For questions on propagation of errors.

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5
votes
1answer
553 views

Multiplication and division of values with geometric standard deviation

What is the geometric standard deviation of a value, which is the result of dividing two independent values, each of which has its own geometric standard deviation ? It is a frequent situation in ...
4
votes
2answers
99 views

Avoiding loss of numerical accuracy

I need to to evaluate the function $f(x) = {1 - (1-A)^x \over A}$, where $0 < A \leq 1$ and $0 \leq x \leq 1$. A straightforward C implementation of $f(x)$ with floating-point arithmetic works ...
4
votes
2answers
3k views

How to estimate variances for Kalman filter from real sensor measurements without underestimating process noise.

As the title says, I want to estimate the variances needed for a Kalman filter from real sensor measurements only. For example we can take a temperature sensor, but the solution shall be as ...
4
votes
2answers
204 views

Regression with error coming from rounding

I am looking at the following model: $c$ is a fixed vector in $\mathbb{R}_+^n$ and for any $x \in \mathbb{R}_+^n$ we obtain a value $y =[c^Tx]$, i.e. rounding $c^Tx$ to the nearest integer. I want ...
3
votes
3answers
962 views

How to calculate the standard deviation of numbers with standard deviations?

I have essentially a propagation-of-error problem I run into frequently with my scientific data. For example, I have three samples, each of which I take two measurements of. So, for each sample, I can ...
3
votes
1answer
53 views

Monte Carlo with error on individual samples

I'm performing a Monte Carlo integration where the individual samples have an error, and I'm wondering how to estimate the final error. Some more detail: The integral E I'm after is estimated in the ...
2
votes
1answer
132 views

Absolute and Relative Error of $x^y$

Suppose that we have two measured values $x$ and $y$ with maximum absolute errors of $e_x$ and $e_y$. Is there a formula to find a good upper bound for absolute and relative error of $x^y$?
2
votes
1answer
67 views

Error propagation, why use variences?

I have been reading up on error propagation and am slightly confused about something. We can the error in $c=f(a,b)$ as the: $$\sigma(c)= f_a \sigma_a+f_b \sigma _b$$ Firstly is this correct and am I ...
2
votes
1answer
202 views

Margin of error for quotient of two measurements

I need to find the total margin of error for calculating velocity, while I have margins of error for time and distance. Actually the margins are the same (as both measurements were based on GPS - but ...
2
votes
1answer
88 views

Confused about the explicit formula for $\psi_0(x)$

In the explicit formula for $\psi_0(x)$ used in the PNT proof : $$\psi_0(x) = x - \sum_{\rho} \frac{x^{\rho}}{\rho} - \frac{\zeta'(0)}{\zeta(0)} - \frac{1}{2} \log (1-x^{-2}) $$ In particular the ...
2
votes
0answers
53 views

Looking analytically if one formula is better than another one

I'm studying errors in functions on numerical methods. On my notes, I've written the Heron's Formula: Let $a\geq b\geq c$:$$A=\sqrt{p(p-a)(p-b)(p-c)}\ \ ,$$ where $$p=\frac{a+b+c}{2}.$$ This formula ...
2
votes
0answers
66 views

Error in distance between points in spherical coordinates

I have two points with spherical coordinates: $a=(r_1,\theta_1,\phi_1)$ and $b=(r_2,\theta_2,\phi_2)$. The cartesian coordinates of the points are: $$ (r_i \cos\theta_i \cos\phi_i, r_i \cos\theta_i \...
2
votes
0answers
193 views

Accuracy of distance and bearing between GPS locations

I'm writing on an Android app that tracks the distance and bearing between two GPS location (each from a different device). Finding the mean distance and angle between the devices is quite easy, and ...
2
votes
1answer
48 views

Relative Error $\frac{x-x_0}{x}$

According to many definitions I've seen the relative error is defined by $$E = \frac{x-x_0}{x}$$ where $x$ is the "true" value. But some people use instead $$\frac{x-x_0}{x_0}. $$ Is this incorrect?
2
votes
0answers
54 views

Show that $\displaystyle\sum_{i=0}^{N-1}|\epsilon_i|\to0, N\to\infty$

Let $I_o=[t_0,t_0+T]\subset\mathbb R, T>0$, If $f\in C^0(I_0\times\mathbb R,\mathbb R)$ and satisfies the Lipschitz condition: $\forall t\in I_0, \forall y,y^{*}\in\mathbb R:|f(t,y)-f(t,y^{*})...
2
votes
0answers
42 views

Yacov Perelman Nepero game

This is my first question, so sorry if I'll make any mistake in using the site formatting. I found this game on a book by Yacov Perelman and I thought it could be nice to introduce Nepero number to ...
2
votes
1answer
55 views

Help in Measuring Error on Estimates of Differential Equations

I am working on a project for class where I have to estimate the solutions to a damped harmonic oscillator ($x''+2 \gamma x'+ \omega^2 x=0$) and compare three methods for doing so (Third Order Runge-...
2
votes
1answer
83 views

Verlet method global error

I was trying to understand the global error calculation for the verlet method on Wikipedia but it's not so clear to me when it goes from: to Shouldn't be considered the error relative to x'' too?
2
votes
0answers
229 views

Problem Condition and Algorithm Stability

Consider 2 mathematical problems: $$ f_1(x) = a - x \\ f_2(x) = e^x -1 $$ The condition number for a function is defined as follows: $$ k(f) = \left| x \cdot \frac{f'}{f} \right| $$ Lets analyze ...
2
votes
1answer
55 views

Expectation/probability question (a bit like Craps, I think)

I'll try to be as brief as possible: I have a number of events that can happen, say e1, e2...eN. N isn't particularly large. Each event has a probability of "failure". (Actually, there will likely ...
1
vote
2answers
41 views

What does it mean to calculate a number to $n$ decimals of exactness.

I was asked to numerically calculate Bessel functions for certain points and report their values to "6 decimal places of exactness". I did this in matlab and there's no truncate function, so I was ...
1
vote
2answers
8k views

Error propagation on weighted mean

I understand that, if errors are random and independent, the addition (or difference) of two measured quantities, say $x$ and $y$, is equal to the quadratic sum of the two errors. In other words, the ...
1
vote
1answer
50 views

Propagation of Uncertainties [closed]

I have five values for the volume of sodium hydroxide needed to neutralise a fixed quantity of hydrochloric acid, each trial with an uncertainty of 0.05 mL. If I take the average of these five values, ...
1
vote
3answers
5k views

Calculate uncertainty of sine function result

I have an angle given in degrees: $$\theta_{\min} = 63^{\circ} \pm 0.5^{\circ}$$ I need to calculate it's sine and still know the uncertainty of the value: $$n = 2\sin(\theta_{\min}) = 1....
1
vote
2answers
33 views

Are the “weights” inside a neural network actually “terms” for a polynomial?

This just hit me today. I am not too experienced with math or neural networks, but I am trying to find out about them in my own way so I can some day understand them well. So I was thinking about how ...
1
vote
2answers
138 views

If $\theta = (59.3\pm 1.2)^{\circ} $, find $\tan(\theta)$ and it´s uncertainty.

How can I find the uncertainty? I need to use this formula: The general formula for uncertanties is: $\sigma_q = \sigma_q (a,b,c,...)$ and $\mathrm{Cov}(a,b,c,...)=0$ $\,\,$then $$\sigma_q ^2 = (\...
1
vote
2answers
52 views

Absolute error in machine-precision terms.

I am trying to wrap my head around errors in floating point calculations. Let me denote absolute error as follows: $e = |x - \hat{x}|$, where $x$ is the exact number and $\hat{x}$ is its floating ...
1
vote
1answer
153 views

Error bound of the Euler method

I am self studying working through the book "A First Course in the Numerical Analysis of Differential Equations" and have come to a deadend on q 1.2. The linear system $y' = Ay, y(0) = y_0$, where $A$...
1
vote
2answers
230 views

How does big-O notation relate to the actual error involved in a numerical differentiation?

Suppose I have some position data ${x_1, x_2, ... x_n}$ that was sampled at an interval $h$. If I wanted the velocity data, I could apply a finite difference scheme: $ v_1 = \frac{x_2 - x_1}{h} + O(h)...
1
vote
2answers
98 views

Propagation of Error

If I have a function $$f(x,y)=\sqrt{x^2+y^2}$$ with error in $x$ be $\Delta x$ and error in $y$ be $\Delta y$, then how do we calculate ${\Delta f}$? I know if we have $$f(x)=x^n$$, then I at least ...
1
vote
1answer
840 views

How to calculate uncertainties?

A question requires me to make a calculation involving variables with uncertainties, giving my answer with its result uncertainty. How do I do that?
1
vote
1answer
847 views

Error in a numerical derivative

I have a graph of data, say temperature ($T$) vs time($t$), I know the error bounds in each $\Delta T$. The range of t is from 0 $\to$ 1600 s, with small steps say 0.001 s. If I numerically take ...
1
vote
1answer
2k views

Error analysis - Bisection algorithm

I have a brief question related to an example in my textbook. In my book, the following theorem on Bisection Method is presented: If $[a_0,b_0], [a_1,b_1],. . .,[a_n,b_n]. . .$ denote the intervals ...
1
vote
1answer
27 views

Rounding error of trapezoidal method

I'm working with the Modified Euler method sometimes called Heun's method or explicit trapezoidal method. I have a book on ordinary differential equations numerical analysis that claims: The ...
1
vote
3answers
35 views

Uncertainties in angle measurement

I wonder why uncertainties in angle measurement MUST be in radians. For example, I want to calculate the uncertainty in measuring the function $y= \sin (\theta)$ when the angle is measured $\theta = ...
1
vote
1answer
22 views

propagation of error from product of Taylor Series

Say I have two functions $f(x)$ and $g(x)$, both of which I will be approximating with Taylor series $T_f(x)$ and $T_g(x)$ respectively. Lets say $f(x)$ is order $O(x^{n_1})$ and $T_f(x)$ has error of ...
1
vote
1answer
114 views

Accuracy of GPS bearing, given two locations and their accuracies

For an Android app I need to determine the accuracy of the bearing returned by the GPS. Android supplies the following data, among others: The GPS position The accuracy of the GPS position, ...
1
vote
1answer
137 views

Simpson's Rule for IVP. Truncation Error proof

Edit: replaced all c's with y's as the c just denotes replacing a series of coupled linear equations Ay with uncoupled equations $\Lambda c$ no biggie. Im working through the lecture notes for a ...
1
vote
2answers
51 views

Prove graphically that the Lambert equation has exactly zero, one or two roots

I need some help on the below problem. Consider the Lambert equation: $xe^x = a$ for real values of x and a (a) Show graphically that the equation has exactly one root $ \xi(a) \ge 0 $ if $ ...
1
vote
1answer
713 views

Error Estimation and Propagation through Trigonometric Functions

I have a problem whereby I have to estimate the absolute error of a value calculated using values read from sensors. The equation used to calculate the value is $T_{xy} = R \tan \theta$. The sensor ...
1
vote
2answers
164 views

Relation between condition number and perturbed matrix

Prove that if $A\vec{x} = \vec{b}$ and $(A+\delta{}A)(\vec{x}+\delta\vec{x}) = \vec{b}$, then $\dfrac{\|\delta\vec{x}\|/\|\vec{x}+\delta\vec{x}\|}{\|\delta{}A\|/\|A\|} \le \kappa{(A)}$, where $\kappa{(...
1
vote
1answer
564 views

Calculating Covariance. (Multiplication of Two Covariance Matrices)

I have an equation T3=T1*T2 where T is 3*3 Transformation matrix representing position of an object in 3D. Now each of these position has some error in the form of 3*3 covariance matrix i-e ∑. My ...
1
vote
1answer
69 views

Too small error on a physics lab

I have this function: $$\lambda=d \sin\left(\arctan\left(\frac{x}{z}\right)\right)$$ and I want to find its absolute error. d is a constant ($10^{-6}$), x is $(0.716 \pm 0.001) m $ , and z is $(1.000 \...
1
vote
1answer
66 views

How to estimate (if/any) displacement/rotations between 2d line segments taken from 2 data sets

I am having set of pair of line segments (2D). Though each pair should be coincided on top of each other they are not so. I derive these two line sets using image based (e.g. CD) and manual method (e....
1
vote
1answer
85 views

Uncertainty in measurements: if $x$ has uncertainty $\pm\epsilon$, what is the uncertainty in $\sin x$?

I have two questions regarding uncertainties in measurements. First, if I have some measured value for $x$ with an uncertainty $\pm e$, what would be the uncertainty in $\sin x$, $\pm\sin e$? ...
1
vote
1answer
101 views

Where does the error propagation formula comes from?

As an engineering student I have come several times across the formula $$\sigma_{f(\vec{x})}=\sqrt{\sum_{i} \big (\dfrac{\partial f}{\partial x_{i}}\sigma_{x_{i}}\big )^{2}}$$ for the propagation of ...
1
vote
1answer
216 views

Angle between two vectors with noisy data?

Given two vectors V1=(x1,y1,z1) and V2=(x2,y2,z2) I can calculate the angle between the two vectors using the dot product of the two vectors and their magnitudes. This approach, however, is only ...
1
vote
1answer
225 views

Derivation of formula for estimating error in bulk-volume

In my textbook, a formula for estimating error in bulk-volume measurements is derived, but I don't quite follow one step in the derivation. The book writes the following: The bulk volume of a porous-...
1
vote
0answers
34 views

Calculating a three sigma limit on data

So I have a list of data points each with their own error (eg. 3 +/- 0.2 units) and I want to determine which of the data points are within 3 sigma. To do this do I just calculate the standard ...
1
vote
0answers
21 views

Uncertainty propagation and division

I am getting confused about a super basic issue. I have two quantities: $6\pm4$ and $4\pm3$ . So let's say $x = 6$, $\Delta x = 4$, $y = 4$, $\Delta y = 3$ Now I want to calculate the uncertainty ...