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!

🐢