Questions tagged [error-propagation]

For questions on propagation of errors.

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What's best, to take the average of 2 measures or the square root of their product (in this case)?

I have a physical system where a real and positive quantity A can be measured by two ratios between 4 measures, $\frac{B}{C}$ and $\frac{D}{E}$ (all of them are positive real numbers). From the ...
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Propagating error through a Fast Fourier transform

I am trying to propagate the error associated with a Fast Fourier transform of $x_{n}$. I know the error (variance) for $x_{n}$. Then, I calculated the following quantity: $$Y=Im\left ( i\omega FFT(x_{...
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How to bound the phase error of the complex number

Given the complex number $e^{i\theta_k}$ and $r' e^{i\theta'}$ for $k=1,2,3,...,L$ and $\sum_{k=1}^{L}p_k=1$ ($1\geq p_k\geq 0$ for each k), where $r',\theta_k,\theta'\in \mathbb{R}$ . Each $\theta_k$ ...
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How to estimate the error of fitting in a simple least squares problem?

Suppose we have estimated the model parameters $m$ of the equation $y=G*m$ from data $y$ as $m=(G'*G)^{-1} * G' * y$. We have the measurement errors in $y$ from which we construct an error co-variance ...
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State error propagation of ODE with uncertain parameters

I have an idea on how to integrate uncertainty of the parameters of an ODE in the state error propagation. But I am unsure if my idea is correct. I have a non-linear ODE of the form: $$ \frac{d}{dt} \...
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5 votes
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Farmer wants to know how wet their field is

Problem A farmer wants a better understanding of rainfall on their field. Assuming rain falls randomly and with equal likelihood over the entire field, the farmer thinks they can model the volume of ...
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Statistical method for assessing experimental and theoretical errors.

Let us suppose I have performed an experiment where the mean value is given by $V^{\rm exp}$ and its std is estimated as $\sigma_{V^{\rm exp}}$. Now, let us also suppose that theoretically, I am ...
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Bounding error propagation in matrix multiplication

For matrices $A$ and $B$ approximated by $\tilde{A}$ and $\tilde{B}$ so that $$\lVert \tilde{A} - A\rVert_2 \le \delta,$$ and $$\lVert \tilde{B} - B\rVert_2 \le \delta.$$ What can be said about $$\...
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Error propagation in compass and straightedge constructions

I was trying to assess the impact of non-idealities on the outcome of a classical geometric construction, performed on paper with actual compass and straightedge. I was thinking of possible approaches,...
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Recurrence relation is not numerically stable.

I have to solve the following Recurrence relation both analytically and numerically. Solving it analytically gives the answer $2a(\frac{1}{2})^n$ which is stable for every value of a, but when I try ...
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Inferring mean(X/Y) from mean(X) and mean(Y)

X and Y are linked variables (i.e. they relate to different measurements in the same individuals), I know the value of mean(X) and of mean(Y) but I do not know the individual level values of X and Y. ...
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For Ax=B, where all variables are matrixes, If A is constant, and B'=1.05B, does x'=1.05x for Ax'=B'?

I feel like the answer to this is is quite a simple yes, but I was asked to prove it on a MATLAB script by solving for x on matrices of increasing sizes, and to my surprise, it was never exactly a 5% ...
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Propagation of error in the solution of a non-linear system of equations

I would like to ask you little question about the title above. Suppose we have a real $n \times n$ non-linear system of equations as follows: $$ f_{1}(x_{1},x_{2},x_{3},x_{4},...,x_{n})=0\\ f_{2}(x_{1}...
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Group of perfect Hamming codes

I read that a group of perfect Hamming code is defined as $[2^r-1,2^r-r-1,3]$ for any integer $r\geq 2$. To be perfect such codes should satisfy the following equivalence (derived from the Hamming ...
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Finding roots of the equation $x^2−40x+1=0$ [closed]

The problem: Given the equation $x^2−40x+1=0$, find its roots to five significant digits. Use $√399≐19.975$, correctly rounded to five digits. Can anyone help me solve this problem? My thoughts: ...
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Differentiation for error propagation

I am computing a quantity, $y$, from a series of measurements, $x$, made over time. The relationship is: \begin{equation} y=c\frac{dx}{dt}(b-x)^{-1} \end{equation} I would like to compute the effect ...
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Upper bound for the error of the gaussian quadrature$\int_{0}^{1} \log (1+\operatorname{sin} x) \mathrm{d} x$

I'm given the integral $\mathrm{I}=\int_{0}^{1} \log (1+\operatorname{sin} x) \mathrm{d} x$. Through the formula of Gaussian quadrature for 3 points, I can find an approximation to this integral. The ...
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How to calculate the uncertainty on a function where the coefficients themselves have their own uncertainties?

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{...
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Error propagation product of sums with dependent variables

I am currently trying to compute / estimate the error of a function $f$ defined as $$ f = \sum_{i=1}^N x_i \cdot \sum_{i=1}^N y_i $$ where I know the estimations for the $2N$ means $\bar{x}_i$ and $\...
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How $\ln{f} = a_{1}\ln{x_{1}} + a_{2}\ln{x_{2}} +...+ a_{k}\ln{x_{k}}$ is differentiated here?

I'm learning about how to evaluate error in measurement and I don't understand how logarithmic derivative method (which is the only method I know about) is derivated. It says: When the function $f$ is ...
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Calculating 1 sigma error for each element

How to calculate the 1 simga error for each count of below data? ...
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Error Propagation (complicated case)

I have a function: $$G=\frac{(\frac{4\pi^2}{T^2}+\frac{1}{\tau^2})\cdot r^2 \cdot d\cdot S}{4M\cdot L}$$ and I need to do error propagation on this function: $$\delta G=\sqrt{(\frac{\partial G}{\...
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Error in first derivative of cubic spline interpolant

Let $f: [a,b] \rightarrow \mathbb{R}$ be a $C^{\infty}$ function, and let $a = x_0 < x_1 < \cdots < x_n = b$ be a partition of the interval $[a,b]$. Let $s(x)$ be a piecewise polynomial ...
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combining two fit results

I have got two different kind of data points $(x_i, y_i)$. Because one kind are the result of several measurements of the same object I know their statistical error $\Delta x_i, \Delta y_i $ for $x_i$ ...
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What is the meaning of $1$ in a relative error?

If we measure a length and is measured as $12.5$ meters long, accurate to $0.1$ of a meter this means the absolute error is $0.05$m. The relative error is: $\frac{0.05}{12.5} = 0.004$. This means that ...
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Runge-Kutta time propagation with large stochastic error

I have to solve a differential equation of the type: $$ \frac{d y}{dt} = f(t,y) $$ $$ y(t=0) = y_0 $$ In general one would solve it with some high order Runge-Kutta method... if one could compute $f(t,...
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Propagation of Error: Tricky Example

I am having a difficult time finding error bars on a particular quantity. There are two random variables, let's call them $price$ and $color$; color can only be either red or blue. We don't have a ...
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Correct Mean Squared Error Function for Neural Network Output

Any help is appreciated. I would like to know if i missed something, or if that would be correct? $m$ is the number of training examples $L$ is the Loss-Function $\hat{\mathbf{y}}^{(i)}$ is the ...
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3 votes
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606 views

How can I find the uncertainty of derivatives?

Suppose I have a quadratic (weighted) least-square fit result obtained from a given set of data: $$ f(x) = \underbrace{-0.243(\pm0.3324)}_{\text{quad}_a}x^2\underbrace{{}-0.921(\pm0.061)}_{\text{quad}...
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How precise is the result after calculating a/b?

I have two numbers, $a$ and $b$, where the precision of $a$ is $n$ bits and the precision of $b$ is $m$ bits. How many bits of precision are preserved after calculating a/b on a normal computer ...
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Why are uncertainties calculated as an average?

Imagine there is a cube with length $l = (10 \pm 0.2) \; \text{cm}$. The volume $V$ would be $l^3 = 1000 \; \text{cm}^3$ and the uncertainty in $V$, $\Delta V$ would be $\frac{10.2^3 - 9.8^3}{2} = 60....
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Floating point arithmetic error propagation

I have a function $$y=ln(\frac{x_1}{x_2})$$, $$x_1, x_2 > 1$$ and two ways to calculate it: $$v = \frac{x_1}{x_2} => y_1=ln(v)$$ $$v_1 = ln(x_1)$$ $$v_2 = ln(x_2)$$ $$y_2 = v_1 - v_2$$ How ...
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Asymptotic propagation of error

Let $\tilde{s}_n$ and $\tilde{p}_n$ be estimators of the quantities $s$ and $p$, respectively ($\mathbb{E}[\tilde{s}_n]=s$ and $\mathbb{E}[\tilde{p}_n]=p$). Imagine we have obtained asymptotic bounds ...
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Error in matrix multiplication

Let's say that I have a matrix where each entry has it's own standard error. $\textbf{A} \pm \textbf{$\delta$ A}$. What would be the error of $Tr(\textbf{A}^2)$? Is the error of $\textbf{A}^2$ just $\...
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Optimize rank order survey experiment

What are methods for describing/comparing rank ordering of many items by multiple evaluators, where evaluators might not have the same items they are evaluating? For example: Suppose I have a dog show ...
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Random Vectors in Uncertainty Propagation

I'm trying to better understand the propagation of uncertainties and read this article on Wikipedia. There the following formula is given for the propagation of uncertainty in the linear case: $$\...
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Propagation of uncertainty for nonlinear combinations

I would like to better understand propagation of uncertainties in case of non-linear combinations. I therefore read this article on Wikipedia. I think I got a good understanding of the first part ...
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Bound the error of numerical ODE simulation

Let's assume, I have an autonomous ODE of the form $$ \frac{dy}{dt} = \sum \limits_{i=1}^n \alpha_i\cdot f_i(y) $$ which I want to solve numerically in $t\in[0, t_{end}]$ with some initial condition $...
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1 answer
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Error propagation on a 5 order polynomial

I have a 5 order polynomial equation which gives log of a chi factor required to convert equivalent widths to normalised H alpha luminosity for M dwarfs, (Reiners et al. 2008). I would like to get the ...
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1 vote
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uncertainty, error propagation, residuals and least squares.

I'm stuck with something, I haven't found a proper discussion about it. Say you have a set of data $\sum_i^n (x_i, y\pm \sigma_i) $ which follows a linear trend, where there's an uncertainty $\sigma_i$...
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Does It Makes Sense to Have the Index of a Sum in the Argument of a Function? Two Examples Given with Integral and Sum.

Question 1: I have recently tried out answering a couple of questions with the method of using the same variable in the argument of a function and a sum for example: $$\int_0^N (\sqrt{x+1}-\sqrt x)^n ...
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-1 votes
1 answer
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Error on the Bin of a Normalised Histogram

Suppose I have a histogram, $N$, each with bins of width $\Delta x$, denoted by bin indices, $i$. The count of a single bin is then $N_{i}$. I wish to estimate the empirical density for a certain bin. ...
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1 vote
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How is SVD better than Gaussian elimination in finding the rank of a matrix?

In Linear Algebra and its Applications, Gilbert Strang, $4^{th}$ ed, one of the applications of SVD is mentioned as finding the effective rank of a matrix. The idea presented in the book is that the ...
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1 vote
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Backward Error Analysis : Determining Errors on Model's Inputs with Known Output Errors

I have an algorithm that takes 5 input parameters $r, i, \Omega, \omega, f$ and returns two outputs $X$ and $Y$. I happen to know the errors $\Delta X$, $\Delta Y$ associated to $X$ and $Y$ (which ...
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Count users with big space between pings logic

please try to be lenient with me because I really have forgotten most of the stuff, so I will probably be making incorrect assumptions, word the problem incorrectly, etc. Context I'm trying to ...
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2 votes
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Error Propagation on Orbital Elements through Non Linear Relation

I'm having a hard time trying to propagate errors for my current work. I have two known positions $X$ and $Y$ that depend on several orbital parameters $r, i, \Omega, \omega, f$ ($r$ is a distance ...
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1 vote
1 answer
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How to relate $|\delta x|$ and $|\tilde\delta x|$ in an inequality $A\le |\tilde\delta x|\le B$ where $A$ and $B$ contain $|\delta x|$?

$|\delta x| = |(\tilde x - x)/x|$ (this is the absolute value of relative error) $|\tilde\delta x| = |(\tilde x - x)/\tilde x|$ (bound for some distinct quantity) I need to find an inequality which ...
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2 answers
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Error propagation of a variable for an integral

I have an integral that depends on two parameters $a\pm\delta a$ and $b\pm \delta b$. I am doing this integral numerically and no python function can calculate the integral with uncertainties. So I ...
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1 answer
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derivate is based on addition, is there a muliplication analogon?

like $$ \operatorname{f^o}(x) = \lim_{h\to 1} \frac{f(x*h)}{f(x)} $$ $$ \operatorname{f}(x)=e^x $$ $$ \operatorname{f^o}(x) = \lim_{h \to 1} e^{x*h}/e^{x} = \lim_{h\to 1} e^{x*h-x}=e^0=1 $$ does it ...
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3 votes
3 answers
158 views

Solving a special rational equation on a very small interval

I need to solve the following equation (for $x$): \begin{equation} \mathcal f(x):=\sum_{i=1}^n b_i \left( \frac{a_i}{1+b_i x}\right)^2-\phi=0, \quad \text{with} \quad -1/b_1< x \le 0. \end{...
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