Dynamic variation and variation decomposition

• Function dynamicVariation assesses the amount of dynamic variation observed across trajectories and the relative contribution of each of them.

• Function variationDecomposition performs a sum of squares decomposition of total variation in three components: (1) across trajectories (entities); (2) across time points; (3) their interaction.

Usage

dynamicVariation(x, ...)

variationDecomposition(x)

Arguments

x

An object of class trajectories (or its children subclasses fd.trajectories or cycles).

...

Additional params to be passed to function trajectoryDistances.

Value

• Function dynamicVariance returns a list with three elements (dynamic sum of squares, dynamic variance and a vector of trajectory relative contributions)

• Function variationDecomposition returns a data frame with results (sum of squares, degrees of freedom and variance estimates) for each variance component and the total.

Details

Function variationDecomposition requires trajectories to be synchronous. The SS sum of temporal and interaction components correspond to the SS sum, across trajectories, of function trajectoryInternalVariation.

See also

defineTrajectories, is.synchronous, trajectoryDistances, trajectoryInternalVariation

Examples

#Description of entities and surveys
entities <- c("1","1","1","1","2","2","2","2","3","3","3","3")
surveys <- c(1,2,3,4,1,2,3,4,1,2,3,4)
  
#Raw data table
xy<-matrix(0, nrow=12, ncol=2)
xy[2,2]<-1
xy[3,2]<-2
xy[4,2]<-3
xy[5:6,2] <- xy[1:2,2]
xy[7,2]<-1.5
xy[8,2]<-2.0
xy[5:6,1] <- 0.25
xy[7,1]<-0.5
xy[8,1]<-1.0
xy[9:10,1] <- xy[5:6,1]+0.25
xy[11,1] <- 1.0
xy[12,1] <-1.5
xy[9:10,2] <- xy[5:6,2]
xy[11:12,2]<-c(1.25,1.0)

d <- dist(xy)

# Defines trajectories
x <- defineTrajectories(d, entities, surveys)

# Assessment of dynamic variation and individual trajectory contributions
dynamicVariation(x)
#> $dynamic_ss
#> [1] 0.7114251
#> 
#> $dynamic_variance
#> [1] 0.3557125
#> 
#> $relative_contributions
#>          1          2          3 
#> 0.51366204 0.06351261 0.42282535 
#> 

# Variation decomposition (entity, temporal and interaction) for synchronous 
# trajectories:
variationDecomposition(x)
#>                    ss df  variance
#> entities     2.489583  2 1.2447917
#> time         7.453125  3 2.4843750
#> interaction  1.718750  6 0.2864583
#> total       11.661458 11 1.0601326

# check the correspondence with internal variation
sum(variationDecomposition(x)[c("time", "interaction"),"ss"])
#> [1] 9.171875
sum(trajectoryInternalVariation(x)$internal_ss)
#> [1] 9.171875