Non Linear Regression for a surface

Question

Hidd_1 2023-6-29

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1989633-non-linear-regression-for-a-surface

编辑： Hidd_1 ，2023-7-27

采纳的回答： S0852306 ，

I have the following matrices, and their dimensions:

X (1*249 double)

Y (1*20 double)

Z (20*249 double)

I know that the relation between X and Z is a Sigmoid function ( Z = 1/exp(b(1).*X + b(2))).

I would like to find a function Z = function(X,Y), with the variables X, and Y.

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Hidd_1 2023-6-29

I think a function like objfun = @(b,X,Y) b(1) + (Y.*b(2))./(b(3) + exp(-b(4).*X + b(5)));

where objfun should fit the Z data.

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Answer 1

S0852306 2023-7-24

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1989633-non-linear-regression-for-a-surface#answer_1277703

编辑：S0852306 2023-7-24

@Hidd_1 "I found a function that fitt the surface with R2 = 0.92, do you have any tips how can I improve it?"

About this problem, you may try this, here's the result:

R-squared: 0.999089

RMSE: 0.1797

MAE: 0.1202

The fitted surface match your data quite well.

The model I used is a neural net, and its mathematical model is just lots of "weighted sigmoid function".

Notation:

is point-wise sigmoid function, and

are weight matrices and bias vectors

The neural net perform the following operation;

Neural nets are universal approximator, so if you want further improvement, use a larger net, but be careful of overfitting.

clear; clc; close all;
load('Z.mat');
Z=Inter_cubic;
% to run this script, download the tool
% at file exchange: https://tinyurl.com/wre9r5uk
%% Reshape the data to required format
X= 1:1:249;
Y = 1:0.2:4.8; 
count=0;
for i=1:numel(X)
    for j=1:numel(Y)
        count=count+1;
        input(1,count)=X(i);
        input(2,count)=Y(j);
        output(count)=Inter_cubic(count);
    end
end
%% model set up
NN.InputAutoScaling='on'; NN.LabelAutoScaling='on'; % perform normalization x'=(x-mean(x))/std(x)
NN.ActivationFunction='Sigmoid'; % 'Gaussian' actually performs better
InSize=2; OutSize=1; % input : x,y ,output : z=f(x,y)
LayerStruct=[InSize,5,5,5,OutSize]; % change the size of network here, e.x. [2,10,10,1];
NN=Initialization(LayerStruct,NN);
%% least square solver
option.MaxIteration=500;
NN=OptimizationSolver(input,output,NN,option);
%% stats
R=FittingReport(input,output,NN);
SST=sum((output-mean(output)).^2);
SSE=sum(R.ErrorVector.^2);
Rsquared=1-SSE/SST;
%% visualization
p=NN.Evaluate(input);
[Xmesh,Ymesh]=meshgrid(X,Y);
pMatrix=reshape(p,size(Z,1),size(Z,2));
figure
scatter3(input(1,:),input(2,:),output,'.')
hold on
s=surf(Xmesh,Ymesh,pMatrix);
s.EdgeColor='none';
legend('data point','fitted surface')

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S0852306 2023-7-26

编辑：S0852306 2023-7-27

@Hidd_1

Type "NN.weight" and "NN.bias" in command window to get

,

, or just click the "NN" struct in your

workspace, like this

you can see all the parameters. here's the code of neural net:

(I think this code explain everything)

function FunctionOutput=ANN(data,NN)
if strcmp(NN.InputAutoScaling,'on')==1
    data=NN.InputScaleVector.*data-NN.InputCenterVector; % linear transform 
end
v=data;
for i=1:NN.depth-1 
    v=NN.weight{i}*v+NN.bias{i};
    v=NN.active(v);
end
v=NN.OutActive(NN.weight{NN.depth}*v+NN.bias{NN.depth});
FunctionOutput=v;

"data" is your input vector, i.e. [x;y] (column vector)

For the output normalization, here's the code and formula:

z=NN.LabelScaleVector.*FunctionOutput+NN.LabelCenterVector;

Notations :

Since I had performed input normalization, you need perform simple linear transform before put data into the net. you can get the coefficients (

) by just typing NN.InputScaleVector,

NN.InputCenterVector...

Summury

the complete computation of this NN model is:

z is the true output of this model.

Hope this answers your question!

If not, I'll upload a live script explaining the math in detail over this weekend :)

btw, could you also leave this question in the discussion section of

the file exchange page ? this way, I don't have to repeat answering this frequently asked

question. Thanks! (I'll post a code and provide a step-by-step numerical example, it

will make these mathematical expressions much easier to understand.)

Hidd_1 2023-7-27

编辑：Hidd_1 2023-7-27

Thank you very much! I have written the question on the file exchange page ? and I would be greatful if you can upload the live script which explain the math in details ;)

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