S+NS*Hot+BMax c = -1*Hot*(S*MS+BMax) Y = (-1*b+sqrt(b*b-4*a*c))/(2*a) I'm only trying to solve for NS, LogKi, and BMax. I have everything else (KdnM, SpAct, Vol, Hot). I would use the simple formula at the bottom and then backsolve for the terms I'm looking for, but the simple formula at the bottom takes out the X term, which is contained within S, which it itself contained in both b and c. So I tried to substitute all the terms back into Y and got the following formula-as.formula(Y ~ (-1*(((NS+1)*((1+10^(X-LogKi))*(KdnM*SpAct*Vol*1000)+Hot))+NS*Hot+BMax)+sqrtNS+1)*((1+10^(X-LogKi))*(KdnM*SpAct*Vol*1000)+Hot))+NS*Hot+BMax)*(((NS+1)*((1+10^(X-LogKi))*(KdnM*SpAct*Vol*1000)+Hot))+NS*Hot+BMax)-4*(-1*(NS+1))*(-1*Hot*(((1+10^(X-LogKi))*(KdnM*SpAct*Vol*1000)+Hot)*NS+BMax/(2*-1*(NS+1))) But trying to use that formula gives me the single gradient message, which I wasn't entirely surprised by. fit-nls(formula=formula,data=data,start=list(NS=.01,LogKi=-7,BMax=33000)) Error in nls(formula = formula, data = all_no_outliers, start = list(NS = 0.01, : singular gradient I've never worked with a formula this complicated in R. Is it even possible to do something like this? Any ideas or points in the right direction would be greatly appreciated. Thanks, Jared [[alternative HTML version deleted]] __ 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] Complicated nls formula giving singular gradient message
Phil,
This is great! I had no idea nls would accept functions in the formula
position. My apologies for not including data to reproduce my issue.
dat-data.frame(X=c(-13.0,-11.0,-10.0,-9.5,-9.0,-8.5,-8.0,-7.5,-7.0,-6.5,-6.0,-5.0,
-13.0,-11.0,-10.0,-9.5,-9.0,-8.5,-8.0,-7.5,-7.0,-6.5,-6.0,-5.0),
Y=c(3146,3321,2773,2415,2183,1091,514,191,109,65,54,50,
3288,3243,2826,2532,2060,896,517,275,164,106,202,53))
With your suggestion, I've changed the formula in nls to the following
function:
myfunc-function(NS,LogKi,BMax)with(dat,{
KdCPM = KdnM*SpAct*Vol*1000
R-NS+1
S-(1+10^(X-LogKi))*KdCPM+Hot
a-(-1*R)
b-R*S+NS*Hot+BMax
c--1*Hot*(S*NS+BMax)
(-1*b+sqrt(b*b-4*a*c))/(2*a)
})
But to get it to compute without errors, I also had to increase the
tolerance level: the step factor keeps being reduced below the min
factor. Looking at the trace of the nls though, I don't see any changes
after the 10th iteration or so; would increasing the tolerance cause any
issue that I'm not thinking of?
KdnM - .8687
SpAct - 4884
Vol - .125
Hot - 10191.0
nls(Y~myfunc(NS,LogKi,BMax),data=dat,start=list(NS=.01,LogKi=-7,BMax=10*max(dat['Y'])),control=nls.control(warnOnly=TRUE,minFactor=1e-5),trace=TRUE)
Also, I've found that if the start value I provide for BMax is too
inaccurate (ex. max(dat['Y']), nls generates the 'singular gradient' error
message, which isn't something I'm used to. Usually nls is kind enough to
inform me that the initial parameter estimates are what caused the problem.
Has the error message changed in a recent update, or is this a different
error message than what I'm thinking about?
Thanks again for all your help,
Jared
On Mon, Dec 13, 2010 at 1:23 PM, Phil Spector spec...@stat.berkeley.eduwrote:
Jared -
nls will happily accept a function on the right hand side
of the ~ -- you don't have to write out the formula in such
detail.
What you provided isn't reproducible because you didn't provide data, and
it's not clear what Y in the formula
represents. Let me provide you with an admittedly simpler
reproducible example.
Suppose we want to estimate the model
response = v * dose / (k + dose)
where response and dose are variables in a data frame called dat,
and v and k are the parameters to be estimated.
Here's the data:
dat = data.frame(dose=c(0.027,0.044,0.073,0.102,0.175,0.257,0.483,0.670),
+ response=c(12.7,16.0,20.4,22.3,26.0,28.2,29.6,31.4))
Normally we would fit such a simple model as
nls(response ~ v*dose / (k + dose),data=dat,start=list(v=30,k=.05))
Nonlinear regression model
model: response ~ v * dose/(k + dose)
data: dat
vk 32.94965 0.04568
residual sum-of-squares: 1.091
Number of iterations to convergence: 4 Achieved convergence tolerance:
8.839e-07
Alternatively, I can write a function like this:
myfunc = function(v,k)with(dat,v * dose / (k + dose))
and use the following call to nls:
nls(response ~ myfunc(v,k),data=dat,start=list(v=30,k=.05))
Nonlinear regression model
model: response ~ myfunc(v, k)
data: dat
vk 32.94965 0.04568
residual sum-of-squares: 1.091
Number of iterations to convergence: 4 Achieved convergence tolerance:
8.839e-07
which gets the identical results.
Admittedly this function is trivial, but perhaps in your case
you could use the segments from prism to construct a function
for the right-hand side of your nls call.
Hope this helps.
- Phil Spector
Statistical Computing Facility
Department of Statistics
UC Berkeley
spec...@stat.berkeley.edu
On Mon, 13 Dec 2010, Jared Blashka wrote:
I'm attempting to calculate a regression in R that I normally use Prism
for,
because the formula isn't pretty by any means.
Prism presents the formula (which is in the Prism equation library as
Heterologous competition with depletion, if anyone is curious) in these
segments:
KdCPM = KdnM*SpAct*Vol*1000
R=NS+1
S=(1+10^(X-LogKi))*KdCPM+Hot
a=-1*R
b=R*S+NS*Hot+BMax
c = -1*Hot*(S*MS+BMax)
Y = (-1*b+sqrt(b*b-4*a*c))/(2*a)
I'm only trying to solve for NS, LogKi, and BMax. I have everything else
(KdnM, SpAct, Vol, Hot).
I would use the simple formula at the bottom and then backsolve for the
terms I'm looking for, but the simple formula at the bottom takes out the
X
term, which is contained within S, which it itself contained in both b and
c.
So I tried to substitute all the terms back into Y and got the following
formula-as.formula(Y ~
(-1*(((NS+1)*((1+10^(X-LogKi))*(KdnM*SpAct*Vol*1000)+Hot))+NS*Hot+BMax)+sqrtNS+1)*((1+10^(X-LogKi))*(KdnM*SpAct*Vol*1000)+Hot))+NS*Hot+BMax)*(((NS+1)*((1+10^(X-LogKi))*(KdnM*SpAct*Vol*1000)+Hot))+NS*Hot+BMax)-4*(-1*(NS+1))*(-1*Hot*(((1+10^(X-LogKi))*(KdnM*SpAct*Vol*1000)+Hot)*NS+BMax/(2*-1*(NS+1)))
But trying to use that formula
Re: [R] Complicated nls formula giving singular gradient message
Jared -
Actually I didn't realize that nls would accept a formula
until I tried my simple example in reaction to your post :-)
I don't recall nls() ever reporting the cause of the singular
gradient as being bad starting values -- I know I spend a lot
of time in my lectures on non-linear regression emphasizing that
bad starting values are the usual culprit when the dreaded
singular gradient message rears its ugly head.
I think your function has a fairly large flat region, wherein
changes in some of the parameters don't really effect the residual
sums of squares that much. I think you can visualize it like this:
therss = function(NS,LogKi,BMax)sum((dat$Y - myfunc(NS,LogKi,BMax))^2)
tst = expand.grid(NS=seq(.007,.009,by=.0005),
+ LogKi=seq(-9.5,-8.5,by=.05),
+ BMax =seq(1.8e05,2.8e05,by=.1e05))
tst$rss = apply(tst,1,function(x)therss(x[1],x[2],x[3]))
library(lattice)
wireframe(rss~BMax+NS|factor(LogKi),data=tst,as.table=TRUE)
If you look at the panels where LogKi is around -8.9 (the reported
maximum), the residual-sums-of-squares surface is pretty flat.
I think you can also see regions where there isn't much change in
the residual sums of squares in this plot:
wireframe(rss~LogKi+BMax|factor(NS),data=tst,as.table=TRUE)
I also ran your data through proc nlp in sas (I know there are a lot of
SAS-bashers on this list, but I worked there many years ago and I know
the quality of their software), and got the following results:
Optimization Results
Parameter Estimates
Gradient
Objective
N Parameter EstimateFunction
1 NS0.006766 -0.121333
2 LogKi-8.966402 -0.000509
3 BMax2370131.109368E-11
The message that nlp reported was
NOTE: At least one element of the (projected) gradient is greater than 1e-3.
Finally, I ran the the same model and data using nlfit in matlab, with all
values set to their defaults. It reported the following without warning:
ans =
1.0e+05 *
0.00086522054 -0.8987006 2.371354822440646
which agrees almost exactly with R.
Hope this helps.
- Phil
On Mon, 13 Dec 2010, Jared Blashka wrote:
Phil,
This is great! I had no idea nls would accept functions in the formula
position. My apologies for not including data to reproduce my issue.
dat-data.frame(X=c(-13.0,-11.0,-10.0,-9.5,-9.0,-8.5,-8.0,-7.5,-7.0,-6.5,-6.
0,-5.0,
-13.0,-11.0,-10.0,-9.5,-9.0,-8.5,-8.0,-7.5,-7.0,-6.5,-6.0,-5.0),
Y=c(3146,3321,2773,2415,2183,1091,514,191,109,65,54,50,
3288,3243,2826,2532,2060,896,517,275,164,106,202,53))
With your suggestion, I've changed the formula in nls to the following
function:
myfunc-function(NS,LogKi,BMax)with(dat,{
KdCPM = KdnM*SpAct*Vol*1000
R-NS+1
S-(1+10^(X-LogKi))*KdCPM+Hot
a-(-1*R)
b-R*S+NS*Hot+BMax
c--1*Hot*(S*NS+BMax)
(-1*b+sqrt(b*b-4*a*c))/(2*a)
})
But to get it to compute without errors, I also had to increase the tolerance
level: the step factor keeps being reduced below the min factor. Looking at
the trace of the nls though, I don't see any changes after the 10th iteration
or so; would increasing the tolerance cause any issue that I'm not thinking
of?
KdnM - .8687
SpAct - 4884
Vol - .125
Hot - 10191.0
nls(Y~myfunc(NS,LogKi,BMax),data=dat,start=list(NS=.01,LogKi=-7,BMax=10*max(
dat['Y'])),control=nls.control(warnOnly=TRUE,minFactor=1e-5),trace=TRUE)
Also, I've found that if the start value I provide for BMax is too inaccurate
(ex. max(dat['Y']), nls generates the 'singular gradient' error message,
which isn't something I'm used to. Usually nls is kind enough to inform me
that the initial parameter estimates are what caused the problem. Has the
error message changed in a recent update, or is this a different error
message than what I'm thinking about?
Thanks again for all your help,
Jared
On Mon, Dec 13, 2010 at 1:23 PM, Phil Spector spec...@stat.berkeley.edu
wrote:
Jared -
nls will happily accept a function on the right hand side
of the ~ -- you don't have to write out the formula in such
detail.
What you provided isn't reproducible because you didn't provide
data, and it's not clear what Y in the formula
represents. Let me provide you with an admittedly simpler
reproducible example.
Suppose we want to estimate the model
response = v * dose / (k + dose)
where response and dose are variables in a data frame called
dat,
and v and k are the parameters to be estimated.
Here's the data:
dat =
data.frame(dose=c(0.027,0.044,0.073,0.102,0.175,0.257,0.483,0.670),
+
Re: [R] Complicated nls formula giving singular gradient message
I always enjoy these direct comparisons between different software packages. I coded this up in AD Model Builder which is freely available at http://admb-project.org ADMB calculates exact derivatives via automatic differentiation so it tends to be more stable for these difficult problems. The parameter estimates are # Number of parameters = 3 Objective function value = 307873. Maximum gradient component = 1.45914e-06 # NS: 0.00865232633386 # LogKi: -8.98700621813 # BMax: 237135.365156 The objective function is just least squares. So it looks like SAS did pretty well before dying. __ 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.
