Minimize distance between curves

I have a dataset with values of multiple curves. An example plot is shown below. I want to scale the curves so that all curves overlap or get closer to the target curve. The following is the plot of a sample dataset

I want to scale the function values in the curves so that all curves overlap. Say, green curve is the target (g) and I want to move the other curves (f) towards target curve by scaling (scale facotor is a).

The function that I want to minimize is:

Expected result:

Data:

data = {{{157, 60}, {140, 57}, {121, 52}, {103, 47}, {86, 39}, {67, 28}, {50,21}, {32, 15}}, {{159, 70}, {140, 67}, {122, 63}, {105, 58}, {85, 45}, {68, 42}, {50, 34}, {33, 29}}, {{157, 74}, {140, 71}, {121, 70}, {105, 65}, {86, 55}, {69, 52}, {52, 43}, {32, 37}}, {{158, 100}, {141, 100}, {123, 95}, {105, 90}, {88, 78}, {69, 72}, {51, 57}, {33, 46}}, {{160, 143}, {140, 146}, {123, 146}, {105, 145}, {86, 136}, {70, 121}, {51, 97}, {34, 60}}}

Could someone please help me with the function that has to be used for solving this problem in MATLAB ?

Answer 1

[Torsten]

Differentiating with respect to a, setting the derivative to 0 and solving for "a" gives

a = sum_i (f_i*g_i) / sum_i (f_i^2)

I leave it to you to insert the data points.

[Deepa Maheshvare]

@Torsten

Thank you, this helps.

Could you please give suggestions on how this scale factor `a`can be determined if one wants to minimize the absolute error instead of the squared error?

[Torsten]

What is the "absolute error" ?

sum_i abs(a*f_i - g_i)

?

If this is the case, use "linprog" to solve

min: sum_i u_i
s.c.
f_i * a - u_i <= g_i
-f_i * a - u_i <= -g_i

with unknowns a and u_i (i=1,...,number of data points).

[Deepa Maheshvare]

@Torsten

Thanks so much. Yes, I mean sum_i abs(a*f_i - g_i)

% Loss Function (absolute error loss)

%% input curves
scale =  1.5;
x1 = [0,4,6,10,15,20]*scale;
y1 =  [18,17.5,13,12,8,10];
x2 = [0,10.5,28]*scale;
y2= [18.2,10.6,10.3];

%% y2 is the target function and y1 is the function to be shifted
f = y1;
g = interp1(x2,y2,x1);

%% linprog inputs
% x = linprog(f,A,b)
% i = 1:length(x1)
A = [fi - 1; -fi -1];
x = [a ui];
b = [gi -gi];
% obj = sum_i u_i (function to minimize)

%% solve system

I am not sure how to set this up in MATLAB's linprog

Could you please help?