Questions tagged [error-propagation]

For questions on propagation of errors.

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How to propagate the uncertainty of momentum through the relativistic velocity equation?

I have the standard deviation of momentum as $σ_p$ and I am trying to find $σ_v$. The equation I want to propogate uncertaintiy through is the relativistic velocity equation: $v=\sqrt{(p/m)^2/(1+(p/m)^...
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52 views

How to find the error of the Pythagorean Theorem equation?

I am trying to find the length of a hypotenuse with error, when the measurements of the two legs have an error. So for this equation $$C = \sqrt{A^2+B^2}$$ when $A$ and $B$ each have an error of $\...
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29 views

Bayesian propagation of classification uncertainty to estimator

I have 10 objects which can be classified as A,B,C,D with some probability. For example: object 1: 40% A, 20% B, 30% C, 10% D object 2: 10% A, 30% B, 40% C, 20% D ... The question I want to ask is ...
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21 views

Simulations and floating point error?

Dear math stack exchange, I'm curious about what would floating point error would look as i've been hoping to conduct some simulations of gravitational phenomenon. Is it highly random or somewhat ...
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Simple error bound of inverse function

Let's suppose that we are given a number $a$ with some absolute error $\epsilon$, i.e. we know that $|a-\overline{a}|\leq \epsilon$. The values of $a$ might rage from 1 to some upper bound $K$. What ...
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33 views

Cancellation error of $\frac{1}{z-w}$

I have a problem where I need to calculate $\frac{1}{z-w}$ where $z$ an $w$ are complex numbers that are very close in the euclidean norm sense. However, when I use this formula in my code, it seems ...
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1answer
38 views

How do error bars change for variables squared?

Let's say I have some variable $\mu$ with an uncertainty estimate: $$\mu = 2 \pm .5$$ Let's say I have another variable $\nu = \mu^2$. Is the uncertainty estimate in $\nu$ equal to the the ...
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5 views

Error bound on square root of relative error bounds

Can you correct me if I am wrong. If $|d^2 - \tilde d^2| \leq \epsilon d^2$, what is the error on $|d - \tilde d|$? I think is $\sqrt \epsilon d$.
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18 views

How to calculate uncertainty in an angle given two vectors with relative uncertainty

So I want to calculate the relative uncertainty in an angle given two vectors with a given relative uncertainty. For the angle calculation I'm using the following formula. $$\cos{\alpha} = \frac{\...
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11 views

How to calculate propagation of uncertainties with asymmetric errors?

Suppose we have a function $f(x,y)$, and we have values for $x$ and $y$ with asymmetric errors, $$ x = \hat x\,^{+A}_{-B} \qquad y = \hat y\,^{+C}_{-D} $$ For the case $A=B=\sigma_x$, $C=D=\sigma_y$, ...
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Should errors need to follow any pattern, or they can be random?

The context of the question: I had plotted a certain 2D function in hydrodynamics by plotting a set of 25 points. Then, I tried to discretize the underlying equations of hydrodynamics to obtain the ...
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Introducing time to calculate relative percentage error using differentials

I solved a problem asking for the maximum corresponding percentage error in computing a cube's (a) surface area and (b) volume assuming the edge $x$ is measured with an error of at most $0.5\%$ using ...
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Error in linear least squares

If I have an overdetermined system of linear equations $Ax=b$, I can solve by least squares. The error vector is: $e=Ax-b$, which is the error that is being minimised by the least squares method. ...
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How to quantify error distribution between dataset 1 and dataset 3?

thanks in advance for looking into this. I have 3 datasets: two estimations (estimation 1 and estimation 2) and one measurement campaign (3). I have included some dummy data here. I’m able to ...
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How to calculate the error bars on a log base 2 scale

I have (x,y) data with with some error bars in y (represented by dy). I can log transform the data in y using $z = \log_{10}y$ For log base 10, the error bars can be log transformed using $dz = 0....
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Estimating error under Taylor expansion of denominator

I have the following function $$f = \frac{a^2bc}{(\sqrt{b^2c^2 + abc\delta}- bc)^2}$$ and am given that $0 < \delta \ll \frac{bc}{a}$ and $a, b, c > 0$. This motivates me to consider the ...
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29 views

Error Propagation in Floating-Point Multiplication

Wikipedia (Machine epsilon) tells me that the result of a multiplication between 2 floating-point numbers, with a rounding induced relative error ϵ, still only has the relative error ϵ. Why do the ...
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Basic Error Propagation: Do we use the formula of the original equation to calculate uncertainty?

This is a very basic question. I have been asked to find the uncertainty for $F_{c,th}=\frac{4\pi^2mr}{T^2}$. What would be the formula for $\delta F_{c,th}$ given that $m$, $r$ and $T$ are measured ...
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Correlated error propagation

Suppose I have a model fitting function $f = f(a,b)$ and I would like to find the error on $f$ by propagating the errors on $a$ and $b$. Denote $a_0$ and $b_0$ as some best fit values. Suppose I also ...
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Calculate the absolute and relative errors in $a+5b/c-3b\cdot c$ where $a=3.5435$, $b=0.2588$, $c=1.0150$ are correct up to 4 decimal places.

I only know how to find the absolute and relative errors when, let say $a=0.123$ and $b=12.37$. I first of all computed the maximum absolute error using rounding and in base 10. And use it to compute ...
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Stability/backwards error of argmax vs “softmax” or Gibbs mean?

Consider \begin{align*} a^* &= \text{arg} \max_a J(a, y) \\ a^*_{soft}(k) &= \frac{1}{Z}\int a e^{kJ(a, y)} \text{d} a \\ Z &= \int e^{kJ(a, y)} \text{d} a \end{align*} I think $a^*_{...
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How to divide by a number with error

How do I divide a constant number by a number with error? For example... $$\frac{1}{(101 \pm 0.0058)} = 0.0099 \space\pm\space ???$$ Please help!
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24 views

Distribution of measurements given the underlying physical process distribution and the measurement uncertainty distribution

How to express the distribution of measurements ($X_m \sim \mathcal{D}_\mathrm{unknown}$) given that the underlying physical process being measured follows a Gaussian distribution ($X_p \sim \mathcal{...
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44 views

How do I avoid significant rounding error in evaluating $(\ln(x) - \sin(\pi x))(1-x)^{-1}$?

How do I avoid significant rounding error in evaluating $$\frac{\ln(x) - \sin(\pi x) }{1-x}$$ This function causes error as $x\to 1$. How can this be avoided? I tried using taylor's expansion but I ...
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42 views

Error of coefficient term in a chi^2 linear regression without an intercept

What is the standard error of the coefficient in a linear regression model performed by a $\chi^2$ regression, without an intercept present? I've determined $b$ and $\sigma_b$ as follows: We want to ...
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how error from taylor approximation is propagated to values that must be fit?

Suppose we have a function f(x) that we can experimentally calculate. Suppose that theoretically, we can describe f(x) with an analytical function g(x), which will depend on a couple of parameters. ...
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22 views

How to estimate numerical bin for Integro-differential equation system?

I try to solve an integro-differential equation system numerically (with LSODA.) The system is following: ( $'$ is the derivation w.r.t. $x$) $y'(x,s_{1}) = f(x,y(x,s_{1}),I[y(x,s)])\\ y'(x,s_{2}) = ...
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How is it possible that the relative error between two points can shift when taking the logarithms?

As a concrete example, suppose I want to calculate something that depends on two variables: $f(x,y)$. I also have an approximation to this function, $g(x,y)$. I want to know how good my approximation ...
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28 views

Sampling error propagation

Imagine you have a population of size N, and a sample of size n. The population comes in three kinds ('red', 'black' and 'blue') and we want to know the proportions of each kind. For 'red' we would ...
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How to find derivative with least uncertainty from data?

I'm a bit lost because I'm unfamiliar with numerical analysis. I have a set of data that relates two variables. I wish to find the derivative of the function that relates the two variables at one ...
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38 views

Error in triangulation calculation

I have a sensor that measures angles. If I have 2 sensors I can calculate a position. I have written the position and angle measured at each sensor in the form $y=mx + c$ and solved the equations to ...
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1answer
74 views

Is $\approx$ an equivalence relation? If $\approx$ is transitive, then does the error inherent in the approximation accumulate?

I was doing some physics calculations that involved approximations such as the small-angle approximation. I then started to wonder about how the relation $\approx$ can be used in comparison to the ...
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Can errors be added in quadrature if they are not random but independent?

For example, if we wish to know the error in density if we're measuring the temperature and pressure of a gas: $$\rho = \frac{P\,M}{R\,T}$$ If the error in the temperature and pressure sensors were ...
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1answer
102 views

Finding the maximum absolute error

Given, $c=15300 \pm 100$. Then what is the maximum absolute error in $c^3$? My attempt: Let $u = c^3$, then maximum absolute error in $u$ is $\Delta u = \frac{du}{dc}\times \Delta c = 3c^2\times \...
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68 views

Calculating relative error in subtracting two approximate numbers

My numerical analysis textbook says: Let $u=x_1+x_2$ Then, absolute error in $u$ is $\Delta u= \Delta x_1+ \Delta x_2$ Relative error in $u$ is $\frac{\Delta u}{u}=\frac{\Delta x_1+ \Delta x_2}{x_1+...
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Calculating error bounds when $\max f^{(2)}(x)$ is undefined

The error bound for Trapezoidal rule is: ${\left|E_{T}\right| \leq k \frac{(b-a)^{3}}{12 n^{2}}} $ I am trying to calculate the error for $x^x$ in the interval $[0,1]$ and let $n = 500$. The problem ...
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Derivation of error bounds for methods of approximating definite integrals

I am reading about error bounds with respect to methods of approximating definite integrals such as the trapezoidal rule, simpsons rule, etc. I am trying to derive the error bound for any of the ...
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31 views

Does uncertainty in a linear fit decrease with number of samples?

I am trying to prove that for a linear fit relating variables $x$ and $y$, if we have more samples of data pairs ($x_i$, $y_i$) between the range we wish to do a fit, the uncertainty of the slope $B$ ...
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46 views

Finding The Absolute And Relative Error Of A System Of Linear Equations

let there be $$ \begin{cases} 3x+ay=10\\ 5x+by=20\\ \end{cases} $$ Where $a=2.1$ and $b=3.3$ and there are rounded to $3$ digits. Find the absolute and relative error of $x,y,x+y$ and that there is a ...
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23 views

Big O Subtraction: $g(x) = f(x) + O(x) \rightarrow f(x) = g(x) + O(x)$

Is this true, that: $$ g(x) = f(x) + O(x) \rightarrow f(x) = g(x) + O(x)$$ In other words, is the order relation symmetric? How can I prove this? I think this is true, since: $$ g(x) = f(x) + O(x) ...
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Error Analysis for a Fraction

Suppose I have the following functions $f(x)$ and $g(x)$, such that: $$ f(x) = \tilde{f}(x) + O(x^{-p}) $$ $$ g(x) = \tilde{g}(x) + O(x^{q}) $$ where $p,q \in \mathbb{Z}_{++}$ and f,g are positive ...
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28 views

Calculating propagation of uncertainty

I need to compute the propagation of uncertainty $u(N')$. It is known that $N' = \frac{N}{t}$ and: $u(N) = \sqrt{N}$ $u(t) = 0.1$ My attempt at the task: $$u(N') = \sqrt{\Bigg[ \frac{\partial N'}{\...
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Finding error of a function

How do I effectively calculate how much polynomial functions deviate from the actual polynomial of interest when the value of $x$ is changed by a small amount that is : $$|f(x+\delta x)-f(x)|$$ Is ...
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Error Propagation confusion

let $x$ be a set of measured values with uncertainty $\sigma_x$ and $y$ be a set of measured values with uncertainty $\sigma_y$ by the formula $\sigma_f^2=\sum_{i=1}^{N}(\dfrac{\partial f}{\partial ...
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25 views

Minimize compounded error in forward difference formula

I am trying to write some code to do a Taylor approximation for a given function, and, as actually using differentiation strategies is very complicated for a computer program, I have used the forward ...
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41 views

Error in Fourier transform integration

This is a very applied question, but I hope it belongs here. When you do numerical integration, you introduce some error, for example in the trapezium rule you cannot have infinite trapeziums, and so ...
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62 views

How do I calculate the uncertainty of this percentage difference?

So, I am working with a data set that is 'supposed to' follow a Poisson Distribution, so that $\langle N \rangle = \sigma^2$, where $\langle N \rangle$ is the mean of my measurements $\{n_i\}$ of the ...
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1answer
26 views

Propagation of Errors - Upper Bound

Let $p, q$ and $r$ be $3$ quantities in a calculation. Assuming that values are rounded 3DP. Calculate the upper bound on the absolute error when calculation $$p+q+r$$ I don’t even know where to ...
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27 views

Propagation of Errors with Time Derivatives

I've been searching the internet for a few weeks and picking the brains of colleagues in person without success. As to what happens to error propagating down a derivative when the error in a ...
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16 views

Calculate error of fit parameter for nonlinear model fit

I have a given data set with error bars and the results from my simulations to reconstruct this data (one free parameter). I already found the parameter with the least square but now I want to ...

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