LsqFit curve fit does not consider bounds on intercepts correctly

General Usage
1

I am trying to get some fits with LsqFit with bounds, but I am observing strange results which seem to imply that LsqFit is minimizing the correlation coefficient but not the residue of the fit. This is a minimal example, with a linear fit, in which I add a bound to the intercept coefficient:

using LsqFit
x = collect(1:10); y = copy(x);
@. model(x,p) = p[1]*x + p[2]
p0 = [ 0. , -5. ];
fit = curve_fit(model,x,y,p0); # no bounds
fit_bounded = curve_fit(model,x,y,upper=[+Inf,-5.]); # with bounds
scatter(x,y)
plot!(x,model(x,fit.param),label="no bounds")
plot!(x,model(x,fit_bounded.param),label="with bounds")

Result:

fit

Of course that is not the line with the minimal residue sum, it is just the same fit, but shifted downwards.

Am I doing or interpreting something wrong?

2

You’re constraining the parameter vector to not allow an intercept greater than -5,so how could you possibly get anything else? Did you mean to wrote +5?

3

I was expecting a greater slope, such that the fit cross the data, that would have a smaller residue relative to the data, while satisfying the bound.

Edit: which can be demonstrated if the model is defined with that parameter equal to -5 from start:

julia> @. model(x,p) = p[1]*x - 5.
model (generic function with 1 method)

julia> x = collect(1:10); y = x;

julia> fit = curve_fit(model, x, y, p0);

julia> scatter(x,y)

julia> plot!(x,model(x,fit.param))

Results in (the expected result):

fit

4

Ah, good point! The gradient for the slope parameter is probably zero in the point in the figure?

5

No, it is not. The result is independent on the initial point as well.

Also, this kind of problem occurs with any model, for example:

julia> x = collect(1:10); y = x.^2; scatter(x,y)

julia> @. model(x,p) = p[1]*x^2 + p[2]
model (generic function with 1 method)

julia> p0 = [ 5. , -20. ];

julia> fit = curve_fit(model, x, y, p0);

julia> plot!(x,model(x,fit.param),label="no bounds")

julia> fit = curve_fit(model, x, y, p0, upper=[+Inf,-20.]);

julia> plot!(x,model(x,fit.param),label="with bounds")

Result:

fit

Edit: I have opened an issue: variable for intercept is being ignored during optimization? · Issue #167 · JuliaNLSolvers/LsqFit.jl · GitHub

If that is the expected result for some reason, please let me know.

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