Database Reference
In-Depth Information
9.2.8
Exercises for Section 9.2
EXERCISE
9.2.1 Three computers,
A
,
B
, and
C
, have the numerical features listed below:
Feature
A
B
C
Processor
Speed
3.06 2.68 2.92
Disk Size
500 320 640
Main-
Memory Size
6
4
6
We may imagine these values as defining a vector for each computer; for instance,
A
's vec-
tor is [3.06
,
500
,
6]. We can compute the cosine distance between any two of the vectors,
but if we do not scale the components, then the disk size will dominate and make differen-
ces in the other components essentially invisible. Let us use 1 as the scale factor for pro-
cessor speed,
α
for the disk size, and
β
for the main memory size.
(a) In terms of
α
and
β
, compute the cosines of the angles between the vectors for each
pair of the three computers.
(b) What are the angles between the vectors if
α
=
β
= 1?
(c) What are the angles between the vectors if
α
= 0.01 and
β
= 0.5?
!
(d) One fair way of selecting scale factors is to make each inversely proportional to
the average value in its component. What would be the values of
α
and
β
, and what
would be the angles between the vectors?
EXERCISE
9.2.2 An alternative way of scaling components of a vector is to begin by nor-
malizing the vectors. That is, compute the average for each component and subtract it from
that component's value in each of the vectors.
(a) Normalize the vectors for the three computers described in
Exercise 9.2.1
.
!!
(b) This question does not require difficult calculation, but it requires some serious
thought about what angles between vectors mean. When all components are non-
negative, as they are in the data of
Exercise 9.2.1
,
no vectors can have an angle
greater than 90 degrees. However, when we normalize vectors, we can (and must)
get some negative components, so the angles can now be anything, that is, 0 to 180
