Database Reference
In-Depth Information
5. MUSCLES
MUSCLES (MUlti-SequenCe LEast Squares)
[58] tries to predict the
value of one stream,
x
t,i
based on the previous values from all streams,
x
t−l,j
,
l>
1, 1
≤
j
≤
n
and current values from other streams,
x
t,j
,
j
=
i
. It uses multivariate autoregression, thus the prediction
x
t,i
for a
given stream
i
is, similar to Eq. 5.2
x
t,i
=
φ
1
,
0
x
t,
1
+
φ
1
,
1
x
t−
1
,
1
+
...
+
φ
1
,W
x
t−W,
1
+
...
+
φ
i−
1
,
0
x
t−
1
,i−
1
+
φ
i−
1
,
1
x
t−
1
,i−
1
+
...
+
φ
i−
1
,w
x
t−W,i−
1
+
φ
i,
1
x
t−
1
,i
+
...
+
φ
i,w
x
t−W,i
+
φi
+1
,
0
x
t,i
+1
+
φ
i
+1
,
1
x
t−
1
,i
+1
+
...
+
φ
i
+1
,w
x
t−W,i
+1
+
...
+
φ
n,
0
x
t,n
+
φ
n,
1
x
t−
1
,n
+
...
+
φ
n,W
x
t−W,n
+
t
.
and employs RLS to continuously update the coecients
φ
i,j
such that
the prediction error
t
x
τ,i
)
2
(
x
τ,i
−
τ
=1
is minimized. Note that the above equation has one dependent variable
(the estimate
x
t,i
)and
v
=
W
1 independent variables (the past
values of all streams plus the current values of all other streams except
i
).
∗
n
+
n
−
Exponentially forgetting MUSCLES
employs a forgetting factor 0
<
λ
≤
1 and minimizes instead
t
λ
t−τ
(
x
τ,i
− x
τ,i
)
2
.
τ
=1
For
λ<
1, errors for old values are down-weighted by an exponential fac-
tor, hence permitting the estimate to adapt as sequence characteristics
change.
5.1 Selective MUSCLES
In case we have too many time sequences (e.g.,
n
= 100
,
000 nodes
in a network, producing information about their load every minute),
even the incremental version of MUSCLES will suffer. The solution
to this problem is based on the conjecture that we do not really need
information from every sequence to make a good estimation of a missing
value. Much of the benefit of using multiple sequences may be captured
