For questions on propagation of errors.

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2
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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
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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 \...
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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
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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
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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
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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 ...
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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 ...
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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 ...
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25 views

How should I compensate for numerical inaccuracy in fixed-point multiplication?

I am experimenting with fixed point arithmetic. I found a library for doing fixed-point math using vectors of 32-bit integers . At the end of the mulfpu (unsigned fixed point multiplication) ...
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13 views

Do theorems involving chaotic mappings hold in a finite precision context?

Chaotic mappings are known as highly sensitive to their initial state. It is well known that the first type of Chebyshev polynomials is chaotic, e.g. this. This mapping is defined recursively as $T_n(...
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66 views

Calculating percentage error for sums when there aren't absolute values?

So, let's say that I measured the lengths of two objects which measured 20 mm and 30 mm. I used the same ruler for both measurements and it has an absolute error of ±1 mm. If I wanted to calculate ...
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20 views

To what set do measured values belong?

This question is more conceptual than practical. It seems that when we apply mathematics to measured values, we treat them like real numbers. When measured values take error into account using ± ...
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33 views

Calculating absolute error. Teacher distributes the abs value signs.

Ill illustrate my confusion with an example: It can be shown, assuming $E_xE_y=0$ that the error in an arithmetical multiplication will be: $E_{xy}=xE_y+yE_x+\mu$ Where $\mu$ is the so called '...
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40 views

Uncertainty in distance from uncertainty in coordinates

I know this is basic, but I've managed to get myself confused. So, I have an object at location $(x,y,z)$ with uncertainty in the location of $(\delta x, \delta y, \delta z)$. What is the ...
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48 views

Is the Taylor Expansion a good approximation

Say I use a computer to sum the first 26 terms of $e^{-5}$ (degree 25), will this taylor expansion provide a good approximation? It summed to $.0067$ To me this seems like a good approximation, but I ...
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22 views

Link between total diffential and error caculation?

I earlier asked this question Error propagation, why use variences?. And am now slightly confused about the link between error propagation and total diffentials. As mentioned in the linked question, ...
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44 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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108 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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64 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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59 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 ...
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86 views

When examining global error bounds for Euler method, can I rescale the domain limits?

I'm looking at provable global error bounds of the Euler method for the first time and I was surprised to find that the bound grows exponentially in the amount of time (the domain size) propagated i.e....
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60 views

How to estimate error pattern of a set of line segments with respect to given reference segments (2D case)

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 have been the reference data and then I extracted other line segments via ...
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24 views

Significance and Poisson processes

Must I include the Poisson error on a new observation when considering whether it is consistent with a distribution? For example, the number of rain drops that fall into a cup per minute are recorded....
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8 views

How to make a covariance matrix from multiple observations of different objects?

I have $N$ objects. From each object, I sample $M$ values $(x,y)$ like so: ...
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10 views

Input Error estimation

I was wondering what are the methods used to detect the input's error when having the output's error of a model. I thoroughly searched on google, but I failed to find a well explined method, or ...
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18 views

Taylor series approximation of inverse trigonometric function

Suppose we have a function of three variables $a,b,c$ defined as, $f(x,y,z)=\arctan\left(\frac{\sqrt{x^2y^2-z^2}}{y^2-z}\right)$. Suppose $x=a, y=b, z=c$ satisfy the following property: (1) $a,b,c>...
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57 views

What does this notation mean? $∂df^2 / ∂x$

I am given variance $\sigma_x$ and function $y=f(x)$ According to my book, the following equation gives the new variance of $y=f(x)$. But I'm not sure what this notation means, as in what this ...
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26 views

Nonlinear equations algorithm - Newton method

Some time ago I posted a question regarding the simple case of finding the intersection point when I have only two functions, and with your help I found an answer. It was this case: $f(x) = a + \...
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16 views

Error propagation in complex formula

I'm currently trying to use error propagation formulae to calculate an estimate for the error in the following molecular dynamics formula: $C_v^* = \frac{3}{2}\bigg[1-\frac{2}{3NT^{*2}}\big\langle (\...
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13 views

Main differences between Monte carlo and Law of propagation of variance

I need to know the main differences/limitations between/of Monte-Carlo simulation technique and law of propagation of variance. Can someone briefly describe it or is there any good reference or link ...
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22 views

Propagation of Errors, Variance of a $N$ dimensional vector

I am working on a calculation and I'm trying to see if the last step is true. I tried to simplify all the details of the problem. Let $\mathbf{W}$ be a $N\times N$ matrix, $a=\{a_{1,}a_{2},a_{3}\}$ ...
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14 views

Two questions about the propagation of uncertainty: e^x, and a complex iterative equation

I'm trying to do some calculations involving the propagation of uncertainties in complex equations as part of my PhD research, and I'm running into brick walls. One advisor isn't around and the other ...
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18 views

Error propagation: add errors in quadrature, or use a weighted standard deviation?

I have a measurement $x$ with a known uncertainty $\sigma_m$. I have a black box that can take an error-free measurement $x$ and produce a value $y$ with a known uncertainty $\sigma_{b}$ (which is ...
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39 views

Find the error in the Maclaurin series for $\ln\left(\frac{1+x}{1-x}\right)$.

I have already that the series is, $$\ln\left(\frac{1+x}{1-x}\right)\approx 2\left(x+\frac{x^3}{3}+\frac{x^5}{5}+...+\frac{x^{2n+1}}{2n+1}\right).$$ The remainder is equal to, $$f^{n+1}(c)\frac{(x)^{...
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15 views

Inverse Propagation of Uncertainty

Can any one help me find a reference under the title of " Inverse Propagation of Uncertainty". I am starting a research on this topic after I had studied the forward propagation of uncertainty. ...
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24 views

Composite Spherical Harmonics expansion and error propagation

Let's assume that $f$, $g$ and $t$ are three functions defined over the surface of a sphere. In particular $f$ is defined using $g$, $t$ and the integral operator as follows: $$f(\omega)=\int_{\...
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17 views

What are disadvantages of solving difference equations backwards?

Say you have the initial value of 100th and 99th term and you want to see how much error is in 1st term instead of picking values from the table for 1st and 2nd term and propagating to 99th and 100th ...
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21 views

error calculation when the error is not constant

I have to calculate the error on the following quantity: $$f(\epsilon^M,\epsilon^S)= \sqrt{ \frac{1}{N}\sum_{i=1}^N (\log{\epsilon_i^M} - \log{\epsilon_i^S})^2 } $$ Usually I would use this standard ...
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31 views

Error propagation when function contains (or is) a derivative

I am familiar with the usual error propagation formula, but am unsure how to proceed when the function itself contains a derivative. I will present the simplest case here: My function f(x) is simply ...
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19 views

Do Monte Carlo perturbations capture all the uncertainty in prediction?

I have a model $M$ that I use to predict a value $y = M(\vec x)$. I have known one-$\sigma$ error bars on each input $x_i \in \vec x$. I want to know the one-$\sigma$ error bar on my prediction $y$. ...
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14 views

Statistical error on the inverse of a variable

Doing a report for my studies I had to calculate error of $\frac{1}{T}$, where T is the temperature I measured with a systematic error of 0.1. I came up with a formula $\frac{1}{T-0,1} - \frac{1}{T}$ (...
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41 views

How to propagate uncertainty into the prediction of a neural network?

I have inputs $x_1\ldots x_n$ that have known $1\sigma$ uncertainties $\epsilon_1 \ldots \epsilon_n$. I am using them to predict outputs $y_1 \ldots y_m$ on a trained neural network. How can I obtain ...
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32 views

Calculate Standard Deviation in Multiplication

I have a question that asks to find the result (in sig figs) as well as the standard deviation for the following $$(23\pm 8)\times(99\pm 11)\times(11\pm 3)\times(8\pm 4)$$ I began this question by ...
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24 views

Finding the weighted mean with varying errors

Simply enough, I've a set of numbers with varying errors: 2.02 +- 0.10 , 2.26 +-0.30 , 3.24 +- 0.30 , 3.33 +- 0.30 , 3.92 +- 0.30 , 4.02 +- 0.10 I'm certain there's a simple formula to compute the ...
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71 views

Uncertainty of Multiplication using Absolute Uncertainties

I am about to calculate the uncertainty of a measurement device. The formula of the result can be written as $$P=\frac{V}{R \times G}$$ All three values are uncertain; for $R$ and $G$, both the ...
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14 views

Error propagation with repeated values in the numerator and detonator

How would you do error propagation on an equation like this: $C=\left(\frac{A-B}{A B}\right)$ where A has error σa and B has error σb I would guess you use standard text book relations but since A ...
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28 views

Error propagation for multiparameter-non-linear fit with covariance matrix.

I have created a multiparameter-fitted-function for some data: $𝒇(𝒙)=𝑨𝒙^{(𝑪+𝟏)}𝒆^{−(\dfrac{(𝑪+𝟏)𝒙}{𝑩})}+𝑬𝒙^{(𝑮+𝟏)} 𝒆^{−(\dfrac{(𝑮+𝟏)𝒙}{𝑭})}$ I know the values and absolute ...
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26 views

What is the source of the error in the Sherman-Morrison formula application?

The Sherman-Morrison formula results in small errors in relation to the standard matrix inverse operation after each application, as shown here. From what I understand, the Sherman-Morrison identity ...
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58 views

Geometric Mean of Random Variables

I measure a series of $n$ objects [O_1, O_2, O_3, ..., O_n]. Because those measurements are quite hard to perform, I have quite a lot of measurement error and ...
0
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69 views

Propagation of standard deviation for random variable with Markov Property

I have a discrete random variable, $X \in \{0,1,2,3\}$. Define the indicator function: $$ 1_{k}\left(x\right) = \begin{cases} 1, & \text{if $x=k$} \\ 0, & \text{otherwise} \\ \end{cases}$$ ...