Some functions for fitting

Today, I put data into Matlab and tried to find some functions useful for my research.

Summary:
T2distribution(:,1)=[]
T2distri=T2distribution
T2row1=T2distri(1,:)
T2row2=T2distri(2,:)
T2row3=T2distri(3,:)
T2row4=T2distri(4,:)
subplot(2,2,1)
area(T2row1)
subplot(2,2,2)
area(T2row2)
subplot(2,2,3)
area(T2row3)
subplot(2,2,4)
area(T2row4)

They are the first 4 distributions of T2. I just plot 4 of them to see how the distributions are like. They are mainly twomodals.

The following are two kind of fitting functions.

1.
General model Fourier3:
     f(x) =  a0 + a1*cos(x*w) + b1*sin(x*w) +
               a2*cos(2*x*w) + b2*sin(2*x*w) + a3*cos(3*x*w) + b3*sin(3*x*w)
       where x is normalized by mean 32.5 and std 18.62
Coefficients (with 95% confidence bounds):
       a0 =   0.0007262  (0.0006442, 0.0008081)
       a1 =   0.0002028  (3.608e-05, 0.0003696)
       b1 =  -0.0009519  (-0.001061, -0.0008427)
       a2 =  -0.0006422  (-0.0007004, -0.000584)
       b2 =  -0.0003332  (-0.000459, -0.0002074)
       a3 =  -0.0001401  (-0.0002752, -4.879e-06)
       b3 =  -1.008e-05  (-5.636e-05, 3.62e-05)
       w =       1.857  (1.655, 2.059)

Goodness of fit:
  SSE: 9.065e-07
  R-square: 0.9812
  Adjusted R-square: 0.9789
  RMSE: 0.0001272

2.
General model Weibull:
     f(x) = a*b*x^(b-1)*exp(-a*x^b)
Coefficients (with 95% confidence bounds):
       a =    0.001374  (-0.0002114, 0.00296)
       b =      0.8532  (0.5792, 1.127)

Goodness of fit:
  SSE: 4.666e-05
  R-square: 0.03477
  Adjusted R-square: 0.0192
  RMSE: 0.0008675

As shown in the figure, the single Weibull is  not effective for fitting. In contrast, the Fourier of 3 terms is effective.

Tomorrow, I will continue to find more effective functions for fitting.