References¶
This section of the documentation is devoted to providing references for the algorithms implemented in scikitguess. Each paper comes with a link, a PDF where permitted, and any additional materials.
List of References
Régressions et équations intégrales¶
The ideas proposed in this paper by Jean Jacquelin were the seed for this scikit.
Concept¶
The concept behind this paper is that integrals and derivatives can be estimated through differentials and cumulative sums. The goal is to set up an integral or differential equation whose solution is the model function. If the righthand side of such an equation can be expressed as a linear combination of integrals and derivatives of itself multiplied by some other predetermined functions, we can estimate the numerical values of the equation’s terms. While the coefficients that make the equation work depend on the fitting parameters, the functions themselves do not. It is therefore possible to set up a simple linear regression for the coefficients based on the numerical approximations of the integrals and derivatives. The numerical approximation of integrals by cumulative sums tend to be more robust than the approximations of derivatives by differentials, so integral equations are generally preferred.
Translation¶
The original paper is mostly in French, so an English translation is provided as part of the documentation of scikitguess. The translation can be read here:
 REGRESSIONS et EQUATIONS INTEGRALES
 Translator’s Note
 Regressions and Integral Equations
 Abstract
 1. Introduction
 2. Principle of Linearization Through Differential and/or Integral Equations
 3. Example: Illustration of the Gaussian Probability Density Function
 4. Discussion
 Appendix 1: Review of Linear Regression
 Appendix 2: Linear Regression of the Gaussian Cumulative Distribution Function
 NonLinear Regression of the Types: Power, Exponential, Logarithmic, Weibull
 Regression of Sinusoids
 1. Introduction
 2. Case Where is Known APriori
 3. Linearization Through an Integral Equation
 4. A Brief Analysis of Performance
 5. Further Optimizations Based on Estimates of and
 6. Final Steps and Results
 7. Discussion
 Appendix 1: Summary of Sinusoidal Regression Algorithm
 Appendix 2: Detailed Procedure for Sinusoidal Regression
 Application to the Logistic Distribution (Three Parameters)
 Application to the Logistic Distribution (Four Parameters)
 Mixed Linear and Sinusoidal Regression
 Generalized Sinusoidal Regression
 Damped Sinusoidal Regression
 Double Exponential Regression & Double Power Regression
 Multivariate Regression
 Supplementary Materials
Citation¶
[Jacquelin] 

Available online at https://www.scribd.com/doc/14674814/Regressionsetequationsintegrales.
A PDF is available with this documentation: Régressions et équations intégrales
.
Circle Fitting by Linear and Nonlinear Least Squares¶
This paper by Ian Coope demonstrates a way to linearize a nonlinear least squares problem.
Concept¶
Rather than solving the traditional nonlinear least squares problem for ndimensional circles, this paper proposes a change of variable that reduces the problem to a simple linear least squares. The change appears to yield more robust results in some cases. This is one of the multidimensional optimizations offered in the scikit.
Citation¶
[Coope]  I. D. Coope, “Circle fitting by linear and nonlinear least squares,” Journal of Optimization Theory and Applications, vol. 76, no. 2, pp. 381–388, 1993. 
Preprint available online at https://ir.canterbury.ac.nz/bitstream/handle/10092/11104/coope_report_no69_1992.pdf.
Image Analysis with Rapid and Accurate TwoDimensional Gaussian Fitting¶
This is one of two papers used to seed the idea for the ndimensional Gaussian estimators.
Concept¶
This paper introduces the idea of linearizing least squares fit to a the a two dimensional Gaussian function. It suggests using a weighted fit and performing thresholding on the image. The supplemental materials show a suggested asymmetrical weighting of
where is the standard deviation of the noise.
The paper also implies, but does not show, a crosscoupling term for rotated elliptical Gaussians.
Citation¶
[AnthonyGranick] 

Available online at http://groups.mrl.illinois.edu/granick/publications/pdf%20files/2009/Image_Analysis_with_2D_Gaussian_Fit_la900393v.pdf
Supplemental materials (MATLAB implementation) available at https://pubs.acs.org/doi/10.1021/la900393v
Supplement¶
A supplement deriving the linearized regression for the NDimensional case is provided for this paper and Star Centroiding Based on Fast Gaussian Fitting for Star Sensors. See the corresponding Star Centroiding Based on Fast Gaussian Fitting for Star Sensors section.
Star Centroiding Based on Fast Gaussian Fitting for Star Sensors¶
This paper uses a similar linearized regression to Image Analysis with Rapid and Accurate TwoDimensional Gaussian Fitting, but with materially different suggestions for weighting and thresholding, as well as a somewhat different intended application.
Concept¶
This paper suggests a twopass approach using a similar regression to Image Analysis with Rapid and Accurate TwoDimensional Gaussian Fitting, but with an initial pass to find highSNR pixels followed by a second pass to fine tune the results.
The suggested weighting derived here is the value of the pixel itself. This is the weighting used by the scikit by default.
This paper too only deals with ellipses without cross coupling and does not generalize to more than two dimensions.
Citation¶
[WanWangWeiLiZhang] 

Available online at https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6163372/
Supplement¶
The two papers on Gaussian centroiding provide the basic idea for Ndimensional
Gaussian fitting without quite getting there. The writeup below fills in a
couple of the missing steps. The math is fairly rudimentary, with the main
attraction being the vectorized implementation provided by skg.ngauss
.