Total Least squares error help. I tried a couple different things here, but i'm not exactly sure how to do a total least squares error. The numbers aren't making sense to me.

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Rian Sullivan 2024-2-17

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评论： Rian Sullivan 2024-2-17

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clear all

% A represents the sample measured in days

A = [1, 0; 1, 1; 1, 2; 1, 3; 1, 4; 1, 5; 1, 6; 1, 7]

A = 8×2

1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7

% b represents the amount of radium measured

b = [100; 82.7; 68.3; 56.5; 46.7; 38.6; 31.9; 26.4]

b = 8×1

100.0000 82.7000 68.3000 56.5000 46.7000 38.6000 31.9000 26.4000

% Least Squares

c = lsqr(A,log(b))

lsqr converged at iteration 2 to a solution with relative residual 9.5e-05.

c = 2×1

4.6051 -0.1903

g = A.'*A

g = 2×2

8 28 28 140

rhs = A.'*log(b);

x = g\rhs

x = 2×1

4.6051 -0.1903

thalf = -x(2)^-1*log(2)

thalf = 3.6417

m0 = exp(x(1))

m0 = 99.9942

m = @(t) m0*exp(x(2)*t);

predictionvalues = m([0, 1, 2, 3, 4, 5, 6, 7]);

squareddifferences = (predictionvalues - b).^2;

totalerror = sum(squareddifferences)

totalerror = 1×8

1.0e+04 * 1.9858 1.0169 0.5788 0.4646 0.5396 0.7175 0.9436 1.1847

t = [0, 1, 2, 3, 4, 5, 6, 7];

plot(t, x(1) + x(2)*t, t, log(b),'o')

%%plot(t, abs(exp(y_approx(t)-b)))

xlabel("t")

ylabel("error")

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Dyuman Joshi 2024-2-17

What seems to be the problem/issue?

Rian Sullivan 2024-2-17

Is the total least squares function okay? I was just making sure I even did the total least squares error code right. The total error seemed really small, but maybe im misunderstanding what it actually means. totalerror =

1.0e+04 *

Is what pops up along with all the other columns and their errors. It makes sense I believe.. but I'm just struggling with thinking my number of error doesn't look right.

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Answer 1

David Goodmanson 2024-2-17

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编辑：David Goodmanson 2024-2-17

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Hi Rian,

the problem is that b is a column vector, but predictionvalues is a row vector. Using

predictionvalues = m([0, 1, 2, 3, 4, 5, 6, 7])';         % column vector
squareddifferences = (predictionvalues - b).^2;
totalerror = sum(squareddifferences)
totalerror =
0.0033

although depending on circumstances, you may want to use sqrt(totalerror/8), the rms value, as the final result