Information Technology Reference
In-Depth Information
The
h-index
of a population
X
over a set
A
(also called
Hirsch index
, introduced
in the context of bibliometrics), is the maximum number
n
of elements of
A
having in
X
a multiplicity greater than or equal to
n
. For example, the h-index of the population
2
a
+
+
+
,
,
3
b
4
c
5
d
is 3, because the multiplicities of 3 of them,
b
c
d
, are equal to
or greater than 3.
Let us consider the statistical distribution of values provided by a variable
Z
as-
suming values in correspondence to the individuals of a population. Let us order the
values of this distribution in a decreasing order. In many real cases there is a law
relating the ordering position of a value
v
with the percentage of elements
x
of the
population for which
Z
v
. For example, in the statistical distribution of the rich-
ness (expressed in money) of a population, it is very common that around the 80%
of the total richness of the population is covered by around the 20% richest people
(in the richness order). This is the famous
Pareto law
of 20/80 (from the Italian
economist who discovered this regularity). Similar laws occur in many situations of
resource distributions, and provide patterns of general relevance.
(
x
)
≥
7.7
Least Square Approximation
The method of least squares was introduced by Carl Friedrich Gauss around 1794
as an approach to the approximate solution of overdetermined systems, that is, sets
of equations with more equations than unknowns. The term “least squares” is used,
because the overall solution minimizes the sum of the squares of the errors made
with respect to an exact solution of the system.
A very elegant analysis of this method can be developed in terms of vector spaces
[225]. In a vector space over the field
R
of real numbers an internal scalar product
(
|
)
(
|
)
∈
R
. In this case we say that the
vector space is a
pre-Hilbert
real space. The scalar product of a pre-Hilbert real
space defines a
norm
by
of vector pairs is defined such that
v
w
||
||
=
(
|
)
and a notion of convergence with respect
to this norm can be given which generalizes the convergence of real functions. A
pre-Hilbert real space is
complete
if any Cauchy sequence of vectors converges to
a vector in the space. A pre-Hilbert space which is complete is said to be a
Hilbert
space
.
A very general and crucial result about pre-Hilbert spaces is the
projection the-
orem
given in Table 7.4. This theorem is the basis for the
Least Square Estimation
given in Table 7.5.
v
v
v
Ta b l e 7 . 4
Existence and unicity of the approximating vector with orthogonal error
The projection theorem
.Let
H
a Hilbert space and
M
a closed subspace of
H
.There
exists a unique vector
m
0
∈
M
such that
||
x
−
m
0
||≤||
x
−
m
||
for any pair
x
∈
H
,
m
∈
M
,
and the error vector
e
=
x
−
m
0
is orthogonal to
M
,thatis,
(
e
|
m
)=
0forevery
m
∈
M
.
