Database Reference
In-Depth Information
tern on a sine wave with period T = 50. As expected, for window size
w = T = 50 the approximation error drops sharply and essentially cor-
responds to the Gaussian noise floor. Naturally, for windows w = iT
that are multiples of T the error also drops. Finally, observe that the
error for all windows is proportional to w , since it is per window element .
Eventually, for window size equal to the length of the entire time series
w = m (notshownin Figure5.5 ,where m = 2000), we get π ( m ) =0
since first pattern is the only singular vector, which coincides with the
series itself, so the residual error is zero.
Formally, the squared approximation error of the time-delay matrix
X ( w ) is
( w ) := t
x ( w )
( t ) x ( w )
X ( w )
2 =
X ( w )
2
F ,
where X ( w ) := P ( w ) ( V ( w ) ) T is the reconstruction and A
( t )
F := i,j a ij
denotes the Frobenius norm of A (sum of squares of matrix entries a ij ).
From the definition of the SVD-based approximation error (see Sec-
tion 3), as well as the fact that the sum of squares of the singular values
of a matrix is equal to the sum of squares of its values, we have
i =1 σ ( w )
2 .
P ( w )
( w ) =
X ( w )
F
F
2
x
i
Based on this, we define the power, which is an estimate of the error per
window element.
Definition 5.8 (Power profile π ( w ) ) For a given number of pat-
terns ( k =2 or 3) and for any window size w ,thepowerprofileis
the sequence defined by
π ( w ) := ( w w . (5.6)
More precisely, this is an estimate of the variance per dimension, as-
suming that the discarded dimensions correspond to isotropic Gaussian
noise (i.e., uncorrelated with same variance in each dimension) [52]. As
explained, this will be much lower when w = T ,where T is the period
of an arbitrary main trend.
The following lemma follows from the above observations. Note that
the conclusion is valid both ways, i.e., perfect copies imply zero power
and vice versa. Also, the conclusion holds regardless of alignment (i.e.,
the periodic part does not have to start at the begining of a windowed
subsequence). A change in alignment will only affect the phase of the
discovered local patterns, but not their shape or the reconstruction ac-
curacy.
Observation 8.1 (Zero power) If x R
t consists of exact copies of
a subsequence of length T then, for every number of patterns k =1 , 2 ,...
and at each multiple of T , we have π ( iT ) =0 , i =1 , 2 ,... , and vice versa.
Search WWH ::




Custom Search