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Related Concept Videos

Calibration Curves: Linear Least Squares01:20

Calibration Curves: Linear Least Squares

A calibration curve is a plot of the instrument's response against a series of known concentrations of a substance. This curve is used to set the instrument response levels, using the substance and its concentrations as standards. Alternatively, or additionally, an equation is fitted to the calibration curve plot and subsequently used to calculate the unknown concentrations of other samples reliably.
For data that follow a straight line, the standard method for fitting is the linear...
Residuals and Least-Squares Property01:11

Residuals and Least-Squares Property

The vertical distance between the actual value of y and the estimated value of y. In other words, it measures the vertical distance between the actual data point and the predicted point on the line
If the observed data point lies above the line, the residual is positive, and the line underestimates the actual data value for y. If the observed data point lies below the line, the residual is negative, and the line overestimates the actual data value for y.
The process of fitting the best-fit...
Estimating Population Mean with Unknown Standard Deviation01:22

Estimating Population Mean with Unknown Standard Deviation

In practice, we rarely know the population standard deviation. In the past, when the sample size was large, this did not present a problem to statisticians. They used the sample standard deviation s as an estimate for σ and proceeded as before to calculate a confidence interval with close enough results. However, statisticians ran into problems when the sample size was small. A small sample size caused inaccuracies in the confidence interval.
William S. Gosset (1876–1937) of the Guinness...
One-Compartment Open Model: Wagner-Nelson and Loo Riegelman Method for ka Estimation01:24

One-Compartment Open Model: Wagner-Nelson and Loo Riegelman Method for ka Estimation

This lesson introduces two critical methods in pharmacokinetics, the Wagner-Nelson and Loo-Riegelman methods, used for estimating the absorption rate constant (ka) for drugs administered via non-intravenous routes. The Wagner-Nelson method relates ka to the plasma concentration derived from the slope of a semilog percent unabsorbed time plot. However, it is limited to drugs with one-compartment kinetics and can be impacted by factors like gastrointestinal motility or enzymatic degradation.
On...
Estimating Population Mean with Known Standard Deviation01:16

Estimating Population Mean with Known Standard Deviation

To construct a confidence interval for a single unknown population mean μ, where the population standard deviation is known, we need sample mean as an estimate for μ and we need the margin of error. Here, the margin of error (EBM) is called the error bound for a population mean (abbreviated EBM). The sample mean is the point estimate of the unknown population mean μ.
The confidence interval estimate will have the form as follows:
(point estimate - error bound, point estimate + error bound)
The...
Linearization and Approximation01:26

Linearization and Approximation

Linearization is a mathematical technique used to approximate complex, nonlinear functions with simpler linear models in the vicinity of a chosen reference point. The method is based on the idea that, although a function may be difficult to evaluate exactly, its behavior near a specific input value can often be closely approximated by the tangent line at that point. This approach is particularly useful when small deviations from a known value are involved.Consider the square root function, for...

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Related Experiment Video

Updated: Jun 6, 2026

ARL Spectral Fitting as an Application to Augment Spectral Data via Franck-Condon Lineshape Analysis and Color Analysis
07:11

ARL Spectral Fitting as an Application to Augment Spectral Data via Franck-Condon Lineshape Analysis and Color Analysis

Published on: August 19, 2021

Least squares estimation without priors or supervision.

Martin Raphan1, Eero P Simoncelli

  • 1Howard Hughes Medical Institute, Center for Neural Science, and Courant Institute of Mathematical Sciences New York University, New York, NY 10003, USA. raphan@cims.nyu.edu

Neural Computation
|November 26, 2010
PubMed
Summary
This summary is machine-generated.

This study introduces a new nonparametric empirical Bayes least squares (NEBLS) estimator for unsupervised measurements. This method optimizes estimators without prior knowledge, generalizing Stein's unbiased risk estimator (SURE) for improved accuracy.

Related Experiment Videos

Last Updated: Jun 6, 2026

ARL Spectral Fitting as an Application to Augment Spectral Data via Franck-Condon Lineshape Analysis and Color Analysis
07:11

ARL Spectral Fitting as an Application to Augment Spectral Data via Franck-Condon Lineshape Analysis and Color Analysis

Published on: August 19, 2021

Area of Science:

  • Statistical Learning
  • Machine Learning
  • Estimation Theory

Background:

  • Optimal estimator selection typically requires supervised data or prior models.
  • Unsupervised measurements from an unknown distribution pose a challenge for traditional estimation.

Purpose of the Study:

  • Develop a general expression for a nonparametric empirical Bayes least squares (NEBLS) estimator.
  • Generalize Stein's unbiased risk estimator (SURE) for unsupervised learning scenarios.
  • Introduce a SURE-optimized parametric least squares (SURE2PLS) estimator.

Main Methods:

  • Derived a general expression for NEBLS estimators based on measurement density.
  • Showcased existence conditions and specific forms for various measurement processes.
  • Generalized SURE to express mean squared error using only measurement density.

Main Results:

  • Introduced NEBLS and SURE2PLS estimators for unsupervised least squares estimation.
  • Demonstrated generalization of SURE for broader applicability.
  • Developed an incremental learning form for linear parameterizations.
  • Showed a generalization of score-matching for density estimation.

Conclusions:

  • NEBLS and SURE2PLS estimators offer robust performance comparable to supervised methods with sufficient data.
  • The generalized SURE provides a powerful tool for risk estimation in unsupervised settings.
  • This work advances nonparametric and empirical Bayes approaches in statistical estimation.