Database Reference
In-Depth Information
Table 5.1.
Description of notation.
Symbol Description
x
,...
Column vectors (lowercase boldface).
A
,...
Matrices (uppercase boldface).
x
t
The
n
stream values
x
t
:= [
x
t,
1
···x
t,n
]
T
at time
t
.
n
Number of streams.
w
i
The
i
-th participation weight vector (i.e., principal direction).
k
Number of hidden variables.
y
t
Vector of hidden variables (i.e., principal components) for
x
t
, i.e.,
y
t
≡
[
y
t,
1
···y
t,k
]
T
:= [
w
T
1
x
t
···
w
k
x
t
]
T
.
x
t
Reconstruction of
x
t
from the
k
hidden variable values, i.e.,
x
t
:=
y
t,
1
w
1
+
···
+
y
t,k
w
k
.
E
t
Total energy up to time
t
.
E
t,i
Total energy captured by the
i
-th hidden variable, up to time
t
.
f
E
,
F
E
Lower and upper bounds on the fraction of energy we wish to maintain via
SPIRIT's approximation.
best computed through the singular value decomposition (SVD) of
X
t
.
Space requirements also depend on
t
. Clearly, in a stream setting, it
is impossible to perform this computation at every step, aside from the
fact that we don't have the space to store all past values. We will ad-
dress this problem in Section 6, where we present a solution that works
without buffering
any
past values.
4. Auto-Regressive Models and Recursive Least
Squares
In this section we review some of the background on popular forecast-
ing methods for time series.
4.1 Auto-Regressive (AR) Modeling
Auto-regressive models are the most widely known and used—more
information can be found in, e.g., [7]. The main idea is to express
x
t
as
a function of its previous values, plus (filtered) noise
t
:
x
t
=
φ
1
x
t−
1
+
...
+
φ
W
x
t−W
+
t
,
(5.1)
where
W
is a the forecasting window size. Seasonal variants (SAR,
SAR(I)MA) also use window offsets that are multiples of a single, fixed
period (i.e., besides terms of the form
y
t−i
, the equation contains terms
of the form
y
t−Si
where
S
is a constant).
If we have a collection of
n
time series
x
t,i
,1
n
then multivariate
AR simply expresses
x
t,i
as a linear combination of previous values of
≤
i
≤
