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Questions tagged [error-propagation]

For questions on propagation of errors.

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Understanding calculation of gradient w.r.t. Weights in a many to one sequence (RNN) network

I am thinking of a many to one sequence such as a sentiment classifier, where a sequence of text tokens are passed and the RNN returns 1 or 0 depending on whether it thinks the text expresses a ...
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Why we sum up the derivatives of the loss w.r.t. Weights at each time step in RNN back-propagation?

I am reading a paper explaining the derivations of the back-propagation equations in RNNs. There I read 'Note that the Weight Matrix remains the same across all time sequence so we can differentiate ...
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Follow error propagation in a neural network

I am currently trying to study the impact of approximate training of an encrypted neural network. For this, my weights are all encrypted with a fully Homomorphic encryption scheme. For my purpose, ...
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Using series to approximate definite integrals.

Use series to approximate the value of $\int_0^1{cos(x^2)dx}$ so that the error in your approximation is less than $\frac{1}{100}$. My work: $$f(x)-T_n(x)=R_n$$ $$R_n=|\frac{f^n (c) x^n}{n!}|$$ $f^...
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Uncertainty in the output when 10 is raised to the power of a value with an associated uncertainty?

For the purpose of error propagation, I need to raise 10 to the power of a value with an uncertainty. How would this error be propagated to the new 'out' value? Thanks in advance.
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Handling negative variances on the derivative of Gaussian processes

The variance of the derivative of a Gaussian process, $f$, is given by (9.1): $$ Var(\frac{\partial f}{\partial x}) =\frac {\partial ^2 k(x,x)}{\partial x^2},$$ where $k(·, ·)$ is both a positive-...
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Error propagation in cone width for kinematic neutron imaging

I'm trying to figure out the error in the opening angle for a cone created with kinematic neutron imaging. The angle is defined as: $$\theta = \sin^{-1}\sqrt{\frac{Ep}{E}}$$ And I want to find the ...
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Loss of significance in $a-b$

When we have two given numbers $a$ and $b$ ,if $a$ is really close to $b$,when performing $ a - b$ , we lose many significant figures,and the relative error gets really big. But why care about the ...
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Trivial derivations about error analysis (combination of errors)

My book deals with the combination of errors for addition, multiplication and for exponents. I have understood the derivation of each but I am struggling to extend the same for division and negative ...
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Propagation of Error for Sum of Reciprocals

Given a set of $n$ complex numbers $a_i$ and error magnitude $\epsilon_i$ with $|a_i| > \epsilon_i$. We define the error disk $D(\epsilon_i) = \{ z \mid |z| \leq \epsilon_i\}$. We have the set $$ ...
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Understanding error propagation in fixed-point multiplication

I'm writing a test suite that checks the correctness of a fixed-point arithmetic library that I wrote. Specifically, it deals with Q4.4 numbers, i.e. 4 integer bits and 4 fractional, so its precision ...
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Estimate approximation error of a function

It is supposed to approximate a function $f$ on the interval $[a, b]$ by a function $p$ that fits piecewise a polynom of degree $n$. I've noted the following steps: Decompose $[a, b]$ into $N$ ...
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analytical propagation of error

I have this function: $\rm \mu(R) = \mu_{\rm eff} + \frac{2.5 b_{\eta}}{ln(10)} \left[ \left( \frac{R}{R_{\rm eff}}\right)^{1/ \eta} - 1 \right] $ where there are associated errors with $\rm \mu(R)$,...
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How does the error in the mean, of the mean, of the mean, etc. propagate?

This is my first question here in the Mathematics section, so please forgive my transgressions. I'm working with a large body of time series temperature measurements from weather stations. The data ...
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exponential fit error propagation

I am stuck with error propagation of exponential fit parameters: $y = a*e^{(b*x)}+c$ I have available errors of fit: $\sigma_a$, $\sigma_b$, and $\sigma_c$ I tried to split the problem to ...
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What is the error of a $\ln(x + R)$? [closed]

I am trying to calculate the error of a $\ln(x)$ function, given my parameter $x$ has an error $R$.
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Propagation of error vs. direct computation

Suppose I have a set $\{x_i\}_{i = \overline{1,N}}$ of data that characterizes some parameter $X$, and I want to compute the standard deviation of some parameter $Y = f(X)$. I can think of two ...
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Computation of marginal distribution for uncertainty quantification of dependent variables

In a few words, I have some dynamics with uncertainties in the initial conditions. I am using the Liouville equation and the method of characteristics to propagate in time the distribution of these ...
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Calculating the standard deviation of calculated values that have uncertainty

I am performing tensile testing and for each sample I test, the yield strength is determined by (yield stress) = (yield load) / (original area). The load and area have uncertainties associated with ...
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22 views

Absolute error and inverse trig functions

I am doing a physics project, and am using the sine law to find the angle of my final momentum vector. Using absolute error arithmetic (i.e. add absolute error when adding/subtracting, add relative ...
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56 views

Drawing random realisation from quantity with poisson error

How would I draw a random realisation of a variable with an upper and lower error determined from Poisson statistics using the Gehrels 1986 formula? See: http://adsabs.harvard.edu/abs/1986ApJ...303.....
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What is the formula of the linear regression with an error propagation

I am in Physics Licenciature and a day the teacher showed me a formula for the linear regression with error propagation, and time after, I was searching this formula and I didn't find it. Then I am ...
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Effect of angle error over distance (error propagation) [closed]

An angle of "$\theta$" has been measured with an error of "$s$" (in degrees). How the position error which is caused by this angle observation error at a distance of "$d$" (in meters) can be estimated?...
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Taylor Series Expansion on error propagation.

I am reading through Stoer and Bulirsch's Introduction to Numerical Analysis. In their section on error propagation they are describing a derivation of the Jacobian as it related to a problems ...
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Extrapolation error of linear regression lines for a two-cluster data set

I am studying on some linear regression problems using the least-squares method and stumbled upon a problem regarding the error when extrapolating far-away datapoints. About the problem: For a point $...
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propagating error analysis

Tutor asked me a question about propagating error analysis: Here is the question: if for $t_0<t_1<\ldots<t_{l}<\ldots<T,$ $$e_{t_1}\leq e_1+e_2e_{t_{l+1}}+e_3,$$ and $$e_{T}\leq e_1+...
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Simple error propagation calculation

I have two elevation models that I subtract from each other, to find out how the topography changed over time. For the elevation models I have given the RMSE error of the horizontal and vertical ...
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Trying to understand an inconsistency within basic error analysis equations

While doing a physics lab, I noticed that the error analysis equation for multiplication $$R = \frac{X*Y}{Z}$$ $$ \delta R = |R|\sqrt{\left(\frac{\delta X}{X}\right)^2+\left(\frac{\delta Y}{Y}\right)^...
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numeric integration, error component of trapezoid rule

Given a list of x values and a list of f(x) values, I understand how to calculate the integral of the lists via the trapezoid rule -> summation of (b-a)*(f(a)+f(b))/2. However I am not sure how to ...
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Low and High frequencies in numerical methods (Multigrid)

I'm kind of having a lot of problems understanding the concept behind error field frequency. Now, I found these concepts studying the multigrid method, in particular here is attached a link where it ...
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Relative error of division

How can I proof that $Rel(\frac{x}{y})$ $\leq$ $Rel(x)+Rel(y)$ where $Rel(x)$ is relative error of $x$
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Calculating the uncertainty of a set of measurements of which maximum is taken.

In the most simple mathematical sense, say I take a number of measurements and average them, the noise reduces by the square root of the repetitions. This is elementary. Now I take the maximum of the ...
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Confidence interval of an average coefficient calculated from curve fits, with each curve fit using a parameter which has its own confidence interval

I am curve fitting a model to data sets in order to determine a coefficient for each curve fit/data set. I then calculate an average coefficient and the 95% confidence interval for this average. ...
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What is the benefit of using forward difference approximation in newton's method of root finding?

I am trying to think of when using forward difference approximation to $$f'(x) = \frac{f(x+\delta) - f(x)}{\delta}$$ in Newton's root finding method of $$f(x_{n+1})=x_n-\frac{f(x_n)}{f'(x_n)}$$ is ...
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New covariance matrix after variable transformation

Apologies if this is a repeat, I have looked around, but couldn't quite find anything that answers my question I have a set of three variables $a_1,a_2,$ and $a_3$ for some data. This is accompanied ...
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Error propagation for complex numbers from real/imaginary to modulus/phase

Suppose I have a complex number $Z = a+bi$ which I want to add relative Gaussian noise to. I want to add noise to the real and imaginary component independently. I write this as: $$Z_n = [a(1+\...
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How to calculate uncertanty based on the uncertanty of a data set?

Suppose you have a data set of experimental measurements were for each measurement you have the uncertainty associated to it. What is the correct way to calculate the uncertainty associated with the ...
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What is the error propagation for $r = \sqrt{a^2+b^2}$ if the error in $\delta a$ and $\delta b$ are known?

Suppose I have some value $r$ which is defined as $$r = \sqrt{a^2+b^2}$$ where $a$ and $b$ have errors of $\delta a $ and $\delta b$, respectively. What is $\delta r$? Using "standard" error ...
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Fourier Transform of function with measurement uncertainty

I'm attempting to create a classifier that will classify two different types of functions, the problem is, typically the functions have significant noise. (This noise is from a dark current, and is ...
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Approximation method of moments (delta) method - worked out example

I am self-studying the delta method, but i cannot understand it. Here i provide an worked out example from Rice in hos book "Mathematical Statistics and Data Analysis". How are the calculations ...
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What is standard error (any approximation) in a variable when only Mean value and percentiles (16th and 84th) of the PDF over variable are known?

I have variable 'X' for which I know only the MEAN value along with 16th and 84th percentile of its PDF. Now this variable (X) is used in another variable calculation (let's say 'Y') for which I have ...
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Why does the formula for the uncertainty due to bias contain the square-root of 3?

I have been tasked with using standard gage blocks to calculate the uncertainty of a super micrometer and must consider the uncertainty due to bias in my calculation. The formula I am using for that ...
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Error estimation with boundary condition

Given a multiset of elements M. I would like to compute the relative likelihood of the elements within M. Let c be the count of an element, meaning how often it is contained in M. Then the relative ...
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Error propagation in pose composition

Pαc and Pβc are the poses of α and β with respect to the ...
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Reducing the error by increasing the number of equations

Supposing the following linear system of equations has always a single solution: $$ a_1X + b_1Y +c_1Z = d_1 $$ $$ a_2X + b_2Y +c_2Z = d_2 $$ $$ a_3X + b_3Y +c_3Z = d_3 $$ Where $a_i,b_i,c_i$ are ...
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Propagation of Error very strange (matlab)

So, I am using this common formula https://en.wikipedia.org/wiki/Propagation_of_uncertainty#Simplification to compute the uncertainty of dl = l2-l1 with matlab. <...
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Uncertainty due to counting statistics in binned data

I am interested in calculating the uncertainty due to counting statistics such that I can obtain vertical error bars for a histogram. I have a probability density function that tells me the ...
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Should I use powers of $2$ when possible in computations?

For example, if I want the most accuracy and efficiency when performing millions of iterations wouldn't it be better to use $2^k$ -- as opposed to some other nearby number -- whenever possible, since ...
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Condition Number and Sensitivity of Matrix-Vector Multiplication

Consider a non-singular matrix $A \in \mathbb{R}^{n \times n}$ and vectors $x,b \in \mathbb{R}^n$ such that $Ax = b.$ Intuitively the condition number $\kappa(A)$ captures the rate at which $x$ will ...
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Numerical Analysis - Rounding after calculating uncertainty of a function

I successfully calculated the uncertainty associated to a group of functions: f(x) +- 0,0104 g(x) +- 0,0155 ...