Questions tagged [error-propagation]
Methods for calculating errors of a function whose arguments have individual errors.
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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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0answers
26 views
How to estimate the uncertainty of a function that has to be solved numerically
Consider I have a function:
$PV_m^3 - PbV_m^2 +aV_m+ab=RT$
In this example I have measured the uncertainties in $P$ and $T$ experimentally and the errors in $a$ and $b$ can be assumed to be zero. ...
1
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26 views
What is the probability a confidence interval will contain the sample mean from future samples?
Suppose I observe $X_1, ..., X_n$ independent and identically distributed random variables and I calculate a confidence interval $(L, U)$ from this data. Now I take a second sample $Y_1, ..., Y_n$ ...
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18 views
Covariances and correlations in curve fitting
I have a set of data that I am trying to curve fit and I'm ultimately interested in the errors on my fit coefficients. I take my errors on each fit coefficient as the on-diagonal elements of the ...
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56 views
Estimate for the error of an error?
Searching through Wikipedia and StackExchange I managed to understand that, for a set of $N$ normally distributed values, the unbiased variance $Var[s^2]$ of the unbiased variance $s^2$ of the ...
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12 views
Error propagation and how to reduce it
I have several analysis but I am unsure how to propagate error, or reduce it.
The method itself has a large error, ±1.5 lets say.
I obtained total 5 analysis on 5 different samples, which yielded for ...
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1answer
33 views
Calculating an overall standard error/confidence interval across multiple datasets
I am using five different datasets (with 250,000+ samples in each) to calculate an overall average across them. I also would like to calculate an overall standard error/confidence interval for them.
...
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37 views
Error propagation in Gaussian processes
I am trying to predict a time-series measurements, $x_{t+1},x_{t+2},...$ using a Gaussian process model, where the kernel is a Gaussian kernel with small noise-level $\sigma_w^2$, i.e. $k_{ij}=e^{-\...
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1answer
25 views
Propagation of asymmetric uncertainties
I have two (fully independent) measurements of the same quantity X.
Each of them reports a measurement $X_{\sigma_L}^{\sigma_R}$ where $\sigma_L$ and $\sigma_R$ are the left and right uncertainties (...
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13 views
Include standard error of values of an independent variable?
Values for one of my independent variables include reported standard errors. Is there some way to include these standard errors (of the values of the IV, not of its coefficient) in a regression model -...
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1answer
30 views
Propagation of uncertainty when summation symbol is involved
I am unsure how to estimate the propagation of uncertainty when there is a summation symbol involved.
I have the formula (its used to calculate the sauter mean diameter but I will give a simpler ...
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1answer
59 views
Standard error of mean vs error propagation
I'm confused on how to correctly calculate the final uncertainty from averaging measurements that each have their own internal errors.
Say I have 3 voltage measurements: (1.232 ± 0.001) V, (1.197 ± 0....
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81 views
Cross-Correlation Propagation of Uncertainty
I would like to calculate the uncertainty of the cross-correlation of two functions (in two dimensions but even one-dimension is a great start). Experimentally, I have discrete values of f and g, and ...
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1answer
20 views
Error propagation in dividing the averages of two data sets
Let's say I have two data sets each containing 1000 points. I want to get the respective averages of both data sets, then divide the resulting averages. What is the best way to propagate the error (e....
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37 views
Fitting model with error on independent variable
This is the second part of a question started here. However, as this touches a different problem inside the same overall issue I decided do separate it into two questions.
I've made a series of ...
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161 views
Converting standard deviation of percent change to absolute change
I have the following data:
However, I wish to know the standard deviation of the absolute change rather than the percentage change.
To clarify, for Cloz: absolute change = 0.14*53 = 7.42, and we are ...
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37 views
Covariance in an error propagation leading to negative variance
Does it ever make sense that, propagating the error from a set of data, the covariance term being very negative makes the variance go to a negative value (which by itself makes no sense)?
Context:
...
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125 views
Calculating the covariance between 1-D arrays for incorporation into propagation of uncertainty
I have four 1-D arrays of dependent variables. They contain hundreds of data points but I have cropped them to 20 in this example. Each point represents a grid cell on a map.
...
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1answer
125 views
Confused about h-step ahead forecasts
I have 12 months of in-sample data and 12 months of out-of-sample data. I'm trying to calculate the scaled error for an h-step ahead forecast where h=1, 6 and 12. Do I just calculate the error at ...
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611 views
Confidence Intervals with Propagation of Uncertainty
Lets say I'm trying to make a measurement of the area, $A$ of an object imaged in a large number of noisy gray-scale image, and I want to include uncertainty quantification to some confidence interval,...
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22 views
Error propagation through an integral with normal distribution weighting
I have the following equation:
$I(a) = \int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi \sigma^2}}e^{-(x/\sigma)^2} rect(a) dx = \int_{-a}^{a}\frac{1}{\sqrt{2\pi \sigma^2}}e^{-(x/\sigma)^2} dx = \frac{1}{\...
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31 views
Error Propagation in R in ANOVA
I have a set of categorical x variables mapped to numerical y values. The y values have a known error (from measurement by an instrument) that I want to take into account when running an ANOVA. I am ...
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13 views
Calculation of the acceleration correction for BCa bootstraps for data with known errors
I have paired measures for a set of data, and these measures have associated uncertainties. I'm attempting to correlate them using Kendall's tau-b (as we have reason to believe that while they are ...
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1answer
177 views
How do I propagate error for a Poisson-binomial distribution (sum of probability estimates with standard deviations)
My question is about quantifying the uncertainty associated with a sum of several probabilities, in a case where the probabilities are unequal and are themselves estimates with associated uncertainty (...
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56 views
Uncertainty of an integral estimate
I am making the following estimate:
\begin{equation}
\langle f \rangle \simeq \int_0^1 \left[ f(x) \cdot K(x) \right] \; \text{d}x
\end{equation}
where
$f$ is an unknown function, possibly $f(...
3
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1answer
650 views
Propagation of error in log ratios
I have experimental "before" and "after" measurements that I need to compare. The standard way in the field is to look at the log ratio, $y = \log_2(\frac{a}{b})$, so no change is $0$. I'm trying to ...
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1answer
528 views
calculate standard deviation for cumulative sum [closed]
I have data which is x and y as shown below. x and y have standard deviation x_sd and y_sd.
I have calculated cumulative sum for x, and x*y as shown below (x_cumsum & xy).
Now I need to calculate ...
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1answer
64 views
Error propagation in combined linear models
I have a set of observed values ($y_{1Obs}$) and 3 predictive variables (n = 27). I use multiple linear regression to create a linear model:
$Z_1=\alpha_0 + \alpha_1 W_1 + \alpha_2 X_1 + \alpha_3 Y_1$...
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27 views
Exploiting the joint density when averaging measurements
I have two measurements of a quantity from two different sensors, each with their own variance based on the sensors' specifications, etc. From collecting a lot of training data, I have access to the ...
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0answers
33 views
group average taking into account individual uncertainties
I am interested in the mean and variance over n random variables $x$ where each $x_{i}$ is independently distributed with an unique mean and standard deviation, $x_{i} ~ \sim \mathcal{N}(\mu_{i}, \...
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0answers
79 views
Propagation of Absolute Error in Weighted Linear Regression
This is probably relatively trivial, but I am having a difficult time searching for what I want.
I have a set of measurements with a well-defined variance, and I'm performing a weighted least squares ...
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1answer
243 views
Is standard error propagation appropriate in this situation, and if so how, and why?
I have a real-world problem which has left me puzzled in terms of how to approach it with any degree of rigour. I'm not a statistician,, and although I can understand it enough to produce a vague ...
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0answers
17 views
Error of weigted average of elements with errors
I have a set of data with errorbars ($x_i, \sigma_i$), and I need to compute a weigted average and the error. I used the formulas from https://ned.ipac.caltech.edu/level5/Leo/Stats4_5.html
For the ...
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0answers
112 views
error propagation of oscillation removal of time series
I have a time series that that has the form:
$$
F(t) = C + \sum_i A_i\sin(2\pi f_i t +p_i)
$$
Where $C$ is a constant, $A_i,f_i,p_i$ are amplitude, frequency, and phase of the $i^{th}$ oscillation, ...
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2answers
145 views
Multiplying means and calculating variance
I have multiplied together two means and now want to calculate the overall standard deviation. The two means and standard deviation are here: 13.7 +/- 12.7 (1SD) and 4.0 +/- 2.6 (1SD). So the answer ...
3
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0answers
72 views
So many ways to calculate the t-statistic - is this the 'super-formula' I need to yield all the different forms?
I've noticed that the formula for the t-statistic takes on a number of forms depending on the nature of the test being performed, i.e.,:
one sample
independent samples (equal variances)
independent ...
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0answers
60 views
Error bars for sum of Monte Carlo integrations
I want to calculate a quantity $I$ which is the sum of integrals $I_k$,
$$
I = \sum_k I_k\;,\quad I_k = \int_{\Omega} f_k(x)\,\textrm d x\;,
$$
Every integral $I_k$ is evaluated independently by a ...
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1answer
60 views
uncertainty in error propagation of normally distributed data?
I have two normally distributed random variables that add together to form a third normally distributed variable. The means in all cases are 0. Thus, the standard deviations of the first 2 variables ...
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0answers
113 views
Error propagation through a nonlinear model with error on constants as well as observations
I'm using the following equation to invert (non-linearly) for three parameters, L Vr and f.
$$T_r=\frac{L}{V_r} - \frac{(L\,\cos( \theta )\,\cos( \alpha - f))}{V} $$
I have observational errors on $...
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0answers
4k views
Extrapolating an error rate from a sample audit to a larger measurement
I have a report being generated from some transaction data (basically a list of items that should have been purchased) and I can also randomly sample the true transactions as they happen as an audit ...
2
votes
1answer
92 views
Error propagation through two consecutive regressions
I have one linear regression between a variable of interest (z) and measured values of another variable (y):
z = ay + b
and another regression between y and a proxy variable (x):
y = cx + d
I want ...
0
votes
1answer
55 views
Propagation of errors for a sum of fractions of random variables
Say I have a random variable $Z$ that is itself a function of two independent random variables $g(X,Y)$, where
$$Z = g(X,Y) = \frac{X}{Y}$$
I know that the propagation of uncertainty gives a good ...
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0answers
33 views
Find the error in an orientation angle for noisy points in $R^2$
Assume I have two points in $R^2$:
$P_{t=0} = (x_1, y_1)$ and
$P_{t=1} = (x_2, y_2)$
at $t=1$ I want to calculate the heading angle/orientation of the particle.
Both points have measurement noise ...
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1answer
20 views
Suspicion about the errorback propagation formula in PRML
From PRML:
Do variations in $a_j$ give rise to variations in $E$ only via $a_k$? True. Is $h'(a_j)$ the same across all $k$? I'm afraid not.
Let's write down what $\partial a_k/\partial a_j$ are....
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1answer
81 views
propagate uncertainty in mass balance
I have been using the Libre Office Solver tool (similar to Excel Solver) to mass balance some chemical systems. I know the chemical compositions of the various phases in the system and the chemical ...
2
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1answer
425 views
Unsure if my implementation of a Convolutional layer doesn't learn or it's the correct behaviour
So for the last week or so I've tried to implement a Feed Forward network with multiple types of layers (Fully Connected, MaxPool, Convolution), multiple types of non-linear functions (tanh, sigmoid, ...
5
votes
2answers
144 views
Combining uncertain measurements
I have a ball on a table located in position $x', y'$.
I am using many different rulers to measure the coordinates $x_i, y_i$ of the ball. I do this with $N$ different rulers, so $i = 1\ldots N$. ...
1
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1answer
1k views
Uncertainty of a weighted mean of uncertain observations
I have measured $x_i$, $i=1\ldots N$ with independent uncertainties $\sigma_i$.
I have calculated the weighted mean $\bar x$ with
$\bar x = \dfrac{\sum_i \dfrac{1}{\sigma_i} x_i }{ \sum_i \dfrac{1}...
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0answers
64 views
Sample number for confidence intervals of derived quantities
I have two related questions:
Assuming (as the extreme case) I measured a quantity (A) just twice, but I know that the values would "become" normally distributed if I measured the quantity repeatedly....
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50 views
Uncertainty propagation and categorical data
Suppose that you have a model in which you want to perform uncertainty propagation. For example, consider a model of temperature in an area of the world. To simplify, in this model, Temperature will ...
