translate(expression = "", snames = "", noflevels, data, ...)compute(expression = "", data, separate = FALSE)findRows(expression = "", ttobj, remainders = FALSE)sop(expression = "", snames = "")
These functions interpret an expression written in a SOP (sum of products) form,
for both crisp and multivalue QCA. The function translate()
translates
the expression into a standard (canonical) SOP form using a matrix of implicants, while
compute()
uses the first to compute the scores based on a particular
dataset.
The function findRows()
takes a QCA expression written in SOP
form, and applies it on a truth table to find all rows that match the pattern in the
expression.
For crisp sets notation, upper case letters are considered the presence of that causal condition, and lower case letters are considered the absence of the respective causal condition. Tilde is recognized as a negation, even in combination with upper/lower letters.
Functions similar to translate()
and compute()
have initially been written by Jirka Lewandowski (2015) but the actual code in these functions
has been completely re-written to integrate it with the package QCA, and expanded with more
extensive functionality (see details and examples below).
The function sop()
transforms any expression (most notably
a POS product of sums) into a sum of products, minimizing it to the simplest equivalent
logical expression.
A SOP ("sum of products") is also known as a DNF ("disjunctive normal form"), or in other
words a "union of intersections", for example A*D + B*c
.
The same expression can be written in multivalue notation: A{1}*D{1} + B{1}*C{0}
.
Both types of expressions are valid, and yield the same result on the same dataset.
For multivalue notation, causal conditions are expected as upper case letters, and they will be
converted to upper case by default. Expressions can contain multiple values to translate, separated
by a comma. If B was a multivalue causal condition, an expression could be:
A{1} + B{1,2}*C{0}
.
In this example, all values in B equal to either 1 or 2 will be converted to 1, and the rest of the (multi)values will be converted to 0.
The function automatically detects the use of tilde "~" as a negation for a particular
causal condition. ~A
does two things: it identifies the presence of causal
condition A
(because it was specified as upper case) and it recognizes that it
must be negated, because of the tilde. It works even combined with lower case names:
~a
, which is interpreted as A
.
To negate a multivalue condition using a tilde, the number of levels should be supplied
(see examples below). Improvements in version 2.5 allow for intersections
between multiple levels of the same condition. For a causal condition with 3 levels (0, 1 and 2)
the following expression ~A{0,2}*A{1,2}
is equivalent with A{1}
,
while A{0}*A{1}
results in the empty set.
The number of levels, as well as the set names can be automatically detected from a dataset via
the argument data
. Arguments snames
and noflevels
have
precedence over data
, when specified.
The use of the product operator *
is redundant when causal snames' names are single
letters (for example AD + Bc
), and is also redundant for multivalue data, where
product terms can be separated by using the curly brackets notation.
When causal snames are binary and their names have multiple letters (for example
AA + CC*bb
), the use of the product operator *
is preferable but the
function manages to translate an expression even without it (AA + CCbb
) by searching
deep in the space of the conditions' names, at the cost of slowing down for a high number of causal
conditions. For this reason, an arbitrary limit of 7 causal snames
is imposed, to write an
expression with.
translate()
, a matrix containing the implicants on the
rows and the set names on the columns, with the following codes:
0 | absence of a causal condition |
1 | presence of a causal condition |
-1 | causal condition was eliminated |
Jirka Lewandowski (2015) QCAtools: Helper functions for QCA in R. R package version 0.1
translate("A + B*C")A B C A 1 B*C 1 1# same thing in multivalue notation translate("A{1} + B{1}*C{1}")A B C A{1} 1 B{1}*C{1} 1 1# using upper/lower letters translate("A + b*C")A B C A 1 b*C 0 1# the negation with tilde is recognised translate("~A + b*C")A B C ~A 0 b*C 0 1# even in combination of upper/lower letters translate("~A + ~b*C")A B C ~A 0 ~b*C 1 1# and even for multivalue variables # in multivalue notation, the product sign * is redundant translate("C{1} + T{2} + T{1}V{0} + C{0}")C T V C{1} 1 T{2} 1 T{1}V{0} 1 0 C{0} 0# negation of multivalue sets requires the number of levels translate("~A{1} + ~B{0}*C{1}", noflevels = c(2, 2, 2))A B C ~A{1} 0 ~B{0}*C{1} 1 1# multiple values can be specified translate("C{1} + T{1,2} + T{1}V{0} + C{0}")C T V C{1} 1 T{1,2} 1,2 T{1}V{0} 1 0 C{0} 0# or even negated translate("C{1} + ~T{1,2} + T{1}V{0} + C{0}", snames = "C, T, V", noflevels = c(2,3,2))C T V C{1} 1 ~T{1,2} 0 T{1}V{0} 1 0 C{0} 0# if the expression does not contain the product sign * # snames are required to complete the translation translate("AB + cD", snames = "A, B, C, D")A B C D AB 1 1 cD 0 1# otherwise snames are not required translate("PER*FECT + str*ing")FECT ING PER STR PER*FECT 1 1 str*ing 0 0# snames are required translate("PERFECT + string", snames = "PER, FECT, STR, ING")PER FECT STR ING PERFECT 1 1 string 0 0# it works even with overlapping columns # SU overlaps with SUB in SUBER, but the result is still correct translate("SUBER + subset", "SU, BER, SUB, SET")SU BER SUB SET SUBER 1 1 subset 0 0# error because combinations of condition names clash (not run) translate("SUPER + subset", "SUP, ER, SU, PER, SUB, SET") # to print _all_ codes from the standard output matrix (obj <- translate("A + b*C"))A B C A 1 b*C 0 1print(obj, original = TRUE) # also prints the -1 codeA B C A 1 -1 -1 b*C -1 0 1# for compute() data(CVF) compute("natpride + GEOCON", data = CVF)[1] 0.95 0.35 0.35 0.78 0.40 0.78 0.78 0.78 0.78 0.17 0.78 0.35 0.95 0.95 0.71 [16] 0.95 0.78 0.35 0.95 0.49 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95data(CVF) compute("natpride + GEOCON", data = CVF, separate = TRUE)natpride GEOCON 1 0.23 0.95 2 0.12 0.35 3 0.09 0.35 4 0.20 0.78 5 0.40 0.35 6 0.34 0.78 7 0.04 0.78 8 0.13 0.78 9 0.30 0.78 10 0.17 0.05 11 0.46 0.78 12 0.07 0.35 13 0.14 0.95 14 0.25 0.95 15 0.71 0.35 16 0.75 0.95 17 0.13 0.78 18 0.05 0.35 19 0.40 0.95 20 0.49 0.35 21 0.38 0.95 22 0.62 0.95 23 0.14 0.95 24 0.25 0.95 25 0.12 0.95 26 0.94 0.95 27 0.66 0.95 28 0.57 0.95 29 0.59 0.95# for findRows() data(LC) ttLC <- truthTable(LC, "SURV") findRows("DEV*ind*STB", ttLC)[1] 18 22 26 30findRows("DEV*ind*STB", ttLC, remainders = TRUE)[1] 18 26 30# for sop() sop("(A + B)(A + ~B)")[1] "A"# to force a certain order of the set names sop("(URB + LIT*~DEV)(~LIT + ~DEV)", snames = "DEV, URB, LIT")[1] "URB*~LIT + ~DEV*URB + ~DEV*LIT"# multilevel conditions can also be specified (and negated) sop("(A{1} + ~B{0})(B{1} + C{0})", snames = "A, B, C", noflevels = c(2, 3, 2))[1] "A{1}*C{0} + B{1} + B{1,2}*C{0}"