`Intersect()` or `Jaccard()` compute the kernel functions of the same name, which are useful for set data. Their input is a matrix or data.frame with dimension NxD, where N>1, D>0. Samples should be in the rows and features in the columns. When there is a single feature, `Jaccard()` returns 1 if the elements of the set are exactly the same in two given samples, and 0 if they are completely different (see Details). Instead, in the multivariate case (D>1), the results (for both `Intersect()` and `Jaccard()`) of the D features are combined with a sum, a mean, or a weighted mean.
Usage
Jaccard(X, elements = LETTERS, comp = "sum", coeff = NULL)
Intersect(
X,
elements = LETTERS,
comp = "sum",
coeff = NULL,
feat_space = FALSE
)Arguments
- X
Matrix (class "character") or data.frame (class "character", or columns = "factor"). The elements in X are assumed to be categorical in nature.
- elements
All potential elements (symbols) that can appear in the sets. If there are some elements that are not of interest, they can be excluded so they are not taken into account by these kernels. (Defaults: LETTERS).
- comp
When D>1, this argument indicates how the variables of the dataset are combined. Options are: "mean", "sum" and "weighted". (Defaults: "mean")
"sum" gives the same importance to all variables, and returns an unnormalized kernel matrix.
"mean" gives the same importance to all variables, and returns a normalized kernel matrix (all its elements range between 0 and 1).
"weighted" weights each variable according to the `coeff` parameter, and returns a normalized kernel matrix.
- coeff
(optional) A vector of weights with length D.
- feat_space
(not available for the Jaccard kernel). If FALSE, only the kernel matrix is returned. Otherwise, the feature space is returned too. (Defaults: FALSE).
Details
Let \(A,B\) be two sets. Then, the Intersect kernel is defined as:
$$K_{Intersect}(A,B)=|A \cap B| $$
And the Jaccard kernel is defined as:
$$K_{Jaccard}(A,B)=|A \cap B| / |A \cup B|$$
This specific implementation of the Intersect and Jaccard kernels expects that the set members (elements) are character symbols (length=1). In case the set data is multivariate (D>1 columns, and each one contains a set feature), elements for the D sets should come from the same domain (universe). For instance, a dataset with two variables, so the elements in the first one are colors c("green","black","white","red") and the second are names c("Anna","Elsa","Maria") is not allowed. In that case, set factors should be recoded to colors c("g","b","w","r") and names c("A","E","M") and, if necessary, 'Intersect()' (or `Jaccard()`) should be called twice.
References
Bouchard, M., Jousselme, A. L., and Doré, P. E. (2013). A proof for the positive definiteness of the Jaccard index matrix. International Journal of Approximate Reasoning, 54(5), 615-626.
Ruiz, F., Angulo, C., and Agell, N. (2008). Intersection and Signed-Intersection Kernels for Intervals. Frontiers in Artificial Intelligence and Applications. 184. 262-270. doi: 10.3233/978-1-58603-925-7-262.
Examples
# Sets data
## Generating a dataset with sets containing uppercase letters
random_set <- function(x)paste(sort(sample(LETTERS,x,FALSE)),sep="",collapse = "")
max_setsize <- 4
setsdata <- matrix(replicate(20,random_set(sample(2:max_setsize,1))),nrow=4,ncol=5)
## Computing the Intersect kernel:
Intersect(setsdata,elements=LETTERS,comp="sum")
#> 1 2 3 4
#> 1 14 0 2 1
#> 2 0 11 2 3
#> 3 2 2 18 2
#> 4 1 3 2 14
## Computing the Jaccard kernel weighting the variables:
coeffs <- c(0.1,0.15,0.15,0.4,0.20)
Jaccard(setsdata,elements=LETTERS,comp="weighted",coeff=coeffs)
#> 1 2 3 4
#> 1 1.00000000 0.0000000 0.05357143 0.0500000
#> 2 0.00000000 1.0000000 0.05000000 0.1666667
#> 3 0.05357143 0.0500000 1.00000000 0.0600000
#> 4 0.05000000 0.1666667 0.06000000 1.0000000