Can't fit nonlinear function including '^' in LsqFit

Specific Domains Optimization (Mathematical)
1

Hi there,

When I try to fit to a concentration-response curve (Hill equation) and use the exponent as a free parameter, Julia throws me an error complaining that the base should be a complex number. Even using lower bounds for curve_fit that prohibit a negative base or an exponent < 1, it still throws an error. Can anyone help? Thanks in advance.

@. H(x,p) = p[1] - (p[1]-p[2]) / ( 1. + (x/p[3])^p[4] )
Hfit = LsqFit.curve_fit(H,xdata,ydata,[1.,0.,0.5,1.],lower=[0.,0.,0.,1.])

gives


DomainError with -...:
Exponentiation yielding a complex result requires a complex argument.
Replace x^y with (x+0im)^y, Complex(x)^y, or similar.
2

It works fine for me. Since you don’t give example data, I generated some:

fx(x) = 1.5 - 1.3 / (1. + (x/0.7)^1.3)
xdata = range(0,3,length=50)
ydata = fx.(xdata).*(1 .+0.1randn(50))
plot(xdata,ydata,legend=:bottomright)

leads to:

image
image1612×1108 62.1 KB

Next, doing model fitting:

julia> @. H(x,p) = p[1] - (p[1]-p[2]) / ( 1. + (x/p[3])^p[4] )
julia> Hfit = LsqFit.curve_fit(H,xdata,ydata,[1.,0.,0.5,1.],lower=[0.,0.,0.,1.])
julia> par = coef(Hfit)
4-element Array{Float64,1}:
 1.4779674751492053
 0.21988771220240352
 0.6945565635031009
 1.6532411388662613

julia> plot(xdata,ydata,label="data")
julia> plot!(xdata,x-> par[1] - (par[1]-par[2])/(1+(x/par[3])^par[4]),label="fitted model",legend=:bottomright)

leads to:

image
image1633×1061 86.4 KB

Of course, if you have negative elements in xdata, you will have problems as x/p[3] will be negative, and raising this to a number that is not an even integer will produce a complex number.

Another problem is that you allow p[3] to have zero as lower bound, hence allows the fitting algorithm to divide by zero in x/p[3].

So, I’d check the minimal value in xdata, and change the lower value of p[3] to 1e-2 or something.

You do indicate that xdata contains concentrations, which should be positive at the outset. Maybe there are some errors in the measurements so that some of them are (slightly) negative??

3

There were indeed a few negative values in xdata. Thanks!