Singular gradient error during loading nls corresponds to bad data

Question

I have a dataset containing an independent variable and a set of dependent variables. I would like to fit a function to each set of independent variables using a bootable non-linear least-squares procedure. In some cases, independent variables are "good quality", that is, they are good enough for the function. In other cases, they are noisy.

In all cases, I can use nls() to get parameter estimates. However, when the data is noisy, the bootstrap generates an Error in nls(...) : singular gradient error. I can understand why nls installation on noisy data will fail, for example. due to the inability to converge after too many iterations, but I don’t understand why this is a singular gradient error, and why I get only oversampled low quality datasets.

the code:

 require(ggplot2) require(plyr) require(boot) # Data are in long form: columns are 'enzyme', 'x', and 'y' enz <- read.table("http://dl.dropbox.com/s/ts3ruh91kpr47sj/SE.txt", header=TRUE) # Nonlinear formula to fit to data mmFormula <- formula(y ~ (x*Vmax) / (x + Km)) 

nls great for setting data (even if in some cases, like a , I doubt that the model is suitable for data.

 # Use nls to fit mmFormula to the data - this works well enough fitDf <- ddply(enz, .(enzyme), function(x) coefficients(nls(mmFormula, x, start=list(Km=100, Vmax=0.5)))) # Create points to plot for the simulated fits xGrid <- 0:200 simFits <- dlply(fitDf, .(enzyme), function(x) data.frame(x=xGrid, y=(xGrid * x$Vmax)/(xGrid + x$Km))) simFits <- ldply(simFits, identity) ggplot() + geom_point(data=enz, aes(x=x, y=y)) + geom_line(data=simFits, aes(x=x, y=y)) + facet_wrap(~enzyme, scales="free_y") + aes(ymin=0) 

Bootstrapping works great for good quality data:

 # Function to pass to bootstrap; returns coefficients of nls fit to formula nlsCoef <- function(df, i) { KmGuess <- median(df$x) VmaxGuess <- max(df$y) dfSamp <- df[i,] nlsCoef <- coefficients(nls(mmFormula, dfSamp, start=list(Km=100, Vmax=0.5))) } eBoot <- boot(subset(enz, enzyme=="e"), nlsCoef, R=1000) #No error 

But not for low quality data

 dBoot <- boot(subset(enz, enzyme=="d"), nlsCoef, R=10) > Error in nls(mmFormula, dfSamp, start = list(Km = KmGuess, Vmax = VmaxGuess)) : singular gradient 

What causes this error? And what should I do with this, given that I would like to use plyr to simultaneously execute many bootstrap syntaxes?

Answer 1

This allows you to check what happens:

 #modified function #returns NAs if fit is not sucessfull #does global assignment to store bootstrap permutations nlsCoef <- function(df, i) { KmGuess <- median(df$x) VmaxGuess <- max(df$y) dfSamp <- df[i,] fit <- NULL try(fit <- nls(mmFormula, dfSamp, start=list(Km=100, Vmax=0.5))) if(!is.null(fit)){ res <- coef(fit) } else{ res <- c(Km=NA,Vmax=NA) } istore[k,] <<- i k <<- k+1 res } n <- 100 istore <- matrix(nrow=n+1,ncol=9) k <- 1 dat <- subset(enz, enzyme=="d") dBoot <- boot(dat, nlsCoef, R=n) #permutations that create samples that cannot be fitted nais <- istore[-1,][is.na(dBoot$t[,1]),] #look at first bootstrap sample #that could not be fitted samp <- dat[nais[1,],] plot(y~x,data=samp) fit <- nls(mmFormula, samp, start=list(Km=100, Vmax=0.5)) #error 

You can also use a self-starting model:

 try(fit <- nls(y ~ SSmicmen(x, Vmax, Km), data = dfSamp)) 

With this error message it becomes a little more informative. For example, one error:

 too few distinct input values to fit a Michaelis-Menten model 

which means that some bootstrap samples contain less than three different concentrations. But there are other errors:

 step factor 0.000488281 reduced below 'minFactor' of 0.000976562 

which you could avoid by reducing minFactor .

Following troubles. You can try different installation algorithms or initial values ​​using these parameters:

 singular gradient matrix at initial parameter estimates singular gradient