# EasyFit

Easy interface for obtaining fits of 2D data.

The purpose of this package is to provide a very simple interface to obtain some of the most common fits of 2D data. Currently, simple fitting functions are available for linear, quadratic, cubic, exponential, and spline fits.

On the background this interface uses the LsqFit and Interpolations, which are already quite easy to use. Additionally, EasyFit contains a simple globalization heuristic, such that good non-linear fits are obtained often.

Our aim is to provide a package for quick fits without having to think about the code.

## Installation

julia> ] add EasyFit

julia> using EasyFit


## Contents

Read the Linear fit section first, because all the others are similar, with few specificities:

## Linear fit

To perform a linear fitting, do:

julia> x = sort(rand(10)); y = sort(rand(10)); # some data

julia> fit = fitlinear(x,y)

------------------- Linear Fit -------------

Equation: y = ax + b

With: a = 0.9245529646308137
b = 0.08608398402393584

Pearson correlation coefficient, R = 0.765338307304594

Predicted y = [-0.009488291459872424, -0.004421217036880542...
Residues = [-0.08666886144188027, -0.12771486789744962...

--------------------------------------------


The fit data structure which comes out of fitlinear contains the output data with the same names as shown in the above output:

julia> fit.a
0.9245529646308137

julia> fit.b
0.08608398402393584

julia> fit.R
0.765338307304594


The fit.x and fit.y vectors can be used for plotting the results:

julia> using Plots

julia> scatter(x,y) # the original data

julia> plot!(fit.x,fit.y) # the fit


Use the fitquad function:

julia> fitquad(x,y)  # or fitquadratic(x,y)

Equation: y = ax^2 + bx + c

With: a = 0.935408728589832
b = 0.07985866671623199
c = 0.08681962205579699

Pearson correlation coefficient, R = 0.765338307304594

Predicted y = [0.08910633345247763, 0.08943732276526263...
Residues = [0.07660191693062993, 0.07143385689027287...

-----------------------------------------------


## Cubic fit

Use the fitcubic function:

julia> fitcubic(x,y)

------------------- Cubic Fit -----------------

Equation: y = ax^3 + bx^2 + cx + d

With: a = 1.6860182468269271
b = -2.197790204605215
c = 1.431666717127516
d = -0.10389199522825227

Pearson correlation coefficient, R = 0.765338307304594

Predicted Y: ypred = [0.024757602237563042, 0.1762724543346461...
residues = [-0.021614675301217884, 0.0668157091306878...

-----------------------------------------------


## N-th degree polynomial fit

Use the fitndgr function:

julia> fitndgr(x,y,4)

------------- n-th degree polynomial degree Fit -------------

Equation: y = sum(p[i] * x^(i-1) for i in n+1:-1:1)

With: p = [1.0000000000011207, 1.99999999996782, 3.0000000001850315, 3.999999999655522, 6.000000000197637]

Pearson correlation coefficient, R = 1.0
Average square residue = 2.2956403558488966e-25

Predicted Y: ypred = [ 1.036097252072566, 1.23390364829286, ...]
residues = [ 6.104006189389111e-13, -5.706546346573305e-13, ...]

-------------------------------------------------------------


## Exponential fits

Use the fitexp function:

julia> fitexp(x,y) # or fitexponential

------------ Single Exponential fit -----------

Equation: y = A exp(x/b) + C

With: A = 0.08309782657193134
b = 0.4408664103095801
C = 1.4408664103095801

Pearson correlation coefficient, R = 0.765338307304594

Predicted Y: ypred = [0.10558554154948542, 0.16605481935145136...
residues = [0.059213264010704494, 0.056598074147493044...

-----------------------------------------------


or add n=N for a multiple-exponential fit:

julia> fit = fitexp(x,y,n=3)

-------- Multiple-exponential fit -------------

Equation: y = sum(A[i] exp(x/b[i]) for i in 1:3) + C

With: A = [2.0447736471832363e-16, 3.165225832379937, -3.2171314371600785]
b = [0.02763465220057311, -46969.25088088338, -4.403370258345724]
C = 3.543252432454542

Pearson correlation coefficient, R = 0.765338307304594

Predicted Y: ypred = [0.024313571992034433, 0.1635108558614995...
residues = [-0.022058705546746493, 0.05405411065754118...

-----------------------------------------------


## Splines

The fitting of splines requires the use of the Interpolations package (this explicit requirement was introduced in version 0.6 of EasyFit, and depends on julia >= 1.9).

To use the fitspline function, do:

julia> using EasyFit, Interpolations

julia> fit = fitspline(x,y)

-------- Spline fit ---------------------------

x spline: x = [0.10558878272489601, 0.1305310750202113...
y spline: y = [0.046372277538780926, 0.05201906296544364...

-----------------------------------------------


Use plot(fit.x,fit.y) to plot the spline.

## Moving Averages

Use the movavg (or movingaverage) function:

julia> ma = movavg(x,50)

------------------- Moving Average ----------

Number of points averaged: 51 (± 25 points)

Pearson correlation coefficient, R = 0.9916417123050962

Averaged Y: y = [0.14243985510210114, 0.14809841636897675...
residues = [-0.14230394758154755, -0.12866864179092025...

--------------------------------------------


Use plot(ma.x,ma.y) to plot the moving average.

## Density function

Use the fitdensity to obtain the density function (continuous histogram) of a data set x:

julia> x = randn(1000)

julia> density = fitdensity(x)

------------------- Density -------------

d contains the probability of finding data points within x ± 0.25

-----------------------------------------


Options are the step size (step=0.5) and normalization type (probability by default, with norm=1 or number of data points, with norm=0).

Example:

julia> x = randn(1000);

julia> density = fitdensity(x,step=0.5,norm=0);

julia> histogram(x,xlabel="x",ylabel="Density",label="",alpha=0.3,framestyle=:box);

julia> plot!(density.x,density.d,linewidth=2);



## Bounds

Lower and upper bounds can be set to the parameters of each function using the l=lower() and u=upper() input parameters. For example:

julia> fit = fitlinear(x,y,l=lower(a=5.),u=upper(a=10.))

julia> fit = fitexp(x,y,n=2,l=lower(a=[0.,0]),u=upper(a=[1.,1.]))


Bounds to the intercepts or limiting values are not supported, but it is possible to set them to a constant value. For example:

julia> fit = fitlinear(x,y,b=5.)

julia> fit = fitexp(x,y,n=2,c=0.)


## Example output:

This figure was obtained using Plots, after obtaining a fit of each type, with

julia> scatter!(x,y) # plot original data
julia> plot!(fit.x,fit.y) # plot the resulting fit


## Options

It is possible to pass an optional set of parameters to the functions. Use, for example:

julia> fitexp(x,y,options=Options(maxtrials=1000))


Available options:

Keyword Type Default value Meaning
fine Int 100 Number of points of fit to smooth plot.
p0_range Vector{Float64,2} [-100*(maximum(Y)-minimum(Y)), 100*(maximum(Y)-minimum(Y))] Range of generation of initial random parameters.
nbest Int 5 Number of repetitions of best solution in global search.
besttol Float64 1e-4 Similarity of the sum of residues of two solutions such that they are considered the same.
maxtrials Int 100 Maximum number of trials in global search.
debug Bool false Prints errors of failed fits.