Conceptual Scaling

Source: vignettes/conceptual-scaling.Rmd 
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Introduction and Motivation

Let us consider the following formal context about intern students:
Grant Intern Agreement Attitude Score
Student 1 yes 7 agree working 2.7
Student 2 yes 10 strongly agree hard working 4.1
Student 3 no 5 neither agree nor disagree working 3.6
Student 4 yes 8 agree working 4.0
Student 5 no 4 disagree lazy 3.6

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:
Grant
Student 1 yes
Student 2 yes
Student 3 no
Student 4 yes
Student 5 no
and can be transformed into the following one:
Grant = no Grant = yes
Student 1 0 1
Student 2 0 1
Student 3 1 0
Student 4 0 1
Student 5 1 0
The mapping from the previous attribute values to the derived context is:
Grant = no Grant = yes
no 1 0
yes 0 1

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:
Intern
Student 1 7
Student 2 10
Student 3 5
Student 4 8
Student 5 4
and can be transformed into:
Intern <= 4 Intern <= 5 Intern <= 7 Intern <= 8 Intern <= 10
Student 1 0 0 1 1 1
Student 2 0 0 0 0 1
Student 3 0 1 1 1 1
Student 4 0 0 0 1 1
Student 5 1 1 1 1 1
using the following scale:
Intern <= 4 Intern <= 5 Intern <= 7 Intern <= 8 Intern <= 10
4 1 1 1 1 1
5 0 1 1 1 1
7 0 0 1 1 1
8 0 0 0 1 1
10 0 0 0 0 1

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:
Agreement
Student 1 agree
Student 2 strongly agree
Student 3 neither agree nor disagree
Student 4 agree
Student 5 disagree
and can be transformed into:
Agreement <= strongly disagree Agreement <= disagree Agreement <= neither agree nor disagree Agreement <= agree Agreement <= strongly agree Agreement >= strongly disagree Agreement >= disagree Agreement >= neither agree nor disagree Agreement >= agree Agreement >= strongly agree
Student 1 0 0 0 1 1 1 1 1 1 0
Student 2 0 0 0 0 1 1 1 1 1 1
Student 3 0 0 1 1 1 1 1 1 0 0
Student 4 0 0 0 1 1 1 1 1 1 0
Student 5 0 1 1 1 1 1 1 0 0 0
using the following scale:
Agreement <= strongly disagree Agreement <= disagree Agreement <= neither agree nor disagree Agreement <= agree Agreement <= strongly agree Agreement >= strongly disagree Agreement >= disagree Agreement >= neither agree nor disagree Agreement >= agree Agreement >= strongly agree
strongly disagree 1 1 1 1 1 1 0 0 0 0
disagree 0 1 1 1 1 1 1 0 0 0
neither agree nor disagree 0 0 1 1 1 1 1 1 0 0
agree 0 0 0 1 1 1 1 1 1 0
strongly agree 0 0 0 0 1 1 1 1 1 1

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:
Attitude
Student 1 working
Student 2 hard working
Student 3 working
Student 4 working
Student 5 lazy
and can be transformed into:
Attitude <= hard working Attitude <= working Attitude >= lazy Attitude >= very lazy
Student 1 0 1 0 0
Student 2 1 1 0 0
Student 3 0 1 0 0
Student 4 0 1 0 0
Student 5 0 0 1 0
using the following scale:
Attitude <= hard working Attitude <= working Attitude >= lazy Attitude >= very lazy
hard working 1 1 0 0
working 0 1 0 0
lazy 0 0 1 0
very lazy 0 0 1 1

Interval scaling

Interval scales group continuous attributes in intervals or bins.

In our example, the attribute Score is continuous:
Score
Student 1 2.7
Student 2 4.1
Student 3 3.6
Student 4 4.0
Student 5 3.6

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 \((2, 3]\).
    Score is C Score is B Score is A
    Student 1 1 0 0
    Student 2 0 0 1
    Student 3 0 1 0
    Student 4 0 1 0
    Student 5 0 1 0
    using the following scale:
    Score is C Score is B Score is A
    2.7 1 0 0
    3.6 0 1 0
    4 0 1 0
    4.1 0 0 1

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()
#> [1] "Nominal"      "Ordinal"      "Interordinal" "Biordinal"    "Interval"    
#> [6] "Implication"

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
#> FormalContext with 5 objects and 6 attributes.
#> # A tibble: 5 × 6
#>   `Grant = no` `Grant = yes` Intern Agreement                  Attitude    Score
#>          <dbl>         <dbl>  <dbl> <chr>                      <chr>       <dbl>
#> 1            0             1      7 agree                      working       2.7
#> 2            0             1     10 strongly agree             hard worki…   4.1
#> 3            1             0      5 neither agree nor disagree working       3.6
#> 4            0             1      8 agree                      working       4  
#> 5            1             0      4 disagree                   lazy          3.6

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
#> FormalContext with 5 objects and 10 attributes.
#> # A tibble: 5 × 10
#>   `Grant = no` `Grant = yes` `Intern <= 4` `Intern <= 5` `Intern <= 7`
#>          <dbl>         <dbl>         <dbl>         <dbl>         <dbl>
#> 1            0             1             0             0             1
#> 2            0             1             0             0             0
#> 3            1             0             0             1             1
#> 4            0             1             0             0             0
#> 5            1             0             1             1             1
#> # ℹ 5 more variables: `Intern <= 8` <dbl>, `Intern <= 10` <dbl>,
#> #   Agreement <chr>, Attitude <chr>, Score <dbl>

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 19 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"))
#> $Grant
#> FormalContext with 2 objects and 2 attributes.
#>      Grant = no  Grant = yes  
#>   no      X                   
#>  yes                  X       
#> 
#> $Score
#> FormalContext with 4 objects and 3 attributes.
#>      Score is C  Score is B  Score is A  
#>  2.7      X                              
#>  3.6                  X                  
#>    4                  X                  
#>  4.1                              X

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()
#> Implication set with 20 implications.
#> Rule 1: {} -> {Agreement <= strongly agree, Agreement >= strongly disagree,
#>   Intern <= 10}
#> Rule 2: {Intern <= 7} -> {Intern <= 8}
#> Rule 3: {Intern <= 5} -> {Intern <= 7}
#> Rule 4: {Intern <= 4} -> {Intern <= 5}
#> Rule 5: {Agreement >= strongly agree} -> {Agreement >= agree}
#> Rule 6: {Agreement >= agree} -> {Agreement >= neither agree nor disagree}
#> Rule 7: {Agreement >= neither agree nor disagree} -> {Agreement >= disagree}
#> Rule 8: {Agreement <= agree, Agreement >= strongly agree} -> {Agreement <=
#>   strongly disagree}
#> Rule 9: {Agreement <= neither agree nor disagree} -> {Agreement <= agree}
#> Rule 10: {Agreement <= neither agree nor disagree, Agreement >= agree} ->
#>   {Agreement <= strongly disagree, Agreement >= strongly agree}
#> Rule 11: {Agreement <= disagree} -> {Agreement <= neither agree nor disagree}
#> Rule 12: {Agreement <= disagree, Agreement >= neither agree nor disagree} ->
#>   {Agreement <= strongly disagree, Agreement >= strongly agree}
#> Rule 13: {Agreement <= strongly disagree} -> {Agreement <= disagree}
#> Rule 14: {Agreement <= strongly disagree, Agreement >= disagree} -> {Agreement
#>   >= strongly agree}
#> Rule 15: {Attitude >= very lazy} -> {Attitude >= lazy}
#> Rule 16: {Attitude <= working, Attitude >= lazy} -> {Attitude <= hard working,
#>   Attitude >= very lazy}
#> Rule 17: {Attitude <= hard working} -> {Attitude <= working}
#> Rule 18: {Score is A, Score is B} -> {Score is C}
#> Rule 19: {Score is A, Score is C} -> {Score is B}
#> Rule 20: {Score is B, Score is C} -> {Score is A}

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
#> Implication set with 26 implications.
#> Rule 1: {} -> {Agreement >= disagree}
#> Rule 2: {Agreement >= disagree, Score is A} -> {Grant = yes, Agreement >=
#>   strongly agree, Attitude <= hard working}
#> Rule 3: {Agreement >= disagree, Score is B} -> {Intern <= 8, Agreement <= agree}
#> Rule 4: {Agreement >= disagree, Score is C} -> {Grant = yes, Intern <= 7,
#>   Agreement <= agree, Agreement >= agree, Attitude <= working}
#> Rule 5: {Agreement >= disagree, Attitude >= very lazy} -> {Grant = no, Grant =
#>   yes, Intern <= 4, Agreement <= strongly disagree, Attitude <= hard working,
#>   Score is B, Score is A}
#> Rule 6: {Agreement >= disagree, Attitude >= lazy} -> {Grant = no, Intern <= 4,
#>   Agreement <= disagree, Score is B}
#> Rule 7: {Agreement >= disagree, Attitude <= working} -> {Agreement >= neither
#>   agree nor disagree}
#> Rule 8: {Agreement >= disagree, Attitude <= hard working} -> {Grant = yes,
#>   Agreement >= strongly agree, Score is A}
#> Rule 9: {Agreement >= strongly agree} -> {Grant = yes, Attitude <= hard working,
#>   Score is A}
#> Rule 10: {Agreement >= agree} -> {Grant = yes, Attitude <= working}
#> Rule 11: {Agreement >= neither agree nor disagree} -> {Attitude <= working}
#> Rule 12: {Agreement <= agree, Agreement >= disagree} -> {Intern <= 8}
#> Rule 13: {Agreement <= neither agree nor disagree, Agreement >= disagree} ->
#>   {Grant = no, Intern <= 5, Score is B}
#> Rule 14: {Agreement <= disagree, Agreement >= disagree} -> {Grant = no, Intern
#>   <= 4, Attitude >= lazy, Score is B}
#> Rule 15: {Agreement <= strongly disagree, Agreement >= disagree} -> {Grant =
#>   no, Grant = yes, Intern <= 4, Attitude <= hard working, Attitude >= very lazy,
#>   Score is B, Score is A}
#> Rule 16: {Intern <= 8, Agreement >= disagree} -> {Agreement <= agree}
#> Rule 17: {Intern <= 7, Agreement >= disagree} -> {Agreement <= agree}
#> Rule 18: {Intern <= 7, Agreement <= agree, Agreement >= disagree, Score is B} ->
#>   {Grant = no, Intern <= 5, Agreement <= neither agree nor disagree}
#> Rule 19: {Intern <= 5, Agreement >= disagree} -> {Grant = no, Agreement <=
#>   neither agree nor disagree, Score is B}
#> Rule 20: {Intern <= 4, Agreement >= disagree} -> {Grant = no, Agreement <=
#>   disagree, Attitude >= lazy, Score is B}
#> Rule 21: {Grant = yes, Agreement >= disagree} -> {Agreement >= agree, Attitude
#>   <= working}
#> Rule 22: {Grant = yes, Intern <= 8, Agreement <= agree, Agreement >= strongly
#>   agree, Attitude <= hard working, Score is A} -> {Grant = no, Intern <= 4,
#>   Attitude >= very lazy, Score is B}
#> Rule 23: {Grant = yes, Intern <= 7, Agreement <= agree, Agreement >= agree,
#>   Attitude <= working} -> {Score is C}
#> Rule 24: {Grant = no, Agreement >= disagree} -> {Intern <= 5, Agreement <=
#>   neither agree nor disagree, Score is B}
#> Rule 25: {Grant = no, Intern <= 4, Agreement <= disagree, Agreement >= neither
#>   agree nor disagree, Attitude <= working, Attitude >= lazy, Score is B} ->
#>   {Grant = yes, Score is A}
#> Rule 26: {Grant = no, Grant = yes, Intern <= 5, Agreement <= neither agree nor
#>   disagree, Agreement >= agree, Attitude <= working, Score is C, Score is B} ->
#>   {Intern <= 4, Attitude <= hard working, Attitude >= very lazy}

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

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")
ring OS nitro
Benzene hex 0
Furan penta O 0
Imidazole penta 2
1-3-Oxazole penta O 1
Pyrazine hex 2
Pyrazole penta 2
Pyridine hex 1
Pyrimidine hex 2
Pyrrole penta 1
Thiazole penta S 1
Thiophene penta S 0
1-3-5-Triazine hex 3

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:
ring = hex ring = penta OS = O OS = S nitro >= 1 nitro >= 2 nitro >= 3
Benzene 1 0 0 0 0 0 0
Furan 0 1 1 0 0 0 0
Imidazole 0 1 0 0 1 1 0
1-3-Oxazole 0 1 1 0 1 0 0
Pyrazine 1 0 0 0 1 1 0
Pyrazole 0 1 0 0 1 1 0
Pyridine 1 0 0 0 1 0 0
Pyrimidine 1 0 0 0 1 1 0
Pyrrole 0 1 0 0 1 0 0
Thiazole 0 1 0 1 1 0 0
Thiophene 0 1 0 1 0 0 0
1-3-5-Triazine 1 0 0 0 1 1 1

The implications derived from the scales are:

fc$background_knowledge()
#> Implication set with 2 implications.
#> Rule 1: {} -> {nitro >= 1}
#> Rule 2: {nitro >= 3} -> {nitro >= 2}

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

fc$find_implications()
fc$implications
#> Implication set with 7 implications.
#> Rule 1: {nitro >= 3} -> {ring = hex}
#> Rule 2: {OS = S} -> {ring = penta}
#> Rule 3: {OS = O} -> {ring = penta}
#> Rule 4: {ring = penta, OS = S, nitro >= 2} -> {ring = hex, OS = O, nitro >= 3}
#> Rule 5: {ring = penta, OS = O, nitro >= 2} -> {ring = hex, OS = S, nitro >= 3}
#> Rule 6: {ring = penta, OS = O, OS = S} -> {ring = hex, nitro >= 3}
#> Rule 7: {ring = hex, ring = penta} -> {OS = O, OS = S, nitro >= 3}

and the concepts:

fc$concepts
#> A set of 15 concepts:
#> 1: ({Benzene, Furan, Imidazole, 1-3-Oxazole, Pyrazine, Pyrazole, Pyridine, Pyrimidine, Pyrrole, Thiazole}, {})
#> 2: ({Imidazole, 1-3-Oxazole, Pyrazine, Pyrazole, Pyridine, Pyrimidine, Thiazole}, {nitro >= 1})
#> 3: ({Imidazole, Pyrazine, Thiazole}, {nitro >= 1, nitro >= 2})
#> 4: ({Furan, Imidazole, 1-3-Oxazole, Pyridine, Pyrimidine, Pyrrole}, {ring = penta})
#> 5: ({Imidazole, 1-3-Oxazole, Pyridine, Pyrimidine}, {ring = penta, nitro >= 1})
#> 6: ({Imidazole}, {ring = penta, nitro >= 1, nitro >= 2})
#> 7: ({Pyrimidine, Pyrrole}, {ring = penta, OS = S})
#> 8: ({Pyrimidine}, {ring = penta, OS = S, nitro >= 1})
#> 9: ({Furan, 1-3-Oxazole}, {ring = penta, OS = O})
#> 10: ({1-3-Oxazole}, {ring = penta, OS = O, nitro >= 1})
#> 11: ({Benzene, Pyrazine, Pyrazole, Thiazole}, {ring = hex})
#> 12: ({Pyrazine, Pyrazole, Thiazole}, {ring = hex, nitro >= 1})
#> 13: ({Pyrazine, Thiazole}, {ring = hex, nitro >= 1, nitro >= 2})
#> 14: ({Thiazole}, {ring = hex, nitro >= 1, nitro >= 2, nitro >= 3})
#> 15: ({}, {ring = hex, ring = penta, OS = O, OS = S, nitro >= 1, nitro >= 2, nitro >= 3})