MATLAB: How to fit the equation with that curve..?? please help..!!!

Question

Here's my equation

and the plot

I have to Fit this equation with the given plot and find the value of parameters Vo, Ro, ao, V1, D1 and R1. Please someone help me.. I have attached a pdf file of my problem.

Here's a lot of data..

R-----------V(R)

0.0000  -373.7902

0.0500  -373.7437

0.1000  -373.6783

0.1500  -373.5867

0.2000  -373.4592

0.2500  -373.2830

0.3000  -373.0417

0.3500  -372.7141

0.4000  -372.2737

0.4500  -371.6875

0.5000  -370.9155

0.5500  -369.9093

0.6000  -368.6124

0.6500  -366.9592

0.7000  -364.8755

0.7500  -362.2792

0.8000  -359.0819

0.8500  -355.1907

0.9000  -350.5117

0.9500  -344.9538

1.0000  -338.4338

1.0500  -330.8817

1.1000  -322.2471

1.1500  -312.5056

1.2000  -301.6648

1.2500  -289.7705

1.3000  -276.9108

1.3500  -263.2197

1.4000  -248.8777

1.4500  -234.1107

1.5000  -219.1859

1.5500  -204.4052

1.6000  -190.0958

1.6500  -176.5985

1.7000  -164.2542

1.7500  -153.3897

1.8000  -144.3026

1.8500  -137.2475

1.9000  -132.4235

1.9500  -129.9640

2.0000  -129.9300

2.0500  -132.3065

2.1000  -137.0031

2.1500  -143.8585

2.2000  -152.6480

2.2500  -163.0943

2.3000  -174.8807

2.3500  -187.6647

2.4000  -201.0935

2.4500  -214.8176

2.5000  -228.5041

2.5500  -241.8472
2.6000  -254.5773

2.6500  -266.4667

2.7000  -277.3327

2.7500  -287.0382

2.8000  -295.4905

2.8500  -302.6369

2.9000  -308.4606

2.9500  -312.9743

3.0000  -316.2143

3.0500  -318.2342

3.1000  -319.0989

3.1500  -318.8797

3.2000  -317.6501

3.2500  -315.4821

3.3000  -312.4442

3.3500  -308.5996

3.4000  -304.0061

3.4500  -298.7159

3.5000  -292.7760

3.5500  -286.2300

3.6000  -279.1186

3.6500  -271.4815

3.7000  -263.3587

3.7500  -254.7919

3.8000  -245.8250

3.8500  -236.5055

3.9000  -226.8845

3.9500  -217.0168

4.0000  -206.9606

4.0500  -196.7768

4.1000  -186.5282

4.1500  -176.2782

4.2000  -166.0896

4.2500  -156.0234

4.3000  -146.1371

4.3500  -136.4843

4.4000  -127.1128

4.4500  -118.0646

4.5000  -109.3748

4.5500  -101.0719

4.6000  -93.1772

4.6500  -85.7055

4.7000  -78.6650

4.7500  -72.0582

4.8000  -65.8825

4.8500  -60.1306

4.9000  -54.7916

4.9500  -49.8513

5.0000  -45.2933

5.0500  -41.0991

5.1000  -37.2493

5.1500  -33.7236

5.2000  -30.5013

5.2500  -27.5620

5.3000  -24.8854

5.3500  -22.4519

5.4000  -20.2426

5.4500  -18.2394

5.5000  -16.4252

5.5500  -14.7840

5.6000  -13.3006

5.6500  -11.9612

5.7000  -10.7526

5.7500  -9.6628

5.8000  -8.6809

5.8500  -7.7966

5.9000  -7.0007

5.9500  -6.2846

6.0000  -5.6407

6.0500  -5.0618

6.1000  -4.5416

6.1500  -4.0742

6.2000  -3.6545

6.2500  -3.2776

6.3000  -2.9393

6.3500  -2.6357

6.4000  -2.3632

6.4500  -2.1188

6.5000  -1.8995

6.5500  -1.7027

6.6000  -1.5263

6.6500  -1.3681

6.7000  -1.2262

6.7500  -1.0991

6.8000  -0.9850

6.8500  -0.8828

6.9000  -0.7911

6.9500  -0.7090

7.0000  -0.6354

7.0500  -0.5694

7.1000  -0.5102

7.1500  -0.4572

7.2000  -0.4097

7.2500  -0.3671

7.3000  -0.3289

7.3500  -0.2947

7.4000  -0.2641

7.4500  -0.2366

7.5000  -0.2120

7.5500  -0.1900

7.6000  -0.1702

7.6500  -0.1525

7.7000  -0.1367

7.7500  -0.1225

7.8000  -0.1097

7.8500  -0.0983

7.9000  -0.0881

7.9500  -0.0789

8.0000  -0.0707

8.0500  -0.0634

8.1000  -0.0568

8.1500  -0.0509

8.2000  -0.0456

8.2500  -0.0408

8.3000  -0.0366

8.3500  -0.0328

8.4000  -0.0294

8.4500  -0.0263

8.5000  -0.0236

8.5500  -0.0211

8.6000  -0.0189

8.6500  -0.0170

8.7000  -0.0152

8.7500  -0.0136

8.8000  -0.0122

8.8500  -0.0109

8.9000  -9.7904e-3

8.9500  -8.7719e-3

9.0000  -7.8593e-3

9.0500  -7.0417e-3

9.1000  -6.3091e-3

9.1500  -5.6528e-3

9.2000  -5.0647e-3

9.2500  -4.5378e-3

9.3000  -4.0657e-3

9.3500  -3.6427e-3

9.4000  -3.2638e-3

9.4500  -2.9242e-3

9.5000  -2.6200e-3

9.5500  -2.3474e-3

9.6000  -2.1032e-3

9.6500  -1.8844e-3

9.7000  -1.6884e-3

9.7500  -1.5127e-3

9.8000  -1.3553e-3

9.8500  -1.2143e-3

9.9000  -1.0880e-3

9.9500  -974.82e-6

10.0000  -873.40e-6

10.0500  -782.54e-6

10.1000  -701.13e-6

10.1500  -628.19e-6

10.2000  -562.83e-6

10.2500  -504.28e-6

10.3000  -451.82e-6

10.3500  -404.81e-6

10.4000  -362.70e-6

10.4500  -324.96e-6

10.5000  -291.16e-6

10.5500  -260.87e-6

10.6000  -233.73e-6

10.6500  -209.41e-6

10.7000  -187.62e-6

10.7500  -168.11e-6

10.8000  -150.62e-6

10.8500  -134.95e-6

10.9000  -120.91e-6

10.9500  -108.33e-6

11.0000  -97.059e-6

11.0500  -86.962e-6

11.1000  -77.915e-6

11.1500  -69.809e-6

11.2000  -62.546e-6

11.2500  -56.039e-6

11.3000  -50.209e-6


            
            

    

        

	
    

        
Best Answer
    

    

								

								
using cftool:
General model:
     f(R) = -V0/(1+exp((R-R0)/a0)) + V1*exp(-((R-D1)/R1)^2)
Coefficients (with 95% confidence bounds):
       D1 =    -0.06323
       R0 =    -0.07497
       R1 =       -4.55  (-5.504, -3.596)
       V0 =      -12.93  (-26.51, 0.6514)
       V1 =      -350.6  (-392.5, -308.6)
       a0 =    -0.00316  (-1.442e+07, 1.442e+07)
And the graph looks pretty bad.
Notice that the confidence interval for V0 crosses 0, so it is having trouble figuring out whether the first term should be added or subtracted. This almost always implies that the fit is nonsense.
Notice too that a0 is large and both positive and negative. a0 is a divisor in exp(), so a large positive or negative divisor implies that the fit is trying to drive the effect of the exp term to 0, effectively making the first term with V0 into a linear term instead of an exponential: that might lead to a better fit mathematically, but clearly it would lead to a plot with only one hump.
						    
						
		
    
        

    

Related Solutions
MATLAB: Unexpected results from ‘exp2’ least squares fit

								

								
Sums of exponential fits are difficult to solve. They are HIGHLY susceptible to starting value choices.
1. When you start with the wrong starting values, then expect crap.
2. Fit by default starts from some very arbitrary starting values. So what should you expect, at least some of the time? (See 1.)
A better scheme to estimate that nonlinear model is to use my fminspleas . It is on the file exchange. Follow the link.
mdlterms = {@(coef,X) exp(-coef(1)*X), @(coef,X) exp(-coef(2)*X)};
format long g
[nlcoef,lincoef] = fminspleas(mdlterms,[0.001,0.002],x,y)
nlcoef =
     -0.000869991132855897       0.00532565665197467
lincoef =
          -3.4116826458322
          6.61061671877021
Fminspleas survives here because it uses a completely different scheme to estimate the linear versus nonlinear coefficients in the model. It reduces the nonlinear search space.
yhat = lincoef(1)*exp(-nlcoef(1)*x) + lincoef(2)*exp(-nlcoef(2)*x);
plot(x,y,'ro',x,yhat,'b-')
To be honest, I really don't like the choice of model here at all. What did I say above? They are difficult to fit. But worse, look at the coefficients!!!!!!!!! Look at them carefully!
One of the nonlinear rate coefficients is positive, the other negative. So if we tried to extrapolate this model, see what happens.
xextrap = linspace(-150,3000,500);
yextrap = lincoef(1)*exp(-nlcoef(1)*xextrap) + lincoef(2)*exp(-nlcoef(2)*xextrap);
plot(x,y,'ro',xextrap,yextrap,'b-')
So, the question is, do rate coefficients of completely different sign make ANY sense in context of why you chose that model? Be honest. Why did you choose that model in the first place? If your only reason was because you thought it might have the right shape, that is a bad reason.
An alternative model is just to use a spline fit. Using my SLM toolbox (also on the file exchange)
slm = slmengine(x,y,'decreasing','on','concaveup','on','plot','on');
The fit looks at least as good a fit as the sum of exponentials.
						    
						
MATLAB: How to do an exponential fit for the data

								

								
This is how I would do it:
A=[ 0.734 2.956 132.351 6.1775 1.385 1.79887 5.122 12.2774 5.1001 6.07515 10.232 20.3542 9.688 0.983 51.3152 7.42856 16.5368 6.44541 6.19359 7.7353 10.5964 5.5078 55.8428 4.35722 2.83427 1.88594 19.367 1.89 24.5355 30.6559 34.2349 7.1513 2.2275 30.2956 3.16718 18.1018 ];
C=[ 4039 1896 2322 2961 3443 3020 1729 7097 1942 1956 2774 2787 2840 4313 1206 2204 2909 2454 2694 3224 2189 2686 2591 3210 3751 3321 1884 2780 1185 1358 1331 1960 3563 2483 4451 2589];
expfit = @(b,x) b(1) + b(2).*exp(b(3).*x + b(4));
SSECF = @(b) sum((C - expfit(b,A)).^2);
B0 = ones(4, 1);
B_est = fminsearch(SSECF, B0);
xpv = linspace(min(A), max(A));
figure(1)
plot(A, C, 'bp')
hold on
plot(xpv, expfit(B_est, xpv), '-r')
hold off
grid
xlabel('Volume of GTV(cc)') 
ylabel('Total MU') 
title('Volume of GTV v. Total MU')
text(mean(xlim)*0.9, mean(ylim), sprintf('\\Sigma MU = %.2f %+.2f\\cdote^{%+.4f\\cdotV_{GTV} %+.2f}', B_est))

						    
						
        
        

            
            
            
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Answer 1

using cftool:

General model:
     f(R) = -V0/(1+exp((R-R0)/a0)) + V1*exp(-((R-D1)/R1)^2)
Coefficients (with 95% confidence bounds):
       D1 =    -0.06323
       R0 =    -0.07497
       R1 =       -4.55  (-5.504, -3.596)
       V0 =      -12.93  (-26.51, 0.6514)
       V1 =      -350.6  (-392.5, -308.6)
       a0 =    -0.00316  (-1.442e+07, 1.442e+07)

And the graph looks pretty bad.

Notice that the confidence interval for V0 crosses 0, so it is having trouble figuring out whether the first term should be added or subtracted. This almost always implies that the fit is nonsense.

Notice too that a0 is large and both positive and negative. a0 is a divisor in exp(), so a large positive or negative divisor implies that the fit is trying to drive the effect of the exp term to 0, effectively making the first term with V0 into a linear term instead of an exponential: that might lead to a better fit mathematically, but clearly it would lead to a plot with only one hump.