Transcript measure

Measure
Measure
Definition:
It is the demonstration of the existence of an
homomorphism between an empirical relational
structure and a numerical relational structure.

Measure
Empirical relational
structure

Numerical relational
structure
<M, R>
Homomorphism
<N, O>
M = objects
N = numbers
R = relations
O = operations
Empirical relational structure
Example: Length
c
b
c
a
« If we place chalks b and c over each
other, then the result will be greater
than chalk a »
a
b
c
M = (chalk a, chalk b, chalk c)
O = ( « concatenation »,
« longer than »)
Numerical relational structure
Example: Length
N = (x, y, z) = (7, 5, 3)
O = (+ « addition », > « greater than »)
y+z>x = 5+3>7
N = (x, y, z) = (9, 4, 2)
O = (+ « addition », > « greater than »)
y+z>x = 4+2>9
Assumptions
Homomorphism
To link objects with numbers
To link relations with operations
Order
Additive
 (a)  (b)  x  y
 (a b)   (a)   (b)  x  y
Example
 (a)  5  x
 (b)  2  y
(Order)
5  2 (Ordre)
7  5  2 (additif)
(Additive)
Homomorphism
To link objects with numbers
To link relations with operations
M,R
a, b  M ;  ,   R
relationnelle
empirique
Empirical
relational structure
a, b ,  ,  = Structure
N,O
 x, y  N ; ,   O
relationnelle
numérique
Numerical
relational structure
x, y , ,  = Structure
a, b ,  , 

 x, y , , 
Scales
The freedom available to construct my scale will
determine its type.
The less the freedom in choice of scale, the more
powerful it will be
Interval
Power
Ratio
Nominal
Ordinal
Parametric
Non parametric
Ordinal scale
a
b
c
 (a )  20
 (b)  15
 (c )  2
 (a)  3
 (b)  2
 (c )  1
20  15  2
3  2 1
Definition: uses number to order objects
 (a)   (b)   (c)
Nonlinearity assumption
Ordinal scale
Performance
Example
Time
Interval scale
Definition: uses number to order objects and the distance
between each attribute is constant.
Linearity assumption: f(x)=mx+b
Example: conversion of
Celsius (x) into
Fahrenheit (y)
y=9/5*x+32
Interval of 5ºC
x1=5 and x2=10
Or
x1=20 and x2=25
Interval scale
Example: conversion of Celsius (x) into Fahrenheit (y)
y=9/5*x+32
Interval of 5ºC
x1=5 and x2=10
(x2-x1=10-5=5)
=> y1=41 and y2=50
(y2-y1=50-41=9)
Or
x1=20 and x2=25
(x2-x1=25-20=5)
=> y1=68 and y2=77
(y2-y1=77-68=9)
Warning
If we double the ºC we do not double the ºF
Ratio
Definition: uses number to order objects, the distance between
each attribute is constant and the zero is “meaningful”.
Example: the distance
traveled (y) in function of
time (x)
y=100*x
Distance (Km)
Linearity assumption: f(x)=mx
Time (hours)