Concept in probability theory
For the notion in quantum mechanics, see scattering matrix . In multivariate statistics and probability theory , the scatter matrix is a statistic that is used to make estimates of the covariance matrix , for instance of the multivariate normal distribution .
Definition
Given n samples of m -dimensional data, represented as the m-by-n matrix,
X
=
[
x
1
,
x
2
,
… … -->
,
x
n
]
{\displaystyle X=[\mathbf {x} _{1},\mathbf {x} _{2},\ldots ,\mathbf {x} _{n}]}
sample mean is
x
¯ ¯ -->
=
1
n
∑ ∑ -->
j
=
1
n
x
j
{\displaystyle {\overline {\mathbf {x} }}={\frac {1}{n}}\sum _{j=1}^{n}\mathbf {x} _{j}}
where
x
j
{\displaystyle \mathbf {x} _{j}}
j -th column of
X
{\displaystyle X}
[ 1]
The scatter matrix is the m -by-m positive semi-definite matrix
S
=
∑ ∑ -->
j
=
1
n
(
x
j
− − -->
x
¯ ¯ -->
)
(
x
j
− − -->
x
¯ ¯ -->
)
T
=
∑ ∑ -->
j
=
1
n
(
x
j
− − -->
x
¯ ¯ -->
)
⊗ ⊗ -->
(
x
j
− − -->
x
¯ ¯ -->
)
=
(
∑ ∑ -->
j
=
1
n
x
j
x
j
T
)
− − -->
n
x
¯ ¯ -->
x
¯ ¯ -->
T
{\displaystyle S=\sum _{j=1}^{n}(\mathbf {x} _{j}-{\overline {\mathbf {x} }})(\mathbf {x} _{j}-{\overline {\mathbf {x} }})^{T}=\sum _{j=1}^{n}(\mathbf {x} _{j}-{\overline {\mathbf {x} }})\otimes (\mathbf {x} _{j}-{\overline {\mathbf {x} }})=\left(\sum _{j=1}^{n}\mathbf {x} _{j}\mathbf {x} _{j}^{T}\right)-n{\overline {\mathbf {x} }}{\overline {\mathbf {x} }}^{T}}
where
(
⋅ ⋅ -->
)
T
{\displaystyle (\cdot )^{T}}
matrix transpose ,[ 2] outer product . The scatter matrix may be expressed more succinctly as
S
=
X
C
n
X
T
{\displaystyle S=X\,C_{n}\,X^{T}}
where
C
n
{\displaystyle \,C_{n}}
n -by-n centering matrix .
Application
The maximum likelihood estimate, given n samples, for the covariance matrix of a multivariate normal distribution can be expressed as the normalized scatter matrix
C
M
L
=
1
n
S
.
{\displaystyle C_{ML}={\frac {1}{n}}S.}
[ 3] When the columns of
X
{\displaystyle X}
S
{\displaystyle S}
Wishart distribution .
See also
References
![{\displaystyle X=[\mathbf {x} _{1},\mathbf {x} _{2},\ldots ,\mathbf {x} _{n}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8606fdf0c878042f9299bc226f0941abce110a0)
