How to fit a cumulative normal distribution into a smooth curve?

I have computed the normal cumulative distribution values for a certain set of data. The data is as follows:

x = [0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4];

prob = [0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0007 0.0018 0.0041 0.0080 0.0138 0.2011 0.4663 0.6518 0.8110 0.9028 0.9470 0.9728 0.9851 0.9920 0.9963 0.9975 0.9983 0.9990];

where prob contains the probability of failure at corresponding value of x already obtained using normcdf on the reliability index.

I want to make a smooth CDF using this data (prob vs x) by passing this data to a function. How do I do it?

I've already used a function available at https://in.mathworks.com/matlabcentral/answers/345595-fitting-cumulative-normal-distribution-function-to-data but using this is leading to a problem that even though the first probability values are zero, the fit does not start from zero. I'm in need of something similar

Answer 1

[Sam Chak]

Hi @Rohit Sinha

The Error function is essentially identical to the standard normal cumulative distribution function, and so I attempted to fit the Error function to the data, with the aim to connect as many points as possible.

myfittype = fittype('0.5*erf(6.6*(x - c)) + 0.5',...
    'dependent', {'prob'}, 'independent', {'x'},...
    'coefficients', {'c'})

[myfit, gof] = fit(x', prob', myfittype)

which produces

myfit =

General model:

myfit(x) = 0.5*erf(6.6*(x - c)) + 0.5

Coefficients (with 95% confidence bounds):

c = 0.3302 (0.3111, 0.3493)

gof =

struct with fields:

sse: 0.0520

rsquare: 0.9895

dfe: 23

adjrsquare: 0.9895

rmse: 0.0476

plot(myfit, x, prob)

[Rohit Sinha]

@Sam Chak This is great, is there a better fit available? Because as you can see, the curve is going below 0 as well as not becoming 1 towards the end. I'm expecting the curve to fit in a manner like the blue line here (I've erased other distributions to stay specific, ignore any other ones even slightly visible). Maybe like a smoothing spline?

[Sam Chak]

@Rohit Sinha, your data points are not uniformly disturbuted. So, the standard algorithm attempts to fit the proposed model to the data for the highest R-square value.

The adjusted Error function model in my edited Answer will fit as many points as possible.

[Rohit Sinha]

@Sam Chak thanks for updating your answer, however, the data I provided is just 1 of many. Even in the same plot, I have as many as 6 datasets each of which I need to put in the same figure, I cannot use the coefficients you provided for all of them so I wanted a generalised thing that would work best, just like the figure I shared. That is the exact thing I want. Hope you understand

[Sam Chak]

@Rohit Sinha, If your other data are bounded between 0 and 1 like most normal cdfs do, then maintain the model structure like this

0.5*erf(b*(x - c)) + 0.5

but adjust the b > 0 value manually (like I did). The Algorithm is not 'smart enough' to know that you want to fit as many points as possible.

[Rohit Sinha]

@Sam Chak, yes, all my data is bounded between 0 and 1. However, 0.5*erf(b*(x - c)) + 0.5 results in the line take an offset from 0. That means, the fit does not start from 0 but a significant (visible) distance above it. Hoever I believe, as you say, playing with the value of 'b' must do the trick. Thanks a lot

[Sam Chak]

@Rohit Sinha, I have not seen all your data. And I hope you understand that the CDF (aka biased Error function) only achieves true 0 at .

By manipulating the b value, you can practically shape the curve you want to fit as many data points as possible.

You will need some time to explore which CDF/Error function model fits the data set. If my Answer solves your original problem and you gain something useful and knowledge from my Comments, please support by voting and accept it.

We can still continue with the discussion, if you are still interested to fit all 6 data sets.