Questions tagged [error-propagation]

Methods for calculating errors of a function whose arguments have individual errors.

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Error-Propoagation: Combine confidence intervals of two subsequent regressions

I have two subsequent regressions and want to combine the results of these two regressions. But I also want to show the increased uncertainty by somehow combining the confidence intervals of these two ...
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What is the error of the mean of data that have uncertainty values attached to them?

Given a set of $n$ values, the error associated with their average will be $$\text{standard deviation}/\sqrt{n}.$$ But if the values themselves have an uncertainty attached to them, such as $100\pm 1,$...
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Can I make replicates of an herbal extract, three measurements of the content for each replicate and measure the deviation based on the averages?

I made the extraction of flavonoids from a plant in triplicate. For each replicate of the extraction, I measured the total phenolic content in triplicate and averaged this value, as performed in ...
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incorporating error in best fit line equation

I have two sets of data {X} and {Y} each element in those two sets has its own unique +/- error. I graphed x vs y in a scatter plot along with every point's error bar (both in x and y (x1 +/- a1 , y1 +...
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T-test with replicate measurements, how to quantify error in the measurements?

I want to compare two tools, A and B. I use tool A to measure 10 very different objects, making 5 repeated measurements for each object so I can record an accurate value. I also do this with tool B. ...
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Statistical uncertainty on an Poisson distributed value estimated using Poisson distributed variables

Let's say I have the number of events in four regions in phase space. Where the regions are named: $A, B, C$, and $D$. These are Poisson distributed and not correlated with one another, and the number ...
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Error propagation for the median of time-series

Let's assume that we have a set of different measurements from an instrument of a parameter of a parameter P that changes with time during a time period t. Let's also assume that the measurements ...
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How to properly (extrapolate?) information from samples to wider areas?

I have 12 plots, which I want to compare in terms of biomass production (the dry weight of the plants that grow there) in g per m2 and in kg per hectare. Imagine that 6 of these plots are full covered ...
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Finding average error

Consider the following data: 20 +- 2, 25+-1, 30+- 3, 40+-1 If I have to find the average error, I can follow two methods: (2 + 1 + 3 + 1)/4 = 1.75 Sqrt (4 + 1+ 9 + 1 )= 3.9 Since, (20 + 25 + 30 + ...
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Error on X having Y(X) greater equal to some constant A

There is a variable $Y(X)$ as a monotone increasing function of $X$, and there is an error data of $Y$. I want to find the minimum $X$, satisfying $Y(X)\geq A$. What will be the error on $X$ in this ...
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Best estimate of a variable from two observations with different efficiencies

Problem Suppose to measure the frequency of a certain rare event (e.g. particle count) with two instruments $I_1$ and $I_2$ for a time $\bar{t}$, the same for both instruments. We expect the same ...
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Uncertainty of `max` and `argmax` of values with uncertainty

Given a sequence $y = [y_1 \pm \Delta y_1, y_2 \pm \Delta y_2, ..., y_N \pm \Delta y_N]$, how can you compute the maximum of $y$ with its associated uncertainty, and the value of the index $i$ at ...
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How to calculate the ratio of two means

I have ten subjects and they have e.g. a measure of height and a measure of mass. So for each individual there is a value for height/mass. I have group level statistics but not the individual data ...
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What is the relationship between the residuals of an objective function and the uncertainties of the minimizer values?

Consider I have some optimization problem and an objective function $f(x, y, z)$. $f$ is defined using the sum of squared residuals, i.e. for some function $g$, we have $f(x, y, z) = \sum[g(x, y, z) - ...
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Prediction and confidence intervals - large number of predictions

I'm using a regression model to predict one quantity, $y$, given another, $x$. I'm trying to estimate the error in future predictions of $y$, but I'm wondering in which scenarios I can fairly use the ...
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How to propagate standard error of medians

I am interested in median-to-median propagation of standard error: The values in my variable (say, $Y$) are all medians (of something, say, $X$), and are each associated with a standard error. Next, I ...
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Calculate uncertainty of the slope when dependent variable in a linear regression has substantial error?

I have a dataset in which the dependent variable (y) has known and substantial error, and yet the observations happen to line up quite well along a line when plotted against the independent variable (...
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Linear regression with double error bars

I am trying to perform linear regression between $y'$ and $x'$ (independent) for some experimental data. Each of the variables has a non-constant random measurement uncertainty, and the error bars do ...
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Linear regression with statistical and systematic error in y and statistical error in x

I was wondering what is the best way to perform a linear regression if I have a data with uncertainties in $y_i$ and $x_i$ (which are not necessarily the same). And suppose that $y_i$ has a systematic ...
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Error propagation for slope of linear regression

I want to calculate the standard deviation of the slope $k$ of $$\ln y = kx + d$$ where the error of $y$ is assumed to be 10% and the error of $x$ is neglected. How to perform a graphical ...
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Uncertainty when substracting average from the same data set

I have a data set with its own measurement uncertainties. Then I do averaging of the population and use standard error of the mean as the uncertainty for the average. My question is, I need to ...
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Overall uncertainty in a set of numbers that are normalised to 1

If I have multiple sets of predictions for $A$, $B$, $C$ subject to the constraint $A+B+C = 1$ I believe I can get standard deviations $\sigma_a$, $\sigma_b$, $\sigma_c$ from the variance over the ...
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Directly discarding big term in the proof of error propagation formula of variance from random variable $x$ to $f(x)$?

I read the error propagation formula scanario said that, the connection between the variance of a random variable $x$ and $f(x)$ is $\frac{var(f(x))}{|\partial_xf(x)|_{x=\bar{x}}|^2}=var(x)$. While I'...
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Correct SD for Area Under Curve (AUC)

I need to calculate the area under curve (AUC) with SD for several repeats of an experiment. I am using the trapeze method to estimate the area, where $$ S = \sum(S_i) $$ and $$ S_i = (a+b)*d/2 = (f(...
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How to propagate error in a regression calculation

My dataset is shown below. In it, I have 4 time points (T0-T3) where I measure counts for n different samples as well as a control. From this, I calculated standard error for each sample with the ...
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Error propagation for cubic relationship

I have the cubic relationship between two variables, x and y, and I need to find the error in x. y = ax^3 + bx^2 + cx + d I have the values for the coefficients and their respective uncertainties. I ...
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What is the difference between chi square and coefficient of determination (R square)?

What is the difference between chi square and coefficient of determination (R square)? What is their significance, limitations, and applicability as compared to another?
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Gaussian distribution with error in both parameters: trying to create a Gaussian distribution with no error in either parameter

I am trying to create a Gaussian distribution where the parameters are not precisely known. I have mean $\mu$ and standard deviation of the mean $\sigma_\mu$ (modeled as an uncorrelated Gaussian), and ...
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Uncertainty on integrated function fitted to data

I have some data that has been fitted with a function of several parameters (using scipy.optimize.curve_fit). The function describes a two-component spectral energy distribution: $$S (\nu) = \frac{2 h ...
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Standard error of the mean and fits

I am trying to understand what kind of statistics I should use to correctly report a mean value. Essentially, I have a number of curves that belong to the same population, and I use the same model to ...
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Propagation of uncertainties in correlated measurements (Inverse of a measurement)

Suppose that you have $n$ uncorrelated measurements $A_{1}= a_{1} \pm \sigma_{1}$ , $A_{2}= a_{2} \pm \sigma_{2}$ ... $A_{n} = a_{n} \pm \sigma_{n}$, where $a_{m}$ is the mean value of $A_{m}$ and $\...
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Uncertainty when taking the difference between a measurement and a subset measurement

Lets say I collected $\mathbf{N_t}$ data points for an experiment and this resulted in a measurement of $A\pm\sigma_A$. Then I'm interested in how a particular variable effects the study and remove $\...
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How to conduct error propagation for an uncertainty budget?

At my job I'm attempting to build an uncertainty budget for an experiment in which repeated measurements of multiple elements' masses were taken. Then I am finding certain mass ratios. I calculated my ...
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How to include standard deviations in diversity calculations?

I have community abundance data where each species has a mean and a standard deviation (determined by replicate samples). I am comparing the diversity of two communities. The various diversity metrics ...
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Error Propagation through Iterated Functions (variable star data)

I'm an astronomer trying to smooth variable star data, and one way I'm doing this is using a 7-point, second-order Savitzky-Golay filter. I iteratively apply the filter 51 times (i.e. I apply the ...
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Average of compound value or compound of averages?

Suppose I'm measuring the flow rate of a tube. I measure the time it takes to fill it twice. Each time, I also measure its final volume. So what is the best approach to calculate the average flow? ...
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Log and exponential uncertainty propogation

I am processing data on a radioactive decay experiment, and need to find the errors on some quantities that I can get from some fit parameters. I would like to obtain: $\sigma A$ from $A = e^{a}$, ...
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Median error propagation

Assuming we have a set of measurements $x_i, i=1,\dots,N$ each with an error of $\Delta x_i$, I then normalize the data by the median of $x$. Can an error for the median value be calculated with the ...
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Estimate Variance-Covariance matrix via Error Propagation of Weighted Least Squares Equation

Given a linear system $b_{obs} = Ax$, how can I derive the covariance of $x$ (i.e. $C$) from the weighted least square solution equation: $$x = (A^TS^{−1}A)^{-1}A^TS^{−1}b_{obs}$$ With $C$ being the ...
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1 answer
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How to report a +/- standard error for logarithmic data?

I have some data $y$ which varies logarithmically which I am fitting with MCMC methods. It makes more sense to do my analysis in the logarithmic space $z = \mathrm{log}(y)$. As I result, I end up with ...
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Taylor expansion of Gaussian process function with input noise

I am reading "Gaussian Process Training with Input Noise" by Andrew McHutchon and Carl Edward Rasmussen, where it is assumed that the inputs $x$ are noisy measurements of the actual latent ...
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Adding of two confidence intervals

Given two variables $\bar{A}$ with 95% confidence interval [$A_0$,$A_1$], and $\bar{X}$ with 95% confidence interval [$X_0$,$X_1$], how would one calculate the 95% confidence interval of a composite ...
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likelihood as random function wrt data

Suppose we have some dataset $x= \{ x_1, \dots, x_n\}$ where every datapoint is i.i.d., $x_i \sim P(\cdot|\theta^*)$ for some known distribution $P$ and true parameter $\theta^*$. Then, for this ...
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MLE like method for uncertain data

I have a rather basic question for which I could not find an answer for. I want to use a similar technique as Maximum Likelihood Estimator (MLE) or even MLE itself for data which has uncertainties for ...
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Bias of mean in toy Monte Carlo sampling of $A\times B$ for $A=a\pm\sigma_A$ and $B=b\pm\sigma_B$

I am trying to do some toy Monte Carlo sampling, to calculate the uncertainty of the product $A\times B$ of two random variables $A=a\pm\sigma_A$ and $B=b\pm\sigma_B$. I also assume that these ...
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1 answer
185 views

Linear regression on data with associated errors in the x and y direction

I have experimental data that I need to linearise in the form $\ln(b/x)$ vs. $x$, where $x$ is an experimentally-derived quantity and the value of $x$ is dependent on $b$ (which varies). $x$ has some ...
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3 votes
1 answer
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Estimating the errors in parameters in the ordinary least square

I am reading the book An Introduction to Error Analysis by John R. Taylor. In Ch8: Least-Squares Fitting, he has derived expressions for parameters $A$ and $B$ in fitting the line $A+Bx$ to the set of ...
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Calculating error from covariance of graph and errors of data?

I have some data, with known error, and have used a linear regression algorithm to make a fit of this data and plot it. I have also used the algorithm to calculate an associated covariance. How would ...
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1 vote
1 answer
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Covariance matrix of element-wise quotient of two sets of measurements with known covariance matrices

I have two sets of measurements, $x_i$ and $y_i$, both with the same number of elements, $N$. For each of these sets, I have known $N\times N$ covariance matrices, $\Sigma_x$ and $\Sigma_y$, ...
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How do I calculate error margins for difference in proportions?

I have data that captures responses to stimuli and categorises both the cue stimuli and response stimuli for both male and female, as follows: Cue Stimulus Response Stimulus Gender Cue Count Cue ...
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