# Fitting Data with a Discontinuity

This example shows how to fit an equation to data which has a sudden discontinuity using the Curve Fitting Toolbox.

## Contents

## Setting Up the Data

Load in the sample data, which is noisy and contains a discontinuity.

load sample_data sample; sample = sample'; x = (1:numel(sample))'; clf plot(x,sample) set(gca,'Ylim',[-.05 .05]) legend({'data'})

## Smoothing the Data

Smooth the data using a moving average filter of length 200. Please note that CUMSUM can be used instead of using a moving average filter.

y = smooth(sample, 10, 'rlowess'); filt_len = 200; a = 1; b = (1/filt_len)*ones(1,filt_len); mov_avg = filter(b,a,y); hold on plot(mov_avg,'r','LineWidth',2) legend({'data','moving average of length 200'})

## Finding the Discontinuity

Find the difference between element(i) and element (i+100) because the changes are very gradual, but there is continuous decrease when there is sudden drop as shown in the red line.

diff100 = mov_avg(1:end-100) - mov_avg(101:end); [~,p] = max(abs(diff100)); disp(p)

1498

## Creating the Heaviside Step Function

The Heaviside step function is defined:

It's implemented here as heaviside.m:

```
type heaviside
```

function result = heaviside(input) result = input; result(input > 1) = 1; result(input < 1) = 0; result = result(:);

## Creating the Custom Equation

Create a custom equation which incorporates the Heaviside tuned to `p`.

equation = ['a*heaviside(x-', num2str(p),')*exp(-b*(x-', num2str(p),'))+c']; disp(equation)

a*heaviside(x-1498)*exp(-b*(x-1498))+c

## Fitting the Data

st_ = [0 0 0]; outliers = excludedata(x,sample,'range',[-0.03 0.03]); fo_ = fitoptions('method','NonlinearLeastSquares',... 'Lower',[-1 -1 -1],... 'Upper',[1 1 1], ... 'Startpoint',st_, ... 'exclude', outliers); ft_ = fittype(equation, ... 'dependent',{'y'},'independent',{'x'},... 'coefficients',{'a', 'b', 'c'}); % Fit this model using new data [cf, gof] = fit(x,sample,ft_,fo_); plot(x, cf(x), 'g', 'linewidth',2); legend({'data', 'moving average of length 200','fitted curve'})

## An Alternative: Using an Continuous Approximation of the Heaviside

Fitting to the Heaviside function is tough because it produces a discontinuous result. An approach to model it could be to use a continuous replacement.

```
alternativeEquation = 'a*(1+.5*erf((x-p)/s))*exp(-b*(x-p))+c';
disp(alternativeEquation)
```

a*(1+.5*erf((x-p)/s))*exp(-b*(x-p))+c

Notice that `heaviside(x-p)` is replaced with `(1+.5*erf((x-p)/s)))`.

This also goes from 0 up to 1, it rises most quickly at `p`, and it approaches the Heaviside function as `s` approaches 0. A large starting value for `s` will help the algorithm along.

ft2 = fittype(alternativeEquation, ... 'dependent',{'y'},'independent',{'x'}, ... 'coefficients',{'a','b','c','p','s'}); fo2 = fitoptions('method','NonlinearLeastSquares',... 'Lower',[-1 -1 -1],... 'Upper',[1 1 1], ... 'Startpoint',[0 0 0 1000 1000], ... 'exclude', outliers); [cf2, gof2] = fit(x,sample,ft2,fo2); clf plot(x, sample); hold on plot(x,cf2(x),'g','linewidth',2); set(gca,'Ylim',[-.05 .05]) legend({'data','fitted curve'})