Curve fitting with more than 3 variables

Good morning,

I'm trying to find an equating to corelate four variables.

So i've done a set of simulations and i obtained the following results.

where x, y and z are the inputs and w the output.

What i wanted to do is to have an equation that giving x, y and w i can calculate the Z.

Is this possible in matlab? I've tried to use curve fitting tool and get 3 different polynomial equations but the results are not ok...

Thank you in advance

Answer 1

[Sam Chak]

Hi @Adriana Mar de Jesus

It has been quite some time since I've done surface fitting. Are you looking for an illustration like the one shown below? It is easier for me to visualize the relationship between two inputs and one output. I think it is possible for you to group the clusters according to the number of houses.

Update: I think that the general equation for the grid energy would probably look like a nested polynomial function:

where , , are third-order polynomial functions given by

.

%% Data for 10 Houses
figure(1)
x10 = [12e3 12e3 12e3 24e3 24e3 24e3 32e3 32e3 32e3]';  % PV Capacity
y10 = [ 3e4  6e4 10e4  3e4  6e4 10e4  3e4  6e4 10e4]';  % Battery
z10 = [37240 23430 5840 15980 698 698.4 12810 0 0]';    % Grid Energy
plot3(x10, y10, z10, 'rp'), grid on, 
hold on

% Surface (manual fitting)
XX = linspace(x10(1), x10(end), 51);
YY = linspace(y10(1), y10(end), 51);
[X, Y] = meshgrid(XX, YY);

% z(x, y) = f(x)*y^2 + g(x)*y + h(x)
f10 = (-3.645e-14)*X.^2 + 1.894e-09*X - 1.719e-05;
g10 =  (3.997e-09)*X.^2 - 0.0002003*X + 1.341;
h10 = (-1.842e-05)*X.^2 +   0.06133*X + 5.35e+04;
Z10 = f10.*Y.^2 + g10.*Y + h10;
s  = surf(X, Y, Z10, 'FaceAlpha', 0.25); s.EdgeColor = 'none';
view(145, 30)
hold off

% Labels
xlabel('PV Capacity / (Watt)'), ylabel('Battery / (Wh)'), zlabel('Grid Energy / (Wh)')
title('Number of Houses: 10')

%% Data for 20 Houses
figure(2)
x20 = [24e3 24e3 24e3 48e3 48e3 48e3 64e3 64e3 64e3]';          % PV Capacity
y20 = [ 3e4  6e4 10e4  3e4  6e4 10e4  3e4  6e4 10e4]';          % Battery
z20 = [93680 78670 57030 61840 35960 8150 56370 32280 7308]';   % Grid Energy
plot3(x20, y20, z20, 'rp'), grid on, 
hold on

% Surface (manual fitting)
XX  = linspace(x20(1), x20(end), 51);
YY  = linspace(y20(1), y20(end), 51);
[X, Y] = meshgrid(XX, YY);

% z(x, y) = f(x)*y^2 + g(x)*y + h(x)
f20 = (-2.845e-15)*X.^2 +  (3.287e-10)*X - 6.832e-06;
g20 =  (7.266e-10)*X.^2 + (-7.856e-05)*X + 1.019;
h20 =  (5.354e-06)*X.^2 +     (-1.035)*X + 1.293e+05;
Z20 = f20.*Y.^2 + g20.*Y + h20;
s   = surf(X, Y, Z20, 'FaceAlpha', 0.25); s.EdgeColor = 'none';
view(145, 30)
hold off

% Labels
xlabel('PV Capacity / (Watt)'), ylabel('Battery / (Wh)'), zlabel('Grid Energy / (Wh)')
title('Number of Houses: 20')

%% Data for 30 Houses
figure(3)
x30 = [36e3 36e3 36e3 72e3 72e3 72e3 96e3 96e3 96e3]';              % PV Capacity
y30 = [ 3e4  6e4 10e4  3e4  6e4 10e4  3e4  6e4 10e4]';              % Battery
z30 = [120500 106600 90520 106400 74190 46690 93280 75360 42610]';  % Grid Energy
plot3(x30, y30, z30, 'rp'), grid on, 
hold on

% Surface (manual fitting)
XX  = linspace(x30(1), x30(end), 51);
YY  = linspace(y30(1), y30(end), 51);
[X, Y] = meshgrid(XX, YY);

% z(x, y) = f(x)*y^2 + g(x)*y + h(x)
f30 = (-8.176e-15)*X.^2 + 1.012e-09*X - 2.496e-05;
g30 =  (1.349e-09)*X.^2 - 0.0001742*X + 3.982;
h30 = (-3.565e-05)*X.^2 +     4.197*X + 3.11e+04;
Z30 = f30.*Y.^2 + g30.*Y + h30;
s   = surf(X, Y, Z30, 'FaceAlpha', 0.25); s.EdgeColor = 'none';
view(145, 30)
hold off

% Labels
xlabel('PV Capacity / (Watt)'), ylabel('Battery / (Wh)'), zlabel('Grid Energy / (Wh)')
title('Number of Houses: 30')

%% Data for 50 Houses
x50 = [6e4 6e4  6e4 12e4 12e4 12e4 16e4 16e4 16e4]';                      % PV Capacity
y50 = [3e4 6e4 10e4  3e4  6e4 10e4  3e4  6e4 10e4]';                      % Battery
z50 = [279300 262100 211600 193100 186400 131100 187200 164200 133500]';  % Grid Energy
plot3(x50, y50, z50, 'rp'), grid on, 
hold on

% Surface (manual fitting)
XX  = linspace(x50(1), x50(end), 51);
YY  = linspace(y50(1), y50(end), 51);
[X, Y] = meshgrid(XX, YY);

% z(x, y) = f(x)*y^2 + g(x)*y + h(x)
f50 =  (5.256e-15)*X.^2 - 1.058e-09*X + 3.471e-05;
g50 = (-6.672e-10)*X.^2 +  0.000136*X - 5.445;
h50 =  (2.818e-05)*X.^2 -     6.886*X + 5.905e+05;
Z50 = f50.*Y.^2 + g50.*Y + h50;

s  = surf(X, Y, Z50, 'FaceAlpha', 0.25); s.EdgeColor = 'none';
view(145, 30)
hold off

% Labels
xlabel('PV Capacity / (Watt)'), ylabel('Battery / (Wh)'), zlabel('Grid Energy / (Wh)')
title('Number of Houses: 50')

[Adriana Mar de Jesus]

Thank you for your comment :D

Yes something like that! I tried something similar to it but my problem is to find a final equation thar corelates four variables in order to, for exemple, say that i have 20 houses, with 10kW of PV capacity and i only want to take from main grid 20% of Available power. What should be the Battery i should use?

[Sam Chak]

% Data for 50 Houses
x = [6e4 6e4  6e4 12e4 12e4 12e4 16e4 16e4 16e4]';                      % PV Capacity
y = [3e4 6e4 10e4  3e4  6e4 10e4  3e4  6e4 10e4]';                      % Battery
z = [279300 262100 211600 193100 186400 131100 187200 164200 133500]';  % Grid Energy
plot3(x, y, z, 'rp'), grid on, 
hold on

% Surface (manual fitting)
XX = linspace(6e4, 16e4, 51);
YY = linspace(3e4, 10e4, 51);
[X, Y] = meshgrid(XX, YY);

% z(x, y) = f(x)*y^2 + g(x)*y + h(x)
f  =  (5.256e-15)*X.^2 - 1.058e-09*X + 3.471e-05;
g  = (-6.672e-10)*X.^2 +  0.000136*X - 5.445;
h  =  (2.818e-05)*X.^2 -     6.886*X + 5.905e+05;
Z  = f.*Y.^2 + g.*Y + h;

s  = surf(X, Y, Z, 'FaceAlpha', 0.25); s.EdgeColor = 'none';
view(145, 30)
hold off

% Labels
xlabel('PV Capacity / (Watt)'), ylabel('Battery / (Wh)'), zlabel('Grid Energy / (Wh)')
title('Number of Houses: 50')

[Sam Chak]

Hi @Adriana Mar de Jesus

I have updated the answer to display the model for each cluster of the Number of Houses after dedicating a few hours to identifying the model. Additionally, I have formulated the general equation for grid energy.

[Adriana Mar de Jesus]

Thank you a lot for your answer!!

It was very helpful

Answer 2

[Torsten]

Do you really think that there is a causal correlation between the housenumber and the energy from the grid ? So if the house number were 500, the energy from the grid were the tenfold of housenumber 50, e.g ?

Take a look here:

https://en.wikipedia.org/wiki/Spurious_relationship

Or do you mean "number of houses" instead of "house number" ?

In any case, you will have to assume how w is related with x, y and z. E.g. if you think a linear approach is sufficient, you can try

w = a0 + a1*x + a2*y + a3*z

and estimate a0,..,a3 using linear regression.

[Adriana Mar de Jesus]

Thank you for your comment :)