Conceptual Scaling

In fcaR: Formal Concept Analysis

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

library(fcaR)

# Nominal
A_nom <- matrix(c("yes", "yes", "no", "yes", "no"), ncol = 1)
colnames(A_nom) <- "Grant"
rownames(A_nom) <- paste0("Student ", seq(nrow(A_nom)))

# Ordinal
A_ord <- matrix(c(7, 10, 5, 8, 4), ncol = 1)
colnames(A_ord) <- "Intern"
rownames(A_ord) <- paste0("Student ", seq(nrow(A_ord)))

# Interordinal
A_int <- matrix(c("agree", "strongly agree", "neither agree nor disagree", "agree", "disagree"), ncol = 1)
colnames(A_int) <- "Agreement"
rownames(A_int) <- paste0("Student ", seq(nrow(A_int)))

# Biordinal
A_bi <- matrix(c("working", "hard working", "working", "working", "lazy"), ncol = 1)
colnames(A_bi) <- "Attitude"
rownames(A_bi) <- paste0("Student ", seq(nrow(A_bi)))

# Interval
A_interv <- matrix(c(2.7, 4.1, 3.6, 4, 3.6), ncol = 1)
colnames(A_interv) <- "Score"
rownames(A_interv) <- paste0("Student ", seq(nrow(A_interv)))

# Aposition
A <- data.frame(A_nom, A_ord, A_int, A_bi, A_interv)
fc <- FormalContext$new(A)

Introduction and Motivation

Let us consider the following formal context about intern students:

fc$incidence() %>% 
  knitr::kable(format = "html",
               align = "rcccc")

The attributes are:

• Grant: a binary attribute that expresses whether the candidate is has received a grant.
• Intern: elapsed time as intern.
• Agreement: 1:strongly disagree, 2:disagree, 3:neither agree nor disagree, 4:agree and 5:strongly agree, with the work dynamics in the internship.
• Attitude: 1:hard working, 2:working, 3:lazy, 4:very lazy.
• Score: the score obtained in the internship procedure, with a maximum of 5.

This is a many-valued context, where we have mixed categorical and numerical attributes.

In such formal context, we cannot derive concepts and implications directly, we need to transform the formal context into one where each attribute is in the interval $[0, 1]$.

Types of Scaling

The transformations to the attributes of a many-valued formal context are called scaling. Depending on the meaning of each attribute, different types of scaling can be applied.

Nominal scaling

Nominal scales are used for scaling attributes whose values exclude each other.

For instance, in the example above, the attribute Grant is categorical:

fc_nom <- FormalContext$new(A_nom)
fc_nom$incidence() %>% 
  knitr::kable(format = "html",
               align = "r")

and can be transformed into the following one:

fc_nom$scale("Grant", type = "nominal")
fc_nom$incidence() %>% 
  knitr::kable(format = "html",
               align = "cc")

The mapping from the previous attribute values to the derived context is:

fc_nom$get_scales("Grant")$incidence() %>% 
  knitr::kable(format = "html",
               align = "cc")

Ordinal scaling

Ordinal scales are used for attributes with ordered values, where each value implies the smaller values (e.g. the number of children of an individual).

In our example, the attribute Intern is ordinal:

fc_nom <- FormalContext$new(A_ord)
fc_nom$incidence() %>% 
  knitr::kable(format = "html",
               align = "c")

and can be transformed into:

fc_nom$scale("Intern", type = "ordinal")
fc_nom$incidence() %>% 
  knitr::kable(format = "html",
               align = "cc")

using the following scale:

fc_nom$get_scales("Intern")$incidence() %>% 
  knitr::kable(format = "html",
               align = "cc")

Interordinal scaling

Interordinal scales are used in attributes that express different degrees (for instance, the Likert scale: strongly disagree, disagree, neither agree nor disagree, agree and strongly agree).

In our example, the attribute Agreement is interordinal:

fc_nom <- FormalContext$new(A_int)
fc_nom$incidence() %>% 
  knitr::kable(format = "html",
               align = "c")

and can be transformed into:

fc_nom$scale("Agreement", 
             type = "interordinal",
             values = c("strongly disagree",
                        "disagree",
                        "neither agree nor disagree",
                        "agree",
                        "strongly agree"))
fc_nom$incidence() %>% 
  knitr::kable(format = "html",
               align = "cc")

using the following scale:

fc_nom$get_scales("Agreement")$incidence() %>% 
  knitr::kable(format = "html",
               align = "cc")

Biordinal scaling

Biordinal scales are used when the attribute values express a degree in one of two poles (for instance, 1:very silent, 2:silent, 3:loud, 4:very loud, where 1:very silent implies 2:silent, and 4:very loud implies 3:loud, but neither silent implies loud nor loud implies silent).

In our example, the attribute Attitude is biordinal:

fc_nom <- FormalContext$new(A_bi)
fc_nom$incidence() %>% 
  knitr::kable(format = "html",
               align = "c")

and can be transformed into:

fc_nom$scale("Attitude", 
             type = "biordinal", 
             values_le = c("hard working", "working"),
             values_ge = c("lazy", "very lazy"))
fc_nom$incidence() %>% 
  knitr::kable(format = "html",
               align = "cc")

using the following scale:

fc_nom$get_scales("Attitude")$incidence() %>% 
  knitr::kable(format = "html",
               align = "cc")

Interval scaling

Interval scales group continuous attributes in intervals or bins.

In our example, the attribute Score is continuous:

fc_nom <- FormalContext$new(A_interv)
fc_nom$incidence() %>% 
  knitr::kable(format = "html",
               align = "c")

and can be categorized into intervals corresponding to the following marks:

• A: score in the interval $(4, 5]$.
• B: score in the interval $(3, 4]$.
• C: score in the interval $(2, 3]$.

fc_nom$scale("Score", 
             type = "interval", 
             values = c(2, 3, 4, 5), interval_names = c("C", "B", "A"))
fc_nom$incidence() %>% 
  knitr::kable(format = "html",
               align = "cc")

using the following scale:

fc_nom$get_scales("Score")$incidence() %>% 
  knitr::kable(format = "html",
               align = "cc")

Scaling in fcaR

Let us see how we can perform scaling with fcaR.

Available scales

The available scales are stored in a registry object called scalingRegistry, which keeps the functions to perform the different types of scaling:

scalingRegistry$get_entry_names()

Applying scales

In order to scale an attribute of a FormalContext, we use the method scale.

This method has two mandatory arguments: attribute (the attribute to scale) and type (the type of scaling). Additional arguments can be supplied for certain types of scaling, as we will se shortly.

For instance, if we call fc the formal context of the example at the beginning of this document, we can replicate the above scales:

fc$scale("Grant", type = "nominal")
fc

Note that the attribute Grant has been substituted by Grant = yes and Grant = no.

In order to apply the ordinal scaling to the Intern attribute, we do:

fc$scale("Intern", type = "ordinal")
fc

Now, the Agreement attribute can be scaled using and interordinal scaling:

fc$scale("Agreement", 
         type = "interordinal",
         values = c("strongly disagree",
                    "disagree",
                    "neither agree nor disagree",
                    "agree",
                    "strongly agree"))

The resulting formal context has r fc$dim()[2] columns, thus it is difficult to print.

Note that the call to scale() has an additional optional argument, values, that indicate, for character attributes, the order between the different attribute values. In this case, the order is "strongly disagree" < "disagree" < "neither agree nor disagree" < "agree" < "strongly agree".

The Attitude attribute can be scaled using a biordinal scale, since it represents two poles: the first is given by values "hard working" and "working" and the other is given by "very lazy" and "lazy". Then, we can do:

fc$scale("Attitude", 
         type = "biordinal", 
         values_le = c("hard working", "working"),
         values_ge = c("lazy", "very lazy"))

The full order of attribute values is "hard working" < "working" < "lazy" < "very lazy".

Note that we have listed the attribute values in two different arguments, values_le (for values that are compared using <=) and values_ge (values compared using >=). Also note that all attributes must appear ordered in the arguments, that is, in the same order as in the full order defined above.

The last attribute was the Score, that can be grouped in intervals using:

fc$scale("Score", 
         type = "interval", 
         values = c(2, 3, 4, 5), 
         interval_names = c("C", "B", "A"))

The additional arguments are values (the endpoints of the intervals) and interval_names (an optional name indicating how to call a given interval).

Note that all of these scales can be applied to numerical and string attributes.

Scale contexts

In order to see the transformation (the scale context) used for any of the attributes, we use the get_scales() method of a FormalContext:

fc$get_scales(c("Grant", "Score"))

With no arguments, it prints all the scales that have been used in a FormalContext.

Background knowledge

Some scalings, particularly of the ordinal family, assume an implicit knowledge (e.g., if attribute Intern is lower than 5, then it is also lower than 7).

This background knowledge is computed after each scaling is performed, and can be printed with:

fc$background_knowledge()

It is a simple ImplicationSet that stores all the implications that can be derived from the scale contexts.

Concepts and implications

As usual, for a given FormalContext, binary or fuzzy, one can compute its concept lattice and its basis of implications. In the case of many-valued contexts, one has first to scale all necessary attributes, such that the resulting context is binary or fuzzy.

Once scaled, the same find_concepts() and find_implications() methods can be used to compute the lattice and the basis.

In the case of the basis of implications, the NextClosure algorithm is used, as in the binary/fuzzy case, but the resulting implications have been post-processed to remove redundant information with respect to the background knowledge.

In our example, the resulting implications are:

fc$find_implications()
fc$implications

Note that, in this case, this is not the basis of implications, since the background knowledge has to be incorporated. That is, these implications are valid, but to be complete they need the implications derived from the scales.

Another example

filename <- system.file("contexts",
                        "aromatic.csv",
                        package = "fcaR")

fc_orig <- FormalContext$new(filename)

Let us consider the following context of attributes of aromatic molecules:

filename <- system.file("contexts",
                        "aromatic.csv",
                        package = "fcaR")

fc <- FormalContext$new(filename)

fc$incidence() %>% 
  knitr::kable(format = "html",
               align = "c")

The meaning of the attributes is as follows:

• ring represents the shape (pentagon or hexagon) of the molecule ring.
• OS means the presence of oxygen (O) or sulfur (S) atoms.
• nitro represents the number of nitrogen atoms.

We can transform this context into a binary one by means of the following scalings:

fc$scale(attributes = "nitro", 
         type = "ordinal", 
         comparison = `>=`, 
         values = 1:3)
fc$scale(attributes = "OS", 
         type = "nominal", 
         c("O", "S"))
fc$scale(attributes = "ring", 
         type = "nominal")

The final formal context is:

fc$incidence() %>% 
  knitr::kable(format = "html",
               align = "c")

The implications derived from the scales are:

fc$background_knowledge()

Also, we can compute the implications that are not represented by the background knowledge:

fc$find_implications()
fc$implications

and the concepts:

fc$concepts

fcaR documentation built on April 28, 2023, 1:11 a.m.