Forcing nls to fit a curve passing through a specified point

Question

I\'m trying to fit a Boltzmann sigmoid 1/(1+exp((x-p1)/p2)) to this small experimental dataset:

xdata <- c(-60,-50,-40,-30,-20,-10,-0,10)
yda         


        

Answer 1

Building on @Cleb's answer, here's a way to pick a specified point the function must pass through and solve the resulting equation for one of the parameters:

dd <- data.frame(x=c(-60,-50,-40,-30,-20,-10,-0,10),
                 y=c(0.04, 0.09, 0.38, 0.63, 0.79, 1, 0.83, 0.56))

Initial fit (using plogis() rather than 1/(1+exp(-...)) for convenience):

fit <- nls(y ~ plogis(-(x-p1)/p2),
           data=dd,
           start=list(p1=mean(dd$x),p2=-5))

Now plug in (x3,y3) and solve for p2:

y3 = 1/(1+exp((x-p1)/p2))
logit(x) = qlogis(-x) = log(x/(1-x))
e.g. plogis(2)=0.88 -> qlogis(0.88)=2
qlogis(y3) = -(x-p1)/p2
p2 = -(x3-p1)/qlogis(y3)

Set up a function and plug it in for p2:

p2 <- function(p1,x,y) {
    -(x-p1)/qlogis(y)
}
fit2 <- nls(y ~ plogis(-(x-p1)/p2(p1,dd$x[3],dd$y[3])),
    data=dd,
    start=list(p1=mean(dd$x)))

Plot the results:

plot(y~x,data=dd,ylim=c(0,1.1))
xr <- data.frame(x = seq(min(dd$x),max(dd$x),len=200))
lines(xr$x,predict(fit,newdata=xr))
lines(xr$x,predict(fit2,newdata=xr),col=2)