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Questions tagged [error-propagation]

For questions on propagation of errors.

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How can we use analysis to determine the error in computing a multidimensional function (Zorich)?

What is the relative error $\delta = \frac{|\Delta f|}{|f|}$ in computing the value of a function $f(x,y,z)$ at a point $(x,y,z)$ whose coordinates have absolute errors $\Delta x, \Delta y, \Delta z$? ...
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How to measure the error between modeled and observed data?

Consider a scenario where observed data is represented in grey and modelled data in red, as below Here, the x-axis is a position, and the y-axis is an expected time, so that the slope defines, in a ...
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Is there any way to compact the propagation of uncertainty formula in terms of vectors?

The uncertainty associated to $\xi=f(\mathbf x)$ with $\mathbf x\in\mathbf R^n$ is $$\delta\xi=\sqrt{\sum_{i=1}^n\left(\frac{\partial \xi}{\partial x_i}\delta x_i\right)^2}$$ What I was wondering was ...
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Uncertainty propagation when using the Trapezium rule

I have some recorded data s(x), and need to compute the value and uncertainty in $$S = \int_{x_{1}}^{x_{2}} s(x) dx$$ I am using the trapezium rule for the integration, and need to estimate the ...
Rustony's user avatar
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How do I go "backwards" with error propagation?

Hopefully this is appropriate for Math SE given that, ultimately, this questions stems from a math textbook (Zorich's Mathematical Analysis). Zorich has shown using simple arguments with absolute ...
EE18's user avatar
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Error Propagation in calculating Binary logarithm by hand

I am interested in the idea of arithmetic performed in binary done without electronics. In particular, I’d like to try calculating the binary logarithm of x (where 10 > x >= 1) using the square/...
MMLgamer's user avatar
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Lop sided error due to non-negative data

I am currently working on a project where we are administering different treatments to subjects, but with some potential side effects. I am trying to measure the effect each method has on the well ...
JMB's user avatar
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Attributing error to each term through a product

I have a relatively straightforward problem. I have a value that gets multiplied by a bunch of correction factors to give a final answer. $Final = Base * CF_1 * CF_2 * CF_3 * CF_4$ I am trying to re-...
user2731076's user avatar
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1 answer
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How to partially differentiate an equation with sums?

I'm working on a simple linear regression model in a physics course, where we are doing measurements of the round trip speed of light, over increasing distances. We are using the Least Squares method ...
THH's user avatar
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How to find the margin of error of a value consisting of the sum of 5 values? [closed]

I have a $X$ value that is the sum of 5 different $x1,x2,x3,x4$, and $x5$ values. Eacn $x_i$ has their error in same scale. I want to calculate the error of X. How can i do that? Basically, i tried to ...
Ege Tunç's user avatar
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Find variance of random variable $W\cos\theta+L\sin\theta$ where $L,W,\theta$ are normal random variables. Uncertainty propagation problem.

I have three variables - these are length ($L$), width ($W$) and angle ($\theta$). Each has a known mean and variance. $\theta$ is independent, but $L$ and $W$ are correlated, and I know their ...
Mike Woods's user avatar
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Difference between Absolute and Relative differences

I was reading a USACO Guide Page on data types and I came across two formulas for the absolute and relative difference in the desired answer and the actual answer that is outputted. This is meant to ...
Shubhankar Dixit's user avatar
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Confused about error propagation

I am confused about the results I am getting for an apparently simple situation. I have 2 measurements (counts), call them $S_+$ and $S_-$. Based on these I build an asymmetry defined as: $$A = \frac{...
Bobo's user avatar
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How to estimate error in a composition of functions?

Suppose I have functions $f_1,f_2: X \to Y$ such that $|f_1(x) - f_2(x)| < \varepsilon_1$ for all $x \in X$. And suppose similarly that $g_1,g_2: Y \to Z$ with $|g_1(y) - g_2(y)| < \varepsilon_2$...
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Error on trace of quadratic forms from Frobenius error bound on central matrix

$\DeclareMathOperator{\Tr}{Tr}$ If a Frobenius error on an estimate of covariance $\|\tilde{\boldsymbol{M}}-\boldsymbol{M}\|_F$ is known to be in the order of $\tilde{O}(f(n,d))$ as some function of $...
hearse's user avatar
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Deriving the Variance of the product of two normally distributed variables

i am refreshing my knowledge on error analysis and just as a little exercise i was trying to algebraically derive the product rule of error propagation \begin{equation}\tag{1}\label{error} \frac{\...
vreithinger's user avatar
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Error propagation and Gradient Descent

I was looking at error propagation (or propagation of uncertainty in wikipedia: https://en.wikipedia.org/wiki/Propagation_of_uncertainty) My primary concern is getting an estimate of error of ...
ponir's user avatar
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Calibration Resolution

I have a flow meter that I calibrate by doing timed catches. The best resolution I have on my scale is down to 0.01 lb. The meter I'm comparing against goes to 0.001 lb. Usually, I'll have something ...
user267587's user avatar
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Is there a "chain rule" for error propagation?

I am investigating certain functions $\mu_1$ and $\mu_2$ which represent a physical quantity $\mu$ where $\mu_i(T,\rho)$; but also $\rho = \rho(T,P)$ so that also $\mu_i = \mu_i(T,P)$. Maybe another ...
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Error in linear interpolation of $n$-dimensional curves

Let's assume we are given an $n$-dimensional smooth curve $\gamma:[a,b] \rightarrow \mathbb{R}^n$ and $N$- sampled points $\{x_1,...,x_N\}$ of that curve. Now we use linear interpolation (or a higher ...
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Calculating limits and integration in an Error-Accumulating Real Numbers field structure...

I am interested to see if anyone can direct me to a text on using limits and integration in the "Error-Accumulating Real Numbers" field structure. This field consists of the real numbers, ...
Spanki's user avatar
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What are the rules wrt. sig figs when multiplying or dividing a number with an uncertainty and an exact number?

The answer sheet for my math book is full of errors and the book does not explain this concept at all. According to it : (28,3 ± 0,05) x 4 = 113,2 ± 0,2 So one sig fig is added to the value and one is ...
user253195's user avatar
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Algorithmic error computing $\frac{e^x -1}{x}$

It is well know that in order to calculate the algorithmic error of a function one can use backward analysis with using the visual representation of a graph with nodes the $i-$th step of the algorithm ...
jacopoburelli's user avatar
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1 answer
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Does using smaller floating-point numbers decrease rounding errors?

I started learning about floating point by reading "What Every Computer Scientist Should know About Floating-Point Arithmetic" by David Goldberg. On page 4 he presents a proof for the ...
Thanks for flying Vim's user avatar
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Error propagation in diagonalization of real symmetric matrix

I am trying to figure out how absolute errors of matrix elements in a real symmetric matrix propagate over to its eigenvalues. My attempt: Let $A$ be a real symmetric $n\times n$ matrix. Let $$\...
Rasmus's user avatar
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Discrepancy in value of $\int_0^\frac\pi2 \delta(\tan(t)-xt)dt$

The goal would have been for the smallest positive solution of $y\cot(y)=x$ using a Dirac $\delta(x)$ Fourier series and the Bateman function: $$y\cot(y)=x\mathop\implies^?\frac1{\sec^2(y)-x}=\int_0^\...
Тyma Gaidash ٠'s user avatar
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1 answer
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How can I properly solve the Black Scholes PDE numerically?

I have a problem numerically solving the following PDE with boundary conditions: $$ u_t + \frac{x^2\sigma^2}2u_{xx} + rxu_x - ru = 0 \quad (x,t) \in (0,N) \times (0,T) $$ with $$ u(x,T) = \max\{0,x-K\}...
julian2000P's user avatar
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Why does the fractional/relative change concept by differentiation $dx/x$ not work on big changes?

There's a term that I've studied about called 'Fractional/relative change'. Basically, if say: $$P = x^ay^bz^c$$ then, $$\log P = a\log x + b\log y + c\log z$$ By differentiation w.r.t $dp$: $$\frac 1 ...
Ani's user avatar
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How accurately do we need to know an approximation of $\pi$ to compute $\sqrt{\pi}$ with four correct decimal?

An approximation $\overline{x}$ of $x$ is said to have $t$ correct decimals if and only if $|\overline{x}-x| \leq \frac{1}{2} 10^{-t}$. Suppose that $\overline{x} = x \pm \frac{1}{2} 10^{-k}$ and $f(x)...
Albelaski's user avatar
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Weighted sum of a series of values, weighted based on the scale of the error.

I need to find the value and error of a weighted sum of values all with their own respective error. Say I have three data points: $ x_1 = 7 \pm 1$ $ x_2 = 9 \pm 0.8$ $ x_3 = 10 \pm 0.7$ I need to find ...
Allentro's user avatar
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2 answers
121 views

Propagation of error for an integral using Leibnitz' rule

I'm computing integrals $u_i$, where $$u_i = \int_a^b s^i G(s) ds$$ and $a$ and $b$ are constants. Now I want to estimate the error in $u_i$; I have estimates for the error $\sigma_G$ of $G$, and ...
Oliver's user avatar
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1 answer
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Propagation of error for an integral

I'm computing integrals $u_i$, where $$u_i = \int_a^b s^i G(s) ds$$ $G$ and $s$ are given numerically; in other words, I have a number of ($s$, $G(s)$) pairs. The integral is evaluated approximately ...
Oliver's user avatar
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How can I simulate uncertainty propagation

I am learning about uncertainty propagation, but it is all very obscure and I am not sure I understand it very well. I have a couple of exercises, but I want to run some simulations just to make sure ...
user17004502's user avatar
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Counterintuitive result in estimating distances

In short, when I try to estimate the positioning error of a point from initial position estimates and true distances from other points, I get an estimate with a much higher error than the initial ...
James's user avatar
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1 answer
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Having difficulties with pendulum theory and percentage error homework problem.

The pendulum theory: $$t=2 \pi \sqrt{l/g},$$ where $t$ is the time of period, $L$ is the length of the pendulum, $G$ is the acceleration due to the gravity (~9.81 m/s²). Calculate the expected ...
FWTH-HK WU's user avatar
1 vote
1 answer
124 views

How to propagate error through a Heaviside step function $H(x\pm\epsilon)$

Suppose we have a set of data points $\{x_1,\cdots,x_n\}$ with a corresponding set of errors $\{\epsilon_1,\cdots,\epsilon_n\}$. Using the standard definition of the Heaviside step function, $$H(x)=\...
nebula's user avatar
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0 answers
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How to calculate error propagation in spherical astronomy formula?

Distance (r) as the angle between two stars are calculated by following formula: $$ \cos(r) = \sin(\delta_1) \sin(\delta_2) - \cos(\delta_1)\cos(\delta_2)\cos(\alpha_1-\alpha_2) $$ $\delta$ and $\...
Bluerose's user avatar
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Uncertainty of inverse cosine function

I am propagating uncertainties for a lab report in Physics. Anyways, I have a function: $$cos⁡(θ_0 )= \frac{0.725\space m-h}{0.675\space m} $$ The uncertainty of both the numerator and ...
aayush's user avatar
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Summing series - Error propagation

Why sum down and sum up given different results? I know that this has a relation with error propagation, but I can't figure why. ( You can use whatever programming language you want, if you choose a ...
Vithor's user avatar
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Error analysis of hardware or software implementation of the 2D DCT

The cosine terms are constants. These are multiplied with fyx for 2D transform. When this is implemented in hardware/software we deal with following issues: fixed point vs floating point ...
gyuunyuu's user avatar
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1 answer
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How do I propagate relative uncertainty through atan2?

I've got $y = \sin(\theta)$ and $x = \cos(\theta)$ with some relative error on both. If I compute $\theta$ with $\theta = \operatorname{atan2}(y,x)$, how do I propagate the relative error from the ...
gct's user avatar
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Uncertainty corresponding to bin population.

I have around 200 data points. I have separated these point into 7 bins. Some of these bins contain up to 50 points while some contain around 3-6 points. Some of these 200 points belong to a group C. ...
Bobasheto's user avatar
3 votes
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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 ...
user2934303's user avatar
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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 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 ...
mng's user avatar
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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} \...
MarkFloatingInSpace's user avatar
5 votes
0 answers
104 views

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 ...
Greedo's user avatar
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6 votes
1 answer
136 views

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,...
lesath82's user avatar
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4 votes
2 answers
152 views

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 ...
nothatcreative5's user avatar
1 vote
0 answers
34 views

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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