Formulas used by fit_gaussian_2D

Overview

This document will provide specific details of 2D-Gaussian equations used by the different method options within gaussplotR::fit_gaussian_2D().

method = "elliptical"

Using method = "elliptical" fits a two-dimensional, elliptical Gaussian equation to gridded data.

\[G(x,y) = A_o + A * e^{-U/2}\]

where G is the value of the 2D-Gaussian at each \({(x,y)}\) point, \(A_o\) is a constant term, and \(A\) is the amplitude (i.e. scale factor).

The elliptical function, \(U\), is:

\[U = (x'/a)^{2} + (y'/b)^{2}\]

where \(a\) is the spread of Gaussian along the x-axis and \(b\) is the spread of Gaussian along the y-axis.

\(x'\) and \(y'\) are defined as:

\[x' = (x - x_0)cos(\theta) - (y - y_0)sin(\theta)\] \[y' = (x - x_0)sin(\theta) + (y - y_0)cos(\theta)\] where \(x_0\) is the center (peak) of the Gaussian along the x-axis, \(y_0\) is the center (peak) of the Gaussian along the y-axis, and \(\theta\) is the rotation of the ellipse from the x-axis in radians, counter-clockwise.

Therefore, all together:

\[G(x,y) = A_o + A * e^{-((((x - x_0)cos(\theta) - (y - y_0)sin(\theta))/a)^{2}+ (((x - x_0)sin(\theta) + (y - y_0)cos(\theta))/b)^{2})/2}\]

Setting the constrain_orientation argument to a numeric will optionally constrain the value of \(\theta\) to a user-specified value. If a numeric is supplied here, please note that the value will be interpreted as a value in radians. Constraining \(\theta\) to a user-supplied value can lead to considerably poorer-fitting Gaussians and/or trouble with converging on a stable solution; in most cases constrain_orientation should remain its default: "unconstrained".

method = "elliptical_log"

The formula used in method = "elliptical_log" uses the modification of a 2D Gaussian fit used by Priebe et al. 2003¹.

\[G(x,y) = A * e^{(-(x - x_0)^2)/\sigma_x^2} * e^{(-(y - y'(x)))/\sigma_y^2}\]

and

\[y'(x) = 2^{(Q+1) * (x - x_0) + y_0}\] where \(A\) is the amplitude (i.e. scale factor), \(x_0\) is the center (peak) of the Gaussian along the x-axis, \(y_0\) is the center (peak) of the Gaussian along the y-axis, \(\sigma_x\) is the spread along the x-axis, \(\sigma_y\) is the spread along the y-axis and \(Q\) is an orientation parameter.

Therefore, all together:

\[G(x,y) = A * e^{(-(x - x_0)^2)/\sigma_x^2} * e^{(-(y - (2^{(Q+1) * (x - x_0) + y_0})))/\sigma_y^2}\]

This formula is intended for use with log2-transformed data.

Setting the constrain_orientation argument to a numeric will optionally constrain the value of \(Q\) to a user-specified value, which can be useful for certain kinds of analyses (see Priebe et al. 2003 for more). Keep in mind that constraining \(Q\) to a user-supplied value can lead to considerably poorer-fitting Gaussians and/or trouble with converging on a stable solution; in most cases constrain_orientation should remain its default: "unconstrained".

method = "circular"

This method uses a relatively simple formula:

\[G(x,y) = A * e^{(-( ((x-x_0)^2/2\sigma_x^2) + ((y-y_0)^2/2\sigma_y^2)) )}\]

where \(A\) is the amplitude (i.e. scale factor), \(x_0\) is the center (peak) of the Gaussian along the x-axis, \(y_0\) is the center (peak) of the Gaussian along the y-axis, \(\sigma_x\) is the spread along the x-axis, and \(\sigma_y\) is the spread along the y-axis.

That’s all!

🐢