How MVQCA works

1
How MVQCA works

• A short Introduction to the Ideas of the
  Algorithm used in MVQCA
• Lasse Cronqvist
• lasse_at_staff.uni-marburg.de

2
MVQCA vs. QCA

• The algorithms used in QCA cant be used for
  medium or large size MV-Datasets
• Size restriction
• All variables are treated as nominal scaled
  variables (but MVQCA cares about ordinal scales)
• Inputs and Outputs are not elegant with Dummy
  Variables.

3
MVQCA vs. QCA

• We could build a MV-Interface for classical
  QCA-Algorithms to ease things up. But this has
  shown to be less successful (Brayton/Khatri 1999
  9)
• Important theoretical work has been made by the
  Electronic Engineers at the University of
  Berkeley. But their procedures are specialised on
  EEApplications
• no ordinal scales possible
• only one solution found
• (no easy to use User Interface)

4
MVQCA

• Due to this, a new algorithm has been developed
  and is included in Tosmana, the Tool for Small-N
  Analysis.

• In this presentation,
• 1. special conditions of MVQCA Analysis in the
  Social Science will be presented.
• 2. the basic Ideas of the Algorithm will be
  introduced
• 3. a sample calculation will be done on a (small)
  data set

5
Social Science and MVQCA

• There are several special conditions of Small-N
  Analysis, which have to be observed when creating
  an Algorithm for MVQCA
• The number of cases is small.
• The number of variables is also rather small (no
  fishing expeditions..)
• Variables may be of ordinal scale
• We want more than one (working) solution.

6
An Algorithm for MVQCA

• The main problem of Multi Value Datasets is their
  complexity.
• Having a data set with ten variable with a range
  of five possible values leaves us with a mass of
  9765625 possible configurations to deal with.
• gt it is not possible to build a configuration
  tree with all configuration to calculate with as
  it is used in some Boolean Minimization
  Algorithms.

7
Calculation in MVQCA

• The Calculation is done on a Matrix in MVQCA

Variables
States of each Variable
8
Calculation in MVQCA (2)

• Each case is placed in the matrix as a link
  between the fields

Variables

• cases with the outcome to minimize are called
  good cases and are drawn with a green line in
  this presentation
• cases with a different outcome to outcome to
  minimize are called bad cases and are drawn
  with a blue line in this presentation

States of each Variable
9
Calculation in MVQCA (3)

• When all cases are placed in the matrix, the
  Implicants are indicated by single fields or
  paths between fields of one or more good cases,
  which are not completely followed by one or more
  bad cases.

Variables

• Some Terms are marked with a red line or a
  filled red circle on the next page.

States of each Variable
10
Calculation in MVQCA (4)

• When all cases are placed in the matrix, the
  Implicants are indicated by single fields or
  paths between fields of one or more good cases,
  which are not completely followed by one or more
  bad cases.

Variables
States of each Variable
11
Calculation in MVQCA (5)

• Only Prime Implicants are of interest All
  term-paths that are non shortable and all single
  field terms are Prime Implicants.

Variables
The red path is not a representing a prime
implicant, as the upper half of the line is a
implicant it self.
States of each Variable
12
Calculation in MVQCA (6)

• Problem Not all (most actually) Terms are
  consisting of states of variables next to each
  other.

Variables
States of each Variable
13
Calculation in MVQCA (7)

• To solve this, a null-field is introduced to all
  variables, and each case has also a path to this
  null-field.

Variables
States of each Variable
14
Calculation in MVQCA (7)

• When all paths are set, a agent is send out from
  a root field to all fields of the first variable
  (including the zero-field)

The agent is having to sets with it A set of
good cases still represented by the agent and a
set of bad cases still represented by the agent.
15
Calculation in MVQCA (8)

• In each field there a various calculations on the
  agent
• The Set of Good Cases is recalculated with the
  Set of good cases passing the field (boolean
  AND).
• The Bad of Good Cases is recalculated with the
  Set of good cases passing the field (boolean
  AND).

16
Calculation in MVQCA (9)

• The agent has following possibilities in each
  field of the Variable-Matrix
• The agents set of good cases is empty go back
• The agents set of bad cases is empty (and the
  set of good cases is not) The path walked by the
  agent is a Term. Report and go back.
• Neither the set of good nor the set of bad cases
  is empty Go on to each field of the next
  variable (if there is one)

17
Calculation in MVQCA (10)

• When the agent has ended its expedition through
  the matrix, all Terms have been collected into a
  set of Terms.
• Terms implied by other Terms are sorted out.
• The remaining Terms (prime implicants) are sorted
  by length.
• Then the Terms are combined to find the Solutions

18
Calculation in MVQCA (11)

• The solutions are found in a simular way, but
  there are some differences in the way of
  calculation
• Terms are used instead of variables in the matrix
• There are just two fields for the each Term
  Used and the zero-field
• The agent has only one set The set of
  non-explained good cases. In each used-field
  this set is recalculated by joining the set of
  the agent with the non-explained set of a Term (
  good-Cases not implied by the Term) with
  boolean AND.

19
Calculation in MVQCA (12)
The agent has following possibilities in the used
field of the Term-Matrix (1).

• The agents new set of non-explained cases is
  simular to the one before visiting the field The
  Term does not explain any case not explained by
  the Terms visited by the agent. Go back (then
  just the path via the zero-field is used).
• The agents set of non- explained cases is empty
  The path walked by the agent is a Solution.
  Report and go back.

20
Calculation in MVQCA (13)
The agent has following possibilities in the used
field of the Term-Matrix (2).

• The length of the Terms on used-field visited
  by the path of the agent is longer than the
  allowed length STOP and go back. The allowed
  length is the size of the shortest solution found
  allowed extra length by the user.
• Otherwise Go on to the zero-field and the
  used-field of the next Term (if there is one).

21
Calculation in MVQCA (14)

• When the agent has ended its expedition through
  the Term-matrix, all Solutions have been
  collected into a set of Solution.
• Solutions longer than allowed are sorted out.
• Solutions implied by other Solutions are sorted
  out.
• The remaining Solution are sorted by length.

22
Calculation in MVQCA (15)

• To speed up the calculation a maximum length of
  prime implicants can be set If a path is
  representing more variables (non null) as
  allowed, the agent is told to go back.
• If this threshold is used, the length of the
  solution must be less or equal this threshold, as
  the completeness of the set of solution may be
  not given else. (In this case the maximum length
  must be raised and the calculation repeated).
• A maximum prime implicant length of 4 has proven
  to be sufficient for datasets with a large number
  of variables.

23
Sample

• The following sample of the agents way of
  working is shown with the following MV Data Set

24
Sample (2)
All fields are loaded with the good and bad cases
represented by the field




g a,b b x
g - b x
g a,b b -
g a b x
g a,b b y
g- b y
g - b x,y
g - b -
g - b -
g b b y
25
Sample (3)
Now the agent is created in the root field and
the sets are filled with all cases.




Agent Sets
g a,b b x
g - b x
g a,b b -
g a b x
Good a,b
Bad x,y
g a,b b y
g- b y
g - b x,y
g - b -
g - b -
g b b y
26
Sample (3)
Now the agent is created in the root field and
the sets are filled with all cases.




Agent Sets
g a,b b x
g - b x
g a,b b -
g a b x
Good a,b
Bad x,y
g a,b b y
g- b y
g - b x,y
g - b -
g - b -
g b b y
27
Sample (4)
On the following calculation pages the steps by
the agent are shown by a red arrow. On left the
agents sets are shown. (The old ones above the
new ones). The Set of found Terms is shown on
the right In this sample we start with the first
real field of the first variable. In real, the
algorithms allways starts with the zero-field.
28
Sample (5)
Now the agent is created in the root field and
the sets are filled with all cases.
V1
V2
V3
V4




Terms found
Ø
Old agent Sets
g a,b b x
g - b x
g a,b b -
g a b x
Good a,b
Bad x,y
g a,b b y
g- b y
g - b x,y
g - b -
New agent Sets
g - b -
g b b y
Good a,b
Bad x
29
Sample (6)
Now the agent is created in the root field and
the sets are filled with all cases.
V1
V2
V3
V4




Terms found
Old agent Sets
Ø
g a,b b x
g - b x
g a,b b -
g a b x
Good a,b
Bad x
g a,b b y
g- b y
g - b x,y
g - b -
New agent Sets
g - b -
g b b y
Good Ø
Bad x
The good-Set is empty. Go Back
30
Sample (7)
Now the agent is created in the root field and
the sets are filled with all cases.
V1
V2
V3
V4




Terms found
V10V21
Old agent Sets
g a,b b x
g - b x
g a,b b -
g a b x
Good a,b
Bad x
g a,b b y
g- b y
g - b x,y
g - b -
New agent Sets
g - b -
g b b y
Good a,b
Bad Ø
The bad-Set is empty. We have found a Term.
Report and Go Back
31
Sample (8)
Now the agent is created in the root field and
the sets are filled with all cases.
V1
V2
V3
V4




Terms found
V10V21
Old agent Sets
g a,b b x
g - b x
g a,b b -
g a b x
Good a,b
Bad x
g a,b b y
g- b y
g - b x,y
g - b -
New agent Sets
g - b -
g b b y
Good a,b
Bad x
Zero-Field. Go on.
32
Sample (9)
Now the agent is created in the root field and
the sets are filled with all cases.
V1
V2
V3
V4




Terms found
V10V21 V10V30
Old agent Sets
g a,b b x
g - b x
g a,b b -
g a b x
Good a,b
Bad x
g a,b b y
g- b y
g - b x,y
g - b -
New agent Sets
g - b -
g b b y
Good a,b
Bad -
The bad-Set is empty. We have found a Term.
Report and Go Back
33
Sample (10)
After the agent is finished, we have found the
following Terms V10V21 V10V30 V10V42
V21V30 V21V40 V30
34
Sample (11)
Two Terms are implied by an other Term and can be
deleted. V10V21 V10V30 V21V30 V30
V10V42 V21V40
35
Sample (12)
In the End, we are having four Terms (and two
Solutions consiting of one Term) Terms Soluti
on V30 yes V10V21 yes V10V42 no
V21V40 no To be complete V10V42
V21V40 is a solution as well.