Non Linear Regression for a surface

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.

[Torsten]

Which variable do you intend to fit by the model in "objfun" ?

[Hidd_1]

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.

Answer 1

[S0852306]

@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')

[Hidd_1]

Really Really Cool! Thank you...

Where can I find the equation that fits the surface?

[S0852306]

@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]

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 ;)