Questions tagged [error-propagation]

For questions on propagation of errors.

Filter by
Sorted by
Tagged with
0
votes
0answers
6 views

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 ...
0
votes
0answers
11 views

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: $$\...
0
votes
0answers
11 views

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 ...
0
votes
0answers
23 views

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 $...
0
votes
1answer
41 views

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 ...
0
votes
0answers
15 views

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 $\sigma_i$...
1
vote
0answers
38 views

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 ...
-1
votes
1answer
47 views

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. ...
1
vote
0answers
34 views

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 ...
0
votes
0answers
11 views

How to pick the axes and the method of linear regression?

I have to make a linear regression in two set of data $x_1,...,x_n$ and $y_1,...,y_n$, with standard deviations $\sigma_{x_1} = ... = \sigma_{x_n}$ and $\sigma_{y_1} \neq ... \neq \sigma_{y_n}$, and I ...
1
vote
0answers
9 views

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 ...
0
votes
1answer
25 views

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 ...
2
votes
0answers
32 views

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 ...
0
votes
0answers
17 views

Propagation of large uncertainties

Unfortunately the error propagation formula breaks down for large errors e.g. $\sigma \approx \mu$ for a gaussian distribution, since it is derived under the assumption of small errors using a Taylor ...
0
votes
0answers
19 views

Question about error analysis for a normally distributed data

Let us suppose we have one constant variable $b \pm \delta b = 20 \pm 1$ and one function that depends on $x$, such as, $a(x) \pm \delta a$ The problem is I want the difference between $a(x)$ and $b$ ...
1
vote
1answer
24 views

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 ...
0
votes
2answers
49 views

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 ...
0
votes
1answer
43 views

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 ...
3
votes
3answers
145 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{...
-1
votes
1answer
22 views

Which of the following lines offers the better fit to the given points in the least-squares?

I have two similar exercises: Which of the following lines, $y = 1 - x$ or $y = 4 - x$, offers the better fit to the points $(1,2),(2,1), (3,1)$ in the least-squares? Justify? My thought process (...
1
vote
0answers
19 views

error on evaluation of line fit at a given x given error on its parameters

Say I have some data points randomly distributed about a linear model. I fit a straight line through it, and obtain best fit parameters $m$ and $b$ with an error $\delta_m$ and $\delta_b$ associated ...
0
votes
0answers
17 views

The trapezoid rule has been used to approximate $\int_{2.3}^{3.1}f(x)dx$, find the maximum absolute error

A previous exercise was Determine $\int_{2.3}^{3.1}f(x)dx$ using the trapezoid rule. I got $0.452557$. Knowing that $E(f) =−h^2(b−a)f′′(η)/12$ with 2.3≤η≤3.1 gives the error of integration of the ...
0
votes
1answer
41 views

How do you propagate uncertainty for the equation $\sqrt{\left( \frac{2 \sin{\theta}}{\sqrt{3}}+\frac{1}{2} \right)^{2}+\frac{3}{4}}$

I am calculating $n(\theta) = \sqrt{\left( \frac{2 \sin{\theta}}{\sqrt{3}}+\frac{1}{2} \right)^{2}+\frac{3}{4}}$. But I have a problem as $n = 59.96 \pm 0.01$. I have tried using a first order Taylor ...
0
votes
1answer
32 views

Show that the error of quadratic interpolation $\xi$ is given by $\xi\le|E|+\frac{5}{4}\epsilon$

Show that the error of quadratic interpolation $\xi$ in equidistant points, the values of which are obtained by rounding and the error is lesser or equal to $\epsilon$, is given by: $$\xi\le|E|+\frac{...
1
vote
1answer
25 views

Find the maximum error of the following polynomial interpolation, but the values are too small

$$f(x) = \log_{10}(x)$$ Error formula: $$e < max[1.35;1.45] |(x-1.35)(x-1.37)(x-1.40)(x-1.45)| max[1.35;1.45] |\frac{-6/\ln(10) * 1/x^4}{4!}| = \\ max[1.35;1.45] |(x-1.35)(x-1.37)(x-1.40)(x-1.45)|*...
0
votes
3answers
34 views

A projectile has been shot into the atmosphere. Find the polynomial that interpolates the speed and acceleration in function of time.

This is the table (tempo = time, velocidade = speed, aceleracao = acceleration) I used Newton's interpolation to get the polynomial (5th degree). I got $$102 + 138(x-15) + (-124/15)(x-15)^2 + (289/...
1
vote
0answers
42 views

Propagation error for central difference approximation

So I know that for a general function $g$ that depends on the variables $a_1,a_2...$, that each have relative uncertainties $\epsilon_1,\epsilon_2...$, the propagation error is given by $$\xi_p=\sum_{...
1
vote
1answer
43 views

Why I am getting this slight deviation

I have solved this equation by two method √3a = 50+a Method 1 √3a-a =50 a(√3-1) = 50 a= 50/(√3-1) a= 50/0.73 = 68.4 Method 2 a(√3-1) = 50 a = 50/(√3-1) a =50(√3+1)/2. [by rationalizing] a =25 * ...
0
votes
0answers
20 views

How to calculate measurement uncertainty for a derived function in respect to data set?

I have a self made pressure sensor for which i am trying to calculate the measurement uncertainty. Im trying to measure Force (in Newtons [N]) and have a +/- ...N result. The Sensor changes its ...
0
votes
1answer
43 views

tail bound of the deviation from sum of functions of random variables to its expectation value

I am struggling at an error propagation recently and I do not know what tools can be used in this problem. Explicitly my problem can be represented as following: I have expression $$\varepsilon=\frac{...
0
votes
0answers
23 views

propagation of error via delta method (without knowing the expected value of random variable)

Suppose there is a Gaussian random variable $x$. An unbiased measurement is made to obtain the estimate $\hat{x}$, which we assume has expected value $x$ and measurement-error standard deviation $s$. ...
0
votes
1answer
24 views

Error Propagation And Multiplication by Zero

So I'm trying to create a python module to be used for Error Propagation just for sake of learning. let: $$A=a \pm \delta a$$ $$B=b \pm \delta b$$ As explained here Error Propagation of a function as ...
2
votes
1answer
59 views

What is the proper way to combine relative error when multiplying/dividing?

I need to multiply two independently-gathered single-variable data points with percent errors. Looking online, I have found two contradictory ways to combine the percent errors. One way is to simply ...
0
votes
0answers
20 views

Find the error of $w_a = \frac{0.2312-123.1}{1.52}$ where all the values were obtained by rounding

$$w_a = \frac{0.2312-123.1}{1.52}$$ My professor told me to use Taylor series/the formula for the propagation of the error: $f(x)-f(x_a) = f'(x_a)*(x-x_a)$ but I am not sure know how to apply it here. ...
0
votes
0answers
29 views

Applied mathematics: Is there a way to characterize transient warmup time to reduce settling time errors?

I don't know if this is the right place to post this but I have an electronic sensor that measures conductance using Conductance = I/V. The results are very good especially with a Newton Raphson ...
0
votes
0answers
29 views

Very Hard : Error Bound on asymptotical approximation [duplicate]

Very Hard: Suppose I have a polynomial $g(x)=x^p $+ lower order terms. Asymptotically the inverse is $g^{-1}(x) \sim x^{1/p}$ for large $|x|.$ What is a bound of the error of this asymptotical ...
1
vote
1answer
40 views

A paradox in the formula for propagation of uncertainty?

My question is the following: what is the uncertainty propagated to area of a square? The usual answer is as follows. Let $(l\pm\sigma_l)$ denote an experimental measure of the sides of the square, ...
0
votes
0answers
12 views

Associative Error of Composite Trapezoidal Rule over 2 Domains

In general, I have found literature suggesting the error is calculated via ${\epsilon} = -\frac{(b-a)^3}{12 n^2} f''(\xi)$ ; $\xi \in(a,b)$ however I have seen varying interpretations of $f''(\xi)$ ...
1
vote
1answer
28 views

Meaning of the cost of constructing a matrix

In a text I saw written, $``$The dominant costs in this method are the construction of the Jacobian matrix $\textbf{f}_\textbf{x}(\textbf{x}_n)$ (typically $\mathcal{O}(m^2)$) and solution of the ...
0
votes
0answers
20 views

Whats the intermediate step of the partial derivatives in this Gaussian error propagation?

I tried it, but I am not sure how to derive frac{\del m}{\del e} partially and get the desired result of \frac{1}{\frac{e}{m}}. What is the intermediate step leading to this derivative? Note that <...
0
votes
0answers
7 views

How do I quantify maximum error in approximations of partial differential systems?

Suppose I have a complicated system of partial differential equations, but I have data suggesting I only need to look at a limited range of values. After linearizing the system with a Jacobian and ...
0
votes
0answers
13 views

Error of sum of taylor expansions

I am interested in the sum over composite functions, and would like to approximate the sum by the sum of the Taylor expansions (to first order to make it simple). Can I make a statement about the ...
0
votes
0answers
14 views

Combining Error Terms to form a General Error Term

Lets say I have 4 error terms: $$e_1,e_2,e_3,e_4$$ Each of these error terms come from different simulations of data using different classification methods. Let $\gamma$ be the number of empirical ...
0
votes
0answers
13 views

Object detection uncertainty due to camera limitations

I have a 2-D image of an object, and am using this to detect an object and estimate the object's position relative to the camera's pose (say, as a vector between the origin of the camera's coordinate ...
0
votes
0answers
9 views

Determine the “share” of each single error in the Gaussian error propagation

I have a set of single errors that are combined using Gaussian error propagation. I'd like to show their "influence" on the total error in a pie chart. But let's start with a simple example ...
0
votes
0answers
52 views

Uncertainty of Integrals

Suppose I have 3 straight lines: $$y_1 = m_1 x + c_1;$$ $$y_2 = m_2 x + c_2;$$ $$y_3 = m_3 x + c_3;,$$ and $$m_1 = 1 \pm 1.1, c_1 = 1 \pm 1.11,$$ $$m_2 = 2 \pm 2.2, c_2 = 2 \pm 2.22,$$ $$m_3 = 3 \pm 3....
0
votes
0answers
41 views

epidemiology and double exponential sigmoid, error bars.

I'm creating a model of the coronavirus in various countries using a combination of a double exponential sigmoid with the SIRD compartmental model. I'm getting some consistent results, but I'm not ...
0
votes
0answers
39 views

Standard Error propagation for correlated variables

The following is a toy problem to illustrate my question. Let $[A_{1}, ... ,A_{n}]$ be a series of $n$ random variables. Each variable has $m$ observation under two experimental conditions resulting ...
0
votes
3answers
113 views

Forward and centered finite difference give same error plot: why?

Let us consider the following standard approximations of the first derivative of a function $$FD = \frac{f(x+h)-f(x)}{h}$$ $$CFD = \frac{f(x+h)-f(x-h)}{2h}$$ The first is first order accurate, while ...
0
votes
1answer
172 views

Roundoff errors and finite difference approximation

Let us consider the centered finite difference approximation of the first derivative of a smooth function $$f'(x_i) = \frac{f(x+h) - f(x-h)}{2h}$$ It's well known that if we do a $\text{loglog}$ plot ...

1
2 3 4 5
7