MATRIX FACTORIZATION FOR MULTIVARIATE TIME SERIES ANALYSIS 3
Example 2.1 (Periodic series ). Assume that T = pτ with p ∈ N
∗
for the sake of
simplicity. To assume that the latent series in t he dictionary W are τ-periodic is
exactly equivalent to writing W = VΛ where V ∈ M
k,τ
(R) and Λ = (I
τ
|. . . |I
τ
) ∈
M
τ,T
(R) is defined by blocks, I
τ
being the indentity matrix in M
τ,τ
(R).
Example 2.2 (Smooth series). We can assume that the series in W are smooth.
For example, say that they belong a a Sobolev space with smoothness β, we have
W
i,t
=
∞
X
n=0
U
i,n
e
n
t
T
where (e
n
)
n∈N
is the Fourier basis (the definition of a Sobolev space is reminded
is Section 3.2 below). Of course, there are infinitely many coefficients U
i,n
and to
estimate them all is not feasible, however, for τ large enough, the approximation
W
i,t
≃
τ −1
X
n=0
U
i,n
e
n
t
T
will be suitable, and can be rewritten as W = UΛ where Λ
i,t
= e
i
(t/T ). More
details will be given in Sect ion 3.2, where we actually cover more general basis of
functions.
So our complete model will finally be written as
(2) X = M + ε = UVΛ + ε,
where U ∈ M
d,k
(R) and V ∈ M
k,τ
(R) are unknown, but τ 6 T and Λ ∈ M
τ,T
(C)
such that rank(Λ) = τ are known (note that the unstructured case corr e sponds to
τ = T and Λ = I
T
).
Note that more constraint can be imposed on the estimator. For example, in
nonnegative matrix factorization [35], one imposes that all the entries in
ˆ
U and
ˆ
W
are nonnegative. Here, we will more generally assume that
ˆ
U
ˆ
V belongs to some
prescribed s ubs e t S ⊆ M
d,T
(R).
In what follows, we will cons ider two norms on M
d,T
. For a matrix A, the
Frobenius norms is given by
kAk
F
= trace(AA
∗
)
1/2
.
and the operator norm by
kAk
op
= sup
kxk=1
kAxk
where k ·k is the Euclidean norm on R
T
.
2.1. Estimation by empirical risk minimization. By multiplying both sides
in (2) by the pseudo-inverse Λ
+
= Λ
∗
(ΛΛ
∗
)
−1
, we obtain the “simplified model”
e
X =
f
M + eε
with
e
X = XΛ
+
,
f
M = UV and eε = εΛ
+
. In this model, the estimation o f
f
M can
be done by empirical risk minimization:
(3)
c
f
M
S
∈ arg min
A∈S
er(A)
where
er(A) = kA −
e
Xk
2
F
; ∀A ∈ M
d,τ
(R).
Therefore, we can define the estimator
c
M
S
=
c
f
M
S
Λ of M.
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