I am trying to compute $\sigma_y$ (the uncertainty), for a 10th order fitted polynomial function to a plot (for Origin 2019 software):
$$y(x)=A+Bx+Cx^2+Dx^3 +Ex^4+Fx^5+Gx^6+Hx^7+Ix^8+Jx^9+Kx^{10}\tag{1}$$
The formula I thought was applicable here is
$$\sigma _y=\left |\frac{dy}{dx}\right|_{x=\bar x}\sigma _x\tag{2}$$
So differentiating,
$$\frac{dy}{dx}=B+2Cx+3Dx^2+4Ex^3+5Fx^4+6Gx^5+7Hx^6+8Ix^7+9Jx^8+10Kx^9$$
The problem is that
$$A=455.94819 \pm 3.1992$$
$$B=-836.25814 \pm 313.04746$$
$$C=28727.35824 \pm 9693.05689$$
$$D=-317119.94747\pm 135699.77149$$
$$E=1565518.7311 \pm 1022991.2689$$
$$F=-2082234.43747 \pm 4548667.1451$$
$$G=-7038931.36659 \pm1.24803E7$$
$$H=3.01618E7 \pm 2.1353E7$$
$$I=-4.60325E7 \pm 2.21761E7$$
$$J=3.29782E7 \pm 1.27857E7$$
$$K=-9275633.98777 \pm 3138360.78468$$
Before asking this question I came across this question earlier asking about a very similar problem for a cubic function, I didn't understand the answer the author gave, I also found this website that deals with finding the uncertainty in a quadratic function, but the explanation I did not understand.
My only possible guess would be to pretend that the $[\mathrm{A}, \mathrm{B}, \mathrm{C},\cdots, \mathrm{K}]$ are actually functions, $y=y(\mathrm{A}, \mathrm{B}, \mathrm{C},\cdots, \mathrm{K}, x)$, then I can use the formula on Wikipedia:
$$\sigma_y=\sqrt{\left(\frac{\partial y}{\partial A}\right)^2{\sigma_A}^2+\left(\frac{\partial y}{\partial B}\right)^2{\sigma_B}^2+\left(\frac{\partial y}{\partial C}\right)^2{\sigma_C}^2+\cdots+\left(\frac{\partial y}{\partial x}\right)^2{\sigma_x}^2}$$
This formula is found in the "Non-linear combinations" section just below "Simplification". I am assuming a covariance of zero in the above formula, as is the case for my function given in $(1)$.
But the above formula does not seem right to me.
So demoting $(1)$ for now to a linear function (just to focus on the theory),
so now we have that
$$y(x)=A + Bx\tag{3}$$
Does anyone understand how to compute $\sigma_y$ for $(3)$, given $A$ and $B$ are uncorrelated?
Best Answer
Propagation of error formulae assume the uncertainty is in the variables, not the coefficients.
Regardless, if we assume you have no error in $x$ and A and B are uncorrelated random variables then you simply apply the usual algebra of random variables to get the variance of $Y$:
$$Var[A+Bx] = Var[A]+x^2Var[B]$$
In general, for a random variable $Z$ and constant $a$:
$$Var[aZ]=a^2Var[Z]$$
I’d you have a linear combination of uncorrellated random variables you can apply this to each:
$$Var\left[\sum a_iZ_i\right]=\sum a_i^2Var[Z_i]$$
I’m your case, the constants are the values of $x, x^2,…$