# expectation.time: calculate the expected (i.e. mean) Euclidean distances... In EvoRAG: Evolutionary Rates Across Gradients

## Description

calculate the expected (i.e. mean) Euclidean distances through time given a rate of evolution, Beta.

## Usage

 ```1 2``` ``` expectation.time(Beta, Alpha="NULL", time.span=c(0, 10), values=TRUE, plot=TRUE, quantile=FALSE) ```

## Arguments

 `Beta` Evolutionary rate parameter to plot `Alpha` Evolutionary constrain parameter tom plot (for OU model only). Leave as "NULL" to implement the BM model `time.span` A vector of length 1 if the expectation is calcuated for a single time; length 2 if to be calculated over a range from 0 to an upper value chosen by the user; or length > 0, where the user supplies 3 or more times over which to calculate the Expectation. `values` TRUE (null) returns the values in matrix form. `plot` Plot the expected (solid line) Euclidean distance and optionally quantiles for a given Beta. `quantile` Calculate (and optionally plot) the expected quantiles (0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.99) for Euclidean distances under a given Beta.

## Details

This function calculates the expectation (i.e. mean value under a half normal distribution) for Eculidean distance across a time range and optionally the quantiles (0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.99).

## Value

Returns a matrix with 3 columns corresponding to L, T and simulated E, and an additional 11 columns with quantiles if qualtiles=TRUE. If plot=TRUE, the expectation (solid line) and optionally the quantiles (dashed lines) are plotted.

Jason T. Weir

## References

Weir JT, D Wheatcroft, & T Price. 2012. The role of ecological constraint in driving the evolution of avian song frequency across a latitudinal gradient. Evolution 66, 2773-2783.

Weir JT, & D Wheatcroft. 2011. A latitudinal gradient in rates of evolution of avian syllable diversity and song length. Proceedings of the Royal Society of London, B 278, 1713-1720.

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19``` ```##Example 1 ###Compare data simulated under BM_null to the expectation and quantiles TIME = c(0:100) * 0.1 GRAD = (0:100)*0 #BM_null does not require GRAD, thus simply make a dummy set of GRAD DATA1 <- sim.sisters(TIME=TIME, GRAD=GRAD, parameters = c(0.1), model=c("BM_null"), MULT=10) expectation.time(Beta=0.1, Alpha="NULL", time.span=c(0, 10), values=FALSE, plot=TRUE, quantile=TRUE) points(DATA1[,3] ~ DATA1[,2], col="black", cex=0.4) ##Example 2 ###Compare data simulated under OU_null to the expectation and quantiles TIME = c(0:100) * 0.1 GRAD = (0:100)*0 #GRAD is not required by these models, so a dummy set of GRAD are provided DATA1 <- sim.sisters(TIME=TIME, GRAD=GRAD, parameters = c(0.1, 1), model=c("OU_null"), MULT=10) expectation.time(Beta=0.1, Alpha=1, time.span=c(0, 10), values=FALSE, plot=TRUE, quantile=TRUE) points(DATA1[,3] ~ DATA1[,2], col="black", cex=0.4) ```