traj: Feature-Based Clustering of Longitudinal Trajectories

Description

Identifies clusters of individual longitudinal trajectories. In the spirit of Leffondre et al. (2004), the procedure involves identifying each trajectory to a point in the space of measures. In this context, a measure is a quantity meant to capture a certain characteristic feature of the trajectory. The points in the space of measures are then clustered using a version of spectral clustering.

Author(s)

Maintainer: Laurence Boulanger laurence.boulanger@umontreal.ca

Authors:

• Marie-Pierre Sylvestre marie-pierre.sylvestre@umontreal.ca

Other contributors:

• Gillis Delmas Tchouangue Dinkou [contributor]

• Dan Vatnik [contributor]

See Also

Useful links:

• https://CRAN.R-project.org/package=traj

Plot trajClusters object

Description

Plots the curves corresponding to (or closest to) the centroids of the clusters and plots a random sample from each groups.

Usage

## S3 method for class 'trajClusters'
plot(x, sample.size = 5, ask = TRUE, which.plots = NULL, ...)

scatterplots(x, ask = TRUE, ...)

CVIplot(x, ...)

Arguments

Examples

## Not run: 
data("trajdata")
trajdata.noGrp <- trajdata[, -which(colnames(trajdata) == "Group")] #remove the Group column

m = trajMeasures(trajdata.noGrp, ID = TRUE)
c3 = trajClusters(m, nclusters = 3)

plot(c3)

#The pointwise mean trajectories correspond to the third and fourth displayed plots.

c4 = trajClusters(m, nclusters = 4)

plot(c4, which.plots = 3:4)


## End(Not run)

Classify the Longitudinal Data Based on the Measures.

Description

Classifies the trajectories by applying a nonparametric clustering algorithm to the measures computed by trajMeasures().

Usage

trajClusters(
  Measures,
  select = NULL,
  fuzzy = FALSE,
  nclusters = NULL,
  nstart = 50
)

## S3 method for class 'trajClusters'
print(x, ...)

## S3 method for class 'trajClusters'
summary(object, ...)

Arguments

Details

The spectral clustering algorithm presented in Meila (2005) is implemented in which the similarity matrix S is built from a binary K nearest neighbors similarity function (S=(W+W^T)/2, where W_{ij}=1 if data point j is among the nearest points to data point i and W_{ij}=0 otherwise).

Value

An object of class trajClusters; a list containing the result of the clustering, as well as a curated form of the arguments. If nclusters is set to NULL, clustering is carried out for each number k of clusters between 2 and (up to) 8 and a plot is produced representing the value of three internal cluster validity indices (C-index, Calinski-Harabasz, Wemmert-Gancarski) as a function of k. As in the 'KmL' package of Genolini et al., these validity indices are presented on a scale from 0 to 1, with 1 corresponding to the highest validity score and 0 corresponding to the lowest. From this, a "best" value of k is determined using a ranked voting system.

References

Genolini, C. et al., kml: K-Means for Longitudinal Data, https://CRAN.R-project.org/package=kml

Meila, M., Spectral Clustering. Handbook of Cluster Analysis, Chapter 7, Chapman and Hall/CRC, 2005.

Examples

## Not run: 
data("trajdata")
trajdata.noGrp <- trajdata[, -which(colnames(trajdata) == "Group")] # remove the Group column

m = trajMeasures(trajdata.noGrp, ID = TRUE, measures = 1:19)

s2.3 <- trajClusters(m, nclusters = 3)
plot(s2.3)

#'s2.4 <- trajClusters(m, nclusters = 4)
plot(s2.4)

#'s2.5 <- trajClusters(m, nclusters = 5)
plot(s2.5)

groups <- s2.4 <- trajClusters(m, nclusters = 4)$partition

## End(Not run)

Compute Measures for Identifying Patterns of Change in Longitudinal Data

Description

trajMeasures computes up to 20 measures for each longitudinal trajectory. See Details for the list of measures.

Usage

trajMeasures(
  Data,
  Time = NULL,
  ID = FALSE,
  measures = c(1:10, 12:20),
  midpoint = NULL,
  cap.outliers = FALSE
)

## S3 method for class 'trajMeasures'
print(x, ...)

## S3 method for class 'trajMeasures'
summary(object, ...)

Arguments

Details

Each trajectory must have a minimum of 3 observations, otherwise it is omitted from the analysis. The 20 measures and their numerical identifiers are listed below. Please refer to the vignette for the specific formulas used to compute them.

1. Maximum
   

2. Minimum
   

3. Range
   

4. Mean value
   

5. Standard deviation
   

6. Slope of the affine approximation
   

7. Intercept of the affine approximation
   

8. Proportion of variance explained by the affine approximation
   

9. Rate of intersection with the best affine approximation
   

10. Net variation per unit of time
    

11. Late variation to early variation contrast
    

12. Total variation per unit time
    

13. Spikiness
    

14. Maximum of the first derivative
    

15. Minimum of the first derivative
    

16. Standard deviation of the first derivative
    

17. First derivative's net variation per unit of time
    

18. Maximum of the second derivative
    

19. Minimum of the second derivative
    

20. Standard deviation of the second derivative
    

If 'cap.outliers' is set to TRUE, Nishiyama's improved Chebychev bound for continuous distributions is used to determine extreme values for each measure, corresponding to a 0.3% probability threshold. Extreme values beyond the threshold are then capped to the 0.3% probability threshold (see vignette for more details).

Value

An object of class trajMeasures; a list containing the values of the measures, a table of the outliers which have been capped, as well as a curated form of the function's arguments.

References

Leffondre K, Abrahamowicz M, Regeasse A, Hawker GA, Badley EM, McCusker J, Belzile E. Statistical measures were proposed for identifying longitudinal patterns of change in quantitative health indicators. J Clin Epidemiol. 2004 Oct;57(10):1049-62. doi: 10.1016/j.jclinepi.2004.02.012. PMID: 15528056.

Nishiyama T, Improved Chebyshev inequality: new probability bounds with known supremum of PDF, arXiv:1808.10770v2 stat.ME https://doi.org/10.48550/arXiv.1808.10770

Examples

## Not run: 
data("trajdata")
trajdata.noGrp <- trajdata[, -which(colnames(trajdata) == "Group")] #remove the Group column

m1 = trajMeasures(trajdata.noGrp, ID = TRUE, measures = 19, midpoint = NULL)
m2 = trajMeasures(trajdata.noGrp, ID = TRUE, measures = 19, midpoint = 3)

identical(m1$measures, m2$measures)

## End(Not run)

Select a Subset of the Measures Using a Similarity Index on the Set of Clusterings

Description

This function examines the effect of reducing the number of measures on which the trajectories are clustered. Specifically, starting from a clustering C in the form of an object of class trajClusters and a choice of a similarity index to compare clusterings, this function finds the subset of measures which results in the clustering most similar to C.

Usage

trajReduce(Measures, Clusters, index = "ARI", keep = 3)

Arguments

Details

The Rand index ranges from 0 to 1 with 0 indicating identical clusters and 1 indicating maximally different clusters. The normalized variation of information distance (nVId) and normalized split-join distance (nSJd) and have the opposite interpretation with 0 indicating maximally different clusters and 1 indicating identical clusters. Therefor, to facilitate comparison, we plot 1 - nVId (resp. 1 - nSJd) instead of nVId (resp. nSJd).

References

Hubert L, Arabie P. Comparing partitions. Journal of Classification 2:193-218, 1985.

Meila M. Comparing clusterings – an information based distance. Journal of Multivariate Analysis, 98, pp 873-895, 2007.

van Dongen S. Performance criteria for graph clustering and Markov cluster experiments. Technical Report INS-R0012, National Research Institute for Mathematics and Computer Science in the Netherlands, Amsterdam, May 2000.

Examples

## Not run: 
data("trajdata")
trajdata.noGrp <- trajdata[, -which(colnames(trajdata) == "Group")] #remove the Group column

m = trajMeasures(trajdata.noGrp, ID = TRUE)
trajReduce(m)

## End(Not run)

trajdata

Description

An artificially created data set with 130 trajectories split into four groups, labelled A, B, C, D according to the data generating process.

Usage

trajdata

Format

This data frame has 130 rows and the following 7 columns:

ID

An identification variable that runs from 1 to 130.

Group

A character variable that's either "A", "B", "C" or "D" depending on which of the four data generating process the trajectory is coming from.

X1

The observation of the trajectory at time t = 1.

X2

The observation of the trajectory at time t = 2.

X3

The observation of the trajectory at time t = 3.

X4

The observation of the trajectory at time t = 4.

X5

The observation of the trajectory at time t = 5.

X6

The observation of the trajectory at time t = 6.