LsqFit.jl

The LsqFit package is a small library that provides basic least-squares fitting in pure Julia under an MIT license. The basic functionality was originally in Optim.jl, before being separated into this library. At this time, LsqFit only utilizes the Levenberg-Marquardt algorithm for non-linear fitting.

There are top-level methods curve_fit() and margin_error() that are useful for fitting data to non-linear models. See the following example. Let's first define the model function:

using LsqFit

# a two-parameter exponential model
# x: array of independent variables
# p: array of model parameters
# model(x, p) will accept the full data set as the first argument `x`.
# This means that we need to write our model function so it applies
# the model to the full dataset. We use `@.` to apply the calculations
# across all rows.
@. model(x, p) = p[1]*exp(-x*p[2])

The function applies the per observation function p[1]*exp(-x[i]*p[2]) to the full dataset in x, with i denoting an observation row. We simulate some data and chose our "true" parameters.

# some example data
# xdata: independent variables
# ydata: dependent variable
xdata = range(0, stop=10, length=20)
ydata = model(xdata, [1.0 2.0]) + 0.01*randn(length(xdata))
p0 = [0.5, 0.5]

Now, we're ready to fit the model.

fit = curve_fit(model, xdata, ydata, p0)
# fit is a composite type (LsqFitResult), with some interesting values:
#	dof(fit): degrees of freedom
#	coef(fit): best fit parameters
#	fit.resid: residuals = vector of residuals
#	fit.jacobian: estimated Jacobian at solution
lb = [1.1, -0.5]
ub = [1.9, Inf]
p0_bounds = [1.2, 1.2] # we have to start inside the bounds
# Optional upper and/or lower bounds on the free parameters can be passed as an argument.
# Bounded and unbouded variables can be mixed by setting `-Inf` if no lower bounds
# is to be enforced for that variable and similarly for `+Inf`
fit_bounds = curve_fit(model, xdata, ydata, p0_bounds, lower=lb, upper=ub)

# We can estimate errors on the fit parameters,
# to get standard error of each parameter:
sigma = stderror(fit)
# to get margin of error and confidence interval of each parameter at 5% significance level:
margin_of_error = margin_error(fit, 0.05)
confidence_inter = confint(fit; level=0.95)

# The finite difference method is used above to approximate the Jacobian.
# Alternatively, a function which calculates it exactly can be supplied instead.
function jacobian_model(x,p)
    J = Array{Float64}(undef, length(x), length(p))
    @. J[:,1] = exp(-x*p[2])     #dmodel/dp[1]
    @. @views J[:,2] = -x*p[1]*J[:,1] #dmodel/dp[2], thanks to @views we don't allocate memory for the J[:,1] slice
    J
end
fit = curve_fit(model, jacobian_model, xdata, ydata, p0)

There's nothing inherently different if there are more than one variable entering the problem. We just need to specify the columns appropriately in our model specification.

using LsqFit

x = collect(range(0, stop=200, length=201))
y = collect(range(0, stop=200, length=201))

xy = hcat(x, y)

function twoD_Gaussian(xy, p)
    amplitude, xo, yo, sigma_x, sigma_y, theta, offset = p
    a = (cos(theta)^2)/(2*sigma_x^2) + (sin(theta)^2)/(2*sigma_y^2)
    b = -(sin(2*theta))/(4*sigma_x^2) + (sin(2*theta))/(4*sigma_y^2)
    c = (sin(theta)^2)/(2*sigma_x^2) + (cos(theta)^2)/(2*sigma_y^2)

    # creating linear meshgrid from xy
    x = xy[:, 1]
    y = xy[:, 2]
    g = offset .+ amplitude .* exp.( - (a.*((x .- xo).^2) + 2 .* b .* (x .- xo) .* (y .- yo) + c * ((y .- yo).^2)))
    return g[:]
end

p0 = Float64.([3, 100, 100, 20, 40, 0, 10])
data = twoD_Gaussian(xy, p0)

# Noisy data
data_noisy = data + 0.2 * randn(size(data))

fit = LsqFit.curve_fit(twoD_Gaussian, xy, data_noisy, p0)

The default is to calculate the Jacobian using a central finite differences scheme if no Jacobian function is provided. The default is to use central differences because it can be more accurate than forward finite differences, but at the expense of computational cost. It is possible to switch to forward finite differences, like MINPACK uses for example, by specifying

fit = curve_fit(model, xdata, ydata, p0; autodiff=:finiteforward)

It is also possible to use forward mode automatic differentiation as implemented in ForwardDiff.jl by using the autodiff=:forwarddiff keyword.

fit = curve_fit(model, xdata, ydata, p0; autodiff=:forwarddiff)

Here, you have to be careful not to manually restrict any types in your code to, say, Float64, because ForwardDiff.jl works by passing a special number type through your functions, to automagically calculate the value and gradient with one evaluation.

It is possible to either use an in-place model, or an in-place model and an in-place Jacobian. It might be pertinent to use this feature when curve_fit is slow, or consumes a lot of memory.

model_inplace(F, x, p) = (@. F = p[1] * exp(-x * p[2]))

function jacobian_inplace(J::Array{Float64,2},x,p)
        @. J[:,1] = exp(-x*p[2])
        @. @views J[:,2] = -x*p[1]*J[:,1]
    end
fit = curve_fit(model_inplace, jacobian_inplace, xdata, ydata, p0; inplace = true)

This package implements optional geodesic acceleration, as outlined by this paper. To enable it, one needs to specify the function computing the directional second derivative of the function that is fitted, as the avv! parameter. It is also preferable to set lambda and min_step_qualityto 0:

curve_fit(model, xdata, ydata, p0; avv! = Avv!,lambda=0, min_step_quality = 0)

Avv! must have the following form:

• p is the array of parameters
• vis the direction in which the direction is taken
• dir_deriv is the output vector (the function is necessarily in-place)

function Avv!(dir_deriv,p,v)
        v1 = v[1]
        v2 = v[2]
        for i=1:length(xdata)
            #compute all the elements of the Hessian matrix
            h11 = 0
            h12 = (-xdata[i] * exp(-xdata[i] * p[2]))
            #h21 = h12
            h22 = (xdata[i]^2 * p[1] * exp(-xdata[i] * p[2]))

            # manually compute v'Hv. This whole process might seem cumbersome, but
            # allocating temporary matrices quickly becomes REALLY expensive and might even
            # render the use of geodesic acceleration terribly inefficient
            dir_deriv[i] = h11*v1^2 + 2*h12*v1*v2 + h22*v2^2

        end
end

Typically, if the model to fit outputs [y_1(x),y_2(x),...,y_m(x)], and that the input data is xdata then Avv!should output an array of size m, where each element is v'*H_i(xdata,p)*v, where H_iis the Hessian matrix of the output y_iwith respect to the parameter vector p.

Depending on the size of the dataset, the complexity of the model and the desired tolerance in the fit result, it may be worthwhile to use automatic differentiation (e.g. via Zygote.jl or ForwardDiff.jl) to determine the directional derivative. Although this is potentially less efficient than calculating the directional derivative manually, this additional information will generally lead to more accurate results.

An example of such an implementation is given by:

using LinearAlgebra, Zygote

function Avv!(dir_deriv,p,v)
    for i=1:length(xdata)
        dir_deriv[i] = transpose(v) * Zygote.hessian(z->model(xdata[i],z),p) * v
    end
end

fit = curve_fit(model, [jacobian], x, y, [w,] p0; kwargs...):

• model: function that takes two arguments (x, params)
• jacobian: (optional) function that returns the Jacobian matrix of model
• x: the independent variable
• y: the dependent variable that constrains model
• w: (optional) weight applied to the residual; can be a vector (of length(x) size or empty) or matrix (inverse covariance matrix)
• p0: initial guess of the model parameters
• kwargs: tuning parameters for fitting, passed to levenberg_marquardt, such as maxIter, show_trace or lower and upper bounds
• fit: composite type of results (LsqFitResult)

This performs a fit using a non-linear iteration to minimize the (weighted) residual between the model and the dependent variable data (y). The weight (w) can be neglected (as per the example) to perform an unweighted fit. An unweighted fit is the numerical equivalent of w=1 for each point (although unweighted error estimates are handled differently from weighted error estimates even when the weights are uniform).

sigma = stderror(fit; atol, rtol):

• fit: result of curve_fit (a LsqFitResult type)
• atol: absolute tolerance for negativity check
• rtol: relative tolerance for negativity check

This returns the error or uncertainty of each parameter fit to the model and already scaled by the associated degrees of freedom. Please note, this is a LOCAL quantity calculated from the Jacobian of the model evaluated at the best fit point and NOT the result of a parameter exploration.

If no weights are provided for the fits, the variance is estimated from the mean squared error of the fits. If weights are provided, the weights are assumed to be the inverse of the variances or of the covariance matrix, and errors are estimated based on these and the Jacobian, assuming a linearization of the model around the minimum squared error point.

margin_of_error = margin_error(fit, alpha=0.05; atol, rtol):

• fit: result of curve_fit (a LsqFitResult type)
• alpha: significance level
• atol: absolute tolerance for negativity check
• rtol: relative tolerance for negativity check

This returns the product of standard error and critical value of each parameter at alpha significance level.

confidence_interval = confint(fit; level=0.05, atol, rtol):

• fit: result of curve_fit (a LsqFitResult type)
• level: confidence level
• atol: absolute tolerance for negativity check
• rtol: relative tolerance for negativity check

This returns confidence interval of each parameter at level significance level.

covar = vcov(fit):

• fit: result of curve_fit (a LsqFitResult type)
• covar: parameter covariance matrix calculated from the Jacobian of the model at the fit point, using the weights (if specified) as the inverse covariance of observations

This returns the parameter covariance matrix evaluated at the best fit point.