ser_dist {seriation}

Dissimilarities and Correlations Between Seriation Orders

Description

Calculates dissimilarities/correlations between seriation orders in a list of type ser_permutation_vector.

Usage

ser_dist(x, y = NULL, method = "spearman", reverse = TRUE, ...)

ser_cor(x, y = NULL, method = "spearman", reverse = TRUE, test = FALSE)

ser_align(x, method = "spearman")

Arguments

x

set of seriation orders as a list with elements which can be coerced into ser_permutation_vector objects.

y

if not NULL then a single seriation order can be specified. In this case x has to be a single seriation order and not a list.

method

a character string with the name of the used measure. Available measures are: "kendall", "spearman", "manhattan", "euclidean", "hamming", "ppc" (positional proximity coefficient), and "aprd" (absolute pairwise rank differences).

reverse

a logical indicating if the orders should also be checked in reverse order and the best value (highest correlation, lowest distance) is reported. This only affect ranking-based measures and not precedence invariant measures (e.g., "ppc", "aprd").

...

Further arguments passed on to the method.

test

a logical indicating if a correlation test should be performed.

Details

ser_cor() calculates the correlation between two sequences (orders). Note that a seriation order and its reverse are identical and purely an artifact due to the method that creates the order. This is a major difference to rankings. For ranking-based correlation measures (Spearman and Kendall) the absolute value of the correlation is returned for reverse = TRUE (in effect returning the correlation for the reversed order). If test = TRUE then the appropriate test for association is performed and a matrix with p-values is returned as the attribute "p-value". Note that no correction for multiple testing is performed.

For ser_dist(), the correlation coefficients (Kendall's tau and Spearman's rho) are converted into a dissimilarity by taking one minus the correlation value. Note that Manhattan distance between the ranks in a linear order is equivalent to Spearman's footrule metric (Diaconis 1988). reverse = TRUE returns the pairwise minima using also reversed orders.

The positional proximity coefficient (ppc) is a precedence invariant measure based on product of the squared positional distances in two permutations defined as (see Goulermas et al 2016):

d_{ppc}(R, S) = 1/h \sum_{j=2}^n \sum_{i=1}^{j-1} (\pi_R(i)-\pi_R(j))^2 * (\pi_S(i)-\pi_S(j))^2,

where R and S are two seriation orders, pi_R and pi_S are the associated permutation vectors and h is a normalization factor. The associated generalized correlation coefficient is defined as 1-d_{ppc}. For this precedence invariant measure reverse is ignored.

The absolute pairwise rank difference (aprd) is also precedence invariant and defined as a distance measure:

d_{aprd}(R, S) = \sum_{j=2}^n \sum_{i=1}^{j-1} | |\pi_R(i)-\pi_R(j)| - |\pi_S(i)-\pi_S(j)| |^p,

where p is the power which can be passed on as parameter p and is by default set to 2. For this precedence invariant measure reverse is ignored.

ser_align() tries to normalize the direction in a list of seriations such that ranking-based methods can be used. We add for each permutation also the reversed order to the set and then use a modified version of Prim's algorithm for finding a minimum spanning tree (MST) to choose if the original seriation order or its reverse should be used. We use the orders first added to the MST. Every time an order is added, its reverse is removed from the possible remaining orders.

Value

• ser_dist() returns an object of class stats::dist.

• ser_align() returns a new list with elements of class ser_permutation.

Author(s)

Michael Hahsler

References

P. Diaconis (1988): Group Representations in Probability and Statistics, Institute of Mathematical Statistics, Hayward, CA.

J.Y. Goulermas, A. Kostopoulos, and T. Mu (2016): A New Measure for Analyzing and Fusing Sequences of Objects. IEEE Transactions on Pattern Analysis and Machine Intelligence 38(5):833-48. doi:10.1109/TPAMI.2015.2470671

See Also

Other permutation: get_order(), permutation_vector2matrix(), permute(), ser_permutation(), ser_permutation_vector()

Examples

set.seed(1234)
## seriate dist of 50 flowers from the iris data set
data("iris")
x <- as.matrix(iris[-5])
x <- x[sample(1:nrow(x), 50), ]
rownames(x) <- 1:50
d <- dist(x)

## Create a list of different seriations
methods <- c("HC_complete", "OLO", "GW", "VAT",
  "TSP", "Spectral", "MDS", "Identity", "Random")

os <- sapply(methods, function(m) {
  cat("Doing", m, "... ")
  tm <- system.time(o <- seriate(d, method = m))
  cat("took", tm[3],"s.\n")
  o
})

## Compare the methods using distances. Default is based on
## Spearman's rank correlation coefficient where reverse orders are
## also considered.
ds <- ser_dist(os)
hmap(ds, margin = c(7,7))

## Compare using correlation between orders. Reversed orders have
## negative correlation!
cs <- ser_cor(os, reverse = FALSE)
hmap(cs, margin = c(7,7))

## Compare orders by allowing orders to be reversed.
## Now all but random and identity are highly positive correlated
cs2 <- ser_cor(os, reverse = TRUE)
hmap(cs2, margin=c(7,7))

## A better approach is to align the direction of the orders first
## and then calculate correlation.
os_aligned <- ser_align(os)
cs3 <- ser_cor(os_aligned, reverse = FALSE)
hmap(cs3, margin = c(7,7))

## Compare the orders using clustering. We use Spearman's foot rule
## (Manhattan distance of ranks). In order to use rank-based method,
## we align the direction of the orders.
os_aligned <- ser_align(os)
ds <- ser_dist(os_aligned, method = "manhattan")
plot(hclust(ds))