Which regression method may give better R^2 values on this scattered data?

I need to use linear regression to estimate a linear relation.

The data is a matrix of 22x5 elements, related to 22 samples and 5 parameters / features for each of the samples, and I need to estimate the coefficients a1 to a5 in this equation:

X * A = Y

(X is 22*5; A is 5*1; Y is 22*1) and y values are the experimental / target value.

I have done the job using fmincon function in MATLAB:

    % constants used in the program
    Number_of_Columns = 5;
    Number_of_Samples = 22;

    % contains the matrix 22*5, as well as the 6th column as the target
    saved_file = 'bank_22.mat';
    load(saved_file);

    X = zeros(Number_of_Samples , Number_of_Columns);
    Y = zeros(Number_of_Samples , 1);

    for  sample = 1 : Number_of_Samples    
        for  col = 1 : Number_of_Columns        
            X(sample , col) = bank_22{sample , col}; % Matrix of unknowns
            Y(sample , 1) = bank_22{sample , 6}; % Target vector (i.e. Experimental values)        
        end    
    end

    tol1 = 1e-12;

    options = optimset('TolFun' , tol1 , 'TolCon' , tol1 , 'TolX' , tol1 , ...
        'MaxIter' , 1e3 , 'MaxFunEvals' , 1e6);

    %  x = fmincon(fun , x0 , A , b , Aeq , beq , lb , ub , nonlcon , options)

    [A , fval] = fmincon(@(A)projMin(A , X , Y) , zeros(Number_of_Columns , 1) , ...
        [] , [] , [] , [] , zeros(Number_of_Columns , 1) , ...
        ones(Number_of_Columns , 1) , [] , options);

(where projMin is a user-defined function to minimized the difference between X*A and Y by norm-1; and the constraint on the values of a1 to a5 coefficients is that they need to be between zero to 1).

After getting the coefficients, I cross-validated the results by using the obtained values for coefficients a1 to a5 and predicting the values for the vector Y.

The max error and standard deviation are fine, but the calculated R-squared is low. So I wonder which other method may be a reasonable option to find this linear relation.

I appreciate your helpful comments.

P.S. For reference, see the result of cross-validation, i.e. predicted vs. target values.

Update: Sorry for confusion! In the following scatterplot, the spatial position of the small circles themselves shows where they are located, while the line shown on the graph is for the ideal linear relation, with R^2 = 1.0.

Answer 1

[Sean de Wolski]

Have you tried using one of the Machine Learning Regression techniques on it?

doc fitlm
doc stepwiselm
doc fitrtree
doc fitensemble

[Trevos16]

Thanks for your answer Sean! I haven't tried any other option yet. So do your suggested techniques make a linear regression? If so, I will continue with applying them on my data. Thanks!

[Trevos16]

I just made an update on the question, and added the last paragraph to clarify the cross-validation graph.

[Sean de Wolski]

fitlm is a multilinear regression, and stepwiselm is a multilinear regression that tries to remove predictors that don't add value.

The regression trees are not linear regressions but are tools used to fitting data that are sometimes superior to linear regression when linear regression is not well posed for that data set.