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Questions tagged [total-least-squares]

A technique to estimate parameters $\beta$ of the linear model $Y=X\beta$ when both $Y$ and $X$ are subject to measurement error. Includes Orthogonal and Deming regression as special cases.

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Correct way to fit a line in 3D (x-position vs y-position vs other quantity)

I have measured the position of light spots $(x,y)$ in an arbitrarily chosen basis and I compare that to some other measured quantity, say the brightness of each spot $B$. Now, in theory all the ...
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Deming regression prediction interval using jackknife resampling

I am trying to write a custom Deming function following the maths in Linnet (1993): https://www.ncbi.nlm.nih.gov/pubmed/2281234 Using jackknife resampling I calculate the standard error for the ...
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Non-negative least squares with errors-in-variables, no repeated measurements

In an experiment, I measured an enzymatic activity $y$ of many solutions containing mixes of different bacterial species. In each of these solutions, I also measured the number of individual cells of ...
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35 views

Standard error of coefficient estimates for model II regression

I'm working with time series data that has error in both the dependent and independent variables, so I'm analyzing each half hour of data with model II linear regression, specifically geometric mean ...
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Selecting appropriate likelihood during non-linear regression

When performing regression to fit a function, $f (x,{\bf \beta})$, to a set of observed data, $y_i(x_i)$, we are seeking to optimize the parameters, $\beta$, of the fitting function, to minimize some ...
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Can total least squares be used to account for uncertainty due to measurement timing?

Suppose we have a dataset in which we wish to perform regression analysis, and where the response/dependent variable is a measurement at time T. But due to pragmatic sampling we do not have ...
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639 views

What's the difference between the SS in the variance and the TSS?

I'm trying to understand how these two statistics differ. My understanding is the variance is the sum of squares of the predictor divided by the degrees of freedom. On the other hand, the sum of ...
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52 views

Regression to estimate parameters

I would like some suggestions to tackle the following problem. Given a system $y = X\beta$ where $y \in \mathcal{R}^m$, $X \in \mathcal{R}^{m \times n}$, $m\geq n$, and $\beta \in \mathcal{R}^n$, ...
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Orthogonal polynomials + cross validation: should subsetting be done prior or after constructing the orthogonal polynomials?

So, just to start... I've just learned of orthogonal polynomial regression today. I've gone through the master's-level linear models courses, and we did not cover that topic. I was always under the ...
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356 views

Why does the total least squares line in 2D pass through the average across all data points?

I have $N$ data points $\mathbf{m_k}$ and I want to fit a line through them with minimal error $$J = \sum_k^N ||\mathbf{m_k^*} - \mathbf{m_k}||^2 = \sum_k^N ||\mathbf{m_0} + a_k\mathbf{e} - \mathbf{...
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Advantage of orthogonal polynomials

What is the sense or background of orthogonal polynomials (regarding using mixed models)? I would like to know why they shall or should be orthogonal. Is it to build independent sample points? On Is ...
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Least Squares Derivation

I come from physics and would like to derive the chi-square function given by the Particle Data Group: \begin{equation} \chi^2 (\boldsymbol\theta) = (\boldsymbol y-\boldsymbol\mu(\boldsymbol \...
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Deming regression implementation: force intercept to 0

I implemented Deming Regression in a known programming language, using the algorithm from here: https://en.wikipedia.org/wiki/Deming_regression However, the algo does not specify what to do in case we ...
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What's the shape of confidence interval for linear regression estimated through TLS (total least squares)?

So, we know the shape of confidence intervals for vanilla linear regression estimated through OLS (ordinary least squares): Shape of confidence interval for predicted values in linear regression. In ...
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666 views

When to use Deming regression

I am currently working on a way to transform two different phosphorus test values into each other. Background There exist many (extraction) methods to measure plant available phosphorus in soil. ...
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Generating Random Data Sets for Linear Regression with Random Slope and Error Term in R

I want to test the effects of sample size on Deming regression using simulated paired data in R. As the data are paired, the expected slope value should be 1 and the intercept 0. The code I have ...
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Specifying the Error Ratio in Deming Regression

We use Deming Regression for method validation in our clinical ELISA laboratory. As we run clinical trials this needs to be performed if we get a new reagent batch, batch of plates, new plate reader ...
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How to compare two Deming regressions?

I have two small data sets. For each data set, I performed a Deming regression. Hence, I have, for each data set, the relationship between X and Y in the form of slope+intercept coefficients. Now, I ...
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small sample approach to simple linear regression with errors-in-variables (measurement errors)

I seek to estimate $b_1$ and $b_0$ from data of the form: $$y_i = b_1x_i + b_0 + e_i, \quad i\in\{0,1,...,N-1\}$$ given $\{y_i\}$ and $\{\tilde{x}_i\}$ where $\tilde{x}_i=x_i + n_i$ (i.e., error-in-...
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Regression of dataset with many x values per y value

I am performing a total least squares regression in which I have many x observations for a given y observation. The x observations are normally distributed. I am aware I could do some sort of ...
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247 views

why regress a y variable on a vector of just ones

I am trying to understand some code somebody has written. I have a y vector which is a factor say book to price for 500 companies. Then I also have a x vector of the same shape which is just ones. ...
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Find best fit diagonal matrix for error minimization

I want a set of input values to be as similar to the output values as possible. I have an input matrix X (m*n) that has m data points and n dimensions for each data point. I also have an output matrix ...
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What weights for weighted Total Least Square (TLS) regression?

I have a dataset with known errors in both the X and Y and want to perform a simple linear regression. From reading other posts, it seems I want to TLS over OLS due to the presence of error in both ...
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Multiple errors-in-variables regression with collinearities

I have a $[k \times N]$ matrix of predictors / independent variables and a $[k \times N]$ matrix of predictands / dependent variables. I have uncertainty estimates for each predictor and each ...
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$X$ and $Y$ at different scales in total least squares?

As this section of the Wiki article says, the best (in total least squares sense) matrix $B$ that projects $X$ to $Y$ is given by $$B=-V_{XY}V_{YY}^{-1},$$ where $V_{XY}$ and $V_{YY}$ come from the ...
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Is quantile regression a better option than total least squares RMA in this case

I have paired cobalt concentrations in bird blood and feathers. Blood levels give me an idea of how recent the cobalt exposure was (<30days), feather give the 6 month accumulated total. Previous ...
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Confidence/prediction intervals for total least squares regression

I am learning the ropes of total least squares regression and I found this thread How to perform orthogonal regression (total least squares) via PCA? where the answer by @amoeba, together with some R ...
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Regression when each point has its own uncertainty in both $x$ and $y$

I made $n$ measurements of two variables $x$ and $y$. They both have known uncertainties $\sigma_x$ and $\sigma_y$ associated with them. I want to find the relation between $x$ and $y$. How can I do ...
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Introductory references about total least squares

I am an engineering student and I've been recently told about the Total Least Square (TLS) method. I am interested in applying it to topographic measurements and in comparing the results with the ...
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Model selection: OLS vs TLS

I have two sets of real-valued data and I am interested in their correlation. From my perspective, there appear to be errors both variables, so I am inclined to perform a regression with TLS (Total ...
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87 views

Total least squares minimization

I've seen some lecture notes about the Total Least Squares, which state the following: Suppose we have a linear system $Y=XB$, which may be inconsistent. Now change this system to $$(Y+\Delta)=(...
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How to calculate confidence/prediction bands for Deming regression

Revised question: My fundamental question seems to be: "I have data with error both in y and x. How do I estimate a value of true x so that y evaluated at that x will be distinguished from y(0) say, ...
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273 views

Is it possible to make a regression with known standard error on y

I want to compare estimate with standard error in function of a continuous variable and a categorial variable . Here an example of what my data look like. ...
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Is it possible to calculate R-squared on a total least squares regression?

I am using the Deming function provided by Terry T. on this archived r-help thread. I am comparing two methods, so I have data that look like this: ...
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841 views

Total least squares with weights [duplicate]

I am looking for a way to perform weighted total least squares in R. I know one can use PCA for this as described nicely in the following post. How to perform orthogonal regression (total least ...
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711 views

Standard error of the intercept in orthogonal regression

I want to perform a univariate regression but with substantial measurement error in both $x$ and $y$. I therefore want to try orthogonal regression with R. The best answer to my question so far have ...
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What version of the AIC should be used in total least squares regression?

If we are carrying out a regression and we have errors in both the x-s and y-s, we can estimate the parameter values using the total least squares method. In that case, I assume we shouldn't use the ...
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What techniques are there to measure goodness of fit of Deming (orthogonal) regression?

Questions: Even if there is no "widely accepted" technique, is there a useful-and-above-average technique for estimating goodness of fit in orthogonal regressions? What are the pros/cons of this ...
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154 views

P-values for regression coefficients in total least squares regression

I want to calculate the p-value for the beta estimated in Total (orthogonal) least squares regression. Do I need to calculate the standard error of the estimates in ...
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Fitting a plane to a set of points in 3D using PCA

I am trying to estimate a midplane of a 3D model using the midpoints of paired landmarks, in order to reconstruct missing data (midplane refers here to the middle/saggital plane of the cranium which ...
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121 views

Total least square intuition

I have yet to find a good intuitive explanation of TLS. Online resources tend to focus on the vertical vs. perpendicular square error pictures (I don't need to see perpendicular lines to understand ...
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Bayesian estimates for Deming regression coinciding with least-squares estimates

Consider the following Deming model with independent replicates : $$x_{i,j} \mid \theta_{i} \sim {\cal N}(\theta_{i}, \gamma_X^2), \quad y_{i,j} \mid \theta_{i} \sim {\cal N}(\alpha+\beta\theta_{i}, \...
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Nonlinear total least squares / Deming regression in R

I've been using nls() to fit a custom model to my data, but I don't like how the model is fitting and I would like to use an approach that minimizes residuals in ...
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Does a correlation matrix of two variables always have the same eigenvectors? [duplicate]

I wanted to conduct a total least squares regression on two variables. My statistical programme does not provide TLS, but TLS luckily equals Principal Component Analysis, as far as I know. Since all ...