Conjunctive classification

What is conjunctive classification? In single-category classification, we want our model to give each test item to be classified a score as a member of each single category. Test item  0.6 as a member of A, 0.5 as a member of B, 0.3 as a member of C. In conjunctive classification, we want our model to give each item a score as a member of each conjunction or ANDED-PAIR of categories. Test item  ??? as a member of A&B (member of both A and B), ??? as a member of A&C (member of both A and C), ??? as a member of A&C (member of both A and C). How do we work out membership in a conjunction A&B?

Conjunction models Independent prototype model Membership in single categories is computed by additive weighted-attribute prototype similarity (as before) To compute an items membership in a conjunction A&B, combine that item’s computed membership in A and its computed membership in B. What functions can be used to combine constituent membership? membership(x, A&B) = product(A,B) = membership(x,A)*membership(x,B).

Other functions membership(x, A&B) = average(A,B) = (membership(x,A)+membership(x,B)) / 2. membership(x, A&B) =minimum(membership(x,A)+membership(x,B)) membership(x, A&B) = sum(A,B) = (membership(x,A) + membership(x,B)) membership(x, A&B) = normalised sum (A,B) = (membership(x,A) + membership(x,B) – membership(x,A) * membership(x,B)) In each of these functions, we’re combining the constituent membership scores to produce the conjunctive membership scores. These are called independent models because the constituent membership scores are computed independently of each other.

Exemplar models Independent exemplar-similarity model Membership in single categories is computed by summing the multiplicative similarity to exemplars of those categories. (as before) To compute an items membership in a conjunction A&B, combine that item’s computed membership in A and its computed membership in B. What functions can be used to combine constituent membership? membership(x, A&B) = product(A, B) = membership(x,A)*membership(x,B). The other four functions are also used. Again, this is an independent approach to conjunction.

Integrative prototype model Membership in a single category is computed by additive weighted- attribute similarity of an item to the prototype of that category. Membership in conjunctions is computed by additive weighted- attribute similarity of an item to the prototype of that conjunction. How do we form the prototype for a conjunction? A prototype is a list of weighted attribute-values. To form prototype for conjunction A&B, combine the lists of attribute weights from its two constituents, prototype A and prototype B. To classify an item in a conjunction, compare it to this integrated prototype as before.

Example of integrative prototype model D1 A  2/2=1.0 B  1/2=0.5 C  0/1=0.0 D2 A  2/3=0.67 B  1/1=1.0 C  0/1=0.0 D3 A  2/2=1.0 B  1/2= 0.5 C  0/1 =0.0 Prototype for A D1 A  0/2=0.0 B  1/2=0.5 C  1/1=1.0 D2 A  1/3=0.33 B  0/1=0.0 C  1/1=1.0 D3 A  0/2=0.0 B  1/2= 0.5 C  1/1 =1.0 Prototype for B 1.0 multiply multiply We need to compute conjunctive prototype attribute weights from constituent prototype attribute weights. Use a function eg multiplication D1 A  1.0*0.0=0.0 B  0.5*0.5=0.25 C  0.0*1.0=0.0 D2 A  0.67*0.33=0.22 B  1.0*0.0 =0.0 C  0.0*1.0 =0.0 D3 A  1.0*0.0=0.0 B  0.5*0.5=0.25 C  0.0*1.0 =0.0 Prototype for A&B

Integrative prototype D1 A  1.0*0.0=0.0 B  0.5*0.5=0.25 C  0.0*1.0=0.0 D2 A  0.67*0.33=0.22 B  1.0*0.0 =0.0 C  0.0*1.0 =0.0 D3 A  1.0*0.0=0.0 B  0.5*0.5=0.25 C  0.0*1.0 =0.0 Prototype for A&B An item (e.g. ) is then classified as an member of the conjunction A&B by similarity to this prototype: by summing the weights of its attributes in that prototoype: = = 0.47 This uses a multiplicative function to compute the attribute weights in the prototype from attribute weights in the constituent categories. The other four functions (sum, average, minimum, normalised sum) can also be used to compute conjunctive attribute weights from constituent attribute weights.

What you should do These slides give some examples of computational model for conjunctive classification. You should come up with your own model of single category and conjunctive category classification, based on how YOU think people’s categories and concepts work. You can use these models as starting points if you like. To test your model, you use it to compute the conjunctive classification scores for test items from that spreadsheet, and compare them to the results given in the spreadsheet. Next week I will spend time talking about the details of this modelling task.