Questions tagged [measurement-error]

Measurement error is the difference between a measured value of a quantity and its true value.

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Distribution of true value when measurement imprecision is non-constant

What I believe to have understood so far (I am not a mathematician or a statistician, so please correct me if I'm wrong.) Say we are making measurements of some phenomenon $X$, and we have a normally ...
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Bayesian model comparison with systematic error

Two parameters $(x,y)$ were measured for 3 different objects, wielding the following results: $$\{ (x,y) \}= \{ (1,3), (3,5), (5,9) \}$$ Knowing that the error in the estimation of the values $y$ is ...
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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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GAM interactions to predict deaths (with measurement error) from weather with distributed lag

I am trying to model the effect of heat (in particular of its constituent components air temperature, humidity, solar radiation and wind speed) on daily number of deaths using a generalized additive ...
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Why is ignoring prediction error not a concern in instrumental variables?

In his book, Statistical Rethinking (2nd edition, p. 137), Richard McElreath states that including parameters with unobserved values, such as residuals, and treating them as if they were perfectly ...
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Estimate of an observable and its uncertainty from two inconsistent measurments

Suppose two independent experiments are set up to measure the same observable $\mathcal{O}$ in the same way and report two results $$ o_1 \pm \delta o_1^{stat}\pm \delta o_1^{sys}, \quad \quad o_2 \pm ...
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Scale-independent error metric for data with many zeros

I've been working on a time series forecasting model. I can't use a scale-dependent error measurement. And my target outputs also occasionally have zeros, meaning I can't use MAPE either. What is the ...
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Dummy variables vs deleting observations

I am trying to figure out whether a general (simple rule) exists for my problem. Specifically, if I have a dataset where some variables (independent/control; not the dependent one) may be measured ...
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How to express the agreement between experiment and theoretical observation?

Let us suppose I have a value measured from experiment and given by $$V_{\text{exp}} \pm \sigma_{V_{\text{exp}}}$$ and a theoretical value given as $$V_{\text{the}} \pm \sigma_{V_{\text{the}}}$$ Is ...
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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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Simplifying probability involving two independent normally distributed variables

Given independent normally distributed random variables $X$ and $Y$ with means $\mu_X = \mu_Y = 0$ and a tolerance $u > 0$, what is $p = P(X+Y\le u\,\text{and}\,X>u)$? Can it be expressed in ...
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Mean of Relative Error Vs. Coefficient of Determination

Consider we have a method that estimates a specific parameter. We want to find the accuracy of this method and we have 10 samples (these samples include the true values of the parameters and their ...
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95 % CI for ratio of variables subject to measurement imprecision

An instrument used for measurements has a known measurement imprecision of $CV = 10 \%$. Thus, if the variable to be measured has a true value $x_{true}$, the range of possible measured values will be ...
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Bayesian data combination?

I have $N$ sensors, where $N$ is typically of order 10. An object comes in, and each sensor activates. Each sensor measures up to $M$ properties of a given object (the same set of possible ...
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Convergence of Percentile in Power Law

I have a probability distribution, that in its tail follows a power law. I've noticed, while I was simulating samples, and determining parameters experimentally, that as I increase the value of a ...
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Representation of error values for large and small raw values

Assume a population where the numbers are in the range of [0:200]. Assuming the numbers represent the performance of a program, e.g. seconds, the numbers look like ...
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Error "propagation" under base change

Imagine you have fitted a curve to data in $x$-$y$-space and obtained errors $\sigma_x$ and $\sigma_y$ for each data point. But now you rotate the coordinate system by $45^\circ$ for example. Can you ...
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Measurement system analysis

Let's say I'm responsible for checking the length of shoes on a production line. Our production system isn't great, so a shoe that should be 25 cm long might actually be a slightly different length (...
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Determining Error Bars on Normal Data from Sensor with Rated Error

You are using a sensor to take temperature measurements at 10 Hz. You sample for 100 seconds and end up with 1000 samples. Upon analyzing a histogram of the data, you see that it appears normal with a ...
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Is this a viable means of diagnosing errors in measurements from data sources?

I recently started a project that involves taking measurements at hundreds of different sources. These measurements, in theory, should all cancel out at each point in time to 0 but that's never the ...
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Multivariate forecast error metrics

I'm working with multivariate models (VAR, VECM, GARCH-M) and I want to measure the forecast error size of each model. An alternative is take individually the forecast of each univariate time series ...
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Is this a reasonable way to determine the reliability of a fit?

Background I have measurements of a trajectory that is parameterized by time. The data consists of points with two spacial coordinates $(\tilde{x}_i, \tilde{y}_i)$ and a time stamp $(t_i)$. I'm using <...
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Error Models beyond Additive/Multiplicative for Observables on Bounded Intervals

Arguably, the two most common error models are Error Model Support Formula Assumptions Constraints Additive $(-∞, +∞)$ $y_\text{obs} = y + ε$ $𝐄[ε]=0$ $𝐕[ε]=σ²$ Multiplicative $(0, +∞)$ $y_\text{...
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Why does regression dilution always bias the slope toward zero?

Regression dilution, in the case of linear regression, is supposed to be what happens when there is noise in the independent variables, namely the slope of the fitted linear regression model becomes ...
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Estimating the error in 3D phase fraction calculations

I am trying to estimate the error of a 3D volume fraction measurement (from X-Ray Microtomography experiments). Kind of like a CT-scan of a small inorganic material with embedded particles. ...
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How to calculate RMSE for two regression lines

I'm struggling on how can I calculate the RMSE for two regressions. Consider the following scenario: I have two linear regressions, and Id like to calculate the joint RMSE for this model. Any hint on ...
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(Co)variance interpretation in Kalman filter

Let's say I have a device which uses Kalman filter to fuse sensor data and produce an optimal estimate of the system parameters. As it should, it also estimates parameter covariance matrices at each ...
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Measurement error and type I. and type II. error

Is there a relationship between measurement error (1.) both systematic and random, (2.) systematic only and (3.) random only in terms of tendency of results of statistical tests to be biased towards ...
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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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How can using standardized residuals as an outcome be a valid approach? And how do results differ from doing one regression only?

The residual approach uses standardized residuals SR from regression of Y on X1 as an outcome, and then regress them on X2 (here is a literature review: https://www.ncbi.nlm.nih.gov/pmc/articles/...
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How to deal with an imperfect reference standard in a comparison of two diagnostic methods?

Summary We discovered after performing our experiment that the method for determining our reference standard was not perfect: it may have a false-negative rate of up to 20%. With no way to confirm ...
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RMSE estimation from model accuracy and RMSE of dependent variable

Given the two variables/measurements, which were used for building up the model (Random forest regressor), I would like to estimate an RMSE of the predictor variable/measurement against the "real&...
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1 Sigma Error Brings Specificity > 1

I am running a classification model on some medical images. When calculating the specificity, we get a value very close to one and then when the 1 sigma standard deviation is calculated its range ...
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Can I include known uncertainty in a regression model?

How can I build a regression model that accounts for the known error ranges (95% confidence intervals) in my independent variable samples? I am not an expert in statistics, but I have the impression ...
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Calculating the quality of text parsing script

I have a python script that picks relevant sentences from a text corpus based on keywords and stopwords and applies some classifiers for the chosen sentences. The context is academic research. The ...
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probability density function of forecast's percent error

Imagine that I am trying to estimate the "number of sales (dollars)" that a client will have when they host a booth at a music festival. The "low forecast" and "high forecast&...
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What is the best way to share the percent error of a forecast to a group of peers?

Let's say that I am trying to roughly forecast the total number of commuting hours for the employees at a client company based on some assumptions (# of business days in a week, # of employees at the ...
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What is the difference between Type A and Type B uncertainty evaluation?

I am trying to evaluate the uncertainty in measurement following the Guide to the expression of uncertainty in measurement (GUM) elaborated by the Joint Committee for Guides in Metrology. Link: https:/...
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Question about Regression Calibration for Measurement Error, Carroll, Ruppert, Stefanski, Crainiceanu Eq 4.4

In the textbook Measurement Error in Nonlinear Models: A Modern Perspective, Eq 4.4 describes the regression calibration approach to finding the best linear approximation for a true predictor given ...
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adding point (0,0) to data set and its uncertainty?

I am conducting a simple experiment to determine the relationship between the force applied to a spring and the displacement of the spring from its rest length. To do so, I hang various masses from ...
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Model performance in time-series forecasting with some outliers

I'm creating forecasts for products where some of them have large seasonal spike during times like Christmas and/or Easter but relatively low sales volume on other times. For this particular product ...
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Least square fit for data with uncertainty [closed]

I have a set of data $\{x_i,y_i\}$ that came from my measurement, so each of them has some uncertainty. I wonder is there a method to fit them using a model (like Gaussian), while taking care of those ...
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Overlapping 95% confidence limits

I cam across these two old blog posts on displayed error bars and tried to work through the result. I believe I am making a mistake somewhere, but I'm not sure where. Let me describe the scenario ...
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Including Ground Truth Uncertainty in Measurement System

I think this might be a basic scenario but I still struggle to find an appropriate methodology for my problem. Consider the following scenario: There is a small moving object on an $xy$-millimeter-...
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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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Weighted least squares fit with measurement uncertainty

I work with geophysics and in my research, some processes are described by power laws $$k=\gamma v^{\alpha} u^{\beta}$$ To find the indices $(\alpha,\beta)$ the equation is rewritten as a logarithm ...
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Accounting for variable errors in independent variables in linear regression

I am trying to estimate value of a dependant variable using linear regression with multiple independent variables. The error in measuring independent variables is not uniform. An analogue to my ...
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Changing the range of data before relative difference computation

I have a correlation coefficient obtained from measurement and a correlation coefficient obtained from theoretical model. I am searching for a difference metric to show how well these two correlation ...
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Maximum Entropy Distributions on Intervals and Domain Consistent Error Models

Given a real-valued observable, without further knowledge about it, the most commly assumed error model is that the error is additive, normally distributed, and independent of the observations. $$X_\...
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error on single measurement from long term data set

I have a data set with a few hundred measurements of the same standard. I also have a single measurement of an unknown using identical methods. If I assume that the error of both types of measurement ...
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