Finding the best fit for a logistic function to data

I have been trying to find the best fit parameters for a logistic regression to data using the following code, however there appears to be an issue. I am not sure if the issue is with my code or with my choice of ranges - I have based the mathematics on the following paper: https://www.math.hmc.edu/~depillis/PCMI2005WEBSITE/logistic_REDWOODS.pdf. I am trying to looop through a range of parameters and define the for each, and then look for the minimum error in order to understand what is the best fit.

Here is my code:

%loading + cleaning up the necessary data

load('trial2vars.mat');

trials = 50;

volpress = sort(volpress);

P = zeros(1,length(trials)); % pre-allocating zeros

%CDF

for t = 1:trials

P(t) = length(volpress(volpress <= volpress(t)))/length(volpress);

end

plot(volpress,P,'o');

hold on

% Fitting a logistic

M= volpress';

x1=0.00004041;

x2= 0.00003957;

y1= 0.68;

y2= 0.46;

dm = (y1 -y2)/(x1 - x2);

K=1;

hold on

r = 4 * dm ;

t0=volpress(20); % guess for the point of inflection

figure(1);

y = logistic_func(volpress,K,r,t0); % defining the guessed function

plot(volpress,y) % first guess/testing

hold off

title('Probability of hearing a puretone before a specific Volume Intensity')

xlabel('Volume Intensity')

ylabel('Probability')

%defining ranges to test each parameter in

num_r=100;

r=linspace(0.000001,0.00003,num_r);

num_t0=100;

t0=linspace(0.000035,0.000045,num_t0);

e=zeros(num_t0,num_r); %pre-allocating zeros in a matrix that depends

%both the size of t0 and r

%assigning each point on the grid its attributed error

for i=1:num_t0 %loops through each t0 value

for j=1:num_r %loops through each r value

%for each t0 value, loop through each r value

e(i,j)=myerror([r(j),t0(i)],volpress,P);

end

end

%making the 3D graph

figure(2)

mesh(r,t0,e)

axis tight

xlabel('r')

ylabel('t_0')

zlabel('Error Function')

%credit: https://web.njit.edu/~goodman/Math227/matlab/algae/

%contour map

figure(3)

contour(r,t0,e)

hold on;

% finding the best spot on the contour plot and marking with a star:

[m1,p1]=min(e);

[m2,p2]=min(m1);

best_r=r(p2);

best_t0=t0(p1(p2));

plot(best_r,best_t0,'*')

hold off

mytitle=sprintf('Best guess is r=%4.2f, t_0=%5.1f',best_r,best_t0);

title(mytitle);

xlabel('r')

ylabel('t_0')

colorbar

% finding the best fit parameters

paramguess=[best_r,best_t0];

params=fminsearch(@myerror,paramguess,[],volpress,P);

r=params(1);t0=params(2);

h = logistic_func(volpress,1,r,t0); %unscaled logistic function

K=dot(h,M)./dot(h,h);

figure(4)

plot(volpress,P,'o',volpress,logistic_func(volpress,K,r,t0));

xlabel('Volume Intensity')

ylabel('Probability')

%defining the functions

function y = logistic_func(t,K,r,t0)

%creates a logistic function with the parameters inputted

y=K./(1+exp(-r*(t-t0)));

end

function z=myerror(params,t,y)

%given a set of parameters and the original data, calculates the error

%params is inputted as an array with 2 values:

r=params(1); %first r

t0=params(2); % then t0

h=1./(1+exp(-r*(t-t0))); % a non-scaled logistic function

%calculates error with formula derived using properties of vectors

numerator = -dot(h,y)^2;

denominator = dot(h,h);

z=numerator/denominator;

end