I want to solve an empirical equation that includes sinus and cosinus and find the coefficients of the equation.

I have and equation as below. I am trying to solve this equation with curvefit toolbox, but sometimes it does not give good results.

 H= H0*q*(a*(1-exp(y^b))+c*y^d+e*cos(y*w)+g*cos(w*x)+h*sin(w*x))

H is the solar radiation, H0 is extraterrestrial radiation. y is the ratio between atmospheric transmittance and relative humidity. x is the ratio between temperature and maximum minimum temperature differences.

I have 3082 lines H, H0, x and y data in Excel format.

I want to find q, a, b, c, d, e, g, h and w empirical coefficients.

I share my Excel file.

y is (100* τ)/RH (metot 1), x is Kelvin in Excel file and H and H0 are given in Excel.

I am very new in MATLAB.

[Massimo Zanetti]

I also want something..

[David Goodmanson]

Seriously, you should not expect to fit 9 adjustable parameters to an equation with any reliability. That's way too many.

[Can Ekici]

I have got 4 meteorological parameters. Others are coefficients that will be found in solution. And I am very new in MATLAB.

[David Goodmanson]

'Coefficients' may be a better term than 'adjustable parameters', so let's call the nine quantities coefficients. You still cannot expect to reliably fit nine coefficients to a nonlinear equation. And this has nothing to do with whether you are using Matlab or some other scientific software. There are just too many ways to vary nine independent quantities and still satisfy the equation fairly well.

[Walter Roberson]

I can find multiple good fits for that formula with that data.

What is the expected range of values for each of the emperical coefficients q, a, b, c, d, e, g, h and w ? What are the constraints on them? For example if we knew that some of them were constrained to be positive then that would help. Are some of them effectively constrained to the range 0 to 2*pi ?

[Walter Roberson]

Sample good fit:

q, a, b, c, d, e, g, h, w
-0.0943476540688531345, -26.4573429943041631, 5.24509337105441986, 24.4040458106928213, 12.8496444415601303, -5.54493121917864684, -2.56697627309664034, -5.7981267837804964, -18.7554412534791304

[Walter Roberson]

Another good fit:

-0.1445690845843673 3.671493473559555 2.996823431481116 29.277484920414 8.193941869997328 -19.67313667145262 -21.9029540793973 7.411103378762792 3.461751091577492

[Walter Roberson]

Another decent fit (not quite as good but could probably be refined)

0.01728974983094427 28.91926055792799 7.458355469953931 23.01964585713372 7.916973007078415 22.73391990966213 6.732196314318775 10.81733666167035 1.592814852302886

[Can Ekici]

I have not got any limitations about these coefficients. I am trying to find a formula that gives a good fit between that data. I think this formula can be a good one to getting results. Would you share the codes for these fits? I will try these codes on my other data sets.

[Walter Roberson]

I used private code that is still under development.

It looks like I might accidentally have fitted only a few data points. I will need to re-test.

[Walter Roberson]

For sure I accidentally tested with only a few data points. The memory requirements for the full dataset are huge; I could never have fit them with my existing program. I will keep testing.

Answer 1

[Walter Roberson]

If you examine your formula

H= H0*q*(a*(1-exp(y^b))+c*y^d+e*cos(y*w)+g*cos(w*x)+h*sin(w*x))

and divide both sides by H0*q you get

H / H0 / q = a*(1-exp(y^b))+c*y^d+e*cos(y*w)+g*cos(w*x)+h*sin(w*x)

Therefore you can reduce your inputs by 1, from H and H0 individually to H ./ H0 .

It also appears to me that you could generate arbitrary values for the parameters other than q, calculate a sum-of-squares of that, and a sum-of-squares of H ./ H0, and then set q to the ratio of the two, and that should be an excellent fit. It is possible, though, that I am misunderstanding some mathematics here (I haven't had any chocolate yet today, so my thought processes are a bit slow!)

[Can Ekici]

I was trying to solve this equation in curvefit toolbox with reducing the inputs. But I think; it does not give satisfactory results. Also, I want to find new equations that express these inputs as well.

[Walter Roberson]

If I set

z = H./H0

and then I use cftool, custom equation

q*(a*(1-exp(y^b))+c*y^d+e*cos(y*w)+g*cos(w*x)+h*sin(w*x))

and I set Fit Options to MaxFunEvals of 10000 and Iters of 4000, then it converges to

General model:
     f(x,y) = q*(a*(1-exp(y^b))+c*y^d+e*cos(y*w)+g*cos(w*x)+h*sin(w*x))
Coefficients (with 95% confidence bounds):
       a =      0.0658  (-8.835e+04, 8.835e+04)
       b =      0.6595  (-0.8893, 2.208)
       c =      0.5135  (-6.895e+05, 6.895e+05)
       d =      0.1803  (-0.2375, 0.598)
       e =     -0.1876  (-2.52e+05, 2.52e+05)
       g =    0.007358  (-9880, 9880)
       h =  -0.0005033  (-675.8, 675.8)
       q =       2.641  (-3.546e+06, 3.546e+06)
       w =        0.34  (0.3298, 0.3502)

Goodness of fit:
  SSE: 40.73
  R-square: 0.556
  Adjusted R-square: 0.5548
  RMSE: 0.1152

[Can Ekici]

Is the curvefit toolbox suitable to solve this equation? If I change the iteration number and mxfunevals to 10000000 then the values of a,b,c... are changing.