Finding the inflection point of a sigmoid function

Question

Iddo Weiner 2017-12-18

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https://ww2.mathworks.cn/matlabcentral/answers/373446-finding-the-inflection-point-of-a-sigmoid-function

编辑： Torsten 2023-1-30

Hi, here's my function:

% define params
p1 = 0.0389;
p2 = 1.9158;
p3 = 162.4272;
p4 = 0.012;
% simulate the function
syms x
f = p1+(p2-p1)/(1+10^((p3-x)*p4));
fplot(f,[1 245])

Now, I would like to find the inflection point (I manually drew where I roughly expect it to be).

Here's what I was trying to do:

double(solve(f,'MaxDegree',3))

but I get this:

ans = 2.1394e+01 - 1.1370e+02i

Which means that there is no real inflection point... Where's my mistake?

Thanks in advance

Iddo

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Answer 1

Star Strider 2017-12-18

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https://ww2.mathworks.cn/matlabcentral/answers/373446-finding-the-inflection-point-of-a-sigmoid-function#answer_296787

在 MATLAB Online 中打开

Try this:

p1 = 0.0389;
p2 = 1.9158;
p3 = 162.4272;
p4 = 0.012;
% simulate the function
syms x
f(x) = p1+(p2-p1)/(1+10^((p3-x)*p4));
inflpt = vpasolve(diff(f,2) == 0);                          % Calculate Inflection Point
figure(1)
fplot(f,[1 245])
hold on
plot(inflpt, f(inflpt), 'pr', 'MarkerSize',10, 'MarkerFaceColor','g')
hold off

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Karan Gill 2017-12-18

Iddo, did you understand why that code works?

Iddo Weiner 2017-12-18

Yes, setting the 2nd derivative to equal zero will give you the inflection point

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Answer 2

Roger Stafford 2017-12-18

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https://ww2.mathworks.cn/matlabcentral/answers/373446-finding-the-inflection-point-of-a-sigmoid-function#answer_296697

编辑：Roger Stafford 2017-12-18

The inflection point occurs at x = p3. You can show it by using the symbolic 'diff' to find the second derivative of f with respect to x and finding the x that makes it zero. That occurs when 10^((p3-x)*p4)) is equal to 1 which forces x to equal p3.

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Iddo Weiner 2017-12-18

在 MATLAB Online 中打开

Hey Roger, thanks for your help.

I understand what you say and it's right for this case, but I would like to have a piece of code that automatically calculates the inflection point.. Looking at the official documentation it looked like this line of code is supposed to do the work:

double(solve(f,'MaxDegree',3))

But, it didn't work.. So do you see a different way to automatically generate the inflection point? Or maybe I'm not using solve() properly?

Thanks, Iddo

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Answer 3

Lalitesh 2023-1-30

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https://ww2.mathworks.cn/matlabcentral/answers/373446-finding-the-inflection-point-of-a-sigmoid-function#answer_1159290

编辑：Torsten 2023-1-30

在 MATLAB Online 中打开

Hi,

I hope you have solved your inflection point problem. Could you please suggest me how to find inflection point in the given below case ?

y = [4941.325135

5373.592952

5341.265151

5109.495841

5164.985178

4910.347906

5036.098753

5210.505118

4897.930074

5303.555152

5059.674066

5294.250906

5045.080549

5139.444791

5184.016273

4989.822273

4879.351675

5037.299623

5193.317198

5126.702477

5114.934361

5163.894699

5250.845909

5236.142352

5425.261686

5026.998422

5181.941373

5043.063249

5226.955717

5045.126978

5285.330253

5147.00929

5089.199187

5433.585096

5259.022363

5135.878393

5075.663506

5068.604444

5240.425345

5013.836134

5235.932975

5293.152198

5362.597535

5288.109111

5034.519472

5095.714828

5267.142594

5168.69091

5050.240489

4921.738751

];

x = [-0.013607785

0.765345224

1.10289259

1.029090207

-0.022931512

-0.017396819

-0.074107114

0.769494196

2.98E-06

0.920557852

0.738470623

-0.025517765

0.820330824

0.561636238

1.02437854

0.748933493

-0.024258105

0.172846729

-0.040833861

1.015883315

0.961682273

0.974008018

-2.782219696

1.02138985

0.556479342

0.411214807

1.056242842

-3.368250299

1.094017765

1.128149932

-0.040686495

-0.013525985

-0.028379761

0.979391037

0.662479735

1.127613475

0.488222296

-0.067298217

1.022925141

-0.119633187

1.04065864

0.286375511

1.195491684

0.957996374

1.103174243

1.049128867

1.080726806

0.99141825

1.198772497

0.864385088

];

fn = @(b,x) b(5).*(b(1) + b(2)./(1 + exp(-b(3).*(x-b(4))))).^(b(6)) % Sigmoid Function (Objective Function)

fn = function_handle with value:

@(b,x)b(5).*(b(1)+b(2)./(1+exp(-b(3).*(x-b(4))))).^(b(6))

SSECF = @(b) sum((y - fn(b,x)).^2); % Sum-Squared-Error Cost Function

B0 = [-0.1; -0.1; -5; 0.1; 0.8; 10]; % Initial Parameter Estimates

[B,SSE] = fminsearch(SSECF, B0) % Estimate Parameters (B), Return Sum-Squared Error (SSE)

B = 6×1

-1.1849 -0.8181 51.3923 0.4650 6.0624 10.1272

SSE = 2.8669e+08 + 1.8180e+08i

xp = linspace(min(x), max(x))

xp = 1×100

-3.3683 -3.3221 -3.2760 -3.2299 -3.1837 -3.1376 -3.0915 -3.0453 -2.9992 -2.9531 -2.9069 -2.8608 -2.8147 -2.7685 -2.7224 -2.6763 -2.6301 -2.5840 -2.5379 -2.4918 -2.4456 -2.3995 -2.3534 -2.3072 -2.2611 -2.2150 -2.1688 -2.1227 -2.0766 -2.0304

figure(1)

plot(xp, fn(B,xp), '-r')

Warning: Imaginary parts of complex X and/or Y arguments ignored.

hold on

plot(x,y,'o')

hold off

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Lalitesh 2023-1-30

I will post this as a new question.

Torsten 2023-1-30

编辑：Torsten 2023-1-30

Hopefully with better data.

Otherwise you can see the question as answered: impossible.

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Finding the inflection point of a sigmoid function

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