How to quickly transform a stepwiselm result into a function for further optimization?

Dear MathWorks Community,

Could you help me in finding an "automized" and quick way to transform the result of a multivariate regression using stepwiselm function of Matlab into a function, which I then use to find a global minimum?

Matlab gives me this for the regression result:

Linear regression model:

y ~ [Linear formula with 19 terms in 7 predictors]

Estimated Coefficients:

Estimate SE tStat pValue

___________ __________ _______ __________

(Intercept) 2.5665 0.36019 7.1252 3.3076e-10

x2 -0.093719 0.018019 -5.2012 1.3745e-06

x3 -0.032841 0.0065094 -5.0451 2.5826e-06

x4 -0.068186 0.013673 -4.9869 3.2601e-06

x5 -0.13995 0.024853 -5.631 2.3202e-07

x6 0.011896 0.0077462 1.5357 0.12837

x7 -0.012796 0.0061187 -2.0913 0.039521

x8 -0.047325 0.01442 -3.2819 0.001503

x2^2 0.0014954 0.00042943 3.4823 0.00079162

x2:x4 0.001958 0.00052507 3.7291 0.000348

x2:x5 0.0022529 0.00051424 4.381 3.3908e-05

x3:x5 0.0022308 0.00048064 4.6414 1.2633e-05

x4:x5 0.0018265 0.00050092 3.6462 0.00046034

x5^2 0.001424 0.00040481 3.5177 0.00070503

x6^2 -0.00066448 0.0002251 -2.9519 0.0040932

x5:x7 0.0015826 0.00040548 3.903 0.00019101

x2:x8 0.0010058 0.00057961 1.7352 0.086365

x4:x8 0.0014672 0.00044572 3.2917 0.0014578

x5:x8 0.0016849 0.00054847 3.072 0.002866

and until now I manually created the function below based on the proprosed regression factors:

>> fun=@(x)+2.5665-0.093719*x(2)-0.032841*x(3)-0.068186*x(4)-0.13995*x(5)+0.011896*x(6)-0.012796*x(7)-0.047325*x(8)+0.0014954*x(2)^2+0.001958*x(2)*x(4)+0.0022529*x(2)*x(5)+0.0022308*x(3)*x(5)+0.0018265*x(4)*x(5)+0.001424*x(5)^2-0.00066448*x(6)^2+0.0015826*x(5)*x(7)+0.0010058*x(2)*x(8)+0.0014672*x(4)*x(8)+0.0016849*x(5)*x(8);

Thanks a lot!!!

Best regards and Merry Christmas!

Philipp

[JClarcq]

did you manage to automate it? if so could you post your solution? thank you.

Answer 1

[Shubham]

Hi Philip,

It seems that you are trying to use the results from a “stepwiselm” function for further optimization. You can access the output of regression model from the “Fitted” parameter in your resultant “Linear Model”. However, if you wish to use the results for further analysis, you can create a function as well.

Have a look at the following example for a simple regression model (y ~ 1 + x1 + x2):

load hald 
mdl = stepwiselm(ingredients,heat,'PEnter',0.06) 

1. Adding x4, FStat = 22.7985, pValue = 0.000576232
2. Adding x1, FStat = 108.2239, pValue = 1.105281e-06
3. Adding x2, FStat = 5.0259, pValue = 0.051687
4. Removing x4, FStat = 1.8633, pValue = 0.2054

mdl = 
Linear regression model:
    y ~ 1 + x1 + x2

Estimated Coefficients:
                   Estimate       SE       tStat       pValue  
                   ________    ________    ______    __________

    (Intercept)     52.577       2.2862    22.998    5.4566e-10
    x1              1.4683       0.1213    12.105    2.6922e-07
    x2             0.66225     0.045855    14.442     5.029e-08


Number of observations: 13, Error degrees of freedom: 10
Root Mean Squared Error: 2.41
R-squared: 0.979,  Adjusted R-Squared: 0.974
F-statistic vs. constant model: 230, p-value = 4.41e-09

coeff_names = mdl.CoefficientNames; 
estimates = mdl.Coefficients.Estimate; 
nRows = mdl.NumObservations; 
len = length(coeff_names); 

%  Initializing output as Intercept 
out = estimates(1); 
for i = 2:len 
    % extracting column number 
    var = coeff_names{i} ;
    var_num = var(2:end) ;
    col_no = str2num(var_num ) ;

    % Calculating prediction 
    out = out + ingredients(:,col_no).* estimates(i) ;
end 
out

out = 13×1

80.0740 73.2509 105.8147 89.2585 97.2925 105.1525 104.0021 74.5754 91.2755 114.5375

Based on your requirements, you can manipulate the “CoefficientNames” as per the “Wilkinson Notation”:

https://www.mathworks.com/help/stats/stepwiselm.html#bt0cpr2-mdl:~:text=of%20Wilkinson%20notation.-,Wilkinson%20Notation,-Terms%20in%20Standard

You can split the string using delimiters based on the operation being performed:

https://www.mathworks.com/help/matlab/ref/split.html

For example, for product of terms, you can try splitting the string using “:” as delimiter:

variable = "x1:x2"; 
ans = split(variable,":")

ans = 2×1 string array
    "x1"
    "x2"

You can try splitting the string based on precedence of operators (operator with least precedence to be accounted first). After splitting the string into tokens, you can perform various operations accordingly. For multiple operators, you can perform this task recursively.

I hope this helps!