For questions on propagation of errors.

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12 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 ...
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1answer
6 views

Error Propagation of parameter in an irreversible equation

I have the following equation that predicts a certain parameter $c_t$: $$c_t = f(x,y_1,y_2,y_3...)$$ The function $f$ is known, continuous, real, differentiable to all arguments and the solution to a ...
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0answers
14 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 ...
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2answers
19 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 ...
2
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1answer
17 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? ...
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3answers
21 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}) = ...
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1answer
56 views

Stopwatch Accuracy and Precision

Let's say I have a stopwatch that can take split times. It's resolution is 1/100th of a second. Let's also say I have a robot that can press the split button exactly 0.244s apart in real time. I have ...
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0answers
24 views

Accounting for drop-outs in clinical trials

'Physical Therapy Review' [Intention to treat analysis, compliance, drop-outs and how to deal with missing data in clinical research: a review Susan Armijo-Olivo, Sharon Warren and David Magee Faculty ...
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0answers
49 views

Expand $\int_{-1}^0 e^{a\cos{\theta}}J_0(b\sin{\theta})\,d\cos{\theta}$ in spherical harmonics.

I want to solve the integral (a probability density function) $$ g(\gamma)=\int_{-1}^0 e^{-f\cos{\theta}\cos{\gamma}}J_0(-if\sin{\theta}\sin{\gamma})\,d\cos{\theta} $$ numerically, everything is ...
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0answers
9 views

Error propagation of non-square Matrix

I measure values of the vector $\vec{P}$ where each value $P_i$ has its own error $E_i$. I'm interested in the vector $\vec{K}$. Both values are linked by $\vec{K} = M \vec{P}$ . Since $\vec{P}$ and ...
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0answers
46 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 ...
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1answer
36 views

Relative error % of cos and sin functions?

I've been searching all over the net and I can't seem to find a definitive answer - perhaps I'm asking the wrong question. How does one calculate the relative error (%) of the cos/sin/tan of an angle ...
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0answers
7 views

Taylor approximation for roots of the quadratic equation

I am working on section 2.2, question 8 of Numerical Analaysis (Ward/Cheney) which asks me to find methods to avoid a loss of significance in the quadratic formula when 4ac is close to b^2. I have a ...
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1answer
41 views

Sum of positive integers estimating sum of fractions

Given $m$ fractions adding up to an positive integer $n$ For example: $m=3\\n=10=\frac{30}{6}+\frac{20}{6}+\frac{10}{6}$ How can we find $m$ positive integers that sum to $n$ (a partition of $n$), ...
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1answer
23 views

Statistics on discrete probability with error?

I need help with this question, am pretty confused on what to do with the error, I can't see how to use it.
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3answers
62 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 ...
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0answers
17 views

Looking for a motivating example for backward error analysis

For many people, it seems self-evident that backward error is a powerful tool in numerical analysis. But for me, it is hard to imagine a situation in which backward error analysis provides any useful ...
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0answers
32 views

Error propagation in pearson correlation

I have two data-sets $X$ and $Y$ with errors $\Delta X$ and $\Delta Y$. I calculated the Pearson Sample Correlation $r$. Is it possible to calculate the error of $r$ using propagation of uncertainty: ...
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2answers
36 views

Order of error of a fraction

If two functions can be written as the sum of some expression and an error term of higher orders of error $\epsilon$: $$f(x+\epsilon)=f_0(x,\epsilon)+O(\epsilon^m)\quad \text{ and} \quad ...
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1answer
190 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 ...
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0answers
17 views

How to express the quantity using significant figures to imply the stated error?

Express the quantity using significant figures to imply the stated error. $$1.77 \pm 0.06$$ I tried to factor out a $6$ and got $6(.295\pm0.01)$ but then when I try to make $.295$ have an error of ...
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2answers
104 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 ...
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1answer
172 views

What is the error in calculated volume of cylinder, given the measurements of length and radius?

Measured length of the rod is $15 \pm 0.4$ cm and the radius of the rod is $6.1 \pm 0.2$ cm. What is the error in calculated volume? (two decimal places) What i've tried is $$V=\pi r^2 l$$ ...
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0answers
14 views

How to find error constant, global and local error of ODE integration method

In literature regarding open channel flows I bumped into strange ODE integration methods: the first one: $$y_{i+1}=y_i+\Delta x\cdot\sqrt{f_i\cdot f_{i+1}}$$ the second one $$y_{i+1}=y_i+\Delta ...
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0answers
27 views

Propagation of variance/covariance for $L_1$ Estimator with no analytic solution

This is my first question on Math Exchange so my apologies if it does not initially fit the format. I have a problem where I'd like to calculate the a-posteriori variance/covariance matrix of ...
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0answers
22 views

what are some good ways to define the errors between two functions?

I have two functions. One is the original function (that contains 4 variables), and the second one is the approximation to the first one (also contains 4 variables). The question is, if I want to ...
0
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1answer
19 views

What is the error on a counting variable?

I don't know if a variable that acts as a counter has an associated error. I don't see it as a measured quantity so I think that it has no error. The problem is this: you have an image of a star and ...
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0answers
6 views

To find error in time histogram

I have a data which is recorded from a detector. Whenever the detector produce signal it records the time. I have recorded the data for several cycles, one cycle is 0 to 1 second. Finally I made the ...
0
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0answers
22 views

Monte carlo error: Combining “experimental” and statistical errors

I'm doing a slightly involved Monte Carlo approximation of a quantity $E$ where I end up with the following formula: $E=\frac{\sum_{i=1}^np_ie_iG_i}{\sum_{i=1}^np_iG_i}\ .\ \ \ \ \ \ \ \ \ $ (1) ...
2
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1answer
101 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
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0answers
33 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 ...
0
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1answer
24 views

Error propagation for ratio data

I'm working with some published ratio data for isotopes of uranium (238U) and lead (208Pb and 206Pb). The data are published in ratios of 238U/208Pb and 206Pb/208Pb, but I need the data to be in the ...
2
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0answers
28 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 ...
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0answers
11 views

looking for asymmetric probability distribution

In physics, quantities are sometimes measured with both an upper and lower error. For example, I might say that an object's mass is $m = 10^{+0.1}_{-0.2} \text{ kg}$. In my case, these arise from ...
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40 views

Standard deviation on absolute square of complex average

I am calculating a quantity $q=|c|^2$ where I obtain $c\in\boldsymbol C$ as an average of a collection of estimates with errors: $\langle c\rangle=\sum_{j=1}^nc_j$, and the question is what error to ...
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1answer
46 views

Mean of data with uncertainties and confidence interval

My question sounds rather simple, but sadly I have not been able to find an answer online (which is rather strange as this seems very basic, so I apologize if I simply haven't looked well enough) The ...
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1answer
36 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 ...
0
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1answer
49 views

Propagation of Error question?

If I have a function $$f(r,V,B)=\frac {2V}{r^2B^2} = 2Vr^{-2}B^{-2}$$ what is the propagation of error? If I use the power rules and multiplication rules described here ...
0
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1answer
52 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 ...
0
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1answer
22 views

Error of function with errors on arguments and size of arguments

EDIT: my question is not how to derive the formula below (I think the derivation is more or less what I guessed, like the answer below supports), but whether it can be made valid for the case where ...
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2answers
56 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 ...
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1answer
46 views

bound on matrix inverse with different elements

I'm hoping that someone can point me to some literature on the following. Is there a way to bound the inverse of a matrix if I change the value of 1 element in that matrix. Let's say I have a matrix ...
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1answer
119 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 ...
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0answers
64 views

Error Propagation - functions of the mean.

Given a number of measurements $\{x_i\}$ with values distributed according to a (known) probability distribution $\rho(x)$ with a theoretical mean $\langle x\rangle = \int dx x\rho(x) = f(y)$ and a ...
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2answers
75 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} + ...
2
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1answer
48 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? ...
1
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1answer
75 views

How is statistical uncertainty calculated for the modulus function?

I know it's an unusual function to calculate an uncertainty for, but I haven't been able to figure out a reasonable means for calculating derivatives for it to do so myself. I know modular arithmetic ...
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0answers
48 views

Finite difference method - implementation and stability

I have a PDE for a function $P(r,t)$ in spherical co-ordinates of the form $P_{t} = D(P_{rr} + (2/r)P_{r}) - a\Omega\left(\frac{P}{P + k} \right) $ where subscripts denote partial derivative with ...
1
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1answer
58 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 ...
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1answer
85 views

How to calculate error size from division of two ratios

How one would calculate an error from division of two rations? I am given 1/y and x/y as their decimal representation (numbers ...