How to solve "Rank deficient, rank = 7, tol = 6.169247e+09." in nlinfit

When the authors use nlinfit to fit the data, they consistently show a rank deficit. Is it because of the initial value setting?

clear;clc

format long

T=[33.2131 42.7043 48.8077 53.6257 57.7245 29.4295 33.0460 35.9052 38.3089...

42.8741 48.9372 53.7321 57.8148 61.4343 64.7215 67.7579 70.5979 73.2786...

75.8258 78.2613 80.5992 82.8526 85.0322 87.1452 89.1993 91.2015 93.1555...

95.0632 96.9314 98.7621 62.3556 71.5780 79.2821 86.1019 92.3337 29.1443...

32.8341 35.7294 38.1597 42.7520 48.8448 53.6546 57.7485 61.3737 64.6661];

% T = xlsread ('T.xlsx','Sheet1');

dT=[12.0335 6.9579 5.2636 4.3835 3.8312 4.0968 3.1274 2.5777 2.2236...

6.9005 5.2363 4.3646 3.8176 3.4396 3.1554 2.9392 2.7668 2.6229...

2.5022 2.3990 2.3119 2.2316 2.1666 2.1046 2.0501 1.9998 1.9544...

1.9143 1.8760 1.8412 10.2210 8.2580 7.1899 6.5020 6.0173 4.1877...

3.1736 2.6077 2.2428 6.9371 5.2576 4.3758 3.8264 3.4432 3.1618];

% dT = xlsread ('dT.xlsx','Sheet1');

Q_0=[90.26328 90.66839 90.47602 90.42722 90.32820 22.95437 22.88944 22.83842 22.81414...

90.65877 90.48850 90.36904 90.26715 90.19663 90.04881 89.98361 89.96804 89.88941...

89.86651 89.73939 89.73409 89.62329 89.61274 89.55704 89.55862 89.49705 89.42748...

89.39781 89.35200 89.39452 274.4339 273.7179 273.2521 272.9280 272.6679 22.89278...

22.88800 22.85388 22.80479 90.60546 90.48330 90.36902 90.26860 90.16708 90.09233];

% Q_0 = xlsread ('Q_0.xlsx','Sheet1');

T1=(2*T-dT)./2;

T2=T1+dT;

%Q_0=(T2-T)*C(1)+(T2.^2-T.^2)*C(2)+(T2.^3-T.^3)*C(3)+(T2.^4-T.^4)*C(4)+(T2.^5-T.^5)*C(5)...

% +(T2.^6-T.^6)*C(6)+(T2.^7-T.^7)*C(7)+(T2.^8-T.^8)*C(8)+(T2.^9-T.^9)*C(9)...

% +(T2.^10-T.^10)*C(10)+(T2.^11-T.^11)*C(11)+(T2.^12-T.^12)*C(12);

Y=Q_0';

X=[T1;T2]';

C0 = [1.070179528057E1,-4.72169505856E-1 ,0.01,1E-6,-1E-6,1E-8,-1E-10,1E-13,-1E-15,1E-18,-1E-21,1E-24]';

Z=dT.\Q_0;

fun=@(C,X) (C(1)*(X(:,2)-X(:,1))+C(2)*(X(:,2).^2-X(:,1).^2)+C(3)*(X(:,2).^3-X(:,1).^3)+...

C(4)*(X(:,2).^4-X(:,1).^4)+C(5)*(X(:,2).^5-X(:,1).^5)+C(6)*(X(:,2).^6-X(:,1).^6)+...

C(7)*(X(:,2).^7-X(:,1).^7)+C(8)*(X(:,2).^8-X(:,1).^8)+C(9)*(X(:,2).^9-X(:,1).^9)+...

C(10)*(X(:,2).^10-X(:,1).^10)+C(11)*(X(:,2).^11-X(:,1).^11)+C(12)*(X(:,2).^12-X(:,1).^12));

%nlinfit

[C,r,J]=nlinfit(X,Y,fun,C0)

plot(T,Z,'r*');

The C values obtained from the same data in the literature are shown in the following figure:

[Star Strider]

Fitting a degree polynomial (without an intercept term) to essentially anything would not surprise me to be rank-deficient. A lower-order polynomial — or better yet a nonlinear model that actually describes the process that created the data — would likely be more appropriate.

Plotting the data (without the fit) —

T=[33.2131 42.7043 48.8077 53.6257 57.7245 29.4295 33.0460 35.9052 38.3089...
    42.8741 48.9372 53.7321 57.8148 61.4343 64.7215 67.7579 70.5979 73.2786...
    75.8258 78.2613 80.5992 82.8526 85.0322 87.1452 89.1993 91.2015 93.1555...
    95.0632 96.9314 98.7621 62.3556 71.5780 79.2821 86.1019 92.3337 29.1443...
    32.8341 35.7294 38.1597 42.7520 48.8448 53.6546 57.7485 61.3737 64.6661];
% T = xlsread ('T.xlsx','Sheet1');
dT=[12.0335 6.9579 5.2636 4.3835 3.8312 4.0968 3.1274 2.5777 2.2236...
    6.9005 5.2363 4.3646 3.8176 3.4396 3.1554 2.9392 2.7668 2.6229...
    2.5022 2.3990 2.3119 2.2316 2.1666 2.1046 2.0501 1.9998 1.9544...
    1.9143 1.8760 1.8412 10.2210 8.2580 7.1899 6.5020 6.0173 4.1877...
    3.1736 2.6077 2.2428 6.9371 5.2576 4.3758 3.8264 3.4432 3.1618];
% dT = xlsread ('dT.xlsx','Sheet1');
Q_0=[90.26328 90.66839 90.47602 90.42722 90.32820 22.95437 22.88944 22.83842 22.81414...
    90.65877 90.48850 90.36904 90.26715 90.19663 90.04881 89.98361 89.96804 89.88941...
    89.86651 89.73939 89.73409 89.62329 89.61274 89.55704 89.55862 89.49705 89.42748...
    89.39781 89.35200 89.39452 274.4339 273.7179 273.2521 272.9280 272.6679 22.89278...
    22.88800 22.85388 22.80479 90.60546 90.48330 90.36902 90.26860 90.16708 90.09233];
% Q_0 = xlsread ('Q_0.xlsx','Sheet1');
T1=(2*T-dT)./2;
T2=T1+dT;
%Q_0=(T2-T)*C(1)+(T2.^2-T.^2)*C(2)+(T2.^3-T.^3)*C(3)+(T2.^4-T.^4)*C(4)+(T2.^5-T.^5)*C(5)...
%    +(T2.^6-T.^6)*C(6)+(T2.^7-T.^7)*C(7)+(T2.^8-T.^8)*C(8)+(T2.^9-T.^9)*C(9)...
%    +(T2.^10-T.^10)*C(10)+(T2.^11-T.^11)*C(11)+(T2.^12-T.^12)*C(12);
Y=Q_0';
X=[T1;T2]';

figure
plot(X, Y, 'p-')
grid
xlabel('X')
ylabel('Y')

I have no idea what the paper reports, however I find it somehow perplexing that a manuscript fitting any continuous function to these data would be accepted for publication.

.

[John D'Errico]

I totally agree with Star here. The reviewers of that article should get a talking to. And the editors of the publication should find better reviewers. Sadly, there are many journals who accept any submissions, with no checking of what was written.

Not only is this just a 12'th degree polynomial fit to that data, but worse, it is a POLYNOMIAL and nlinfit is being employed. SIGH.

[宏宇]

Q_0 is not represented by simple T2 and T1, but also includes power and parameters.

[Star Strider]

@John D'Errico — Thank you!

@宏宇 — The regression is on (X,Y) so that is what I plotted —

[C,r,J]=nlinfit(X,Y,fun,C0)

However the reported plot is —

Z=dT.\Q_0;

plot(T,Z,'r*');

so I have absolutely no idea what is being done here.

The function in this Comment has just now appeared. I assume that this is the ‘Q_0’ vector and it has already been calculated. If not, there is no further calculation for it, and to complicate all this, ‘C’ is being calculated from ‘Q_0’ (being ‘Y’) so this becomes even more confusing.

I just cannot follow whatever this is doing. At least more explanation is necessary. There might be something of interest in all these data, and they have just not been presented or analysed correctly.

.

Answer 1

[Torsten]

There must be something that you misunderstood in the article.

The problem from above is a linear fitting problem for the vector C. The solution can be obtained without nlinfit as below, but the problem with the rank-deficiency remains.

clear;clc
format long
T=[33.2131 42.7043 48.8077 53.6257 57.7245 29.4295 33.0460 35.9052 38.3089...
    42.8741 48.9372 53.7321 57.8148 61.4343 64.7215 67.7579 70.5979 73.2786...
    75.8258 78.2613 80.5992 82.8526 85.0322 87.1452 89.1993 91.2015 93.1555...
    95.0632 96.9314 98.7621 62.3556 71.5780 79.2821 86.1019 92.3337 29.1443...
    32.8341 35.7294 38.1597 42.7520 48.8448 53.6546 57.7485 61.3737 64.6661];
% T = xlsread ('T.xlsx','Sheet1');
dT=[12.0335 6.9579 5.2636 4.3835 3.8312 4.0968 3.1274 2.5777 2.2236...
    6.9005 5.2363 4.3646 3.8176 3.4396 3.1554 2.9392 2.7668 2.6229...
    2.5022 2.3990 2.3119 2.2316 2.1666 2.1046 2.0501 1.9998 1.9544...
    1.9143 1.8760 1.8412 10.2210 8.2580 7.1899 6.5020 6.0173 4.1877...
    3.1736 2.6077 2.2428 6.9371 5.2576 4.3758 3.8264 3.4432 3.1618];
% dT = xlsread ('dT.xlsx','Sheet1');
Q_0=[90.26328 90.66839 90.47602 90.42722 90.32820 22.95437 22.88944 22.83842 22.81414...
    90.65877 90.48850 90.36904 90.26715 90.19663 90.04881 89.98361 89.96804 89.88941...
    89.86651 89.73939 89.73409 89.62329 89.61274 89.55704 89.55862 89.49705 89.42748...
    89.39781 89.35200 89.39452 274.4339 273.7179 273.2521 272.9280 272.6679 22.89278...
    22.88800 22.85388 22.80479 90.60546 90.48330 90.36902 90.26860 90.16708 90.09233];
% Q_0 = xlsread ('Q_0.xlsx','Sheet1');
T1=(2*T-dT)./2;
T2=T1+dT;
%Q_0=(T2-T)*C(1)+(T2.^2-T.^2)*C(2)+(T2.^3-T.^3)*C(3)+(T2.^4-T.^4)*C(4)+(T2.^5-T.^5)*C(5)...
%    +(T2.^6-T.^6)*C(6)+(T2.^7-T.^7)*C(7)+(T2.^8-T.^8)*C(8)+(T2.^9-T.^9)*C(9)...
%    +(T2.^10-T.^10)*C(10)+(T2.^11-T.^11)*C(11)+(T2.^12-T.^12)*C(12);
Y=Q_0';
X=[T1;T2]';
A = [X(:,2)-X(:,1),X(:,2).^2-X(:,1).^2,X(:,2).^3-X(:,1).^3,X(:,2).^4-X(:,1).^4,X(:,2).^5-X(:,1).^5,...
    X(:,2).^6-X(:,1).^6,X(:,2).^7-X(:,1).^7,X(:,2).^8-X(:,1).^8,X(:,2).^9-X(:,1).^9,X(:,2).^10-X(:,1).^10,...
    X(:,2).^11-X(:,1).^11];
b = Y;
C = A\b

Warning: Rank deficient, rank = 6, tol = 4.797514e+07.

C = 11×1

1.0e-06 * 0 0 0 0 0 0.245847633266853 -0.011686042243299 0.000238193643051 -0.000002533319987 0.000000013833084

[宏宇]

Thanks Torsten

This is the formula used in the artical.

When I set all C0 to 0, the resulting C are 0, 0, 0, 0, 0, 1.41529825705046e-08, 6.31675239060952e-09, -3.42374529788427e-10, 7.39232094870089e-12, -8.09513845548424e-14, 4.48443319773084e-16, -1.00170397811213e-18.

The new Q_0 obtained according to C is similar to the old Q_0, but there are still some errors, which are not as good as the C in the original text.

clear;clc

format long

T=[33.2131 42.7043 48.8077 53.6257 57.7245 29.4295 33.0460 35.9052 38.3089...

42.8741 48.9372 53.7321 57.8148 61.4343 64.7215 67.7579 70.5979 73.2786...

75.8258 78.2613 80.5992 82.8526 85.0322 87.1452 89.1993 91.2015 93.1555...

95.0632 96.9314 98.7621 62.3556 71.5780 79.2821 86.1019 92.3337 29.1443...

32.8341 35.7294 38.1597 42.7520 48.8448 53.6546 57.7485 61.3737 64.6661];

dT=[12.0335 6.9579 5.2636 4.3835 3.8312 4.0968 3.1274 2.5777 2.2236...

6.9005 5.2363 4.3646 3.8176 3.4396 3.1554 2.9392 2.7668 2.6229...

2.5022 2.3990 2.3119 2.2316 2.1666 2.1046 2.0501 1.9998 1.9544...

1.9143 1.8760 1.8412 10.2210 8.2580 7.1899 6.5020 6.0173 4.1877...

3.1736 2.6077 2.2428 6.9371 5.2576 4.3758 3.8264 3.4432 3.1618];

Q_0=[90.26328 90.66839 90.47602 90.42722 90.32820 22.95437 22.88944 22.83842 22.81414...

90.65877 90.48850 90.36904 90.26715 90.19663 90.04881 89.98361 89.96804 89.88941...

89.86651 89.73939 89.73409 89.62329 89.61274 89.55704 89.55862 89.49705 89.42748...

89.39781 89.35200 89.39452 274.4339 273.7179 273.2521 272.9280 272.6679 22.89278...

22.88800 22.85388 22.80479 90.60546 90.48330 90.36902 90.26860 90.16708 90.09233];

T1=(2*T-dT)./2;

T2=T1+dT;

Y=Q_0';

X=[T1;T2]';

C0 =[0,0,0,0,0,0,0,0,0,0,0,0]';

Z=dT.\Q_0;

fun=@(C,X) (C(1)*(X(:,2)-X(:,1))+C(2)*(X(:,2).^2-X(:,1).^2)+C(3)*(X(:,2).^3-X(:,1).^3)+...

C(4)*(X(:,2).^4-X(:,1).^4)+C(5)*(X(:,2).^5-X(:,1).^5)+C(6)*(X(:,2).^6-X(:,1).^6)+...

C(7)*(X(:,2).^7-X(:,1).^7)+C(8)*(X(:,2).^8-X(:,1).^8)+C(9)*(X(:,2).^9-X(:,1).^9)+...

C(10)*(X(:,2).^10-X(:,1).^10)+C(11)*(X(:,2).^11-X(:,1).^11)+C(12)*(X(:,2).^12-X(:,1).^12));

[C,r,J]=nlinfit(X,Y,fun,C0)

Q_2=(C(1)*(X(:,2)-X(:,1))+C(2)*(X(:,2).^2-X(:,1).^2)+C(3)*(X(:,2).^3-X(:,1).^3)+...

C(4)*(X(:,2).^4-X(:,1).^4)+C(5)*(X(:,2).^5-X(:,1).^5)+C(6)*(X(:,2).^6-X(:,1).^6)+...

C(7)*(X(:,2).^7-X(:,1).^7)+C(8)*(X(:,2).^8-X(:,1).^8)+C(9)*(X(:,2).^9-X(:,1).^9)+...

C(10)*(X(:,2).^10-X(:,1).^10)+C(11)*(X(:,2).^11-X(:,1).^11)+C(12)*(X(:,2).^12-X(:,1).^12));

S=dT.\Q_2';

plot(T,Z,'r*',T,S,'--');

But why is there more than one T, S curve obtained?

Moreover, the optimal C value cannot be obtained. ：(

[Torsten]

With a little cheating, here is an even better C than in the article.

But most probably the T, dT and Q_0 values were already rounded by @宏宇

Note that the entries of the matrix A are up to the order of 1e23. Simply using A\b seems to lead nowhere in this case.

T=[33.2131 42.7043 48.8077 53.6257 57.7245 29.4295 33.0460 35.9052 38.3089...
    42.8741 48.9372 53.7321 57.8148 61.4343 64.7215 67.7579 70.5979 73.2786...
    75.8258 78.2613 80.5992 82.8526 85.0322 87.1452 89.1993 91.2015 93.1555...
    95.0632 96.9314 98.7621 62.3556 71.5780 79.2821 86.1019 92.3337 29.1443...
    32.8341 35.7294 38.1597 42.7520 48.8448 53.6546 57.7485 61.3737 64.6661];
dT=[12.0335 6.9579 5.2636 4.3835 3.8312 4.0968 3.1274 2.5777 2.2236...
    6.9005 5.2363 4.3646 3.8176 3.4396 3.1554 2.9392 2.7668 2.6229...
    2.5022 2.3990 2.3119 2.2316 2.1666 2.1046 2.0501 1.9998 1.9544...
    1.9143 1.8760 1.8412 10.2210 8.2580 7.1899 6.5020 6.0173 4.1877...
    3.1736 2.6077 2.2428 6.9371 5.2576 4.3758 3.8264 3.4432 3.1618];
Q_0=[90.26328 90.66839 90.47602 90.42722 90.32820 22.95437 22.88944 22.83842 22.81414...
    90.65877 90.48850 90.36904 90.26715 90.19663 90.04881 89.98361 89.96804 89.88941...
    89.86651 89.73939 89.73409 89.62329 89.61274 89.55704 89.55862 89.49705 89.42748...
    89.39781 89.35200 89.39452 274.4339 273.7179 273.2521 272.9280 272.6679 22.89278...
    22.88800 22.85388 22.80479 90.60546 90.48330 90.36902 90.26860 90.16708 90.09233];
T1=(2*T-dT)./2;
T2=T1+dT;
Y=Q_0';
X=[T1;T2]';
C_article = [1.070179528057e1;-4.72169505856e-1;9.985458119236e-3;...
    3.443201289415e-6;-1.486069268038e-6;1.901352098615e-8;...
    -1.300438485128e-10;5.60742395948e-13;-1.572000054992e-15;...
    2.789945522377e-18;-2.854347532609e-21;1.284323931260e-24];
A = [X(:,2)-X(:,1),X(:,2).^2-X(:,1).^2,X(:,2).^3-X(:,1).^3,X(:,2).^4-X(:,1).^4,X(:,2).^5-X(:,1).^5,...
     X(:,2).^6-X(:,1).^6,X(:,2).^7-X(:,1).^7,X(:,2).^8-X(:,1).^8,X(:,2).^9-X(:,1).^9,X(:,2).^10-X(:,1).^10,...
     X(:,2).^11-X(:,1).^11,X(:,2).^12-X(:,1).^12];
b = Y;
format long g
C = lsqnonlin(@(C)A*C-b,C_article);

Local minimum possible.
lsqnonlin stopped because the size of the current step is less than
the value of the step size tolerance.

C(1:3).'

ans = 1×3

10.70179528057 -0.472169505856 0.009985458119236

C(4:6).'

ans = 1×3

1.0e+00 * 3.443201289415e-06 -1.486069268038e-06 1.901352098615e-08

C(7:9).'

ans = 1×3

1.0e+00 * -1.300438485128e-10 5.60742395947993e-13 -1.57200005544683e-15

C(10:12).'

ans = 1×3

1.0e+00 * 2.78991720704103e-18 -4.18123723457899e-21 1.38932757013367e-23

error_article = norm(A*C_article-b)

error_article = 
          1.70199728742361

error = norm(A*C-b)

error = 
          1.45598694473648

[宏宇]

Thanks Torsten

After adjustment, the fitting of yesterday was relatively successful, but after continuing to increase the data of T, dT and Q0, the fitting of C failed again. So not sure if you have a better way to fit the data.

Or use the fitting method you used yesterday, and still can't get a good fit.

clear;clc
format long
b=xlsread('shuju.xlsx');
T=b(:,1);
dT=b(:,2);
Q_0=b(:,3);
T1=(2*T-dT)./2;
T2=T1+dT; 
Y=Q_0;
X=[T1 T2];
C0 = [1.070179528057e1;-4.72169505856e-1;9.985458119236e-3;...
    3.443201289415e-6;-1.486069268038e-6;1.901352098615e-8;...
    -1.300438485128e-10;5.60742395948e-13;-1.572000054992e-15;...
    2.789945522377e-18;-2.854347532609e-21;1.284323931260e-24]';
%C0 is C0 in the literature, but it is impossible to know in the actual process, 
% so how to obtain C0.
Z=Q_0./dT;
fun=@(C,X) (C(1)*(X(:,2)-X(:,1))+C(2)*(X(:,2).^2-X(:,1).^2)+C(3)*(X(:,2).^3-X(:,1).^3)+...
    C(4)*(X(:,2).^4-X(:,1).^4)+C(5)*(X(:,2).^5-X(:,1).^5)+C(6)*(X(:,2).^6-X(:,1).^6)+...
    C(7)*(X(:,2).^7-X(:,1).^7)+C(8)*(X(:,2).^8-X(:,1).^8)+C(9)*(X(:,2).^9-X(:,1).^9)+...
    C(10)*(X(:,2).^10-X(:,1).^10)+C(11)*(X(:,2).^11-X(:,1).^11)+C(12)*(X(:,2).^12-X(:,1).^12));
[C,r,J]=nlinfit(X,Y,fun,C0)
Q_2=0;
for i=1:1:12
    Q_2=Q_2+(T2.^i-T1.^i)*C(i); 
end
S=dT.\Q_2;
plot(T,Z,'r*',T,S,'kx');

[Torsten]

Fits are done for that they can be used for other measurement data as well. So I don't understand why you fit the parameters again. But if you like ...

b = [33.2131	12.0335	90.26328
    42.7043	6.9579	90.66839
    48.8077	5.2636	90.47602
    53.6257	4.3835	90.42722
    57.7245	3.8312	90.3282
    29.4295	4.0968	22.95437
    33.046	3.1274	22.88944
    35.9052	2.5777	22.83842
    38.3089	2.2236	22.81414
    42.8741	6.9005	90.65877
    48.9372	5.2363	90.4885
    53.7321	4.3646	90.36904
    57.8148	3.8176	90.26715
    61.4343	3.4396	90.19663
    64.7215	3.1554	90.04881
    67.7579	2.9392	89.98361
    70.5975	2.7668	89.96804
    73.2786	2.6229	89.88941
    75.8258	2.5022	89.86651
    78.2613	2.399	89.73939
    80.5992	2.3119	89.73409
    82.8526	2.2316	89.62329
    85.0322	2.1666	89.61274
    87.1452	2.1046	89.55704
    89.1993	2.0501	89.55862
    91.2015	1.9998	89.49705
    93.1555	1.9544	89.42748
    95.0632	1.9143	89.39781
    96.9314	1.876	89.352
    98.7621	1.8412	89.29452
    62.3556	10.221	274.4339
    71.578	8.258	273.7197
    79.2821	7.1899	273.2521
    86.1019	6.502	272.928
    92.3337	6.0173	272.6679
    29.1443	4.1877	22.95278
    32.8341	3.1736	22.888
    35.7294	2.6077	22.85388
    38.1597	2.2428	22.80479
    42.752	6.9371	90.60546
    48.8448	5.2576	90.4833
    53.6546	4.3758	90.36902
    57.7485	3.8264	90.2686
    61.3737	3.4432	90.16708
    64.6661	3.1618	90.09233
    300.294	2.066	171.11
    302.419	2.061	171.13
    303.809	0.528	43.93
    275.05	0.549	44.73
    275.881	1.091	88.8
    278.5	4.129	336.76
    279.032	4.446	362.53
    283.489	4.416	361.6
    287.955	4.414	362.85
    292.433	4.387	362.01
    296.923	4.385	362.77
    295.532	4.397	363.52
    300.004	4.357	361.64
    304.486	4.357	362.6
    309.774	4.32	361.27
    314.12	4.301	360.81
    318.482	4.278	360.32
    309.649	4.291	358.79
    314.3	4.298	361.24
    319.013	4.286	361.95
    273.211	4.535	368.33
    277.724	4.472	364.76
    282.215	4.462	365.31
    286.704	4.442	365.09
    291.192	4.423	365.02
    295.681	4.406	364.89
    300.167	4.383	364.34
    304.708	4.369	364.55
    309.206	4.346	363.82
    314.458	5.802	487.91
    319.745	4.306	363.53
    127.08	4.659	276.68
    131.608	4.566	276.97
    136.098	4.47	276.91
    142.318	8.114	515.94
    150.056	7.443	487.31
    131.622	1.69	102.45
    134.09	3.322	203.83
    137.347	3.266	203.25
    139.657	1.623	102.05
    142.81	4.755	302.82
    147.463	4.647	301.24
    152.806	6.141	405.71
    161.128	5.143	348.38
    167.386	7.49	516.45
    174.725	7.355	517.13
    175.502	3.793	267.16
    179.197	3.742	265.77
    182.608	3.703	264.77
    186.241	3.686	265.63
    188.966	1.899	137.84
    190.803	1.878	136.84
    194.4	5.425	397.55
    199.723	5.372	397.3
    202.396	7.062	524.84
    206.831	1.954	146.42
    208.722	1.96	147.17
    212.843	6.685	505.29
    220.458	8.686	664.24
    226.417	3.377	260.27
    229.716	3.365	260.77
    235.624	8.567	668.38
    232.939	5.909	459.71
    238.781	5.872	459.82
    244.575	5.807	457.64
    250.293	5.808	460.72
    256.055	5.756	459.52
    261.79	5.723	459.75
    267.5	5.665	457.91
    273.202	5.657	459.37
    277.804	5.639	459.7
    283.507	5.594	458.47
    289.44	5.567	458.77
    295.144	5.529	457.9
    87.956	4.522	193.92
    92.45	4.502	203.72
    96.923	4.469	212.18
    101.371	4.456	220.68
    105.801	4.442	228.64
    110.252	4.499	239.77
    114.693	4.503	247.67
    119.179	4.519	255.8
    123.684	4.549	264.42
    128.143	4.435	264.04
    276.822	5.616	457.34
    282.678	5.617	459.8
    276.345	5.642	459.05
    281.862	5.606	458.25
    276.552	5.612	456.82
    282.382	5.598	458.11
    275.701	5.604	455.7
    281.212	5.61	458.36];
T=b(:,1);
dT=b(:,2);
Q_0=b(:,3);
T1=(2*T-dT)./2;
T2=T1+dT; 
Y=Q_0;
X=[T1 T2];
C0 = [1.070179528057e1;-4.72169505856e-1;9.985458119236e-3;...
      3.443201289415e-6;-1.486069268038e-6;1.901352098615e-8;...
      -1.300438485128e-10;5.60742395948e-13;-1.572000054992e-15;...
      2.789945522377e-18;-2.854347532609e-21;1.284323931260e-24].';
%C0 is C0 in the literature, but it is impossible to know in the actual process, 
% so how to obtain C0.
Z=Q_0./dT;
A = [X(:,2)-X(:,1),X(:,2).^2-X(:,1).^2,X(:,2).^3-X(:,1).^3,X(:,2).^4-X(:,1).^4,X(:,2).^5-X(:,1).^5,...
     X(:,2).^6-X(:,1).^6,X(:,2).^7-X(:,1).^7,X(:,2).^8-X(:,1).^8,X(:,2).^9-X(:,1).^9,X(:,2).^10-X(:,1).^10,...
     X(:,2).^11-X(:,1).^11,X(:,2).^12-X(:,1).^12];
b = Y;
format long g
C = lsqnonlin(@(C)A*C-b,C0.')

Local minimum possible.
lsqnonlin stopped because the size of the current step is less than
the value of the step size tolerance.

C = 12×1

10.70179528057 -0.472169505856 0.009985458119236 3.443201289415e-06 -1.486069268038e-06 1.901352098615e-08 -1.30043848512968e-10 5.60742386224611e-13 -1.57219671585905e-15 2.79170998059763e-18

norm(A*C-b)

ans = 
          3.62409076105991

%fun=@(C,X) (C(1)*(X(:,2)-X(:,1))+C(2)*(X(:,2).^2-X(:,1).^2)+C(3)*(X(:,2).^3-X(:,1).^3)+...
%    C(4)*(X(:,2).^4-X(:,1).^4)+C(5)*(X(:,2).^5-X(:,1).^5)+C(6)*(X(:,2).^6-X(:,1).^6)+...
%    C(7)*(X(:,2).^7-X(:,1).^7)+C(8)*(X(:,2).^8-X(:,1).^8)+C(9)*(X(:,2).^9-X(:,1).^9)+...
%    C(10)*(X(:,2).^10-X(:,1).^10)+C(11)*(X(:,2).^11-X(:,1).^11)+C(12)*(X(:,2).^12-X(:,1).^12));
%[C,r,J]=nlinfit(X,Y,fun,C0);
%norm(fun(C,X)-Y)
Q_2 = A*C;
S=dT.\Q_2;
plot(T,Z,'r*',T,S,'kx');