Sigmoid function shaping and fitting by curve-fitting toolbox

Hi all,

I am trying to fit a sigmoid function to my data.

Basicaly,

x=1 2 3 4 5 6 7 48

y= 119 163 240 395 617 1461 4310 36955464

first seven data are Electric Vehicles in a country corresponding to year number 1 to 7, and last data (48, 36955464) is the 48th year data.

I have fitted a Logistic Curve as y=36.955.464/(1+119*e^-0,425x)

However, although it fits my data well and ends up showing an expected curve, I need to shape the curve a bit. For instance, parabolic part of the logistic function has to be a bit stronger (neds to increase a bit faster), while exponential part of the sigmoid curve should increase slower.

Is there any other suggestion to fit an adjustable Sigmoid Curve to build up a nicer curve? For example, if I prefer to construct a Hyperbolic Tangent curve (which is also a S-shaped function) how should I construct it ?

Best wishes to all Matlab community

Answer 1

[William Rose]

@Batuhan,

A fairly general sigmoid curve can be expressed with four parameters: ymin, ymax, x0, and slope (dy/dx) at x0. The wikipedia article Sigmoid Function has equations for various sigmoid functions. It has a nice plot comparing different functions. The functions in the wikipedia image all have ymin=-1, ymax=+1, x0=0, and slope=1 at x=0. The image shows that different sigmoid funcitons with the same central slope, and the same upper and lower limits (ymin and ymax) differ in how rapidly they approach the asymptotes. Therefore you can try the different functions and select the one whose results you like.

I do not really understand what you mean by a "nicer curve". You said it "needs to increase a bit faster", etc. When you say "it needs to...", do you means based on the eye-test of what looks good, or some quantifiable goodness of fit criterion? I do not know what you mean by the "parabolic" and "exponential" parts of the curve. Perhaps you mean the left side and the ridght side? Or the middle versus the outpr parts? Typical sigmoid curves are symmetrical about the center.

The equations given for sigmoid functions in the wikipedia article all are centered at x=0. Replace "x" with "(x-x0)" in the wikipedia equations, and set x0 as you wish, to get sigmoid curves that are centered at x=x0.

The hyperbolic tangent funciton is just a shifted and scaled version of the logistic function, so it has the exact same shape as the logistic function, when the parameters are adjusted appropriately. The hyperbolic tangent function is

Replace x with x-x0 to shift the curve horizontally, and add a vertical shift and scale, as you wish. I notice that your logistic function has ymin=0, which is a sensible choice for the number of EVs in a coutry at early times. To shift the hyperbolic tangent function so that it has ymin=0, do this:

In the equation above, k determines the slope at the center. The center is at x=x0. The max value is ymax. You can show with a bit of algebra that the equation above is eqivalent to a logistic function. Notice that the immediately preceding equation has 3 adjustable parameters: ymax, k, and x0. I said earlier that a general sigmoid function has 4 adjustable parameters. The equation above has only three, because we fixed ymin at 0, thus eliminating one adjustable parameter.

Good luck with your work.

[Batuhan]

Hi @William Rose,

Thanks for your answer and sorry about my wording since I am not as experienced as you.

Let me explain what I am trying to do by a visual:

Here blue dots are Logistic Growth Curve which I have fitted to my data. However (a nicer curve :) a better curve which I would like to create is as the black line in above graph.

Yes I meant the left and right part by saying parabolic and exponential parts.

The reason why I said "it needs to..." is that I "forecast" the data to have a behaviour as black line.

The final equation does not have to fit very well to my existing data set because this data set is not efficient for sure. However, in the end, my selected function form should satisfy three criterias:

1. First 7 data should be satisfied "as much as" it can
2. Should be S curve and behave as black line above (first part and second part)
3. Should saturate at 8th data (48, 36955464)

My all in all target is to have a function (with variable x) to predict the data between "known" data.

Thanks again for your kind help and answer.

[William Rose]

@Batuhan, thank you for the image. Can you share the data points or the equation that was used to generate the black line? The black line in your figure looks like it is two curves: one for x<24 and a different curve for x>=24.

You said you want the sigmoid to plateau at y=3.70e+7 (the y-value at x=48). Therefore the script implements that constraint.

I suspect that we will not be able to match the black curve very well with any sigmoid funciton - because the black curve looks quite straight from x=35 to th end (x=48), and a sigmoid funciton will be flattening out.

The black curve looks asymmetric: the section from x=35 to x=48 is very straight (s noted above), and the correspondng part at the other end (x=0 to 13) is not so straight. Also, the slope from x=35 to 48 is greater than the average slope of the curve on the oppostive side of the center. As I said in my initial answer, all the sigmoid functions listed in the wiki[edia article are symmetric about the middle. But we can create an assymetric sigmoid function. There are diferent ways you could do it. One way to fit an assymetric sigmoid is by a nonlinear mapping of x=(0,48) to =(0,48). Then we fit a standard sigmoid function such as the logistic function, but we use instead of x. A nonlinear mapping with a single adjustable parameter would be nice. The nonlinear mapping must map x=0 to =0, and it must map x=48 to =48. Here's a candidate mapping:

where p=0 to +Inf. Here is a plot of how the map works, with different values for p.

More later. I have to go.

[Sam Chak]

Hi @Batuhan

@William Rose is right that no symmetrical sigmoid function can match the desired Black curve. That's why I commented I won't use the sigmoid models. At first glance on the Black curve, the inverse hyperbolic sine function came to my mind:

However, it won't stay saturated at (48, 36955464) as desired, leading to abandon this idea. So, I proposed a non-symmetrical S-shaped Gompertz function in my edited Answer.

[William Rose]

@Batuhan, @Sam Chak has a nice idea with Gompertz, which is a popular funciton in the world of mortality modeling. I will look at your blck curve data in a couple of days but busy for now.

Answer 2

[Sam Chak]

Hi @Batuhan

Edit 2: Since the shape of the Black curve is preferred, Gompertz function is proposed for the fitting.

Click on the link to understand how you can adjust parameters b and c to obtain the fine-tune desired shape.

% Data
x  = [1 2 3 4 5 6 7 48];
y  = [119 163 240 395 617 1461 4310 36955464];

% Proposed non-symmetrical S-shaped Gompertz function
xx = linspace(1, 48, 47001);
a  = 36955464;      % a stands for amplitude (max value of y data)
b  = 5.631e-09;     % adjust the displacement in x-axis (if d is tuned, leave b = 0)
c  = 0.2184;        % growth rate (or the slope)
d  = 17.08;         % adjust the displacement in x-axis (if b is tuned, leave d = 0)
yy = a*exp(-exp(-b - c*(xx - d)));

plot(xx, yy, 'linewidth', 1.5), hold on
plot(x, y, 'ro', 'MarkerSize', 10)
grid on, xlabel('x'), ylabel('y'), hold off
legend('Gompertz', 'data', 'location', 'northwest', 'FontSize', 10)

General Gompertz model:

f(x) = a*exp(-exp(-b - c*(x - d)))

Coefficients (with 95% confidence bounds):

a = 3.7e+07 (3.654e+07, 3.746e+07)

b = 5.631e-09 (fixed at bound)

c = 0.2184 (-0.04142, 0.4782)

d = 17.08 (5.059, 29.09)

Goodness of fit:

SSE: 1.555e+06

R-square: 1

Adjusted R-square: 1

RMSE: 557.7

[Batuhan]

Hi @Sam Chak,

Firstly thanks very much for your informative, clear and useful answer.

My expected curve is as follow:

Actually, Logistic Growth Function fits this data very well.

I have defined L as saturation point (which is the last data in my case as (48, 36955464), and I(0) is the initial data as 119, and I have played with "k" to find the best fit to my data.

After all, I am trying to find another alternative S-shaped function (such as Hyperbolic Tangent) to fit my data with an S shape just as above graph.

In the case you have defined (tanh scenario), if I play with the coefficient b, will I get there ?

[Sam Chak]

No, you won't get there. A large portion of the data in the central region are unavailable. So, there are many Sigmoids can fit these 8 points. However, I don't think that the curve-fitting algorithm can intelligently fit the Sigmoid model without making a few assumptions.

Based on your preference of the Black curve, I probably won't use Sigmoids.