Scalar datasets

Scalar datasets should be have the form shown below, with each variable (time included) contained in each column of the table.

If users wish they can add uncertainty to scalar data using one of two methods: Square root uncertainty or Poisson uncertainty. The square root approach transforms each data point x into an interval [ x + a*sqrt(x), x - a*sqrt(x)] where a is a constant determined by the user (set to 2 by default). Poisson uncertainty assumes a poisson counting process while collecting the data, and calculates the intervals based on a percent confidence level given by the user (set to 95% by default).

Download Example Dataset

Interval datasets

Interval datasets should take the form of a 3D array, consisting of two tables such as the one shown above. The first table should contain the left endpoints of the dataset, and the second should contain the right endpoints of the dataset.

Download Example Dataset

Monte Carlo Simulations

As with interval datasets, Monte Carlo simulation results should take the form of a 3D array, consisting of n tables, where each table is an individual replicate.

Once the data has been uploaded, users can select which variable denotes time, and which (if any) denote comparison variables. Users can then choose to 'Commit' the dataset, in which case they must fill in the required fields (such as title, description, and location). They can also choose to use the dataset privately, in which case they should only press the 'Upload' button. Upon uploading, users should see a preview of the dataset in a table at the bottom of the page.

Download Example Dataset

Converting Scalar Datasets to Interval Data

If a user's dataset is scalar, there are two options for converting it into an interval dataset:

Square Root Method

This method, given a parameter p (usually between 1 and 3), sends each data point x to x +/- p*sqrt(x)

Poisson Confidence Interval Method

This method assumes a Poisson counting process when collecting abundance data, from this it can calculate percent confidence intervals for each of the abundance values. Users can input any value from 1-99.9%.

Interval Datasets

Interval datasets, as explained in the 'Uploading data' pane, consist of left and right endpoints for each abundance value (lower and upper bounds, respectively). Using interval arithmetic, the application can calculate lower and upper bounds on the shape change measures discussed by Spencer.

For example, suppose we want to 'intervalize' the equation for proportional growth rates of a species, given in the following equation: (note that x_i(t) represents the abundance of the i^th species at time t)

$$r_i = \frac{log({x_i}(t + \Delta t)) - log({x_i}(t))}{\Delta t}$$

We'll start by defining an interval x:

$$x = [\underline{x},\bar{x}]$$

Where x represents the lower bound of x, and x represents the upper bound of x.

We see that to minimize r_i, we must take the minimum value of log(x_i(t + dt)) and the maximum value of log(x_i(t)):

$$\underline{r_i} = \frac{log(\underline{x_i}(t + \Delta t)) - log(\bar{x_i}(t))}{\Delta t}$$

Likewise, in order to maximize r_i, we must take the maximum value of log(x_i(t + dt)) and the minimum value of log(x_i(t)):

$$\bar{r_i} = \frac{log(\bar{x_i}(t + \Delta t)) - log(\underline{x_i}(t))}{\Delta t}$$

Using these two formulas for the lower and upper bounts of r_i, we can then calculate the associated interval value of r_i.

Monte Carlo Datasets

Monte Carlo datasets are generated by the user, and uploaded to the application. For each iterate, the shape change measures are calculated, and saved as a separate iterate. At the end, results are displayed for all of the iterates at once. Users can also apply a confidence interval to these results using the Highest Posterior Density interval, effectively converting the Monte Carlo results into Interval-valued results.

Abundances

This category contains abundances for each of the species. The user can select multiple abundances to plot against time alongside one another, or only one species. If users wish to see the log abundances (by default abundances are stored as their real values), they can simply select the 'log transform' option in the sidebar to the right.

Proportional Growth Rates

This category contains the proportional growth rates of each of the species, which gives us an idea of how rapidly a species is increasing or decreasing in abundance. Letting x_i be the abundance of species i, the proportional growth rate r_i of the i^th species at time t is defined as follows:

$$r_i = \frac{log({x_i}(t + \Delta t)) - log({x_i}(t))}{\Delta t}$$

Relative Abundances

This category contains the relative abundances of each species at each point in time. Let p_i(t) represent the relative abundance of the i^th species at time t: (n is the number of species)

$$p_i = \frac{{x_i}(t)}{\sum\limits_{i=1}^n {x_i}(t)}$$

Unit Sphere Projections

This is a slightly more esoteric way of describing species abundances, and was first proposed by Jassby and Goldman (1974)⁴. Let e_i(t) be the sphere projection of the i^th species at time t: (This is used in Jassby and Goldman's shape change measure, the formula for which is given later on in this section)

$$e_i = \frac{{x_i}(t)}{\sqrt{\sum\limits_{i=1}^n {{x_i}(t)}^2}}$$

Shape Change Measures

This category contains the 5 measures that make up the focus of this application. These measures are meant to measure how the 'shape' of a community changes over time.

Spencer

This measure was created by Matthew Spencer¹ and is given by the following formula:

$${s_r}(t) = \sqrt{\frac{1}{n - 1}\sum\limits_{i=1}^n ({r_i}(t) - \bar{r(t)})^2}$$

where

$$r(t) = \sum\limits_{i=1}^n {r_i}(t)$$

Foster and Tilman

This measure was described by Foster and Tilman (2000)³ and is given by the following formula:

$${s_r}(t) = \frac{1}{\Delta t}\sqrt{\sum\limits_{i=1}^n ({p_i}(t + \Delta t) - {p_i}(t))^2}$$

Jassby and Goldman

This measure was described by Jassby and Goldman (1974)⁴ and is given by the following formula:

$${s_r}(t) = \sqrt{\sum\limits_{i=1}^n (\frac{({e_i}(t + \Delta t) - {e_i}(t))}{\Delta t})^2}$$

Lewis

This measure was described by Lewis (1978)⁶ and is given by the following formula:

$${s_r}(t) = \frac{1}{\Delta t}\sum\limits_{i=1}^n |{p_i}(t + \Delta t) - {p_i}(t)|$$

Bray-Curtis Field

This measure was described in Bray-Curtis (Field et al., 1982)⁵ and is given by the following formula:

$${s_r}(t) = \frac{1}{\Delta t}\frac{\sum\limits_{i=1}^n |{k_i}(t + \Delta t) - {k_i}(t)|}{\sum\limits_{i=1}^n |{k_i}(t + \Delta t) + {k_i}(t)|}$$

where

$${k_i}(t) = [{p_i}(t)]^{\frac{1}{4}} $$

Rate of Size Change

This isn't a measure of shape change - rather, it is a measure of how fast the community is growing. It is given by the following formula:

$$s(t) = \frac{\sum\limits_{i=1}^n {p_i}(t) }{n}$$

Surreal

In his paper, Spencer discussed an approach to dealing with extinction events that used surreal numbers. This charting option is only available for Spencer's shape change measure on scalar data, and consists of a line plot, where each point along the line is accompanied by an arrow. The length of the arrow is determined by the real part of the surreal number, and the position of the point is determined by the infinite part of the surreal number. More details on this method can be found in Spencer's paper.