Multi-target data optimization

Question

Alejandro Fernández 2020-2-2

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https://ww2.mathworks.cn/matlabcentral/answers/503213-multi-target-data-optimization

评论： Thiago Henrique Gomes Lobato 2020-2-2

Hi, there. I was wondering if someone could help me with something.

I have 3 data matrices, of which I show you the contour plots (these I show would be just an example, these are from functions but the real ones are not from any kind of functions).

What I need is to be able to define a region of minimum optimums by weighting the matrices, that is, that not all of them have the same importance. The thing is that I only find things related to functions and they are not useful to me...

Thank you very much for everything.

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Mohammad Sami 2020-2-2

What if you just add the 3 weighted matrix and then use the minimum function

Z1 = load('Z1.mat');
Z2 = load('Z2.mat');
Z3 = load('Z3.mat');
w1 = 1;
w2 = 1;
w3 = 1;
Z = w1 * Z1.Z + w2 * Z2.Z + w3 * Z3.Z;
[,idx] = min(Z);

Alejandro Fernández 2020-2-2

Ok, I'll try, thank you so much.

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Answer 1

Thiago Henrique Gomes Lobato 2020-2-2

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https://ww2.mathworks.cn/matlabcentral/answers/503213-multi-target-data-optimization#answer_413329

Could you explain a little more the problem you're having? In the end it really doesn't matter if your matrix are functions or not since you can define the matrix weighting as a function, as an example:

f = @(Z1,Z2,Z3)Z1*2+Z2*1+Z3*3;

But actually what you seems to want is the minimum of the matrix weighted sum, and if you already have the values, it is really as simple as sum them and calculate the min:

w1 = 1;
w2 = 2;
w3 = 3;
Zweighted = Z1*w1+Z2*w2+Z3*w3;
[MinVal,MinPos] = min(Zweighted)

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Thiago Henrique Gomes Lobato 2020-2-2

You don't need the Pareto front in your case. Pareto front is just when you have multiple different objectives and then want to check how to compromise one in respect to the other. In your case all 3 matrix are actually just one variable, more specifically they all represent together a distribution of your result. In this way you will get non-sensical results if you just do a weigted matrix sum. In your case there's at least two main approaches that are reasonable:

The minimum is the minimum of the average matrix: This will produce good results if your goal is to minimize a specific process that will repeat itself for many many times.
The minimum is the minimum of the mean-std*confidence : This will produce good results if your goal is to find the best possible scenario it could possible happens. The idea is to look at the tails of the distributions with some statistic confidence interval and check the minimum. In code it will look like something like this:

Zweighted = Zaverage-(Zstd*Confidence)./sqrt(ZnumberOfParticipants)

Confidence can be, for example, 1.96 for 95% confidence (read here for a more exact definition if you want, http://sphweb.bumc.bu.edu/otlt/MPH-Modules/BS/BS704_Confidence_Intervals/BS704_Confidence_Intervals_print.html)