System and methods for dynamic covariance estimation of a multivariate signal
View Patent ↗A system and methods of capturing rapid changes in complex systems by examining the interaction of the various components within the system. A multivariate signal is decomposed into a dictionary of elements and a corresponding sparse mixing matrix Each dictionary element may provide a rank-1 estimation of a disjoint partition of time points of the signal that may permit estimation of the covariance of each partition using an outer-product of the corresponding dictionary element.
1. A method for capturing dynamic covariance of a multivariate signal comprising the steps of:
receiving an input multivariate signal;
decomposing the input multivariate signal into a dictionary and a sparse mixing matrix, wherein the decomposing step further comprises the step of representing the input multivariate signal as a plurality of data points, wherein each data point is represented by a singleton dictionary element; and
estimating the dynamic covariance of a partition of data-points using a self-outer product, wherein the self-outer product corresponds to the singleton dictionary element up to a scaling value.
2. The method of claim 1 , wherein the method does not require a locality assumption.
3. The method of claim 1 , wherein the dictionary is defined according to:
min
D
,
M
X
(
t
)
-
M
(
t
)
D
2
+
λ
∑
i
[
m
1
…
k
(
i
)
]
0
wherein X(t) is an input matrix with dictionary D∈ k×n and mixing matrix M(t)=[m 1 (t), m 2 (t), . . . , m k (t)]∈ T×k where k is a number of dictionary elements, and m 1 . . . k (i) is an ith row of the mixing matrix M and λ is a regularizer parameter controlling a degree of sparsity.
4. The method of claim 3 , wherein a row of the X(t) is a sparse linear combination of the dictionary D of rows of the mixing matrix M(t).
5. The method of claim 3 , further comprising calculating the dynamic covariance according to the following equation:
min
D
,
M
(
t
)
X
(
t
)
-
M
(
t
)
D
2
s
.
t
.
∀
i
m
1
…
k
(
i
)
0
=
1
wherein ∥m 1 . . . k (i)∥ 0 =1 is a hard constraint.
6. The method of claim 1 , further comprising calculating a rank-1 estimate of the input multivariate signal according to the following:
X ( t )≅ m ( t ) v T
7. The method of claim 1 , further comprising finding disjoint partitions s i of the input multivariate signal according to the following:
∑
i
=
1
k
C
i
where
C
i
=
X
(
t
)
-
m
(
t
)
v
i
T
2
and
t
∈
s
i
.