surface fitting issue when exporting polynomial

Hello there,

I'm using matlab to fit some experimental data. In this case I have the temperature and the pressure as x and y, and I have measured the liquid phase concentration as z, so my data are:

T= [4.4;4.6;4.8;5;5.2;5.4;5.5;5.6;5.7;5.8;]

p= [3;3.2;3.4;3.6;3.8;4;4.2;4.4;4.6;4.8;5;5.2;5.4;5.6;5.8;6;]

and

Z= [0.972604477 0.925657091 0.872451415 0.809828448 0.730361186 0.608428527 0.498661917 0.333396162 0.181062214 0.065747859

0.969339971 0.923431583 0.871870319 0.812179155 0.739118958 0.638373974 0.564120655 0.455628826 0.316002614 0.184750708

0.965913105 0.920933789 0.870794311 0.813502473 0.745169504 0.656675875 0.59785416 0.520540508 0.416267407 0.294691062

0.96234282 0.918201278 0.869307645 0.814026608 0.749361355 0.669013857 0.618744244 0.557085041 0.478281715 0.379522293

0.958645013 0.915264434 0.867474761 0.813910905 0.752195615 0.677731146 0.63293655 0.580321632 0.516453915 0.437614269

0.954833164 0.912148192 0.865346097 0.813271236 0.753993832 0.684017651 0.643058849 0.596310988 0.541630142 0.476299938

0.950918799 0.90887328 0.862961903 0.812194602 0.754975065 0.688559969 0.65045739 0.607835941 0.559271998 0.502902027

0.946911856 0.905457112 0.860354829 0.81074806 0.755295472 0.691790211 0.655909544 0.616361752 0.572149793 0.52197214

0.942820964 0.901914471 0.857551736 0.80898442 0.755070399 0.693997987 0.659903511 0.622744115 0.581790512 0.536103499

0.93865366 0.898258009 0.854574999 0.806946038 0.754387525 0.6953869 0.662765477 0.627522829 0.589106398 0.546816041

0.934416569 0.894498642 0.851443452 0.804667418 0.753315091 0.696105556 0.664723976 0.631059008 0.594679763 0.555049002

0.930115545 0.890645853 0.848173108 0.802177049 0.75190727 0.696265748 0.66594525 0.633605578 0.598902619 0.561413731

0.925755784 0.886707934 0.844777691 0.799498733 0.7502078 0.695953696 0.666553955 0.635346315 0.602050416 0.566326603

0.921341921 0.882692177 0.841269057 0.796652559 0.748252509 0.695237295 0.666645941 0.636418812 0.60432359 0.570082274

0.916878106 0.878605026 0.837657516 0.793655645 0.746071116 0.694170969 0.666296475 0.636928686 0.605872276 0.57289609

0.912368069 0.874452209 0.833952092 0.790522691 0.743688551 0.692799013 0.665565734 0.636958726 0.606811692 0.574929782]

we have a small system where we need to implement a simple fitting function up to degree 5, and cannot use other functions like sin,cos, exp...only simple operations, so we decided to fit the dada with a polynomial function of degree 5 for x and degree 4 for y.

I used cftool, selected "polynomial" and the degrees 5(for x) and 4(for y); Robust "off".

My problem starts when I try to export/use the "fitting polynomial" in the form of f = p00+ p01*x....etc. I mean, if I save it to workspace and use it as a matlab function, I have no problems and passing a value it returns the correct result, but if I want to use the polynomial and its coeff. on their own the results are totally wrong....for instance if I write x=4.8 and y=3 the result I have using the polynomial is 25.88...the real result should be z=0.8724;

Does anyone of you have an idea?

Thanks, Flavio.

The results are these:

Linear model Poly54:

     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
Coefficients (with 95% confidence bounds):
       p00 =        -375  (-842, 91.94)
       p10 =       402.5  (-48.9, 853.9)
       p01 =       -32.2  (-74.96, 10.57)
       p20 =      -168.3  (-343.5, 6.955)
       p11 =        20.3  (-8.7, 49.31)
       p02 =       2.671  (-2.521, 7.863)
       p30 =       34.44  (0.336, 68.54)
       p21 =      -4.569  (-12.37, 3.233)
       p12 =      -1.261  (-3.393, 0.8696)
       p03 =     -0.1928  (-0.697, 0.3114)
       p40 =      -3.401  (-6.727, -0.07578)
       p31 =      0.1891  (-0.7798, 1.158)
       p22 =      0.5681  (0.2349, 0.9013)
       p13 =     -0.2035  (-0.3335, -0.07347)
       p04 =      0.0762  (0.05197, 0.1004)
       p50 =      0.1246  (-0.005309, 0.2546)
       p41 =      0.0365  (-0.01007, 0.08307)
       p32 =    -0.08888  (-0.1089, -0.0689)
       p23 =     0.05173  (0.042, 0.06146)
       p14 =    -0.01618  (-0.02082, -0.01154)

Goodness of fit: SSE: 0.005731 R-square: 0.9988 Adjusted R-square: 0.9986 RMSE: 0.006398

Answer 1

[John D'Errico]

Oh, too bad! You are the 1000001'th person to make this mistake. The person before you got an all expense paid trip to Newark, New Jersey. Ok, most of their expenses were paid. Maybe some of them were paid? :)

Seriously, this is a very common mistake that people make with polynomials, or with curve fitting in general.

You see those coefficients written out to 4 digits, so you try to use them. Bad idea. A really bad idea. For example, the p50 coefficient is 0.1246. Or is it? Not exactly. In fact, there are some more digits past the 4th decimal place that are important.

When you try to evaluate that function at x = 4.8, remember that you are forming terms like this:

(0.1246 + delta)*4.8^5

for some small error delta, on the order of +/- 0.0001. Since

4.8^5
ans =
         2548

is moderately big, when you multiply that by the error you make by representing the coefficient by exactly 0.1246, instead of perhaps 0.1245785675446642 (or whatever it truly was) this introduces a serious amount of error into your prediction.

Now, do that for EVERY coefficient in that model. What happens is instead of a valid prediction, you get random garbage.

You need to use the EXACTLY estimated coefficients in the polynomial model. Not just some 4 digit approximation as it was written in the command window.

Next, I would point out that your predictions with this model are probably worth very little. Look at the predicted uncertainties in those coefficients. For example, consider the constant term.

p00 =        -375  (-842, 91.94)

Yep, -375, plus or minus something well over 400. So in fact, that constant term could easily have been zero. Or it might have been -800. Your z values were all between 0 and 1. In fact, just the error that you make by representing p00 as exactly 375, instead of some floating point number that is close to 375 is as large as the entire variation in your data.

This is just one of the problems with using high order polynomials in curve and surface fitting. And this is a moderately high order polynomial. Most of those coefficients are barely significantly different from zero, if at all. What that means is even if the model does predict your data reasonably well, it has very little meaning, and it will certainly do poorly in trying to extrapolate beyond the data.

[FlavioB]

well...what can I say....just thanks! I tried to dig into other threads but was not easy to find the solution....yes indeed I suspected there was some sort of round off error...but I really don't use often high order polynomials, normally only up to power 2....also because they tend to diverge a lot at the borders... thanks again!!!! ps: I understand about the big fluctuaion of the coefficients, but fortunately this polynomial will be used only for a very rough estimation....for initial values during a liquefaction process.. Cheers again!

[John D'Errico]

I plotted the surface that you got from that polynomial. It does not fit poorly, IF you use the exact coefficients as were returned. This is why you got reasonable predictions from the tool, but poor ones when you tried it essentially by hand. So as long as you stay within the bounds (support) of the data you will not do TOO terribly with that model.

Note that down in the corner where the data falls off a cliff, the model did moderately poorly. It predicted that corner with a fair amount of error.

In general edges and the perimeter points will be problematic with polynomials. Extrapolation using any polynomial past linear is often a waste of time, so try not to go there.

High order polynomials are nice in theory. Effectively one seem them all over the place in things like series approximations. In practice, those high order polynomials tend to have many problems, as you are finding.

If you can find a model for your system that makes physical sense, you are almost always better off using that mode, estimating the coefficients. If you have no model, then you can still use tools like my gridfit (found on the file exchange) to fit a general surface. But then you can only use tools like interp2 to predict values. But gridfit will do a much nicer job of fitting scattered data. In your case, you don't have scattered data.

So if your goal is merely to predict the values of your surface at some intermediate points, then there is no need to fit ANY model at all. Just use interp2 on your data directly! It will predict very nicely!

For example, using NO model at all, we get:

interp2(T,p,Z,4.8,3)
ans =
      0.87245

As you can see interp2 is VERY easy to use.

[FlavioB]

thank you very much! really appreciated!!! and indeed you were so right....coefficients problem....they are much longer....it's just that I trusted what I saw on the display, when cftool shows you fitting polynomial and coeff.... I guess that for our application this fitting is enough, as we are most likely not reaching the edges. I cannot use other functions like (interp2) because this is a dedicated system that allows us to implement only simple operations...*, /, +,-.... I remember many years ago I had a chat with a friend about "never use high order polynomials"...they brings always problems... In any case, thank you, I will download gridfit...seems useful for other things (at least for me)... Cheers!