Trying to create a polynomial formula from xyz chart data where x and y equate to a z value.

I'm trying to create a polynomial formula it could be a 3rd degree to a 6th degree it doesn't matter how long the formula is I just need it to be accurate and be able to extrapolate as accurately as possibly if the x or y data inputted is off the ends of the chart. Attached is the example of the data that I'm referencing I would like to get a code that I could edit and run myself as I have about 20 similar charts that I want to extrapolate a formula for. I need to be only a temp(y) and a humidity(x) and have the formula spit out (z) and it be accurate based off the chart.

Answer 1

[Milan Bansal]

Hi Graham,

I understand that you wish to fit 2D surface to your data which will accept the temperature and humidity and will predict the extrapolated values.

To achieve this, you can use the "fit" function in MATLAB to fit a surface plot. Define the type of fit of using the "fittype" function. For this case you can use fit type as "poly33" (degree of the for the x terms and degree of three for the y terms).

Please refer to the following code snippet to for step wise solution;

% Data preparation
temp = [-2, 2, 5, 8, 10, 13, 15, 18, 22, 26, 28]';
humidity = [35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85];

values = [
    10.7, 11.4, 12.0, 12.7, 13.5, 14.2, 15.0, 15.9, 16.9, 18.0, 19.3;
    10.4, 11.1, 11.8, 12.5, 13.3, 14.0, 14.8, 15.7, 16.7, 17.8, 19.1;
    10.3, 11.0, 11.7, 12.4, 13.1, 13.9, 14.7, 15.5, 16.5, 17.6, 19.0;
    10.1, 10.8, 11.5, 12.2, 13.0, 13.7, 14.5, 15.4, 16.4, 17.5, 18.9;
    10.0, 10.7, 11.4, 12.1, 12.9, 13.6, 14.4, 15.3, 16.3, 17.4, 18.8;
    9.9, 10.6, 11.3, 12.0, 12.7, 13.5, 14.3, 15.2, 16.2, 17.3, 18.7;
    9.8, 10.5, 11.2, 11.9, 12.6, 13.4, 14.2, 15.1, 16.1, 17.2, 18.6;
    9.6, 10.3, 11.0, 11.8, 12.5, 13.3, 14.1, 15.0, 16.0, 17.1, 18.5;
    9.4, 10.2, 10.9, 11.6, 12.3, 13.1, 13.9, 14.8, 15.8, 16.9, 18.3;
    9.3, 10.0, 10.7, 11.4, 12.2, 12.9, 13.8, 14.6, 15.6, 16.8, 18.2;
    9.2, 9.9, 10.6, 11.3, 12.1, 12.8, 13.7, 14.6, 15.6, 16.7, 18.1
];

% Flatten the data for fitting
[X, Y] = meshgrid(humidity, temp);
Z = values;

% Fit a 2D polynomial surface of degree 3 (example)
ft = fittype('poly33'); % poly33 degree of the for the x terms and degree of three for the y terms
f = fit([X(:), Y(:)], Z(:), ft);

% Display the fitted model
disp(f);

     Linear model Poly33:
     f(x,y) = p00 + p10*x + p01*y + p20*x^2 + p11*x*y + p02*y^2 + p30*x^3 + p21*x^2*y 
                    + p12*x*y^2 + p03*y^3
     Coefficients (with 95% confidence bounds):
       p00 =       2.512  (2.05, 2.975)
       p10 =      0.3455  (0.3214, 0.3696)
       p01 =    -0.06088  (-0.07545, -0.0463)
       p20 =   -0.004431  (-0.004838, -0.004024)
       p11 =   5.833e-05  (-0.0003571, 0.0004738)
       p02 =   0.0005309  (9.877e-06, 0.001052)
       p30 =   3.148e-05  (2.923e-05, 3.372e-05)
       p21 =   1.251e-06  (-1.992e-06, 4.494e-06)
       p12 =  -1.737e-06  (-6.865e-06, 3.391e-06)
       p03 =  -5.446e-06  (-1.603e-05, 5.141e-06)

% Plot the fitted surface
figure;
plot(f, [X(:), Y(:)], Z(:));
xlabel('Humidity');
ylabel('Temperature');
zlabel('Values');

To predict the values from this surface, use the feval function as shown in the code snippet below:

newTemp = 34;
newHumidity = 90;
value = feval(f, newHumidity, newTemp)

value = 19.3369

Please refer to the following documentation links to learn more about fit, fittype and fiteval.

• https://www.mathworks.com/help/curvefit/fit.html
• https://www.mathworks.com/help/curvefit/fittype.htm
• https://www.mathworks.com/help/curvefit/feval.html

Hope this helps!

[John D'Errico]

Good answer. I would add only a little:

Extrapolating a polynomial is pretty much always a terrible idea, unless the polynomial is linear, where you cannot get into too much trouble. But even then, you will sometimes get into trouble. But extrapolating a 6th degree polynomial in 2 dimensions? This is a good definition of pure insanity. At least, it will surely drive you there.

And that means to use as low order polynomial as possible, AND it means to extrapolate as little as possible. In fact, I would bet that even a small extrapolation with a 6th degree polynomial in 2 dimensions will do very strange things, even on this well-behaved data. Since fit does not define poly66, I'll try it using poly55.

temp = [-2, 2, 5, 8, 10, 13, 15, 18, 22, 26, 28]';
humidity = [35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85];
values = [
    10.7, 11.4, 12.0, 12.7, 13.5, 14.2, 15.0, 15.9, 16.9, 18.0, 19.3;
    10.4, 11.1, 11.8, 12.5, 13.3, 14.0, 14.8, 15.7, 16.7, 17.8, 19.1;
    10.3, 11.0, 11.7, 12.4, 13.1, 13.9, 14.7, 15.5, 16.5, 17.6, 19.0;
    10.1, 10.8, 11.5, 12.2, 13.0, 13.7, 14.5, 15.4, 16.4, 17.5, 18.9;
    10.0, 10.7, 11.4, 12.1, 12.9, 13.6, 14.4, 15.3, 16.3, 17.4, 18.8;
    9.9, 10.6, 11.3, 12.0, 12.7, 13.5, 14.3, 15.2, 16.2, 17.3, 18.7;
    9.8, 10.5, 11.2, 11.9, 12.6, 13.4, 14.2, 15.1, 16.1, 17.2, 18.6;
    9.6, 10.3, 11.0, 11.8, 12.5, 13.3, 14.1, 15.0, 16.0, 17.1, 18.5;
    9.4, 10.2, 10.9, 11.6, 12.3, 13.1, 13.9, 14.8, 15.8, 16.9, 18.3;
    9.3, 10.0, 10.7, 11.4, 12.2, 12.9, 13.8, 14.6, 15.6, 16.8, 18.2;
    9.2, 9.9, 10.6, 11.3, 12.1, 12.8, 13.7, 14.6, 15.6, 16.7, 18.1
    ];
% Flatten the data for fitting
[X, Y] = meshgrid(humidity, temp);
Z = values;
% Fit a 2D polynomial surface of degree 3 (example)
ft = fittype('poly55'); % poly55 degree of the for the x terms and degree of three for the y terms
f = fit([X(:), Y(:)], Z(:), ft)

Warning: Equation is badly conditioned. Remove repeated data points or try centering and scaling.

f = 
     Linear model Poly55:
     f(x,y) = p00 + p10*x + p01*y + p20*x^2 + p11*x*y + p02*y^2 + p30*x^3 + p21*x^2*y 
                    + p12*x*y^2 + p03*y^3 + p40*x^4 + p31*x^3*y + p22*x^2*y^2 
                    + p13*x*y^3 + p04*y^4 + p50*x^5 + p41*x^4*y + p32*x^3*y^2 
                    + p23*x^2*y^3 + p14*x*y^4 + p05*y^5
     Coefficients (with 95% confidence bounds):
       p00 =     -0.3664  (-7.716, 6.983)
       p10 =      0.7404  (0.08258, 1.398)
       p01 =     -0.1726  (-0.377, 0.03175)
       p20 =    -0.02304  (-0.04612, 3.038e-05)
       p11 =    0.005583  (-0.00784, 0.01901)
       p02 =    0.004086  (-0.003503, 0.01167)
       p30 =   0.0004294  (3.228e-05, 0.0008265)
       p21 =  -7.343e-05  (-0.0004033, 0.0002564)
       p12 =  -0.0002305  (-0.0005239, 6.297e-05)
       p03 =   8.657e-05  (-0.0002481, 0.0004212)
       p40 =   -3.96e-06  (-7.317e-06, -6.04e-07)
       p31 =    6.36e-08  (-3.541e-06, 3.668e-06)
       p22 =    4.12e-06  (-5.464e-08, 8.295e-06)
       p13 =  -8.761e-07  (-6.692e-06, 4.94e-06)
       p04 =  -3.421e-06  (-1.418e-05, 7.335e-06)
       p50 =   1.492e-08  (3.759e-09, 2.608e-08)
       p41 =   2.724e-09  (-1.208e-08, 1.752e-08)
       p32 =  -2.195e-08  (-4.358e-08, -3.161e-10)
       p23 =  -2.629e-09  (-4.053e-08, 3.527e-08)
       p14 =   2.229e-08  (-4.757e-08, 9.215e-08)
       p05 =   4.026e-08  (-1.114e-07, 1.92e-07)

Note the near singularity warning. That alone should be a hint of bad things to come.

figure;
plot(f, [X(:), Y(:)], Z(:));
xlabel('Humidity');
ylabel('Temperature');
zlabel('Values');

methods(f)

Methods for class sfit:

argnames       coeffnames     confint        differentiate  fitoptions     indepnames     numargs        plot           probnames      quad2d         sfit           
category       coeffvalues    dependnames    feval          formula        islinear       numcoeffs      predint        probvalues     setoptions     type           

figure
[T,H] = meshgrid(linspace(-10,40),linspace(0,100))

T = 100x100

-10.0000 -9.4949 -8.9899 -8.4848 -7.9798 -7.4747 -6.9697 -6.4646 -5.9596 -5.4545 -4.9495 -4.4444 -3.9394 -3.4343 -2.9293 -2.4242 -1.9192 -1.4141 -0.9091 -0.4040 0.1010 0.6061 1.1111 1.6162 2.1212 2.6263 3.1313 3.6364 4.1414 4.6465 -10.0000 -9.4949 -8.9899 -8.4848 -7.9798 -7.4747 -6.9697 -6.4646 -5.9596 -5.4545 -4.9495 -4.4444 -3.9394 -3.4343 -2.9293 -2.4242 -1.9192 -1.4141 -0.9091 -0.4040 0.1010 0.6061 1.1111 1.6162 2.1212 2.6263 3.1313 3.6364 4.1414 4.6465 -10.0000 -9.4949 -8.9899 -8.4848 -7.9798 -7.4747 -6.9697 -6.4646 -5.9596 -5.4545 -4.9495 -4.4444 -3.9394 -3.4343 -2.9293 -2.4242 -1.9192 -1.4141 -0.9091 -0.4040 0.1010 0.6061 1.1111 1.6162 2.1212 2.6263 3.1313 3.6364 4.1414 4.6465 -10.0000 -9.4949 -8.9899 -8.4848 -7.9798 -7.4747 -6.9697 -6.4646 -5.9596 -5.4545 -4.9495 -4.4444 -3.9394 -3.4343 -2.9293 -2.4242 -1.9192 -1.4141 -0.9091 -0.4040 0.1010 0.6061 1.1111 1.6162 2.1212 2.6263 3.1313 3.6364 4.1414 4.6465 -10.0000 -9.4949 -8.9899 -8.4848 -7.9798 -7.4747 -6.9697 -6.4646 -5.9596 -5.4545 -4.9495 -4.4444 -3.9394 -3.4343 -2.9293 -2.4242 -1.9192 -1.4141 -0.9091 -0.4040 0.1010 0.6061 1.1111 1.6162 2.1212 2.6263 3.1313 3.6364 4.1414 4.6465 -10.0000 -9.4949 -8.9899 -8.4848 -7.9798 -7.4747 -6.9697 -6.4646 -5.9596 -5.4545 -4.9495 -4.4444 -3.9394 -3.4343 -2.9293 -2.4242 -1.9192 -1.4141 -0.9091 -0.4040 0.1010 0.6061 1.1111 1.6162 2.1212 2.6263 3.1313 3.6364 4.1414 4.6465 -10.0000 -9.4949 -8.9899 -8.4848 -7.9798 -7.4747 -6.9697 -6.4646 -5.9596 -5.4545 -4.9495 -4.4444 -3.9394 -3.4343 -2.9293 -2.4242 -1.9192 -1.4141 -0.9091 -0.4040 0.1010 0.6061 1.1111 1.6162 2.1212 2.6263 3.1313 3.6364 4.1414 4.6465 -10.0000 -9.4949 -8.9899 -8.4848 -7.9798 -7.4747 -6.9697 -6.4646 -5.9596 -5.4545 -4.9495 -4.4444 -3.9394 -3.4343 -2.9293 -2.4242 -1.9192 -1.4141 -0.9091 -0.4040 0.1010 0.6061 1.1111 1.6162 2.1212 2.6263 3.1313 3.6364 4.1414 4.6465 -10.0000 -9.4949 -8.9899 -8.4848 -7.9798 -7.4747 -6.9697 -6.4646 -5.9596 -5.4545 -4.9495 -4.4444 -3.9394 -3.4343 -2.9293 -2.4242 -1.9192 -1.4141 -0.9091 -0.4040 0.1010 0.6061 1.1111 1.6162 2.1212 2.6263 3.1313 3.6364 4.1414 4.6465 -10.0000 -9.4949 -8.9899 -8.4848 -7.9798 -7.4747 -6.9697 -6.4646 -5.9596 -5.4545 -4.9495 -4.4444 -3.9394 -3.4343 -2.9293 -2.4242 -1.9192 -1.4141 -0.9091 -0.4040 0.1010 0.6061 1.1111 1.6162 2.1212 2.6263 3.1313 3.6364 4.1414 4.6465

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H = 100x100

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.0101 1.0101 1.0101 1.0101 1.0101 1.0101 1.0101 1.0101 1.0101 1.0101 1.0101 1.0101 1.0101 1.0101 1.0101 1.0101 1.0101 1.0101 1.0101 1.0101 1.0101 1.0101 1.0101 1.0101 1.0101 1.0101 1.0101 1.0101 1.0101 1.0101 2.0202 2.0202 2.0202 2.0202 2.0202 2.0202 2.0202 2.0202 2.0202 2.0202 2.0202 2.0202 2.0202 2.0202 2.0202 2.0202 2.0202 2.0202 2.0202 2.0202 2.0202 2.0202 2.0202 2.0202 2.0202 2.0202 2.0202 2.0202 2.0202 2.0202 3.0303 3.0303 3.0303 3.0303 3.0303 3.0303 3.0303 3.0303 3.0303 3.0303 3.0303 3.0303 3.0303 3.0303 3.0303 3.0303 3.0303 3.0303 3.0303 3.0303 3.0303 3.0303 3.0303 3.0303 3.0303 3.0303 3.0303 3.0303 3.0303 3.0303 4.0404 4.0404 4.0404 4.0404 4.0404 4.0404 4.0404 4.0404 4.0404 4.0404 4.0404 4.0404 4.0404 4.0404 4.0404 4.0404 4.0404 4.0404 4.0404 4.0404 4.0404 4.0404 4.0404 4.0404 4.0404 4.0404 4.0404 4.0404 4.0404 4.0404 5.0505 5.0505 5.0505 5.0505 5.0505 5.0505 5.0505 5.0505 5.0505 5.0505 5.0505 5.0505 5.0505 5.0505 5.0505 5.0505 5.0505 5.0505 5.0505 5.0505 5.0505 5.0505 5.0505 5.0505 5.0505 5.0505 5.0505 5.0505 5.0505 5.0505 6.0606 6.0606 6.0606 6.0606 6.0606 6.0606 6.0606 6.0606 6.0606 6.0606 6.0606 6.0606 6.0606 6.0606 6.0606 6.0606 6.0606 6.0606 6.0606 6.0606 6.0606 6.0606 6.0606 6.0606 6.0606 6.0606 6.0606 6.0606 6.0606 6.0606 7.0707 7.0707 7.0707 7.0707 7.0707 7.0707 7.0707 7.0707 7.0707 7.0707 7.0707 7.0707 7.0707 7.0707 7.0707 7.0707 7.0707 7.0707 7.0707 7.0707 7.0707 7.0707 7.0707 7.0707 7.0707 7.0707 7.0707 7.0707 7.0707 7.0707 8.0808 8.0808 8.0808 8.0808 8.0808 8.0808 8.0808 8.0808 8.0808 8.0808 8.0808 8.0808 8.0808 8.0808 8.0808 8.0808 8.0808 8.0808 8.0808 8.0808 8.0808 8.0808 8.0808 8.0808 8.0808 8.0808 8.0808 8.0808 8.0808 8.0808 9.0909 9.0909 9.0909 9.0909 9.0909 9.0909 9.0909 9.0909 9.0909 9.0909 9.0909 9.0909 9.0909 9.0909 9.0909 9.0909 9.0909 9.0909 9.0909 9.0909 9.0909 9.0909 9.0909 9.0909 9.0909 9.0909 9.0909 9.0909 9.0909 9.0909

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surf(H,T,f(H,T))
hold on
plot3(X,Y,Z,'o')

Note that while the fit is reasonable within the support of the data, going outside that support, here even with a 5x5 degree polynomial, things start to get squirrely. And your data was actually quite well-behaved.

All of this suggests using a different scheme to interpolate, as well as extrapolate. Perhaps interp2 would be a better choice.

[Graham]

Thank you for the answers, this was exactly what I was looking for! I wont need to extrapolate the data very much outside of the input chart maybe a units, so some error is acceptable!

Cheers

[Steven Lord]

In support of John's comment about extrapolation being potentially dangerous, take a look at these two blog posts written by Cleve Moler: post 1, post 2.

In post 1, the pictures show interpolating census data using four different degrees of polynomials. Degrees 1 and 3 look fairly normal even after the last data point in 2020. Degree 7 looks concerning with its sharp drop after 2020, while degree 12 looks even more concerning with its rapid population explosion.

In post 2 the plot in the Conclusion section for degree 4 also looks concerning, and the error estimates get to be extremely broad.