User:Lothar.brendel: Difference between revisions
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== Polynomial fit \(\hat y=\sum_{p=0}^{m-1}a_p x^p\) == | == Polynomial fit \(\hat y=\sum_{p=0}^{m-1}a_p x^p\) == | ||
vanishing partial derivative with respect to \(a_q\): | |||
\[ | \[ | ||
\begin{align} | \begin{align} | ||
0&=\sum_i \frac{y_i-\sum_p a_p x_i^p}{\sigma_i^2}x_i^q\\ | 0&=\sum_i \frac{y_i-\sum_p a_p x_i^p}{\sigma_i^2}x_i^q\\ | ||
\Leftrightarrow\quad | \Leftrightarrow\quad | ||
\underbrace{\sum_i\frac{y_i x_i^q}{\sigma_i^2}}_{\displaystyle\psi_q}&=\sum_i\sum_p \frac{a_p x_i^{p+q}}{\sigma_i^2}=\sum_p a_p\underbrace{\sum_i\frac{x_i^{p+q}}{\sigma_i^2}}_{\displaystyle X_{pq}} | \underbrace{\sum_i\frac{y_i x_i^q}{\sigma_i^2}}_{\displaystyle\psi_q}& =\sum_i\sum_p \frac{a_p x_i^{p+q}}{\sigma_i^2}=\sum_p a_p\underbrace{\sum_i\frac{x_i^{p+q}}{\sigma_i^2}}_{\displaystyle X_{pq}} | ||
\end{align} | \end{align} | ||
\] | \] | ||
Assuming that the (non-stochastic) matrix \(X\) has an inverse \(\Xi\), the following holds true (from now on, summation convention) | |||
\[ | \[ | ||
a_p = \Xi_{pq}\psi_q | a_p = \Xi_{pq}\psi_q | ||
\] | \] | ||
and, due to linearity and the fact that \(\Xi\) does not depend on the stochastic \(y_i\), also | |||
\[ | \[ | ||
\langle a_p\rangle = \Xi_{pq}\langle\psi_q\rangle | \langle a_p\rangle = \Xi_{pq}\langle\psi_q\rangle | ||
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=== Uncertainty of the fitting parameters === | === Uncertainty of the fitting parameters === | ||
For the correlations, we need the products | |||
\[ | \[ | ||
\langle\psi_q\psi_{q'}\rangle = \frac{x_i^q x_j^{q'}}{\sigma^2_i\sigma^2_j}\langle y_i y_j\rangle | \langle\psi_q\psi_{q'}\rangle = \frac{x_i^q x_j^{q'}}{\sigma^2_i\sigma^2_j}\langle y_i y_j\rangle | ||
\quad\text{ | \quad\text{and}\quad | ||
\langle\psi_q\rangle\langle\psi_{q'}\rangle = \frac{x_i^q x_j^{q'}}{\sigma^2_i\sigma^2_j}\langle y_i\rangle\langle y_j\rangle | \langle\psi_q\rangle\langle\psi_{q'}\rangle = \frac{x_i^q x_j^{q'}}{\sigma^2_i\sigma^2_j}\langle y_i\rangle\langle y_j\rangle | ||
\] | \] | ||
so | so that | ||
\[ | \[ | ||
\langle\psi_q\psi_{q'}\rangle-\langle\psi_q\rangle\langle\psi_{q'}\rangle=\frac{x_i^q x_j^{q'}}{\sigma^2_i\sigma^2_j}\big(\underbrace{\langle y_i y_j\rangle-\langle y_i\rangle\langle y_j\rangle}_{\displaystyle\sigma^2_i\delta_{ij}}\big)=\sum_i\frac{x_i^{q+q'}}{\sigma^2_i}=X_{qq'} | \langle\psi_q\psi_{q'}\rangle-\langle\psi_q\rangle\langle\psi_{q'}\rangle=\frac{x_i^q x_j^{q'}}{\sigma^2_i\sigma^2_j}\big(\underbrace{\langle y_i y_j\rangle-\langle y_i\rangle\langle y_j\ rangle}_{\displaystyle\sigma^2_i\delta_{ij}}\big)=\sum_i\frac{x_i^{q+q'}}{\sigma^2_i}=X_{qq'} | ||
~. | ~. | ||
\] | \] | ||
Thus, and using | |||
\[ | \[ | ||
a_p a_{p'}=\Xi_{pq}\Xi_{p'q'}\psi_q\psi_{q'} | a_p a_{p'}=\Xi_{pq}\Xi_{p'q'}\psi_q\psi_{q'} | ||
\] | \] | ||
we can calculate the variance of the fit parameters (\(p'=p\)) as | |||
\[ | \[ | ||
\langle a_p^2\rangle-\langle a_p\rangle^2=\Xi_{pq}\Xi_{pq'}\big(\langle\psi_q\psi_{q'}\rangle-\langle\psi_q\rangle\langle\psi_{q'}\rangle\big)=x^{p+p'}\Xi_{pq}\Xi_{pq'}X_{qq'}=\Xi_{pq}\delta_{pq}=\Xi_{pp} | \langle a_p^2\rangle-\langle a_p\rangle^2=\Xi_{pq}\Xi_{pq'}\big(\langle\psi_q\psi_{q'}\rangle-\langle\psi_q\rangle\langle\psi_{q'}\rangle\big)=x^{p+p'}\ \Xi_{pq}\Xi_{pq'}X_{qq'}=\Xi_{pq}\delta_{pq}=\Xi_{pp} | ||
~. | ~. | ||
\] | \] | ||
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=== Uncertainty of the fitting function's values === | === Uncertainty of the fitting function's values === | ||
For a given \(x\), the following holds for the predicted \(y=a_p x^p\): | |||
\[ | \[ | ||
\begin{align} | \begin{align} | ||
\langle y\rangle &=\langle a_p\rangle x^p=\langle\psi_q\rangle\Xi_{qp}x^p\\ | \langle y\rangle &=\langle a_p\rangle x^p=\langle\psi_q\rangle\Xi_{qp}x^p\\ | ||
\langle y^2\rangle-\langle y\rangle^2 &= x^{p+p'}\big(\langle a_p a_{p'}\rangle-\langle a_p\rangle\langle a_{p'}\rangle\big)=x^{p+p'}\Xi_{pq}\Xi_{p'q'}\big(\langle\psi_q\psi_{q'}\rangle-\langle\psi_q\rangle\langle\psi_{q'}\rangle\big)\\ | \langle y^2\rangle-\langle y\rangle^2 &= x^{p+p'}\big(\langle a_p a_{p'}\rangle-\langle a_p\rangle\langle a_{p'}\rangle\big)=x^{p+p'}\ \Xi_{pq}\Xi_{p'q'}\big(\langle\psi_q\psi_{q'}\rangle-\langle\psi_q\rangle\langle\psi_{q'}\rangle\big)\\ | ||
&=x^{p+p'}\Xi_{pq}\Xi_{p'q'}X_{qq'}=x^{p+p'}\Xi_{pq}\delta_{p'q}=\Xi_{pp'}x^{p+p'} | &=x^{p+p'}\Xi_{pq}\Xi_{p'q'}X_{qq'}=x^{p+p'}\ \Xi_{pq}\delta_{p'q}=\Xi_{pp'}x^{p+p'} | ||
\end{align} | \end{align} | ||
\] | \] | ||
The variance of \(y(x)\) is thus a polynomial of twice the degree of the fitting polynomial. | |||
=== Uncertainty of a crossing point === | === Uncertainty of a crossing point === | ||
Let \(x_0\) be defined by \(\langle y\rangle(x_0)=c\). We linearize about \(x_0\), i.e., | |||
\[ | \[ | ||
\langle y\rangle(x) = \langle y\rangle(x_0)+(x-x_0)\langle y'\rangle(x_0)+\mathcal O((x-x_0)^2) | \langle y\rangle(x) = \langle y\rangle(x_0)+(x-x_0)\langle y'\rangle(x_0)+\mathcal O((x-x_0)^2) | ||
\] | \] | ||
and ask about the effect of the fluctuations on the measured location: | |||
\[ | \[ | ||
\begin{align} | \begin{align} | ||
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\end{align} | \end{align} | ||
\] | \] | ||
To leading order, the fluctuations of the measured location are thus given by those of \(y(x_0)\), while those of \(y'(x_0)\) contribute only as a higher-order correction: | |||
\[ | \[ | ||
\langle(x-x_0)^2\rangle\approx\frac{\langle y^2\rangle(x_0)-\langle y\rangle^2(x_0)}{\langle y'\rangle^2(x_0)}=\frac{\Xi_{p'q}x_0^{p'+q}}{(p\langle a_p\rangle x_0^{p-1})^2} | \langle(x-x_0)^2\rangle\approx\frac{\langle y^2\rangle (x_0)-\langle y\rangle^2(x_0)} {\langle y'\rangle^2(x_0)}=\frac{\Xi_{p'q}x_0^{p'+q}}{(p\langle a_p\rangle x_0^{p-1})^2} | ||
\] | \] | ||
= Scratch Pad = | = Scratch Pad = | ||
Revision as of 10:58, 24 April 2026
Polynomial fitting and estimation of uncertainty
Background
If the measured values \(y_i\) are statistically independent but follow the same normal distribution centered around a mean \(\langle y_i\rangle\) that depends on \(x_i\), then the parameters \(\{a_k\}\) which minimize \[ \sum_{i=1}^n(y_i-\hat y(x_i;a_1,\dots,a_m))^2 \] maximize the probability density for the occurrence of the data set \(\{y_i\}\); i.e., adjacent data sets are less likely. If the variance of the respective normal distribution, \(\sigma_i\), is also \(x_i\)-dependent, then instead \[ \frac{\chi^2}{2}=\sum_{i=1}^n\frac{(y_i-\hat y(x_i;a_1,\dots,a_m))^2}{2\sigma_i^2} \] must be minimized.
Polynomial fit \(\hat y=\sum_{p=0}^{m-1}a_p x^p\)
vanishing partial derivative with respect to \(a_q\): \[ \begin{align} 0&=\sum_i \frac{y_i-\sum_p a_p x_i^p}{\sigma_i^2}x_i^q\\ \Leftrightarrow\quad \underbrace{\sum_i\frac{y_i x_i^q}{\sigma_i^2}}_{\displaystyle\psi_q}& =\sum_i\sum_p \frac{a_p x_i^{p+q}}{\sigma_i^2}=\sum_p a_p\underbrace{\sum_i\frac{x_i^{p+q}}{\sigma_i^2}}_{\displaystyle X_{pq}} \end{align} \] Assuming that the (non-stochastic) matrix \(X\) has an inverse \(\Xi\), the following holds true (from now on, summation convention) \[ a_p = \Xi_{pq}\psi_q \] and, due to linearity and the fact that \(\Xi\) does not depend on the stochastic \(y_i\), also \[ \langle a_p\rangle = \Xi_{pq}\langle\psi_q\rangle ~. \]
Uncertainty of the fitting parameters
For the correlations, we need the products \[ \langle\psi_q\psi_{q'}\rangle = \frac{x_i^q x_j^{q'}}{\sigma^2_i\sigma^2_j}\langle y_i y_j\rangle \quad\text{and}\quad \langle\psi_q\rangle\langle\psi_{q'}\rangle = \frac{x_i^q x_j^{q'}}{\sigma^2_i\sigma^2_j}\langle y_i\rangle\langle y_j\rangle \] so that \[ \langle\psi_q\psi_{q'}\rangle-\langle\psi_q\rangle\langle\psi_{q'}\rangle=\frac{x_i^q x_j^{q'}}{\sigma^2_i\sigma^2_j}\big(\underbrace{\langle y_i y_j\rangle-\langle y_i\rangle\langle y_j\ rangle}_{\displaystyle\sigma^2_i\delta_{ij}}\big)=\sum_i\frac{x_i^{q+q'}}{\sigma^2_i}=X_{qq'} ~. \] Thus, and using \[ a_p a_{p'}=\Xi_{pq}\Xi_{p'q'}\psi_q\psi_{q'} \] we can calculate the variance of the fit parameters (\(p'=p\)) as \[ \langle a_p^2\rangle-\langle a_p\rangle^2=\Xi_{pq}\Xi_{pq'}\big(\langle\psi_q\psi_{q'}\rangle-\langle\psi_q\rangle\langle\psi_{q'}\rangle\big)=x^{p+p'}\ \Xi_{pq}\Xi_{pq'}X_{qq'}=\Xi_{pq}\delta_{pq}=\Xi_{pp} ~. \]
Uncertainty of the fitting function's values
For a given \(x\), the following holds for the predicted \(y=a_p x^p\): \[ \begin{align} \langle y\rangle &=\langle a_p\rangle x^p=\langle\psi_q\rangle\Xi_{qp}x^p\\ \langle y^2\rangle-\langle y\rangle^2 &= x^{p+p'}\big(\langle a_p a_{p'}\rangle-\langle a_p\rangle\langle a_{p'}\rangle\big)=x^{p+p'}\ \Xi_{pq}\Xi_{p'q'}\big(\langle\psi_q\psi_{q'}\rangle-\langle\psi_q\rangle\langle\psi_{q'}\rangle\big)\\ &=x^{p+p'}\Xi_{pq}\Xi_{p'q'}X_{qq'}=x^{p+p'}\ \Xi_{pq}\delta_{p'q}=\Xi_{pp'}x^{p+p'} \end{align} \]
The variance of \(y(x)\) is thus a polynomial of twice the degree of the fitting polynomial.
Uncertainty of a crossing point
Let \(x_0\) be defined by \(\langle y\rangle(x_0)=c\). We linearize about \(x_0\), i.e., \[ \langle y\rangle(x) = \langle y\rangle(x_0)+(x-x_0)\langle y'\rangle(x_0)+\mathcal O((x-x_0)^2) \] and ask about the effect of the fluctuations on the measured location: \[ \begin{align} y(x) &= \underbrace{\langle y\rangle(x_0)}_c+\delta y(x_0)+(x-x_0)(\langle y'\rangle(x_0)+\delta y'(x_0)) \stackrel{!}{=}c\\ \Leftrightarrow\quad x-x_0 &= \frac{\delta y(x_0)}{\langle y'\rangle(x_0)+\delta y'(x_0)}\\ \Rightarrow\quad (x-x_0)^2 &= \frac{\delta y^2(x_0)}{\langle y'\rangle^2(x_0)}\big(1+\delta y'(x_0)/\langle y'\rangle(x_0)\big)^{-2}\\ &\approx \frac{\delta y^2(x_0)}{\langle y'\rangle^2(x_0)}\big(1-2\delta y'(x_0)/\langle y'\rangle(x_0)+3\delta y'^2(x_0)/\langle y'\rangle^2(x_0)+\dots\big) \end{align} \] To leading order, the fluctuations of the measured location are thus given by those of \(y(x_0)\), while those of \(y'(x_0)\) contribute only as a higher-order correction: \[ \langle(x-x_0)^2\rangle\approx\frac{\langle y^2\rangle (x_0)-\langle y\rangle^2(x_0)} {\langle y'\rangle^2(x_0)}=\frac{\Xi_{p'q}x_0^{p'+q}}{(p\langle a_p\rangle x_0^{p-1})^2} \]
Scratch Pad
2D flow in polar coordinates \((\rho,\phi)\)
- vorticity: \(\omega=(\vec\nabla\times\vec u)\cdot\vec e_z=\partial_\rho u_\phi+\displaystyle\frac{u_\phi-\partial_\phi u_\rho}{\rho }\)
- its gradient: \(\vec\nabla\omega=\vec e_\rho\partial_\rho\omega+\displaystyle\frac{\vec e_\phi}{\rho}\partial_\phi\omega\)
- divergence \(\nabla\cdot\vec u=\displaystyle\frac{\partial_r(ru_r)+\partial_\phi u_\phi}{r}\)
- stress tensor:
- \(\sigma_{rr}=-p+(\zeta-2\mu/3)\displaystyle\frac{\partial_r(ru_r)+\partial_\phi u_\phi}{r}+2\mu\,\partial_r u_r\)
- \(\sigma_{r\phi}=\sigma_{\phi r}=\mu\displaystyle\frac{\partial_\phi u_r-u_\phi}{r}+\mu\,\partial_r u_\phi\)
- \(\sigma_{\phi\phi}=-p+(\zeta-2\mu/3)\displaystyle\frac{\partial_r(ru_r)+\partial_\phi u_\phi}{r}+2\mu\displaystyle\frac{\partial_\phi u_\phi+u_r}{r}\)
Energy equation
\begin{align} E' &= \rho e+\frac{\rho}{2}\vec u^2\\ E &= E' + \rho\phi\\ \partial_t E &= \partial_t E' + \phi\partial_t\rho\\ &= -\vec\nabla\cdot\big((E+P)\vec u\big)\\ &= -\vec\nabla\cdot\big((E'+P)\vec u+\phi\rho\vec u\big)\\ &= -\vec\nabla\cdot\big((E'+P)\vec u\big)-\phi\vec\nabla\cdot(\rho\vec u)-\vec u\cdot\rho\vec\nabla\phi\\ \Leftrightarrow\quad \partial_t E' &= -\vec\nabla\cdot\big((E'+P)\vec u\big)+\vec u\cdot\vec f \end{align}
Florian's eCDF
Let \(p_n\) be the relative frequencies and let \(p_n=0\) for \(n_{i-1}<n<n_i\). Then, being a step-function, the eCDF fulfills $$ c(n_i)-c(n_{i-1})\stackrel{*}{=}c(n_i)-c(n_i-1)=c(n_i)-c(n_i-\epsilon)=p_{n_i}~. $$ Florian's modified eCDF replaces the step-function \(c\) by a continous, piecewise linear function \(\tilde c\) with \(\tilde c(n_i)=c(n_i)\), from which one obtains fictious empirical probabilities \(\tilde p_n=\tilde c(n)-\tilde c(n-1)\) for \(n_{i-1}<n\le n_i\). Due to the "natural" \(\Delta n=1\) of the Steigungsdreieck, those \(\tilde p_n\) (being all the same because of the linearity) coincide with the slope $$ s_{n_i}=\frac{\tilde c(n_i)-\tilde c(n_{i-1})}{n_i-n_{i-1}} = \frac{c(n_i)-c(n_{i-1})}{n_i-n_{i-1}}~, $$ which implies $$ \sum_{n=n_{i-1}+1}^{n_i} \tilde p_n = (n_{i}-n_{i-1})\tilde p_n = (n_{i}-n_{i-1})s_{n_i} = c(n_i)-c(n_{i-1}) = p_{n_i}~. $$ Hence, the effect of using \(\tilde c\) is replacing the "bar" \(p_{n_i}\) and the zeros to its left by smaller "bars" of equal height \(\tilde p_n\) while preserving probability, which of course implies \(\sum_k\tilde p_k=1\). In other words, the scheme is a left-biased coarse graining.
*: even though in general \(n_{i-1}\ne n_i-1\)
Hamiltons principle
- Phys.SE: Modified Hamilton's Principle overconstraining a system by imposing too many boundary conditions
- Phys.SE: Boundary conditions for calculus of variations in phase space and under canonical transformations
Discretizations
\begin{align} f'_0 &= \frac{-f_{-1/2}+f_{1/2}}{h}+\mathcal O(h^2)\\ &= \frac{f_{-3/2}-27 f_{-1/2}+27 f_{1/2}-f_{3/2}}{24h}+\mathcal O(h^4)\\ &= \frac{-9 f_{-5/2}+125 f_{-3/2}-2250 f_{-1/2}+2250 f_{1/2}-125 f_{3/2}+9 f_{5/2}}{1920h}+\mathcal O(h^6)\\[3ex] f_0 &= \frac{f_{-1/2}+f_{1/2}}{2}+\mathcal O(h^2)\\ &= \frac{-f_{-3/2}+9f_{-1/2}+9f_{1/2}-f_{3/2}}{16}+\mathcal O(h^4)\\ &= \frac{3 f_{-5/2}-25 f_{-3/2}+150 f_{-1/2}+150 f_{1/2}-25 f_{3/2}+3 f_{5/2}}{256}+\mathcal O(h^6) \end{align}
Values of ghost cells
Robin \(af(0)+b f'(0)=c\) : \[ f(-x):=\frac{b+ax}{b-ax}f(x)-c\frac{2x}{b-ax} \]
Special cases:
Dirichlet \(f(0)\): \[ f(-x):=2f(0)-f(x) \]
Neumann \(f'(0)\): \[ f(-x):=f(x)-2xf'(0) \]