ch 20 scale types
Post on 05-Jan-2016
58 Views
Preview:
DESCRIPTION
TRANSCRIPT
Ch 20Scale Types
Ch 20Scale Types
May 23, 2011presented by Tucker Lentz
May 23, 2011presented by Tucker Lentz
• Depth disclaimer
• Presentation only goes up to Theorem 7
What is the key to measurement?
1. Rich empirical structure
2. Symmetry
108108
“By symmetry, one means that the structure is isomorphic to itself...”
self-isomorphisms are called automorphisms
Symmetry109109
Stanley Smith Stevens
• 1906-1973
• American psychologist who founded Harvard's Psycho-Acoustic Laboratory
• Stevens’ Power Law in psychophysics
“In most cases a formulation of the rules of assignment discloses directly the kind of measurement and hence the kind of scale involved. If there remains any ambiguity, we may seek the final and definitive answer in the mathematical group structure of the scale form: in what ways can we transform its values and still have it serve all the functions previously fulfilled?”
109109
“Why do not psychologists accept the natural and obvious conclusion that subjective measurements of loudness in numerical terms (like those of length or weight or brightness) ... are mutually inconsistent and cannot be the basis of measurement?”
110110
“...Measurement is not a term with some mysterious inherent meaning, part of which may have been overlooked by physicists and may be in course of discovery by psychologists...we cease to know what is to be understood by the term when we encounter it; our pockets have been picked of a useful coin ....”
110f110f
1. Wasn’t interested in existence and uniqueness theorems
2. Limited his work to only a handful of transformations and didn’t ask what the possible groups of transformations are.
3. Failed to raise the question of “possible candidate representations that exhibit a particular degree of uniqueness” (?)
4. No proper justification for the importance of invariance under automorphisms has been provided.
Problems for Stevens111f111f
Stevens’ Classification of Scale Types
113113
A is a non-empty set (possibly empirical entities, possibly numbers)
J is the index set, non empty, usually integers
∀j ∈ J, Sj is a relation of finite order on A
A = ⟨A, Sj⟩j ∈ J is a relational structure
Formal Definitions115115
If one of the Sj is a weak or total order, we use ≿A = ⟨A, , S≿ j⟩j ∈ J is a weakly or totally ordered relational structure
If A is a subset of Re and the weak or total order Sj we used for is ≥, then we write≿
ℛ = ⟨R, ≥, Rj⟩j ∈ J and call it an ordered numerical structure
Formal Definitions115115
Isomorphism: φ is 1-to-1 mapping between the structuresHomomorphism: φ is onto, but not 1-to-1Automorphism: φ is an isomorphism between A and itselfEndomorphism: φ is a homomorphism between A and itself
Formal Definitions115115
Numerical Representation:
A is a totally ordered structure
ℛ is an ordered numerical structure
A is isomorphic to ℛ
Formal Definitions115115
M-point Homogeneity
M is the size of two arbitrarily selected sets of ordered points that can always be mapped into each other by one of our automorphisms (element in )ℋ
115115
N-1 is the largest number of points at which any two distinct automorphic transformations may agree
N-point Uniqueness116116
Note that we have moved from to ℋ G.
Homogeneous if at least 1-point homogeneous
Unique if there is an upper bound on the number of fixed points distinct automorphisms can agree
Homogeneity and Uniqueness (in general)
116116
M is the largest degree of homogeneity
N is the least degree of uniqueness
(M, N) is the scale type
Scale Type116116
Theorem 1
i) if M-point homogeneous, then (M-1)-point homogeneous
ii) If N-point unique, then (N+1)-point unique
iii) M ≤ N
117117
Theorem 2
M < order of some Sj, or the scale type is (∞,∞)
117117
For theorem 3 we need some more definitions
F is a function or “generalized operation” on An
F is a set of generalized operations
A-invariance
Algebraic closure of B under F
118118
Theorem 3
Another way of getting at N-point uniqueness, relating uniqueness to invariance
118118
Dilation & Translation
Every automorphism is either a dilation or a translation
The identity function is the only automorphism that is both
Dilations have at least one fixed point
118118
Theorem 4
1-point uniqueness means that two translations can have at most 1 point in common.
118f118f
Real Relational Structures
119-119-122122
“[The Archimedean] concept has been defined up to now only in structures for which an operation is either given or readily defined, as in the case of difference or conjoint structures. For general relational structures one does not know how to define the Archimedean property. This may be a reasonable way to do so in general, as is argued at some length in Luce and Narens (in press).
Homogeneous, Archimedean Ordered Translation Groups
123123
Theorem 7123123
Theorem 7124124
Theorem 7124124
“What is clear is that the usual physical representation involving units in no way depends upon extensive measurement or even on having an empirical operation. The key to the representation is for the structure lying on one component of an Archimedean conjoint structure to have translations that form a homogeneous, Archimedean ordered group...”
Theorem 7124f124f
ENDEND
top related