Multiple regression with nonlinear variables

Hello,

I am working with the attached dataset, where the first column represents temperature and the next six columns (2–7) correspond to temperature-dependent properties.

I would like to explore whether it is possible to model temperature as a function of these six properties simultaneously, similar to a multiple regression approach. I have previously done this with linear responses, but in this case, the relationships are nonlinear (sigmoidal, Z-shaped).

I considered using a generalized additive model (GAM), but I have no prior experience with this method and may be overlooking a simpler or more suitable approach.

Could anyone provide insights or suggestions on how to best tackle this?

Thanks in advance! :)

Answer 1

[Star Strider]

I am not certain what you want to do.

Fitting a generalized additive model (GAM) for regression using the fitrgam function would go something like this —

T1 = readtable('temp-vs-properties.txt')

T1 = 9x7 table
    Var1     Var2       Var3       Var4       Var5       Var6       Var7  
    ____    _______    _______    _______    _______    _______    _______

     12      10.941     2.2536      1.029     4.8549     10.633    0.45658
     25      9.8131     2.2364     1.0215     4.3878     9.6068    0.45674
     50      6.7255     2.2287    0.97971     3.0177     6.8649    0.43959
     75      3.6134     2.1326    0.88567     1.6944     4.0799     0.4153
    100      1.3233     1.7712    0.64251    0.74709     2.0595    0.36275
    125     0.51476     1.2776    0.38488    0.40291     1.3374    0.30126
    150      0.2727    0.96724    0.24871    0.28194     1.0965    0.25713
    175     0.19387    0.85354    0.19718    0.22713    0.98321    0.23101
    200     0.16745    0.86259    0.18527    0.19413    0.90385    0.21478

VN = T1.Properties.VariableNames;

figure
plot(T1{:,1}, T1{:,2:end})
grid
xlabel(VN{1})
ylabel('Properties')
legend(VN{2:end}, Location='best')

●

Mdl = fitrgam(T1, 'Var1')

Mdl = 
  RegressionGAM
            PredictorNames: {'Var2'  'Var3'  'Var4'  'Var5'  'Var6'  'Var7'}
              ResponseName: 'Var1'
     CategoricalPredictors: []
         ResponseTransform: 'none'
                 Intercept: 101.3333
    IsStandardDeviationFit: 0
           NumObservations: 9

I am not certain that I am plotting your data correctly.

There are other options, such as fitnlm that might be more appropriate, depending on what you want to do.

.

[Fernando Maturi]

Thanks, @Star Strider

My goal is to model Var1 as a function of Var2 through Var7 simultaneously, similar to a multiple regression approach.

[Star Strider]

This fitrgam call does that.

I also creaztted a function that fits each column as gaussians.

The individual columns are fitted to:

The function works, however the initial parameeter extimates could be more appropriate.

T1 = readtable('temp-vs-properties.txt')

T1 = 9x7 table
    Var1     Var2       Var3       Var4       Var5       Var6       Var7  
    ____    _______    _______    _______    _______    _______    _______

     12      10.941     2.2536      1.029     4.8549     10.633    0.45658
     25      9.8131     2.2364     1.0215     4.3878     9.6068    0.45674
     50      6.7255     2.2287    0.97971     3.0177     6.8649    0.43959
     75      3.6134     2.1326    0.88567     1.6944     4.0799     0.4153
    100      1.3233     1.7712    0.64251    0.74709     2.0595    0.36275
    125     0.51476     1.2776    0.38488    0.40291     1.3374    0.30126
    150      0.2727    0.96724    0.24871    0.28194     1.0965    0.25713
    175     0.19387    0.85354    0.19718    0.22713    0.98321    0.23101
    200     0.16745    0.86259    0.18527    0.19413    0.90385    0.21478

VN = T1.Properties.VariableNames;

figure
plot(T1{:,1}, T1{:,2:end})
grid
xlabel(VN{1})
ylabel('Properties')
legend(VN{2:end}, Location='best')

np = 4;

x = T1{:,2:end};
y = T1{:,1};

Mdl = fitnlm(x, y, @(b,x)gfit(b,x,np), rand(np*size(x,2),1)+5)

Warning: Rank deficient, rank = 15, tol = 3.394038e-14.

Warning: Rank deficient, rank = 6, tol = 2.199340e-14.

Warning: Rank deficient, rank = 6, tol = 2.198351e-14.

Warning: The model is overparameterized, and model parameters are not identifiable. You will not be able to compute confidence or prediction intervals, and you should use caution in making predictions.

Mdl = 
Nonlinear regression model:
    y ~ F(b,x)

Estimated Coefficients:
            Estimate          SE           tStat         pValue   
           ___________    __________    ___________    ___________

    b1         -4142.6    3.2032e-16    -1.2933e+19    1.4311e-150
    b2         -62.858    1.9618e-32    -3.2041e+33    1.0082e-265
    b3          685.64    8.1732e-49     8.3888e+50              0
    b4          17.201        3.6866         4.6659      0.0016112
    b5           5.865    1.3694e-80     4.2829e+80              0
    b6          5.8148    2.4084e-96     2.4144e+96              0
    b7          5.9381    5.785e-113    1.0265e+113              0
    b8          16.291        3.6866         4.4189      0.0022297
    b9          5.9551             0            Inf              0
    b10         5.3737             0            Inf              0
    b11         5.1931             0            Inf              0
    b12         17.173        3.6866         4.6581      0.0016275
    b13         805.24             0            Inf              0
    b14         8.2765             0            Inf              0
    b15         731.55             0            Inf              0
    b16          16.91        3.6866         4.5869      0.0017859
    b17    -8.8137e+05             0           -Inf              0
    b18         8109.5             0            Inf              0
    b19          92794             0            Inf              0
    b20         17.003        3.6866         4.6121      0.0017279
    b21         5.8773             0            Inf              0
    b22         5.1137             0            Inf              0
    b23         5.6659             0            Inf              0
    b24         16.756        3.6866         4.5451      0.0018866


Number of observations: 9, Error degrees of freedom: 8
Root Mean Squared Error: 66.4
R-Squared: 0,  Adjusted R-Squared 0
F-statistic vs. zero model: 21, p-value = 0.0018

parameters = array2table(reshape(Mdl.Coefficients.Estimate, 4, []), RowNames={'b(1)','b(2)','b(3)','b(4)'}, VariableNames=compose('Var%d',2:7))

parameters = 4x6 table
             Var2       Var3      Var4      Var5        Var6         Var7 
            _______    ______    ______    ______    ___________    ______

    b(1)    -4142.6     5.865    5.9551    805.24    -8.8137e+05    5.8773
    b(2)    -62.858    5.8148    5.3737    8.2765         8109.5    5.1137
    b(3)     685.64    5.9381    5.1931    731.55          92794    5.6659
    b(4)     17.201    16.291    17.173     16.91         17.003    16.756

function Var1 = gfit(b,Var2_7,np)
% np = 4;
% size(Var2_7,2)
for k = 1:size(Var2_7,2)
   bref = (1:np)+np*(k-1);
   term(:,k) = b(bref(1)).*exp(-(Var2_7(:,k)-b(bref(2))).^2*b(bref(3))) + b(bref(4));
end
Var1 = sum(term,2);
end

% % for k = 1:7
% %    bref = (1:3)+3*(k-1)
% % end

.

[Fernando Maturi]

That's exactly what I was looking for!

Thanks a lot :)

[Star Strider]

As always, my pleasure!