Author: Vikram B. Baliga
Date: 2021-05-02

# Fitting 2D-Gaussians to Data

#### Vikram B. Baliga

#### 2021-05-02

## Overview

The function `fit_gaussian_2D()` can be used fit 2D-Gaussians to data, and has several methods for how the fitting is implemented. This vignette will run you through what these methods mean with worked examples.

We’ll begin by loading `gaussplotR` and loading the sample data set provided within. The raw data we’d like to use are in columns 1:3, so we’ll shave the data set down to those columns before running through the examples.

    library(gaussplotR)

    ## We'll also use lattice, ggplot2 and metR
    library(lattice); library(ggplot2); library(metR)
    #> Warning: package 'ggplot2' was built under R version 4.0.5
    #> Warning: package 'metR' was built under R version 4.0.5

    ## Load the sample data set
    data(gaussplot_sample_data)

    ## The raw data we'd like to use are in columns 1:3
    samp_dat <-
      gaussplot_sample_data[,1:3]

It generally helps to plot the data beforehand to get a sense of its overall shape. We’ll simply produce a contour plot.

    lattice::levelplot(
      response ~ X_values * Y_values,
      data = samp_dat,
      col.regions = colorRampPalette(
        c("white", "blue")
        )(100),
      xlim = c(-5, 0),
      ylim = c(-1, 4),
      asp = 1
    )

## The `method` and `constrain_orientation` arguments

### `method`

`gaussplotR::fit_gaussian_2D()` has three main options for its `method` argument: 1) `"elliptical"`, 2) `"elliptical_log"`, or 3) `"circular"`.

The most generic method (and the default) is `method = "elliptical"`. This allows the fitted 2D-Gaussian to take an ellipsoid shape. If you would like the best-fitting 2D-Gaussian, this is most likely your best bet.

A slightly-altered method to fit an ellipsoid Gaussian is available in `method = "elliptical_log"`. This method follows Priebe et al. 20031 and is geared towards use with log2-transformed data.

A third option is `method = "circular"`. This produces a very simple 2D-Gaussian that is constrained to have to have a roughly circular shape (i.e. spread in X- and Y- are roughly equal).

An additional argument, `constrain_orientation` gives additional control over the orientation of the fitted Gaussian. By default, the `constrain_orientation` is `"unconstrained"`, meaning that the best-fit orientation is returned.

### `constrain_orientation`

Setting constrain_orientation to a numeric (e.g. `constrain_orientation = pi/2`) will force the orientation of the Gaussian to the specified value, but this is only available when using `method = "elliptical"` or `method = "elliptical_log"`

Note that supplying a numeric to `constrain_orientation` is handled differently by `method = "elliptical"` vs `method = "elliptical_log"`. With `method = "elliptical"`, a `theta` parameter dictates the rotation, in radians, from the x-axis in the clockwise direction. Thus, using `method = "elliptical", constrain_orientation = pi/2` will return parameters for an elliptical 2D-Gaussian that is constrained to a 90-degree (pi/2) orientation. In contrast, the `method = "elliptical_log"` procedure uses a `Q` parameter to determine the orientation of the 2D-Gaussian. Setting `method = "elliptical_log", constrain_orientation = 0` will result in a diagonally-oriented Gaussian, whereas setting `constrain_orientation = -1` will result in horizontal orientation. Again, see Priebe et al. 2003 for more details.

## Example 1: Unconstrained elliptical

Unconstrained ellipticals are the default option and are generally recommended for most purposes. Here’s an example:

    gauss_fit_ue <-
        fit_gaussian_2D(samp_dat)

    gauss_fit_ue
    #> Model coefficients
    #>   A_o   Amp theta X_peak Y_peak    a    b
    #>  0.83 32.25  3.58  -2.64   2.02 0.91 0.96
    #> Model error stats
    #>     rss rmse deviance AIC
    #>  156.23 2.08   156.23 171
    #> Fitting methods
    #>          method       amplitude     orientation
    #>    "elliptical" "unconstrained" "unconstrained"
    attributes(gauss_fit_ue)
    #> $names
    #> [1] "coefs"             "model"             "model_error_stats"
    #> [4] "fit_method"
    #>
    #> $gaussplotR
    #> [1] "gaussplotR_fit"
    #>
    #> $class
    #> [1] "list"           "gaussplotR_fit"

Fitting an unconstrained ellipse returns an object (here: `gauss_fit_ue`) that is a `data.frame` with one column per fitted parameter. The fitted parameters are: `A_o` (a constant term), `Amp` (amplitude), `theta` (rotation, in radians, from the x-axis in the clockwise direction), `X_peak` (x-axis peak location), `Y_peak` (y-axis peak location), `a` (width of Gaussian along x-axis), and `b` (width of Gaussian along y-axis).

Note that the `data.frame` in `gauss_fit_ue$fit_method` indicates the fitting method and whether amplitude and/or orientation were constrained. This `data.frame` is used by `predict_gaussian_2D()` to automatically determine what method (and therefore, identity of parameters) was used and then sample points from that fitted Gaussian.

We can elect to sample more points from the fitted Gaussian by feeding in a grid of x- and y- values on which to predict (via `expand.grid()`.

Then, the fitted object `gauss_fit_ue` along with the grid of points can be supplied to `predict_gaussian_2D` to sample more points from the fit, which can be useful for plotting.

    ## Generate a grid of x- and y- values on which to predict
    grid <-
      expand.grid(X_values = seq(from = -5, to = 0, by = 0.1),
                  Y_values = seq(from = -1, to = 4, by = 0.1))

    ## Predict the values using predict_gaussian_2D
    gauss_data_ue <-
      predict_gaussian_2D(
        fit_object = gauss_fit_ue,
        X_values = grid$X_values,
        Y_values = grid$Y_values,
      )

    ## Plot via ggplot2 and metR
    ggplot_gaussian_2D(gauss_data_ue)

## Example 2: Constrained elliptical

As noted above, the `constrain_orientation` can be used to dictate the orientation. Please note that this will very likely result in a poorer fit, but may be useful for certain types of analyses.

Here we’ll force the Gaussian to be horizontally-oriented.

    gauss_fit_ce <-
        fit_gaussian_2D(samp_dat,
                        constrain_orientation = 0)

    gauss_fit_ce
    #> Model coefficients
    #>   A_o   Amp theta X_peak Y_peak   a    b
    #>  1.26 24.77     0  -2.53   1.95 1.2 1.33
    #> Model error stats
    #>     rss rmse deviance    AIC
    #>  885.83 4.96   885.83 231.47
    #> Fitting methods
    #>          method       amplitude     orientation
    #>    "elliptical" "unconstrained"   "constrained"

We’ll use the same grid of x- and y- points as above

    ## Predict the values using predict_gaussian_2D
    gauss_data_ce <-
      predict_gaussian_2D(
        fit_object = gauss_fit_ce,
        X_values = grid$X_values,
        Y_values = grid$Y_values,
      )

    ## Plot via ggplot2 and metR
    ggplot_gaussian_2D(gauss_data_ce)

## Example 3: Unconstrained elliptical_log

This procedure follows the formula used in Priebe et al. 2003 and is geared towards log2-transformed data (which the example data are).

Parameters from this model include: `Amp` (amplitude), `Q` (orientation parameter), `X_peak` (x-axis peak location), `Y_peak` (y-axis peak location), `X_sig` (spread along x-axis), and `Y_sig` (spread along y-axis).

    gauss_fit_uel <-
        fit_gaussian_2D(samp_dat,
                        method = "elliptical_log")

    gauss_fit_uel
    #> Model coefficients
    #>    Amp     Q X_peak Y_peak X_sig Y_sig
    #>  32.23 -0.19  -2.63   2.01  2.11  1.42
    #> Model error stats
    #>     rss rmse deviance    AIC
    #>  169.88 2.17   169.88 172.02
    #> Fitting methods
    #>           method        amplitude      orientation
    #> "elliptical_log"  "unconstrained"  "unconstrained"

    ## Predict the values using predict_gaussian_2D
    gauss_data_uel <-
      predict_gaussian_2D(
        fit_object = gauss_fit_uel,
        X_values = grid$X_values,
        Y_values = grid$Y_values,
      )

    ## Plot via ggplot2 and metR
    ggplot_gaussian_2D(gauss_data_uel)

## Example 4: Constrained elliptical_log

Similar to the above, but here the `constrain_orientation` can be used to dictate the value of the `Q` parameter used in Priebe et al. 2003. Setting `Q` to 0 will result in a diagonally-oriented Gaussian, whereas setting `Q` to -1 will result in horizontal orientation. `Q` is a continuous parameter, so values in between may be used as well, such as in this example:

    gauss_fit_cel <-
        fit_gaussian_2D(
          samp_dat,
          method = "elliptical_log",
          constrain_orientation = -0.66
        )

    gauss_fit_cel
    #> Model coefficients
    #>    Amp     Q X_peak Y_peak X_sig Y_sig
    #>  40.37 -0.66   -2.6   2.06  1.85  1.25
    #> Model error stats
    #>    rss rmse deviance    AIC
    #>  406.1 3.36    406.1 201.39
    #> Fitting methods
    #>           method        amplitude      orientation
    #> "elliptical_log"  "unconstrained"    "constrained"

    ## Predict the values using predict_gaussian_2D
    gauss_data_cel <-
      predict_gaussian_2D(
        fit_object = gauss_fit_cel,
        X_values = grid$X_values,
        Y_values = grid$Y_values,
      )

    ## Plot via ggplot2 and metR
    ggplot_gaussian_2D(gauss_data_cel)

Again, setting the value of `Q` via `constrain_orientation` will very likely result in poorer-fitting Gaussians. See the analyses in Priebe et al. 2003 to get a sense of useful applications of this approach. Forcing Q = -0.66 in the above example isn’t all that useful, but goes to show that it can be done.

## Example 5: Circular

Using `method = "circular"` constrains the Gaussian to have a roughly circular shape (i.e. spread in X- and Y- are roughly equal).

If this method is used, the fitted parameters are: `Amp` (amplitude), `X_peak` (x-axis peak location), `Y_peak` (y-axis peak location), `X_sig` (spread along x-axis), and `Y_sig`(spread along y-axis).

    gauss_fit_cir <-
        fit_gaussian_2D(samp_dat,
                        method = "circular")

    gauss_fit_cir
    #> Model coefficients
    #>    Amp X_peak Y_peak X_sig Y_sig
    #>  23.18  -2.55   1.81  1.32  1.64
    #> Model error stats
    #>     rss rmse deviance    AIC
    #>  899.61    5   899.61 230.03
    #> Fitting methods
    #>          method       amplitude     orientation
    #>      "circular" "unconstrained"              NA

    ## Predict the values using predict_gaussian_2D
    gauss_data_cir <-
      predict_gaussian_2D(
        fit_object = gauss_fit_cir,
        X_values = grid$X_values,
        Y_values = grid$Y_values,
      )

    ## Plot via ggplot2 and metR
    ggplot_gaussian_2D(gauss_data_cir)

That’s all!

🐢

1. Priebe NJ, Cassanello CR, Lisberger SG. The neural representation of speed in macaque area MT/V5. J Neurosci. 2003 Jul 2;23(13):5650-61. doi: 10.1523/JNEUROSCI.23-13-05650.2003.↩︎