Segmenting Multiple Time Series by
Contemporaneous Linear Transformation:
PCA for time series
Qiwei Yao
Department of Statistics, London School of Economics
q.yao@lse.ac.uk
Joint work with
Jinyuan Chang
Bin Guo
University of Melbourne
Peking University
– p.1
Goal of study:
a high-dim TS → several uncorrelated lower-dim TS
Methodology: PCA for time series
• Transformation via an eigenanalysis
• Permutation
– maximum cross correlations
– FDR based on multiple tests
Real data illustration
Asymptotic properties in 3 settings:
p fixed,
p = o(nc ),
log p = o(nc )
Simulation
Segmenting multiple volatility processes
– p.2
Goal: For p × 1 weakly stationary time series yt , search for a
contemporaneous linear transformation:
yt = Axt ,
or
xt = Byt
(i.e. B = A−1 )
such that


xt = 
(1)
(1)
xt
..
.
(q)
xt


,
(i)
(j)
Cov(xt , xs ) = 0 ∀ i 6= j and t, s.
(q)
Hence, xt , · · · , xt can be modelled separately!
– p.3
Goal: For p × 1 weakly stationary time series yt , search for a
contemporaneous linear transformation:
yt = Axt ,
or
xt = Byt
(i.e. B = A−1 )
such that


xt = 
(1)
xt
..
.
(q)
xt
(1)


,
(i)
(j)
Cov(xt , xs ) = 0 ∀ i 6= j and t, s.
(q)
Hence, xt , · · · , xt can be modelled separately!
• realistic?
RealData
• how to find B and xt ?
– p.3
Observations: y1 , · · · , yn from a p×1 weakly stationary time series
Assumption: yt = Axt , and


xt = 
(1)
xt
..
.
(q)
xt


,
(i) (j)
Cov(xt , xs )
= 0 ∀ i 6= j and t, s.
Without loss of generality: Var(yt ) = Var(xt ) = Ip , and thus
A′ A = Ip ,
i.e. A
is orthogonal, xt = A′ yt
Goal: estimate A = (A1 , · · · , Aq ), or more precisely, M(A1 ), · · · ,
M(Aq ), as
(j)
x
bt
b ′ yt , j = 1, · · · , q .
=A
j
Note. (A, xt ) can be replaced by (AH, H′ xt ) for any
H = diag(H1 , · · · , Hq ) with H′j Hj = Ipj
– p.4
Step 1: Transformation via eigenanalysis
Notation:
Σy (k) = Cov(yt+k , yt ),
Wy =
Wx =
k0
X
Σx (k) = Cov(xt+k , xt ),
Σy (k)Σy (k)′ ,
k=0
k0
X
k0
X
k0
X
Σx (k)Σx (k)′ .
k=0
Σy (k)Σy (k)′ = Ip +
k=1
Σx (k)Σx (k)′ = Ip +
k=1
– p.5
Step 1: Transformation via eigenanalysis
Notation:
Σy (k) = Cov(yt+k , yt ),
Wy =
Wx =
k0
X
Σx (k) = Cov(xt+k , xt ),
Σy (k)Σy (k)′ ,
k=0
k0
X
k0
X
k0
X
Σx (k)Σx (k)′ .
Σy (k)Σy (k)′ = Ip +
k=1
Σx (k)Σx (k)′ = Ip +
k=0
As Σy (k) = AΣx (k)A′ ,
k=1
Wy = AWx A′ .
– p.5
Step 1: Transformation via eigenanalysis
Notation:
Σy (k) = Cov(yt+k , yt ),
Wy =
Wx =
k0
X
Σx (k) = Cov(xt+k , xt ),
Σy (k)Σy (k)′ ,
k=0
k0
X
k0
X
k0
X
Σx (k)Σx (k)′ .
Σy (k)Σy (k)′ = Ip +
k=1
Σx (k)Σx (k)′ = Ip +
k=0
As Σy (k) = AΣx (k)A′ ,
k=1
Wy = AWx A′ .
Eigenanalysis: Wx Γx = Γx D, columns of Γx are eigenvectors
of Wx with the eigenvalues in diagonal matrix D.
Wy AΓx = AWx A′ AΓx = AWx Γx = AΓx D
Thus Γy = AΓx , and Γ′y yt = Γ′x A′ yt = Γ′x xt .
– p.5
zt ≡ Γ′y yt (= Γ′x xt ) is the required transformation upto a permutation
– p.6
zt ≡ Γ′y yt (= Γ′x xt ) is the required transformation upto a permutation
Note Wx = diag(Wx,1 , · · · , Wx,q )
Proposition 1.
(i) Γx can be taken with the same block-diagonal
structure as Wx .
(ii) Any Γx is a column-permutation of Γx described in (i), provided
λ(Wx,i ) 6= λ(Wx,j ) for any i 6= j.
– p.6
zt ≡ Γ′y yt (= Γ′x xt ) is the required transformation upto a permutation
Note Wx = diag(Wx,1 , · · · , Wx,q )
Proposition 1.
(i) Γx can be taken with the same block-diagonal
structure as Wx .
(ii) Any Γx is a column-permutation of Γx described in (i), provided
λ(Wx,i ) 6= λ(Wx,j ) for any i 6= j.
For Γx specified in (i), Γ′x xt is segmented exactly the same as xt
– p.6
zt ≡ Γ′y yt (= Γ′x xt ) is the required transformation upto a permutation
Note Wx = diag(Wx,1 , · · · , Wx,q )
Proposition 1.
(i) Γx can be taken with the same block-diagonal
structure as Wx .
(ii) Any Γx is a column-permutation of Γx described in (i), provided
λ(Wx,i ) 6= λ(Wx,j ) for any i 6= j.
For Γx specified in (i), Γ′x xt is segmented exactly the same as xt
Let
c y = Ip +
W
k0
X
k=1
b y (k)Σ
b y (k)′ ,
Σ
cyΓ
by = Γ
b y D.
b
W
′
b
bt
zt to obtain x
zt = Γy yt – require permute components of b
Then b
– p.6
Permutation
′
b
Goal: put the connected components of b
zt = Γy yt together.
Visual examination of CCF if p is not large!
Two component series of b
zt is connected if the multiple null hypothesis
H0 : ρ(k) = 0 for any k = 0, ±1, ±2, . . . , ±m
is rejected, where ρ(k) is cross correlation between two series at lag k.
Permutation is performed as follows:
i. Start with p groups: each containing one component of b
zt .
ii. Combine the two groups together if one connected pair are split over
the two groups.
iii. Repeat Step ii above until all connected components are within one
group.
– p.7
Permutation Method I: Max CCF
Put b
zt = (zb1,t , · · · , zbp,t )′ .
Let ρbi,j (h) be the sample CCF of (zbi,t , zbj,t ) at lag h, and
bn (i, j) = max |ρbi,j (h)|,
L
|h|≤m
bn (i, j).
reject H0 for the pair (zbi,t , zbj,t ) for large values of L
– p.8
Permutation Method I: Max CCF
Put b
zt = (zb1,t , · · · , zbp,t )′ .
Let ρbi,j (h) be the sample CCF of (zbi,t , zbj,t ) at lag h, and
bn (i, j) = max |ρbi,j (h)|,
L
|h|≤m
bn (i, j).
reject H0 for the pair (zbi,t , zbj,t ) for large values of L
bn (i, j), 1 ≤ i < j ≤ p, in the descending order:
Line up L
Define
b1 ≥ · · · ≥ L
b p0 ,
L
p0 = p(p − 1)/2.
bj /L
bj+1 ,
rb = arg max L
1≤j<c0 p0
c0 ∈ (0, 1).
b1 , · · · , L
brb
Reject H0 for the pairs corresponding to L
– p.8
Graph representation: Let vertexes Vb = {1, · · · , p} stand for the
′
b
components of b
zt = Γ yt , and
bn = {edge connecting i and j : zbi,t , zbj,t are connected}.
E
Let V = {1, · · · , p} stand for the components of zt = Γ′ yt , and
E = {edge connecting i and j :
max
−m≤h≤m
|Corr(zi,t+h , zj,t )| > 0 }.
– p.9
Graph representation: Let vertexes Vb = {1, · · · , p} stand for the
′
b
components of b
zt = Γ yt , and
bn = {edge connecting i and j : zbi,t , zbj,t are connected}.
E
Let V = {1, · · · , p} stand for the components of zt = Γ′ yt , and
E = {edge connecting i and j :
max
|Corr(zi,t+h , zj,t )| > 0 }.
Let ̟ = mini6=j min |λ(Wx,i ) − λ(Wx,j )|,
⇒
max
max |Corr(zi,t+h , zj,t )| > 0,
1≤i<j≤p |h|≤m
−m≤h≤m
min
max |Corr(zi,t+h , zj,t )| = 0,
1≤i<j≤p |h|≤m
b j + δn )/(L
b j+1 + δn ).
rb = arg max (L
1≤j<p0
Proposition 2.
As n → ∞, δn → 0,
1 c
̟ kWy
− Wy k2 = o(δn ), and
bn = E) → 1.
log p = o(nα ). Then P (E
– p.9
Prewhitening
To make CCF for different pairs comparable, prewhiten each
′
b
component series of b
zt = Γy yt separately first.
– p.10
Prewhitening
To make CCF for different pairs comparable, prewhiten each
′
b
component series of b
zt = Γy yt separately first.
(i) If ρi,j (h) = 0, ρbi,j (h) ∼ N (0, 1/n) asymptotically,
provided at least one of xi,t and xj,t is white noise.
(ii) For h 6= k , ρbi,j (h), ρbi,j (k) are asymptotically independent,
and Cov{ρbi,j (h), ρbi,j (k)} = oP (1/n), provided both xi,t
and xj,t are white noise.
Brockwell & Davis (1996, Corollary 7.3.1).
– p.10
Prewhitening
To make CCF for different pairs comparable, prewhiten each
′
b
component series of b
zt = Γy yt separately first.
(i) If ρi,j (h) = 0, ρbi,j (h) ∼ N (0, 1/n) asymptotically,
provided at least one of xi,t and xj,t is white noise.
(ii) For h 6= k , ρbi,j (h), ρbi,j (k) are asymptotically independent,
and Cov{ρbi,j (h), ρbi,j (k)} = oP (1/n), provided both xi,t
and xj,t are white noise.
Brockwell & Davis (1996, Corollary 7.3.1).
In practice, we filter out the autocorrelation for each component
series of b
zt by fitting an AR with the order determined by AIC and not
greater than 5.
– p.10
Permutation Method II: FDR based on multiple tests
To fix the idea, let ξt , ηt be two WN, ρ(k) = Corr(ξt+k , ηt ) = 0,
ρb(k) =
n−k
X
t=1
¯ t − η̄)
(ξt+k − ξ)(η
n
.n X
t=1
(ξt − ξ̄)
2
n
X
t=1
(ηt − η̄)
2
o1/2
.
Asymptotically ρb(k) ∼ N (0, 1/n), and ρb(k), ρb(h) are independent.
– p.11
Permutation Method II: FDR based on multiple tests
To fix the idea, let ξt , ηt be two WN, ρ(k) = Corr(ξt+k , ηt ) = 0,
ρb(k) =
n−k
X
t=1
¯ t − η̄)
(ξt+k − ξ)(η
n
.n X
t=1
(ξt − ξ̄)
2
n
X
t=1
(ηt − η̄)
2
o1/2
.
Asymptotically ρb(k) ∼ N (0, 1/n), and ρb(k), ρb(h) are independent.
√
The P -value for testing ρ(k) = 0 (simple) is pk = 2Φ(− n|b
ρ(k)|)
– p.11
Permutation Method II: FDR based on multiple tests
To fix the idea, let ξt , ηt be two WN, ρ(k) = Corr(ξt+k , ηt ) = 0,
ρb(k) =
n−k
X
t=1
¯ t − η̄)
(ξt+k − ξ)(η
n
.n X
t=1
(ξt − ξ̄)
2
n
X
t=1
(ηt − η̄)
2
o1/2
.
Asymptotically ρb(k) ∼ N (0, 1/n), and ρb(k), ρb(h) are independent.
√
The P -value for testing ρ(k) = 0 (simple) is pk = 2Φ(− n|b
ρ(k)|)
Let p(1) ≤ · · · ≤ p(2m+1) be the order statistics of {pk , |k| ≤ m}.
Simes (1986): For H0 : ρ(k) = 0, ∀ |k| ≤ m, a multiple test
rejects H0 at the level α if
p(j) ≤ jα/(2m + 1) for at least one 1 ≤ j ≤ 2m + 1
– p.11
Permutation Method II: FDR based on multiple tests
To fix the idea, let ξt , ηt be two WN, ρ(k) = Corr(ξt+k , ηt ) = 0,
ρb(k) =
n−k
X
t=1
¯ t − η̄)
(ξt+k − ξ)(η
n
.n X
t=1
(ξt − ξ̄)
2
n
X
t=1
(ηt − η̄)
2
o1/2
.
Asymptotically ρb(k) ∼ N (0, 1/n), and ρb(k), ρb(h) are independent.
√
The P -value for testing ρ(k) = 0 (simple) is pk = 2Φ(− n|b
ρ(k)|)
Let p(1) ≤ · · · ≤ p(2m+1) be the order statistics of {pk , |k| ≤ m}.
Simes (1986): For H0 : ρ(k) = 0, ∀ |k| ≤ m, a multiple test
rejects H0 at the level α if
p(j) ≤ jα/(2m + 1) for at least one 1 ≤ j ≤ 2m + 1
The P -value for the multiple test is
P = min{α > 0 : p(j) ≤ jα/(2m + 1) for some 1 ≤ j ≤ 2m + 1}
=
min
1≤j≤2m+1
p(j) (2m + 1)/j.
– p.11
b ′ yt , we test multiple hypothesis
For each pair components of b
zt = Γ
y
H0 , obtaining P -value Pi,j for 1 ≤ i < j ≤ q.
Arranging those P -values in ascending order:
P(1) ≤ P(2) ≤ · · · ≤ P(p0 ) ,
p0 = p(p − 1)/2
FDR: For a given small β ∈ (0, 1), let
db = max{k : 1 ≤ k ≤ p0 , P(k) ≤ kβ/p0 },
and rejects the hypothesis H0 for the db pairs of the components of zt
corresponding to the P -values P(1) , · · · , P(d)
b
– p.12
b ′ yt , we test multiple hypothesis
For each pair components of b
zt = Γ
y
H0 , obtaining P -value Pi,j for 1 ≤ i < j ≤ q.
Arranging those P -values in ascending order:
P(1) ≤ P(2) ≤ · · · ≤ P(p0 ) ,
p0 = p(p − 1)/2
FDR: For a given small β ∈ (0, 1), let
db = max{k : 1 ≤ k ≤ p0 , P(k) ≤ kβ/p0 },
and rejects the hypothesis H0 for the db pairs of the components of zt
corresponding to the P -values P(1) , · · · , P(d)
b
(i) The P -values pk (|k| < m) are asymptotically independent
(ii) The P -values Pi,j (1 ≤< j ≤ p) are not independent, causing
difficulties in choosing β in FDR.
(iii) Ranking the pairwise dependences among the components of b
zt .
– p.12
Real data examples
Example 1. Monthly temperatures in 1954 - 1998 in p = 7 cities
in Eastern China.
32.5
Dongtai
•
Shanghai
•
30.5
31.0
31.5
Hefei
•
Huoshan
•
Anqing
•
Hangzhow
•
30.0
Latitude
32.0
Nanjing
•
116
118
120
122
Longitude
– p.13
0
15
30
Time series plots of monthly temperatures in 7 cities
1959
1964
1969
1974
1979
1984
1989
1994
1999
1954
1959
1964
1969
1974
1979
1984
1989
1994
1999
1954
1959
1964
1969
1974
1979
1984
1989
1994
1999
1954
1959
1964
1969
1974
1979
1984
1989
1994
1999
1954
1959
1964
1969
1974
1979
1984
1989
1994
1999
1954
1959
1964
1969
1974
1979
1984
1989
1994
1999
1954
1959
1964
1969
1974
1979
1984
1989
1994
1999
0
15
30
0
15
30
0 10
25
0
15
30
0
15
30
0
15
30
1954
– p.14
CCF of monthly temperatures in 7 cities
−5
−10
−5
0
−15
−10
−5
−15
−10
−5
5
1.0
1.0
−1.0 0.0
15
−15
−10
−15
−10
0
15
−5
−5
5
5
10
15
0
5
10
15
10
15
1.0
Y5 & Y7
0
5
10
15
0
5
10
15
Y6 & Y7
0
5
10
15
0
5
Y7 & Y6
0
15
1.0
0
Y6
0
10
Y4 & Y7
Y5 & Y6
10
5
1.0
−1.0 0.0
10
−1.0 0.0
15
Y7 & Y5
0
−1.0 0.0
1.0
−1.0 0.0
1.0
5
1.0
0
0
Y3 & Y7
1.0
−1.0 0.0
0
Y6 & Y5
0
15
−1.0 0.0
−5
10
1.0
10
15
1.0
−10
5
Y4 & Y6
−1.0 0.0
5
Y7 & Y4
0
15
1.0
−15
0
Y5
1.0
−15
0
Y6 & Y4
0
10
10
Y2 & Y7
−1.0 0.0
−5
5
5
10
15
Y7
1.0
−10
−1.0 0.0
1.0
1.0
−1.0 0.0
15
1.0
−15
0
0
Y3 & Y6
−1.0 0.0
0
15
Y4 & Y5
10
15
1.0
−5
10
1.0
5
10
Y2 & Y6
1.0
−1.0 0.0
15
−1.0 0.0
0
Y7 & Y3
0
10
5
−15
−10
−5
0
−1.0 0.0
−10
5
Y6 & Y3
0
−1.0 0.0
1.0
−1.0 0.0
1.0
1.0
−10
5
Y5 & Y4
1.0
−15
0
0
Y3 & Y5
1.0
−15
0
Y4
0
15
−1.0 0.0
−5
15
Y1 & Y7
1.0
−10
10
1.0
−5
10
−1.0 0.0
0
1.0
0
−1.0 0.0
−5
−10
5
1.0
−5
Y7 & Y2
1.0
−10
15
1.0
−15
Y7 & Y1
−15
10
1.0
−10
5
Y5 & Y3
1.0
0
−1.0 0.0
−5
−15
Y6 & Y2
1.0
−10
5
0
Y2 & Y5
−1.0 0.0
0
15
−1.0 0.0
−5
0
Y4 & Y3
−1.0 0.0
−15
Y6 & Y1
−15
0
Y1 & Y6
1.0
−10
10
Y3 & Y4
1.0
−15
5
−1.0 0.0
0
1.0
0
−1.0 0.0
−5
15
−1.0 0.0
−5
Y5 & Y2
1.0
−10
10
−1.0 0.0
−10
−1.0 0.0
0
Y5 & Y1
−15
5
1.0
−15
0
Y3
1.0
−5
−1.0 0.0
−10
0
−1.0 0.0
15
Y1 & Y5
Y2 & Y4
−1.0 0.0
10
Y4 & Y2
1.0
−1.0 0.0
−15
15
−1.0 0.0
5
−1.0 0.0
0
10
1.0
0
1.0
−5
−1.0 0.0
−10
5
Y3 & Y2
1.0
−1.0 0.0
−15
0
Y2 & Y3
−1.0 0.0
0
Y4 & Y1
−1.0 0.0
15
1.0
−5
−1.0 0.0
−10
Y3 & Y1
−1.0 0.0
10
−1.0 0.0
5
Y2
1.0
−1.0 0.0
−15
Y1 & Y4
1.0
0
−1.0 0.0
15
−1.0 0.0
10
−1.0 0.0
5
−1.0 0.0
−1.0 0.0
0
Y2 & Y1
−1.0 0.0
Y1 & Y3
1.0
Y1 & Y2
1.0
Y1
0
5
10
15
– p.15

b t
bt = By
Transformation: x
0.244 −0.066
0.0187 −0.050 −0.313 −0.154
0.200



 −0.703
0.324 −0.617
0.189
0.633
0.499 −0.323 




 0.375
1.544 −1.615
0.170 −2.266
0.126
1.596 




b

B =  3.025 −1.381 −0.787 −1.691 −0.212
1.188 −0.165 




 −0.197 −1.820 −1.416
3.269
.301
−1.438
1.299




 −0.584 −0.354
0.847 −1.262 −0.218 −0.151
1.831 


1.869 −0.742
0.034
0.501
0.492 −2.533
0.339
′
b y (0)−1/2 .
b
b
Note. B = Γy Σ
n = 540, p = 7
Used k0 = 5 (in defining Wy ), hardly changed for 2 ≤ k0 ≤ 36.
– p.16
−4
−2
0
Time series plots for transformed monthly temperatures
1959
1964
1969
1974
1979
1984
1989
1994
1999
1954
1959
1964
1969
1974
1979
1984
1989
1994
1999
1954
1959
1964
1969
1974
1979
1984
1989
1994
1999
1954
1959
1964
1969
1974
1979
1984
1989
1994
1999
1954
1959
1964
1969
1974
1979
1984
1989
1994
1999
1954
1959
1964
1969
1974
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1984
1989
1994
1999
1954
1959
1964
1969
1974
1979
1984
1989
1994
1999
−5
−2
1
0 2 4 6
−4
2 6
0
4
−4
−1 1
−1
1
1954
– p.17
CCF for transformed monthly temperatures
−5
−10
−5
0
−15
−10
−5
−15
−10
−5
5
1.0
1.0
−1.0 0.0
15
−15
−10
Segmentation: {1, 2, 3}, {4}, {5}, {6}, {7}
−15
−10
0
15
−5
−5
5
5
10
15
0
5
10
15
10
15
1.0
X5 & X7
0
5
10
15
0
5
10
15
X6 & X7
0
5
10
15
0
5
X7 & X6
0
15
1.0
0
X6
0
10
X4 & X7
X5 & X6
10
5
1.0
−1.0 0.0
10
−1.0 0.0
15
X7 & X5
0
−1.0 0.0
1.0
−1.0 0.0
1.0
5
1.0
0
0
X3 & X7
1.0
−1.0 0.0
0
X6 & X5
0
15
−1.0 0.0
−5
10
1.0
10
15
1.0
−10
5
X4 & X6
−1.0 0.0
5
X7 & X4
0
15
1.0
−15
0
X5
1.0
−15
0
X6 & X4
0
10
10
X2 & X7
−1.0 0.0
−5
5
5
10
15
X7
1.0
−10
−1.0 0.0
1.0
1.0
−1.0 0.0
15
1.0
−15
0
0
X3 & X6
−1.0 0.0
0
15
X4 & X5
10
15
1.0
−5
10
1.0
5
10
X2 & X6
1.0
−1.0 0.0
15
−1.0 0.0
0
X7 & X3
0
10
5
−15
−10
−5
0
−1.0 0.0
−10
5
X6 & X3
0
−1.0 0.0
1.0
−1.0 0.0
1.0
1.0
−10
5
X5 & X4
1.0
−15
0
0
X3 & X5
1.0
−15
0
X4
0
15
−1.0 0.0
−5
15
X1 & X7
1.0
−10
10
1.0
−5
10
−1.0 0.0
0
1.0
0
−1.0 0.0
−5
−10
5
1.0
−5
X7 & X2
1.0
−10
15
1.0
−15
X7 & X1
−15
10
1.0
−10
5
X5 & X3
1.0
0
−1.0 0.0
−5
−15
X6 & X2
1.0
−10
5
0
X2 & X5
−1.0 0.0
0
15
−1.0 0.0
−5
0
X4 & X3
−1.0 0.0
−15
X6 & X1
−15
0
X1 & X6
1.0
−10
10
X3 & X4
1.0
−15
5
−1.0 0.0
0
1.0
0
−1.0 0.0
−5
15
−1.0 0.0
−5
X5 & X2
1.0
−10
10
−1.0 0.0
−10
−1.0 0.0
0
X5 & X1
−15
5
1.0
−15
0
X3
1.0
−5
−1.0 0.0
−10
0
−1.0 0.0
15
X1 & X5
X2 & X4
−1.0 0.0
10
X4 & X2
1.0
−1.0 0.0
−15
15
−1.0 0.0
5
−1.0 0.0
0
10
1.0
0
1.0
−5
−1.0 0.0
−10
5
X3 & X2
1.0
−1.0 0.0
−15
0
X2 & X3
−1.0 0.0
0
X4 & X1
−1.0 0.0
15
1.0
−5
−1.0 0.0
−10
X3 & X1
−1.0 0.0
10
−1.0 0.0
5
X2
1.0
−1.0 0.0
−15
X1 & X4
1.0
0
−1.0 0.0
15
−1.0 0.0
10
−1.0 0.0
5
−1.0 0.0
−1.0 0.0
0
X2 & X1
−1.0 0.0
X1 & X3
1.0
X1 & X2
1.0
X1
0
5
10
15
– p.18
• Visual examination of CCF: {1, 2, 3}, {4}, {5}, {6} and {7}
• Permutation based on max-CCF: the same grouping with
2 ≤ m ≤ 30
• Permutation based on FDR: the same grouping with
2 ≤ m ≤ 30 and β ∈ [0.001%, 1%].
7-dim time series → 5 uncorrelated time series
Methodology
– p.19
Post-Sample Forecast
Forecasting based on segmentation: fit each subseries of xt with
a VAR model, forecast xt based on the fitted models, the
b −1 xt .
forecasts for yt are obtained via yt = B
Compare with the forecasts based on fitting a VAR and a
restricted VAR (RVAR) directly to yt .
Implementation: using VAR in R-package vars
For each of the last 24 values (i.e. the monthly temperatures in
1997-1998), we use the data upto the previous month for the
fittings. We calculate MSE of one-step-ahead forecasts,
two-step-ahead forecasts (by plug-in) for each of 7 cities.
One-step MSE Two-step MSE
VAR
1.669(2.355)
1.815(2.372)
1.677(2.324)
1.829(2.398)
RVAR
Segmentation
1.381(1.888)
1.543(1.874)
– p.20
Example 2. 8 monthly US Industrial Production indices
in 1947-1993 published by the US Federal Reserve.
8 indices: total index, manufacturing index, durable manufacturing,
nondurable manufacturing, mining, utilities, products, materials.
Nonstationary trends: difference each series
– p.21
−3
0 2
8 differenced monthly US Industrial Indices
1952
1957
1962
1967
1972
1977
1982
1987
1992
1947
1952
1957
1962
1967
1972
1977
1982
1987
1992
1947
1952
1957
1962
1967
1972
1977
1982
1987
1992
1947
1952
1957
1962
1967
1972
1977
1982
1987
1992
1947
1952
1957
1962
1967
1972
1977
1982
1987
1992
1947
1952
1957
1962
1967
1972
1977
1982
1987
1992
1947
1952
1957
1962
1967
1972
1977
1982
1987
1992
1947
1952
1957
1962
1967
1972
1977
1982
1987
1992
−4
0
4
−2
0
2
−5
5
−5
5
−2 0
2
−3
0
3
−3
0
2
1947
– p.22
CCF of 8 differenced monthly US Industrial Indices
−15
−5
−15
−5
0
−15
−5
−15
0
−15
−15
0
−5
−5
−5
0
5
−15
15
−15
10
15
−15
0
5
10
0
5
−5
0
−5
−5
0
5
5
0
5
−15
−15
−0.2 0.6
−0.2 0.6
−0.2 0.6
15
0
10
15
10
0
5
10
15
0
5
−5
0
0
0
−5
0
5
5
5
5
5
15
0
5
−15
15
10
15
10
15
10
15
10
15
Y7 & Y8
10
15
0
5
Y8 & Y7
0
10
Y6 & Y8
Y7
0
15
Y5 & Y8
15
10
10
Y4 & Y8
15
10
5
Y3 & Y8
Y6 & Y7
Y8 & Y6
0
15
Y5 & Y7
Y7 & Y6
0
10
Y4 & Y7
15
10
5
−0.2 0.6
0
Y6
Y8 & Y5
0
15
Y5 & Y6
Y7 & Y5
0
0
Y4 & Y6
Y6 & Y5
0
10
0
Y2 & Y8
−0.2 0.6
5
15
Y3 & Y7
−0.2 0.6
0
10
−0.2 0.6
−5
−0.2 0.6
−0.2 0.6
−0.2 0.6
15
10
15
−0.2 0.6
5
10
−0.2 0.6
0
5
5
Y2 & Y7
Y3 & Y6
Y5
Y8 & Y4
0
−0.2 0.6
−0.2 0.6
15
Y7 & Y4
Y8 & Y3
0
10
Y6 & Y4
Y7 & Y3
0
−15
0
Y4 & Y5
Y5 & Y4
0
10
0
−0.2 0.6
−5
5
5
15
−0.2 0.6
−15
0
15
−0.2 0.6
0
Y6 & Y3
0
0
10
−0.2 0.6
−5
15
Y4
−0.2 0.6
−5
10
10
−0.2 0.6
5
5
−0.2 0.6
0
−0.2 0.6
−15
0
5
Y2 & Y6
Y3 & Y5
−0.2 0.6
15
Y5 & Y3
0
−0.2 0.6
−0.2 0.6
−0.2 0.6
10
0
Y1 & Y8
−5
10
15
Y8
0
−0.2 0.6
0
−15
15
Y3 & Y4
−0.2 0.6
0
10
15
−0.2 0.6
−5
5
10
Y2 & Y5
Y1 & Y7
−0.2 0.6
−15
0
5
−0.2 0.6
−5
5
0
Y1 & Y6
−0.2 0.6
−15
0
15
−0.2 0.6
−5
15
−0.2 0.6
−5
10
−0.2 0.6
5
10
Y2 & Y4
Y4 & Y3
Y8 & Y2
−0.2 0.6
−0.2 0.6
−5
0
5
−0.2 0.6
0
Y8 & Y1
−15
0
Y7 & Y2
−0.2 0.6
−0.2 0.6
−5
−15
0
Y1 & Y5
−0.2 0.6
0
Y7 & Y1
−15
−5
Y6 & Y2
−0.2 0.6
−0.2 0.6
−5
−15
15
−0.2 0.6
0
Y6 & Y1
−15
−15
10
Y3
Y5 & Y2
−0.2 0.6
−0.2 0.6
−5
15
−0.2 0.6
0
Y5 & Y1
−15
10
Y4 & Y2
−0.2 0.6
−0.2 0.6
−5
5
−0.2 0.6
0
Y4 & Y1
−15
0
5
Y3 & Y2
−0.2 0.6
−0.2 0.6
−5
0
Y2 & Y3
−0.2 0.6
0
Y3 & Y1
−15
15
−0.2 0.6
−5
−0.2 0.6
−0.2 0.6
−15
10
Y2
−0.2 0.6
5
Y1 & Y4
−0.2 0.6
0
−0.2 0.6
15
−0.2 0.6
10
Y2 & Y1
−0.2 0.6
5
Y1 & Y3
−0.2 0.6
0
Y1 & Y2
−0.2 0.6
−0.2 0.6
Y1
0
5
10
15
– p.23

5.012

 10.391


 −6.247


 1.162
b =
B

 6.172

 0.868


 3.455

0.902
b t
bt = By
Transformation: x
−1.154
−0.472
−0.880
−0.082
−0.247
−2.69
8.022
−3.981
−3.142
0.186
0.019
−6.949
11.879
−4.8845
−4.0436
0.289
−0.011
2.557
−6.219
3.163
1.725
0.074
−0.823
0.646
−4.116
2.958
1.887
0.010
0.111
−2.542
1.023
−2.946
−4.615
−0.271
−0.354
3.972
−2.744
5.557
3.165
0.753
0.725
−2.331
−2.933
−1.750
−0.123
0.191
−0.265
3.759
−1.463


−4.203 


0.243 


−0.010 
.

−3.961 

1.902 


−1.777 

0.987
– p.24
−2
2
Transformed differenced monthly US Industrial Indices
1952
1957
1962
1967
1972
1977
1982
1987
1992
1947
1952
1957
1962
1967
1972
1977
1982
1987
1992
1947
1952
1957
1962
1967
1972
1977
1982
1987
1992
1947
1952
1957
1962
1967
1972
1977
1982
1987
1992
1947
1952
1957
1962
1967
1972
1977
1982
1987
1992
1947
1952
1957
1962
1967
1972
1977
1982
1987
1992
1947
1952
1957
1962
1967
1972
1977
1982
1987
1992
1947
1952
1957
1962
1967
1972
1977
1982
1987
1992
−4
0
4
−4 0
4
−4
0
3
−4
0
−4
0
4
−3 0
3
−3
0
1947
– p.25
CCF of transformed differenced monthly US Industrial Indices
−15
−5
15
−15
−15
10
−5
−5
5
15
−5
0
0.6
−0.4
0.6
10
15
Segmentation: {1, 2, 3}, {4, 8}, {5}, {6}, {7}
−5
5
0
5
10
15
10
15
0.6
0
5
10
15
0
5
10
15
0.6
X6 & X8
0
5
10
15
0
5
10
15
0.6
X7 & X8
0
5
10
15
0
5
X8 & X7
0
15
X5 & X8
10
15
X8
0.6
−15
10
X4 & X8
0.6
−15
0
X7
X8 & X6
0
15
X6 & X7
10
5
0.6
5
X7 & X6
0
10
0.6
0
0
X5 & X7
15
15
0.6
0
X6
0
5
−0.4
0
−0.4
5
10
X3 & X8
0.6
0
5
0.6
−5
15
−0.4
10
−0.4
−15
10
−0.4
15
5
X4 & X7
−0.4
10
−0.4
0.6
15
X5 & X6
X8 & X5
0
−0.4
0.6
5
0
X2 & X8
0.6
0
X7 & X5
0
10
0.6
5
15
−0.4
15
−0.4
0
0
X4 & X6
−0.4
10
10
0.6
5
−0.4
0
0.6
−5
−0.4
0.6
−0.4
0.6
5
5
X3 & X7
−0.4
0
15
0.6
−5
10
0.6
0
0
X2 & X7
0.6
−0.4
15
0.6
−15
5
X6 & X5
X8 & X4
0
10
X1 & X8
−15
−5
0
−0.4
−5
5
0.6
−15
0
X5
0
15
0.6
−5
10
−0.4
−15
0
0.6
−15
5
X3 & X6
−0.4
15
X7 & X4
0
15
−0.4
10
−0.4
0
10
0.6
−5
5
−0.4
5
0.6
−15
0
0.6
0
0
X4 & X5
−0.4
−5
−0.4
0.6
0.6
−0.4
15
X6 & X4
X8 & X3
0
10
X1 & X7
X2 & X6
0.6
5
−0.4
0
0.6
−15
15
X3 & X5
−0.4
0
10
−0.4
−5
15
−0.4
−5
X7 & X3
0
10
0.6
−15
5
X5 & X4
0.6
−15
5
0.6
−5
0
−0.4
0
0.6
−15
−0.4
0.6
−0.4
0.6
−5
X1 & X6
X2 & X5
0.6
−15
0
X4
−0.4
0
15
0.6
15
0.6
−5
−0.4
−15
0.6
0
−0.4
−5
10
X6 & X3
X8 & X2
0.6
−15
5
−0.4
0
0.6
0
−0.4
−5
X8 & X1
0
0.6
−5
−0.4
−15
10
X3 & X4
X5 & X3
X7 & X2
0.6
−15
15
−0.4
0
0.6
0
−0.4
−5
X7 & X1
10
0.6
−5
−0.4
−15
5
X4 & X3
X6 & X2
0.6
X6 & X1
−15
5
−0.4
0
0.6
0
−0.4
−5
0
−0.4
−5
X1 & X5
X2 & X4
0.6
−15
0
X3
X5 & X2
0.6
−15
15
−0.4
15
0.6
0
−0.4
−5
10
0.6
10
−0.4
5
X4 & X2
X5 & X1
−0.4
0
−0.4
0
0.6
−0.4
−15
5
X2 & X3
0.6
−5
−0.4
0.6
−0.4
−15
0
X3 & X2
X4 & X1
−0.4
15
0.6
0
−0.4
−5
X3 & X1
−0.4
10
−0.4
5
X2
0.6
−0.4
−15
X1 & X4
0.6
0
−0.4
15
−0.4
10
−0.4
5
−0.4
−0.4
0
X2 & X1
−0.4
X1 & X3
0.6
X1 & X2
0.6
X1
0
5
10
15
– p.26
• Visual examining CCF: {1, 2, 3}, {4, 8}
• Permutation with max-CCF
1 ≤ m ≤ 20:
{1, 3}
• Permutation with FDR
m = 20 and β ∈ [10−6 , 0.01], or m = 5 and β ∈ [10−6 , 0.001]:
{1, 3}
m = 5 and β = 0.005:
m = 5 and β = 0.01:
{1, 2, 3}, {4, 8}
{1, 2, 3, 5, 6, 7} and {4, 8}.
Two recommended groupings:
seven groups: {1, 3}
five groups: {1, 2, 3}, {4, 8}
Methodology
– p.27
Post-sample forecast
Forecast 24 monthly indices in Jan 1992 – Dec 1993.
Using the segmentation: {1, 3} and other six single element
groups
Miss some small but significant cross correlations
One-step MSE Two-step MSE
VAR
0.615(1.349)
1.168(2.129)
RVAR
0.606(1.293)
1.159(2.285)
0.588(1.341)
1.154(2.312)
Segmentation
– p.28
Example 3. Weekly notified measles cases in 7 cities in England
(i.e. London, Bristol, Liverpool, Manchester, Newcastle,
Birmingham and Sheffield) in 1948-1965, before the advent
of vaccination.
All the 7 series show biennial cycles, which is the major driving
force for the cross correlations among different cities.
n = 937, p = 7.
– p.29
0
2000
Weekly recorded cases of measles in 7 cities in England in 1948-1965
1955
1960
1965
1950
1955
1960
1965
1950
1955
1960
1965
1950
1955
1960
1965
1950
1955
1960
1965
1950
1955
1960
1965
1950
1955
1960
1965
0
400 800
0
1500
0
300 600
0
400
0
400 800
0
400
1950
– p.30
CCF of weekly recorded cases of measles in 7 cities
20
−5
−5
0
−20 −15 −10
−20 −15 −10
0.6
−0.2
0.6
15
20
10
15
0
−5
−5
10
15
5
10
−20 −15 −10
5
20
−5
−5
10
15
20
0
5
10
15
20
0.6
Y4 & Y7
5
10
15
20
0
5
10
15
20
0.6
Y5 & Y7
5
10
15
20
0
5
10
15
20
Y6 & Y7
0.6
Y6
0
5
10
15
20
0
5
Y7 & Y6
0
20
−0.2
0
0
15
0.6
5
Y5 & Y6
15
10
Y3 & Y7
−0.2
0
Y7 & Y5
−20 −15 −10
0
Y4 & Y6
20
20
−0.2
0
Y6 & Y5
0
20
0.6
0
0
15
0.6
5
15
−0.2
−5
10
Y3 & Y6
20
10
0.6
5
0.6
5
5
−0.2
0
Y5
Y7 & Y4
0
10
0
Y2 & Y7
−0.2
0
Y6 & Y4
20
−0.2
0
20
15
0.6
5
Y4 & Y5
15
10
10
15
20
15
20
Y7
0.6
10
5
−0.2
0.6
−0.2
0
0.6
5
0
Y2 & Y6
−0.2
15
0.6
−20 −15 −10
−0.2
0.6
−0.2
0.6
−0.2
0.6
−0.2
0.6
10
0.6
−20 −15 −10
20
Y3 & Y5
−0.2
5
−20 −15 −10
Y7 & Y3
0
20
0.6
0
0
0.6
−5
15
−0.2
−5
15
−0.2
0
10
Y5 & Y4
−0.2
−20 −15 −10
5
−0.2
0
0.6
−5
0.6
0
−5
10
Y2 & Y5
Y4
−0.2
−20 −15 −10
0
Y6 & Y3
−0.2
0.6
−5
−20 −15 −10
Y7 & Y2
−0.2
−20 −15 −10
−20 −15 −10
0
0.6
0
20
0.6
−5
0
Y5 & Y3
−0.2
0.6
−5
15
−0.2
−20 −15 −10
Y7 & Y1
10
5
−0.2
0
Y6 & Y2
−0.2
−20 −15 −10
5
0
Y3 & Y4
0.6
−5
0.6
0
Y6 & Y1
20
Y4 & Y3
−0.2
0.6
−5
0
Y5 & Y2
−0.2
−20 −15 −10
15
−0.2
−20 −15 −10
20
0.6
0
0.6
0
Y5 & Y1
10
0.6
−5
−0.2
0.6
−5
5
Y3
Y4 & Y2
−0.2
−20 −15 −10
0
−0.2
−20 −15 −10
15
−0.2
20
10
Y2 & Y4
0.6
15
0.6
0
5
−0.2
10
0
0.6
5
−0.2
0.6
−5
Y4 & Y1
20
0.6
0
Y3 & Y2
−0.2
−20 −15 −10
15
−0.2
0
Y3 & Y1
10
−0.2
0.6
−5
5
Y2 & Y3
−0.2
0.6
−0.2
−20 −15 −10
0
0.6
20
Y1 & Y7
−0.2
15
Y2
0.6
10
−0.2
5
0.6
0
Y1 & Y6
−0.2
20
−0.2
15
0.6
10
Y2 & Y1
Y1 & Y5
−0.2
5
−0.2
−0.2
−0.2
0
Y1 & Y4
0.6
Y1 & Y3
0.6
Y1 & Y2
0.6
Y1
−20 −15 −10
−5
0
0
5
10
– p.31







b =
B







b t
bt = By
Transformation: x
−4.898e4
3.357e3
−3.315e04
−6.455e3
2.337e3
1.151e3
7.328e4
2.85e4
−9.569e6
−2.189e3
1.842e3
1.457e3
−5.780e5
5.420e3
−5.247e3
5.878e4
−2.674e3
−1.238e3
−1.766e3
3.654e3
3.066e3
2.492e3
2.780e3
8.571e4
−1.466e3
−7.337e4
−5.896e3
3.663e3
6.633e3
3.472e3
−2.981e4
−8.716e4
6.393e8
−2.327e3
5.365e3
−9.475e4
−7.620e4
−3.338e3
1.471e3
2.099e3
−1.318e2
4.259e3
−1.047e3


1.067e3 


6.280e3 


2.356e3 


−4.668e3 

8.629e3 

6.581e4
Code: aek = a × 10−k
– p.32
0 2 4 6
Transformed weekly recorded cases of measles in 7 cities
1955
1960
1965
1950
1955
1960
1965
1950
1955
1960
1965
1950
1955
1960
1965
1950
1955
1960
1965
1950
1955
1960
1965
1950
1955
1960
1965
−6
0
4
−4
0
4
−6
−2
2
−4
0
4
−4
0
4
−4
0
1950
– p.33
CCF of transformed weekly recorded cases of measles in 7 cities
20
−5
−5
0
−20 −15 −10
−20 −15 −10
0.5
−0.5
0.5
15
20
10
15
0
−5
−5
10
15
5
10
−20 −15 −10
5
20
−5
−5
10
15
20
0
5
10
15
20
0.5
X4 & X7
5
10
15
20
0
5
10
15
20
0.5
X5 & X7
5
10
15
20
0
5
10
15
20
X6 & X7
0.5
X6
0
5
10
15
20
0
5
X7 & X6
0
20
−0.5
0
0
15
0.5
5
X5 & X6
15
10
X3 & X7
−0.5
0
X7 & X5
−20 −15 −10
0
X4 & X6
20
20
−0.5
0
X6 & X5
0
20
0.5
0
0
15
0.5
5
15
−0.5
−5
10
X3 & X6
20
10
0.5
5
0.5
5
5
−0.5
0
X5
X7 & X4
0
10
0
X2 & X7
−0.5
0
X6 & X4
20
−0.5
0
20
15
0.5
5
X4 & X5
15
10
10
15
20
15
20
X7
0.5
10
5
−0.5
0.5
−0.5
0
0.5
5
0
X2 & X6
−0.5
15
0.5
−20 −15 −10
−0.5
0.5
−0.5
0.5
−0.5
0.5
−0.5
0.5
10
0.5
−20 −15 −10
20
X3 & X5
−0.5
5
−20 −15 −10
X7 & X3
0
20
0.5
0
0
0.5
−5
15
−0.5
−5
15
−0.5
0
10
X5 & X4
−0.5
−20 −15 −10
5
−0.5
0
0.5
−5
0.5
0
−5
10
X2 & X5
X4
−0.5
−20 −15 −10
0
X6 & X3
−0.5
0.5
−5
−20 −15 −10
X7 & X2
−0.5
−20 −15 −10
−20 −15 −10
0
0.5
0
20
0.5
−5
0
X5 & X3
−0.5
0.5
−5
15
−0.5
−20 −15 −10
X7 & X1
10
5
−0.5
0
X6 & X2
−0.5
−20 −15 −10
5
0
X3 & X4
0.5
−5
0.5
0
X6 & X1
20
X4 & X3
−0.5
0.5
−5
0
X5 & X2
−0.5
−20 −15 −10
15
−0.5
−20 −15 −10
20
0.5
0
0.5
0
X5 & X1
10
0.5
−5
−0.5
0.5
−5
5
X3
X4 & X2
−0.5
−20 −15 −10
0
−0.5
−20 −15 −10
15
−0.5
20
10
X2 & X4
0.5
15
0.5
0
5
−0.5
10
0
0.5
5
−0.5
0.5
−5
X4 & X1
20
0.5
0
X3 & X2
−0.5
−20 −15 −10
15
−0.5
0
X3 & X1
10
−0.5
0.5
−5
5
X2 & X3
−0.5
0.5
−0.5
−20 −15 −10
0
0.5
20
X1 & X7
−0.5
15
X2
0.5
10
−0.5
5
0.5
0
X1 & X6
−0.5
20
−0.5
15
0.5
10
X2 & X1
X1 & X5
−0.5
5
−0.5
−0.5
−0.5
0
X1 & X4
0.5
X1 & X3
0.5
X1 & X2
0.5
X1
−20 −15 −10
−5
0
0
5
10
– p.34
CCF of prewhitened transformed weekly recorded cases of measles
20
−5
−5
0
−20 −15 −10
−20 −15 −10
1.0
−0.2 0.4
1.0
15
20
10
15
0
−5
−5
10
15
5
10
−20 −15 −10
5
20
−5
−5
10
15
20
0
5
10
15
20
1.0
X4 & X7
5
10
15
20
0
5
10
15
20
1.0
X5 & X7
5
10
15
20
0
5
10
15
20
X6 & X7
1.0
X6
0
5
10
15
20
0
5
X7 & X6
0
20
−0.2 0.4
0
0
15
1.0
5
X5 & X6
15
10
X3 & X7
−0.2 0.4
0
X7 & X5
−20 −15 −10
0
X4 & X6
20
20
−0.2 0.4
0
X6 & X5
0
20
1.0
0
0
15
1.0
5
15
−0.2 0.4
−5
10
X3 & X6
20
10
1.0
5
1.0
5
5
−0.2 0.4
0
X5
X7 & X4
0
10
0
X2 & X7
−0.2 0.4
0
X6 & X4
20
−0.2 0.4
0
20
15
1.0
5
X4 & X5
15
10
10
15
20
15
20
X7
1.0
10
5
−0.2 0.4
1.0
−0.2 0.4
0
1.0
5
0
X2 & X6
−0.2 0.4
15
1.0
−20 −15 −10
−0.2 0.4
1.0
−0.2 0.4
1.0
−0.2 0.4
1.0
−0.2 0.4
1.0
10
1.0
−20 −15 −10
20
X3 & X5
−0.2 0.4
5
−20 −15 −10
X7 & X3
0
20
1.0
0
0
1.0
−5
15
−0.2 0.4
−5
15
−0.2 0.4
0
10
X5 & X4
−0.2 0.4
−20 −15 −10
5
−0.2 0.4
0
1.0
−5
1.0
0
−5
10
X2 & X5
X4
−0.2 0.4
−20 −15 −10
0
X6 & X3
−0.2 0.4
1.0
−5
−20 −15 −10
X7 & X2
−0.2 0.4
−20 −15 −10
−20 −15 −10
0
1.0
0
20
1.0
−5
0
X5 & X3
−0.2 0.4
1.0
−5
15
−0.2 0.4
−20 −15 −10
X7 & X1
10
5
−0.2 0.4
0
X6 & X2
−0.2 0.4
−20 −15 −10
5
0
X3 & X4
1.0
−5
1.0
0
X6 & X1
20
X4 & X3
−0.2 0.4
1.0
−5
0
X5 & X2
−0.2 0.4
−20 −15 −10
15
−0.2 0.4
−20 −15 −10
20
1.0
0
1.0
0
X5 & X1
10
1.0
−5
−0.2 0.4
1.0
−5
5
X3
X4 & X2
−0.2 0.4
−20 −15 −10
0
−0.2 0.4
−20 −15 −10
15
−0.2 0.4
20
10
X2 & X4
1.0
15
1.0
0
5
−0.2 0.4
10
0
1.0
5
−0.2 0.4
1.0
−5
X4 & X1
20
1.0
0
X3 & X2
−0.2 0.4
−20 −15 −10
15
−0.2 0.4
0
X3 & X1
10
−0.2 0.4
1.0
−5
5
X2 & X3
−0.2 0.4
1.0
−0.2 0.4
−20 −15 −10
0
1.0
20
X1 & X7
−0.2 0.4
15
X2
1.0
10
−0.2 0.4
5
1.0
0
X1 & X6
−0.2 0.4
20
−0.2 0.4
15
1.0
10
X2 & X1
X1 & X5
−0.2 0.4
5
−0.2 0.4
−0.2 0.4
−0.2 0.4
0
X1 & X4
1.0
X1 & X3
1.0
X1 & X2
1.0
X1
−20 −15 −10
−5
0
0
5
10
– p.35
Note. When none of the component series are WN, the
√
confidence bounds ±1.96/ n could be misleading.
Segmentation assumption is invalid for this example!
(b) Ratios of maximum cross correlations
1.00
0.08
1.05
0.12
1.10
0.16
1.15
1.20
0.20
(a) Max cross correlations between pairs
5
10
15
20
5
10
15
20
(a) The maximum cross correlations, plotted in descending order, among each of the ( 72 ) = 21 pairs
component series of the transformed and prewhitened measles series. The maximization was taken
over the lags between -20 to 20. (b) The ratios of two successive correlations in (a).
– p.36
The segmentations determined by different numbers of connected pairs
for the transformed measles series from 7 cities in England.
No. of connected pairs
No. of groups
1
6
{4, 5}, {1}, {2}, {3}, {6}, {7}
2
5
{1, 2}, {4, 5}, {3}, {6}, {7}
3
4
{1, 2, 3} , {4, 5}, {6}, {7}
4
3
{1, 2, 3, 7}, {4, 5}, {6}
5
2
{1, 2, 3, 6, 7}, {4, 5}
6
1
{1, · · · , 7}
Segmentation
– p.37
Post-sample forecasting
Forecast the notified measles cases in the last 14 weeks of the
period for all 7 cities
Using the segmentation with four groups: {1, 2, 3}, {4, 5},
{6} and {7}
One-step MSE Two-step MSE
VAR
503.408(1124.213) 719.499(2249.986)
RVAR
574.582(1432.217) 846.141(2462.019)
Segmentation 472.106(1088.17) 654.843(1807.502)
– p.38
Example 4. Daily impressions of 32 keywords for an air ticket booking
website powered by Baidu (www.baidu.com) over 130 days.
⇒
n = 130, p = 32
Data have been coded to protect the confidentiality.
Advertisements are accessed by keywords search from a search engine.
Each display of an advertisement is counted as one impression.
Applying the proposed transformation with m = 10 (or 1 ≤ m ≤ 15), the
transformed 32 time series are segmented into 31 groups with the only
non-single-element group {10, 13}.
– p.39
−0.5
2.0
Time series of 8 randomly selected daily impressions
20
40
60
80
100
120
1
20
40
60
80
100
120
1
20
40
60
80
100
120
1
20
40
60
80
100
120
1
20
40
60
80
100
120
1
20
40
60
80
100
120
1
20
40
60
80
100
120
1
20
40
60
80
100
120
−0.5
1.0
0.5
2.5
−1
1
−0.5
1.5
−1.0 0.0
0.5
2.0
0.5 2.5
1
– p.40
CCF of 8 randomly selected daily impressions
0
−5
0
0
−5
−15
−10
−5
−15
−10
−5
−15
−10
−5
−15
−10
−5
5
10
−15
−10
−5
15
−10
−5
−15
−10
−5
0.8
−0.2
0.8
0.8
−0.2
0.8
−0.2
15
5
10
5
10
−15
−10
−5
−15
−10
−5
0.8
10
15
0
15
5
10
15
0.8
Y21 & Y31
5
10
15
0
5
10
15
Y23 & Y31
0
5
10
15
0
Y24
5
10
15
Y24 & Y31
0
5
10
15
0
5
Y31 & Y24
0
10
0.8
5
Y23 & Y24
0
5
Y18 & Y31
−0.2
0
Y31 & Y23
0
0
Y21 & Y24
15
15
−0.2
0
15
10
−0.2
0.8
−0.2
10
Y24 & Y23
Y31 & Y21
0
15
0.8
0
0
10
0.8
5
Y23
0
5
Y18 & Y24
−0.2
0
Y24 & Y21
−15
0
Y21 & Y23
Y23 & Y21
0
15
5
Y17 & Y31
−0.2
0
0
0.8
0
0
10
Y18 & Y23
15
15
−0.2
0
5
15
0.8
−5
0
10
−0.2
−10
−0.2
0.8
0.8
−0.2
0.8
−0.2
10
10
0.8
−5
Y31 & Y18
0
15
5
Y17 & Y24
0.8
5
0
−0.2
−10
15
−0.2
0
Y24 & Y18
Y31 & Y17
0
−0.2
0.8
0.8
−0.2
0.8
−0.2
−15
0
10
0.8
−15
0
0.8
−10
−0.2
0.8
−0.2
0.8
−0.2
0.8
−0.2
0.8
−0.2
0.8
−0.2
−5
5
Y21
0.8
−10
−0.2
−15
15
10
Y17 & Y23
Y18 & Y21
Y23 & Y18
0.8
−10
0.8
−5
0
−0.2
−15
−0.2
0.8
−10
−15
Y31 & Y9
−0.2
−15
−5
Y24 & Y17
0.8
−5
Y31 & Y1
10
5
5
Y9 & Y31
10
15
Y31
0.8
0
5
0
0
−0.2
−5
−0.2
0.8
−10
−10
0.8
−10
0
Y23 & Y17
Y24 & Y9
−0.2
−15
−15
0
Y21 & Y18
−0.2
−15
15
15
0.8
0
Y24 & Y1
10
10
−0.2
0
15
5
0.8
−5
0.8
−5
0
0.8
−10
−0.2
0.8
−10
−5
−0.2
−15
Y23 & Y9
−0.2
−15
−10
10
0
−0.2
0
Y23 & Y1
5
Y18
Y21 & Y17
0.8
−5
−15
5
Y17 & Y21
0.8
0
0
Y1 & Y31
Y9 & Y24
−0.2
−5
0
15
0.8
−10
−0.2
0.8
−10
15
0.8
−15
Y21 & Y9
−0.2
−15
10
10
−0.2
0
Y21 & Y1
15
−0.2
−5
5
−0.2
0.8
−10
0
Y18 & Y17
−0.2
0.8
−0.2
−15
10
5
0.8
0
5
0
Y9 & Y23
−0.2
−5
Y18 & Y9
15
0.8
−10
0
Y17 & Y18
0.8
−15
Y18 & Y1
10
−0.2
0
15
5
Y9 & Y21
0.8
−5
10
−0.2
0.8
−10
5
Y17
−0.2
0.8
−0.2
−15
0
Y17 & Y9
0
−0.2
15
15
0.8
10
10
−0.2
5
5
Y9 & Y18
0.8
0
Y17 & Y1
0
0.8
0
15
−0.2
−5
10
−0.2
0.8
−10
5
Y9 & Y17
−0.2
0.8
−0.2
−15
0
0.8
15
Y1 & Y24
−0.2
10
Y9
−0.2
5
0.8
0
Y1 & Y23
−0.2
15
0.8
10
Y9 & Y1
Y1 & Y21
−0.2
5
−0.2
−0.2
−0.2
0
Y1 & Y18
0.8
Y1 & Y17
0.8
Y1 & Y9
0.8
Y1
−15
−10
−5
0
0
5
10
15
– p.41
CCF of 8 prewhitened transformed daily impressions (6 randomly selected)
0
−5
0
0
−5
−15
−10
−5
−15
−10
−5
−5
−15
−10
−5
−15
−10
−5
15
−15
10
15
−10
−5
−15
−10
−5
−15
−10
−5
0.8
0.8
−0.2
0.8
−0.2
0
10
5
10
−15
−10
−5
−15
−10
−5
0.8
0
10
15
0
5
10
15
X15 & X28
5
10
15
0
5
10
15
X23 & X28
5
10
15
0
X24
5
10
15
X24 & X28
0
5
10
15
0
5
X28 & X24
0
15
0.8
0
0
10
0.8
5
X23 & X24
15
5
X11 & X28
−0.2
0
X28 & X23
0
15
X15 & X24
15
15
−0.2
0
X24 & X23
0
10
0.8
5
10
−0.2
0.8
−0.2
15
X23
0
5
X11 & X24
−0.2
0
X28 & X15
0
10
5
0.8
5
0
X15 & X23
X24 & X15
0
15
0.8
5
0
X1 & X28
−0.2
0
X23 & X15
0
10
X11 & X23
0.8
0
−0.2
0.8
−0.2
0.8
0.8
−0.2
0.8
−0.2
10
15
−0.2
0
5
15
0.8
−10
0
0.8
5
10
0.8
−5
X28 & X11
0
15
5
X1 & X24
−0.2
0
X24 & X11
0
10
0
−0.2
−10
15
10
−0.2
−15
X28 & X1
0
−0.2
0.8
0.8
−0.2
0.8
−0.2
−15
0
0.8
−10
−0.2
0.8
−0.2
0.8
−0.2
0.8
−0.2
0.8
−0.2
0.8
−0.2
−5
5
X15
0.8
−10
−0.2
−15
15
10
X1 & X23
X11 & X15
X23 & X11
0.8
−10
0.8
−5
0
−0.2
−15
−0.2
0.8
−10
−15
X28 & X13
−0.2
−15
−5
X24 & X1
0.8
−5
X28 & X10
10
5
5
X13 & X28
10
15
X28
0.8
0
5
0
0
−0.2
−5
−0.2
0.8
−10
−10
0.8
−10
0
X23 & X1
X24 & X13
−0.2
−15
−15
0
X15 & X11
−0.2
−15
15
15
0.8
0
X24 & X10
10
10
−0.2
0
15
5
0.8
−5
0.8
−5
0
0.8
−10
−0.2
0.8
−10
−5
−0.2
−15
X23 & X13
−0.2
−15
−10
10
0
−0.2
0
X23 & X10
5
X11
X15 & X1
0.8
−5
−15
5
X1 & X15
−0.2
0
0
X10 & X28
X13 & X24
0.8
−5
0
15
−0.2
−10
−0.2
0.8
−10
15
0.8
−15
X15 & X13
−0.2
−15
10
10
0.8
0
X15 & X10
15
−0.2
−5
5
−0.2
0.8
−10
0
X11 & X1
−0.2
0.8
−0.2
−15
10
5
−0.2
0
5
0
X13 & X23
0.8
−5
X11 & X13
15
−0.2
−10
0
X1 & X11
0.8
−15
X11 & X10
10
0.8
0
15
5
X13 & X15
0.8
−5
10
−0.2
0.8
−10
5
X1
−0.2
0.8
−0.2
−15
0
X1 & X13
0
0.8
15
15
−0.2
10
10
0.8
5
5
X13 & X11
0.8
0
X1 & X10
0
−0.2
0
15
−0.2
−5
10
−0.2
0.8
−10
5
X13 & X1
−0.2
0.8
−0.2
−15
0
X10 & X24
−0.2
15
0.8
10
X13
−0.2
5
0.8
0
X10 & X23
−0.2
15
0.8
10
X13 & X10
X10 & X15
−0.2
5
−0.2
−0.2
−0.2
0
X10 & X11
0.8
X10 & X1
0.8
X10 & X13
0.8
X10
−15
−10
−5
0
0
5
10
15
– p.42
Example 5. Daily sales of a clothing brand in 25 provinces in
China in 1 January 2008 – 16 December 2012.
n = 1812, p = 25
Annual pattern: peak in February
Strong periodicity component with the period 7.
The 25 transformed series are segmented into 24 groups with {15, 16} as
one group.
Permutation is performed using the max-CCF with 14 ≤ m ≤ 30.
– p.43
7
9
11
Time series of daily sales of a clothing brand in 8 provinces
2009
2010
2011
2012
2008
2009
2010
2011
2012
2008
2009
2010
2011
2012
2008
2009
2010
2011
2012
2008
2009
2010
2011
2012
2008
2009
2010
2011
2012
2008
2009
2010
2011
2012
2008
2009
2010
2011
2012
7.5
9.5
7.0
9.0
7
9
7.0
9.0
8.0 9.5
7.5
9.5
8.0
9.5
2008
– p.44
CCF of daily sales of a clothing brand in 8 provinces
−20
−10
−10
20
20
−10
−10
10
10
0.8
0.8
0.0
0.8
0.0
10
20
0
10
20
0
−10
5
10
20
0
5
20
10
20
Y7 & Y8
0.8
Y7
0
5
10
20
0
5
Y8 & Y7
0
10
Y6 & Y8
0.8
5
Y8 & Y6
−20
20
0.0
0
0
10
0.8
5
0.8
−10
20
Y5 & Y8
0.0
−20
5
Y6 & Y7
20
10
Y4 & Y8
0.8
5
5
0.0
0
Y7 & Y6
0
0
0.0
0
0
20
Y5 & Y7
20
20
0.8
5
Y6
0
10
0.8
5
Y8 & Y5
−20
5
0.0
0
10
0.0
0
0.8
−20
0.0
0.8
0.8
0.0
10
5
Y3 & Y8
0.8
5
0.8
−10
0
Y4 & Y7
0.0
−20
20
0.8
0
Y5 & Y6
20
10
0.0
0
Y7 & Y5
0
20
20
0.0
10
5
Y4 & Y6
Y6 & Y5
0
10
10
Y2 & Y8
0.0
5
0.8
5
Y8 & Y4
−20
0.0
0.8
0.8
0
5
Y3 & Y7
0.8
10
0
10
20
Y8
0.8
−10
0
0.0
5
20
0.0
−20
20
0.0
0
0
10
0.8
20
10
Y2 & Y7
0.0
10
5
0.8
−10
5
Y5
0
0
Y3 & Y6
0.8
−10
20
0.0
0.8
0.0
0
0.8
5
Y7 & Y4
0
20
0.0
0
0.8
−20
0
10
0.8
20
10
Y2 & Y6
0.0
0.8
10
5
Y4 & Y5
0.8
−10
0.0
0.8
0.0
0.8
0.8
0.8
0.0
0
0.0
−20
0
0.8
0
20
0.0
−20
5
Y6 & Y4
0.0
−10
5
Y8 & Y3
0.8
−20
10
0.0
−10
0
Y3 & Y5
0.8
0
20
Y1 & Y8
0.0
0
0
0.0
−10
0.8
−10
20
Y5 & Y4
0.0
0.8
0.0
−20
0
Y7 & Y3
0.0
0.8
0
0
0.8
−20
Y8 & Y2
0.0
−10
−10
0.8
0
10
Y4
0.0
−10
5
Y6 & Y3
0.8
0.8
0.8
0
Y8 & Y1
−20
−20
Y7 & Y2
0.0
−10
−20
0
0.0
−20
Y7 & Y1
−20
0
0.0
−10
5
Y5 & Y3
0.8
−20
0
20
10
Y2 & Y5
0.0
0
Y6 & Y2
0.0
−10
10
0.8
−10
0.0
0.8
0
Y6 & Y1
−20
5
Y5 & Y2
0.0
−10
0
5
Y3 & Y4
0.0
−20
Y5 & Y1
−20
0
Y4 & Y3
0.8
0
20
0.8
0
0.0
0.8
−10
10
0.0
−10
Y4 & Y2
0.0
−20
5
Y3
0.8
−20
0
0.0
0
0.0
0.8
0
Y4 & Y1
20
0.8
20
10
Y2 & Y4
0.0
10
5
0.8
5
Y3 & Y2
0.0
−10
0
0.8
0
Y3 & Y1
−20
20
0.0
0
10
0.0
0.8
−10
5
Y2 & Y3
0.0
0.8
0.0
−20
0.0
0
Y1 & Y7
0.0
20
Y2
0.8
10
0.0
5
0.8
0
Y1 & Y6
0.0
20
0.8
10
Y2 & Y1
Y1 & Y5
0.0
5
0.0
0.0
0.0
0
Y1 & Y4
0.8
Y1 & Y3
0.8
Y1 & Y2
0.8
Y1
−20
−10
0
0
5
10
20
– p.45
CCF of 8 transformed daily sales of a clothing brand
−10
10
20
−10
−20
−10
10
20
0.6
−0.2
0.6
10
20
20
−20
−10
−20
−10
0
5
5
5
10
20
0
5
20
10
20
10
20
0.6
X20 & X23
0
5
10
20
0
5
10
20
0.6
X21 & X23
0
5
10
20
0
5
X23 & X21
0
10
0.6
0
X21
0
20
X19 & X23
−0.2
10
10
0.6
5
0.6
5
5
X3 & X23
X20 & X21
−0.2
0
0
X19 & X21
X23 & X20
0
20
−0.2
5
0
0.6
0
X21 & X20
0
10
−0.2
5
0.6
0
20
X2 & X23
−0.2
5
10
10
20
X23
0.6
−20
0
X20
0
20
0.6
−10
10
0.6
0
5
X3 & X21
0.6
−20
5
−0.2
20
−0.2
0.6
0.6
−0.2
20
−0.2
10
0
X19 & X20
X23 & X19
0
10
0
X16 & X23
0.6
5
−0.2
0
−0.2
5
20
X2 & X21
0.6
0
10
−20
−10
0
−0.2
−20
20
−0.2
20
X21 & X19
0
−0.2
0.6
−0.2
0.6
10
5
0.6
−10
10
0.6
5
−0.2
0
0
X3 & X20
0.6
−20
5
0.6
−0.2
20
−0.2
0
X23 & X3
0
10
X15 & X23
X16 & X21
0.6
−10
20
−0.2
−10
5
0.6
−20
0
X20 & X19
0.6
−20
0
−0.2
0
10
X2 & X20
−0.2
−10
X21 & X3
0
20
0.6
−20
5
0.6
−10
10
X19
−0.2
0
−0.2
0.6
0.6
−0.2
−0.2
20
0.6
−20
5
0.6
10
0
X3 & X19
−0.2
−10
0
0.6
5
X23 & X2
0
20
−0.2
0
X21 & X2
0
10
X15 & X21
X16 & X20
−0.2
−10
5
X20 & X3
0.6
−20
0
0.6
−20
20
X2 & X19
−0.2
0
10
0.6
−10
20
−0.2
−10
0.6
−20
10
0.6
−20
5
X19 & X3
−0.2
0
−0.2
0.6
−0.2
0.6
0
X15 & X20
X16 & X19
0.6
−10
0
X3
0.6
−10
5
X20 & X2
−0.2
−20
20
0.6
20
−0.2
0
0.6
0
−0.2
−10
−20
X23 & X16
0.6
−20
10
0.6
−10
0
X19 & X2
−0.2
−20
10
X2 & X3
−0.2
0
X23 & X15
20
−0.2
0
0.6
−10
−0.2
−20
5
X21 & X16
0.6
X21 & X15
10
0.6
−10
−0.2
−20
5
X3 & X2
0.6
0
−0.2
−10
0
X20 & X16
0.6
X20 & X15
−20
5
−0.2
0
X15 & X19
X16 & X3
0.6
−10
−0.2
−20
0
X2
0.6
0
−0.2
−10
0
X19 & X16
0.6
−20
20
−0.2
20
0.6
0
−0.2
−10
X19 & X15
−0.2
10
X3 & X16
0.6
−0.2
−20
10
0.6
5
0.6
0
−0.2
−10
5
X2 & X16
0.6
−0.2
−20
0
X16 & X2
−0.2
0
X3 & X15
−0.2
20
0.6
0
−0.2
−10
X2 & X15
−0.2
10
−0.2
5
X16
0.6
−0.2
−20
X15 & X3
0.6
0
−0.2
20
−0.2
10
−0.2
5
−0.2
−0.2
0
X16 & X15
−0.2
X15 & X2
0.6
X15 & X16
0.6
X15
0
5
10
20
– p.46
CCF of 8 prewhitened transformed daily sales of a clothing brand
−20
−10
−10
−20
−10
−20
−20
−10
−10
0
−20
−10
−20
−10
0
−20
−10
5
10
0
5
10
0
−20
−10
−20
−10
20
−20
−10
0
5
10
20
0
5
10
20
0
0
5
10
5
20
0
5
−20
−10
20
−20
−10
−0.2 0.6
20
0
10
0
5
10
20
0
5
10
20
0
5
0
5
10
5
5
20
0
5
−20
−10
20
10
20
10
20
10
20
X20 & X23
20
0
5
10
20
X21 & X23
20
0
5
X23 & X21
0
10
X19 & X23
X21
0
20
X3 & X23
X20 & X21
X23 & X20
0
10
10
X2 & X23
X19 & X21
X21 & X20
0
0
X3 & X21
X20
0
20
−0.2 0.6
10
−0.2 0.6
5
10
−0.2 0.6
−0.2 0.6
5
5
X2 & X21
X19 & X20
X23 & X19
0
0
X3 & X20
X21 & X19
0
20
−0.2 0.6
0
−0.2 0.6
−0.2 0.6
−0.2 0.6
0
X20 & X19
X23 & X3
0
20
X19
X21 & X3
0
−0.2 0.6
−0.2 0.6
−0.2 0.6
20
X20 & X3
X23 & X2
0
10
10
X3 & X19
0
X16 & X23
−0.2 0.6
0
X21 & X2
0
5
20
−0.2 0.6
−20
0
X19 & X3
X20 & X2
0
20
10
10
−0.2 0.6
5
5
−0.2 0.6
0
0
5
X2 & X20
−0.2 0.6
0
20
X2 & X19
X3
0
X16 & X21
−0.2 0.6
−10
10
10
20
−0.2 0.6
−20
−0.2 0.6
−0.2 0.6
−0.2 0.6
−0.2 0.6
5
X19 & X2
0
5
10
X16 & X20
X15 & X23
10
20
X23
0
−0.2 0.6
0
−10
0
5
−0.2 0.6
−10
0
0
X15 & X21
−0.2 0.6
−20
20
20
−0.2 0.6
−20
20
X2 & X3
−0.2 0.6
0
10
10
−0.2 0.6
−10
5
5
−0.2 0.6
−20
0
−0.2 0.6
−10
10
0
X16 & X19
−0.2 0.6
−20
20
X3 & X2
X23 & X16
−0.2 0.6
−0.2 0.6
−10
5
20
X15 & X20
−0.2 0.6
0
X23 & X15
−20
0
X21 & X16
−0.2 0.6
−0.2 0.6
−10
−10
10
10
−0.2 0.6
0
X21 & X15
−20
0
X20 & X16
−0.2 0.6
−0.2 0.6
−10
−20
5
−0.2 0.6
0
X20 & X15
−20
5
X19 & X16
−0.2 0.6
−0.2 0.6
−10
−10
0
X15 & X19
X16 & X3
−0.2 0.6
−20
20
X2
−0.2 0.6
0
X19 & X15
−20
0
X3 & X16
−0.2 0.6
−0.2 0.6
−10
20
−0.2 0.6
0
X3 & X15
−20
10
10
−0.2 0.6
5
X2 & X16
−0.2 0.6
−0.2 0.6
−10
5
−0.2 0.6
0
−0.2 0.6
0
X2 & X15
−20
0
X16 & X2
−0.2 0.6
−10
20
X16
−0.2 0.6
−0.2 0.6
−20
10
−0.2 0.6
5
X15 & X3
−0.2 0.6
0
−0.2 0.6
20
−0.2 0.6
10
X16 & X15
−0.2 0.6
5
X15 & X2
−0.2 0.6
0
X15 & X16
−0.2 0.6
−0.2 0.6
X15
0
5
10
20
– p.47
Post-sample forecasting
Forecast the daily sales for the last two weeks in each of the 25
provinces.
Segmentation: 24 groups with {15, 16} as one group
One-step MSE Two-step MSE
univariate AR
0.208(1.255)
0.194(1.250)
VAR
0.295(2.830)
0.301(2.930)
0.293(2.817)
0.296(2.962)
RVAR
0.153(0.361)
0.163(0.341)
Segmentation
Cross correlations between provinces are useful information.
However they cannot be used via direct VAR!
– p.48
Asymptotic theory
For p × r matrices H1 and H2 and H′1 H1 = H′2 H2 = Ir , let
r
1
D(M(H1 ), M(H2 )) = 1 − tr(H1 H′1 H2 H′2 ).
r
Then D(M(H1 ), M(H2 )) ∈ [0, 1].
It equals 0 iff M(H1 ) = M(H2 ), and 1 iff M(H1 ) ⊥ M(H2 ).
yt is assumed to be weakly stationary and α-mixing, i.e.
αk ≡ sup
i
sup
∞
i
, B∈Fi+k
A∈F−∞
|P (A ∩ B) − P (A)P (B)| → 0, as k → ∞,
where Fij = σ(yt : i ≤ t ≤ j).
– p.49
p fixed
C1. For some constant γ > 2,
sup max E(|yi,t − Eyi,t |2γ ) < ∞.
t
C2.
P∞
1−2/γ
k=1 αk
1≤i≤p
< ∞.
Theorem 1. Let conditions C1, C2 hold, ̟ be positive and p be
b = (A
b 1, · · · , A
b q ) of which the
fixed. Then there exists an A
b y , such that
columns are a permutation of the columns of Γ
b j ), M(Aj )) = Op (n−1/2 ).
max D(M(A
1≤j≤q
– p.50
p = o(nc ) for some c > 0
C3. (Sparsity of A = (ai,j )) For some constant ι ∈ [0, 1),
p
p
X
X
max
|ai,j |ι ≤ s1
and
max
|ai,j |ι ≤ s2 ,
1≤j≤p
i=1
1≤i≤p
j=1
where s1 and s2 are positive constants which may diverge together
with p.
C4. For some positive constants l > 2 and τ > 0,
sup max P (|yi,t − µi | > x) = O(x−2(l+τ ) ) as x → ∞.
t
1≤i≤p
C5. αk = O(k −l(l+τ )/(2τ ) ) as k → ∞.
Remark. As Σy (k) = AΣx (k)A′ , the sparsity of A and the maximum
block size of Σx (k) provide a measure for the sparsity of Σy (k).
– p.51
Let Smax be the maximum block size in Σx (k), and
ρj = min min |λ(Wx,i ) − λ(Wx,j )|,
j = 1, · · · , q,
1≤i≤q
i6=j
δ = s1 s2
max
k=1,...,k0
kΣx (k)kι∞ ,
κ1 =
min
k=1,...,k0
kΣx (k)k2 ,
κ2 =
max
k=1,...,k0
kΣx (k)k2 .
b y (k) = (b
Thresholding CCVF: Let Σ
σi,j (k)) be CCVF. Define
e y (k) = σ
Σ
bi,j (k)I{|b
σi,j (k)| ≥ u} ,
u = M p2/l n−1/2 .
Theorem 2. Let conditions C3-C5 hold, minj ρj > 0 and p = o(nl/4 ).
b = (A
b 1, · · · , A
b q ) of which the columns are
Then there exists an A
b y , such that
a permutation of the columns of Γ
b j ), M(Aj ))
max ρj D(M(A
1≤j≤q
n Op κ2 (p4/l n−1 )(1−ι)/2 Smax δ , κ−1 (p4/l n−1 )(1−ι)/2 Smax δ = O(1)
1
=
4/l −1 1−ι 2
−1 δ −1 = O(1).
Op (p n ) Smax δ 2 ,
κ2 (p4/l n−1 )−(1−ι)/2 Smax
– p.52
Remarks
(i) Theorem 2 gives the uniform convergence rate for
b j ), M(Aj )). The smaller ρj is, more difficult the
ρj D(M(A
estimation for M(Aj ) is.
(ii) The smaller ι, s1 , s2 and Smax are, more sparse Σy (k) is, and
the faster the convergences are.
(iii) Similar results can be obtained for the cases with
log p = o(nc ) by assuming the sub-Gaussianity for yt and
exponential decay rates for α-mixing coefficients.
– p.53
Simulation
Let A = (A1 , · · · , Aq ), Aj is p × pj ,
P
j
pj = p, and
P
b
b
b
b
bj = p.
A = (A1 , · · · , Aqb), Aj is p × pbj ,
jp
Correct segmentation: qb = q, pbj = pj , and
b j )) = min D(M(Aj ), M(A
b i )),
D(M(Aj ), M(A
1≤i≤q
j = 1, · · · , q.
for H′1 H1 = Ir1 and H′2 H2 = Ir2 ,
d(M(H1 ), M(H2 )) = 1 −
1/2
1
′
′
tr(H1 H1 H2 H2 )
.
min (r1 , r2 )
b j ) is an estimator for
Incomplete segmentation: qb < q, and each M(A
the linear space spanned by one, or more than one Ai
volatility
– p.54
No. of replications: 500 times for each setting.
Let A = (aij ), aij ∼ U (−3, 3) independently.
Example 6. Let p = 6, yt = Axt , and
(1)
(2)
(3)
xi,t = ηt+i−1 for i = 1, 2, 3, xj,t = ηt+j−4 for j = 4, 5, x6,t = ηt .
where
(1)
= 0.5ηt−1 + 0.3ηt−2 + et − 0.9et−1 + 0.3et−2 + 1.2et−3 + 1.3et−4 ,
(2)
= −0.4ηt−1 + 0.5ηt−2 + et + et−1 − 0.8et−2 + 1.5et−3 ,
(3)
= 0.9ηt−1 + et ,
ηt
ηt
ηt
(1)
(1)
(2)
(1)
(3)
(2)
(2)
(1)
(2)
(1)
(2)
(1)
(2)
(1)
(1)
(2)
(3)
(3)
and et , et , et
are indep N (0, 1). Thus
q = 3, i.e. 3 segmented subseries with 3, 2, 1 elements.
– p.55
Relative frequencies of correct and incomplete segmentations
n
100
200
300
400
500
1000
1500
2000
2500
3000
Correct
.350
.564
.660
.712
.800
.896
.906
.920
.932
.945
Incomplete
.508
.386
.326
.282
.192
.104
.094
.080
.068
.055
P
b i )) (with correct segmentations only)
D(M(A
),
M(
A
i
1≤i≤3
0.0
0.2
0.4
0.6
0.8
1.0
Boxplots of
1
3
100
200
300
400
Sample size
500
1000
1500
– p.56
CCF of yt (one instance)
−10
0
0
−10
−10
0
10
15
20
0
1.0
−1.0
1.0
−1.0
0.0
15
20
−20
−15
5
−10
0
5
0
−10
10
15
20
0
5
0
20
10
15
20
10
15
20
0.0
1.0
Y5 & Y6
0
5
10
15
20
0
5
Y6 & Y5
−5
15
1.0
5
Y5
−5
10
0.0
0
1.0
−15
0
Y4 & Y6
0.0
−20
20
1.0
10
1.0
5
15
Y3 & Y6
0.0
0
10
0.0
5
Y6 & Y4
−5
20
−1.0
0
Y5 & Y4
−5
15
Y4 & Y5
1.0
−15
0.0
1.0
1.0
0.0
−1.0
20
0.0
−20
10
1.0
15
1.0
−15
5
0.0
10
0.0
−20
0
−1.0
−5
Y6 & Y3
−5
−1.0
0.0
1.0
1.0
0.0
5
5
Y3 & Y5
1.0
0
0
Y2 & Y6
−1.0
0
20
0.0
−10
20
10
15
20
15
20
Y6
1.0
−5
15
1.0
−15
1.0
−15
10
Y4
0.0
−20
5
0.0
−20
15
1.0
−10
1.0
0
−1.0
−5
20
1.0
−15
0
Y5 & Y3
0.0
1.0
−10
15
10
Y2 & Y5
0.0
0
5
−20
−15
−10
−5
0
−1.0
−5
Y6 & Y2
0.0
−15
10
0
Y3 & Y4
0.0
−20
Y6 & Y1
−20
5
Y1 & Y6
0.0
−10
−1.0
0
20
1.0
−15
1.0
−5
−1.0
−10
15
0.0
−20
0.0
1.0
−15
0
Y5 & Y2
0.0
−20
10
Y4 & Y3
−1.0
0
Y5 & Y1
20
−1.0
0
15
−1.0
−5
1.0
−5
−1.0
−10
5
Y3
0.0
1.0
−15
0
1.0
−10
10
−1.0
20
0.0
−15
5
−1.0
15
−1.0
−20
0
Y2 & Y4
−1.0
10
Y4 & Y2
0.0
−20
20
1.0
5
1.0
0
15
0.0
0
0.0
−5
−1.0
−10
10
Y3 & Y2
1.0
−15
5
−1.0
0
−1.0
−5
0.0
−1.0
−20
0
Y2 & Y3
1.0
−10
−1.0
−15
Y4 & Y1
−1.0
20
0.0
1.0
0.0
−1.0
−20
Y3 & Y1
−1.0
15
Y2
−1.0
10
−1.0
5
Y1 & Y5
0.0
1.0
0.0
0
−1.0
20
Y1 & Y4
−1.0
15
−1.0
10
−1.0
0.0
5
−1.0
0.0
−1.0
0
Y2 & Y1
−1.0
Y1 & Y3
1.0
Y1 & Y2
1.0
Y1
0
5
10
– p.57
−10
0
0
−10
−10
0
Segmentation: {1, 4, 6}, {2, 5}, {3}
0
10
15
20
1.0
1.0
20
−20
−15
5
−10
0
5
0
−10
0
20
10
15
20
1.0
5
10
15
20
0
5
10
15
20
0.0
1.0
Z5 & Z6
0
5
10
15
20
0
5
Z6 & Z5
−5
15
0.0
0
Z5
−5
10
Z4 & Z6
1.0
−15
0
1.0
15
0.0
−20
20
0.0
10
1.0
5
15
Z3 & Z6
0.0
0
10
0.0
−1.0
20
−1.0
5
Z6 & Z4
−5
−1.0
0.0
1.0
−1.0
1.0
−1.0
0.0
0
Z5 & Z4
−5
15
Z4 & Z5
1.0
−15
10
1.0
20
0.0
−20
5
−1.0
−5
Z6 & Z3
−5
15
1.0
−15
0
0.0
10
0.0
−20
0.0
1.0
1.0
0.0
5
5
Z3 & Z5
1.0
0
0
Z2 & Z6
−1.0
0
20
0.0
−10
20
10
15
20
15
20
Z6
1.0
−5
15
1.0
−15
1.0
−15
10
Z4
0.0
−20
5
0.0
−20
15
1.0
−10
1.0
0
−1.0
−5
20
1.0
−15
0
Z5 & Z3
0.0
1.0
−10
15
10
Z2 & Z5
0.0
0
5
−20
−15
−10
−5
0
−1.0
−5
Z6 & Z2
0.0
−15
10
0
Z3 & Z4
0.0
−20
Z6 & Z1
−20
5
Z1 & Z6
0.0
−10
−1.0
0
20
1.0
−15
1.0
−5
−1.0
−10
15
0.0
−20
0.0
1.0
−15
0
Z5 & Z2
0.0
−20
10
Z4 & Z3
−1.0
0
Z5 & Z1
20
−1.0
0
15
−1.0
−5
1.0
−5
−1.0
−10
5
Z3
0.0
1.0
−15
0
1.0
−10
10
−1.0
20
0.0
−15
5
−1.0
15
−1.0
−20
0
Z1 & Z5
Z2 & Z4
−1.0
10
Z4 & Z2
0.0
−20
20
1.0
5
1.0
0
15
0.0
0
0.0
−5
−1.0
−10
10
Z3 & Z2
1.0
−15
5
−1.0
0
−1.0
−5
0.0
−1.0
−20
0
Z2 & Z3
1.0
−10
−1.0
−15
Z4 & Z1
−1.0
20
0.0
1.0
0.0
−1.0
−20
Z3 & Z1
−1.0
15
Z2
−1.0
10
−1.0
5
0.0
1.0
0.0
0
−1.0
20
Z1 & Z4
−1.0
15
−1.0
10
−1.0
0.0
5
−1.0
0.0
−1.0
0
Z2 & Z1
−1.0
Z1 & Z3
1.0
Z1 & Z2
1.0
Z1
bt (one instance)
CCF of x
0
5
10
– p.58
Example 7. Let p = 20, q = 5, (p1 , · · · , p5 ) = (6, 5, 4, 3, 2)
xt is defined similarly as in Example 6.
Relative frequencies of correct and incomplete segmentations
n
400
500
1000
1500
2000
2500
3000
Correct
0.058
0.118
0.492
0.726
0.862
0.902
0.940
Incomplete
0.516
0.672
0.460
0.258
0.130
0.096
0.060
1
5
P
1≤i≤5
b i )) (with correct segmentations only)
D(M(Ai ), M(A
0.0
0.2
0.4
0.6
0.8
1.0
Boxplots of
400
500
1000
1500
Sample size
2000
2500
3000
– p.59
Segmenting multiple volatility processes
Let Ft = σ(yt , yt−1 , · · · ),
E(yt |Ft−1 ) = 0,
Assumption:
Var(yt |Ft−1 ) = Σy (t).
yt = Axt , Var(xt |Ft−1 ) = diag Σ1 (t), · · · , Σq (t) .
Let Var(yt ) = Var(xt ) = Ip , then A is orthogonal.
Let Bt−1 be a π -class and σ(Bt−1 ) = Ft−1 . put
X
X
[E{xt x′t I(B)}]2 .
[E{yt yt′ I(B)}]2 , Wx =
Wy =
B∈Bt−1
B∈Bt−1
For any B ∈ Bt−1 ,
E{xt x′t I(B)} = E[I(B)E{xt x′t |Ft−1 }] = E[I(B)diag(Σ1 (t), · · · , Σq (t))]
is a block diagonal matrix, so is Wx .
– p.60
Since
Wy = AWx A′ ,
the proposed method continues to apply.
In practice we estimate Wy by
cy =
W
k0
XX
B∈B k=1
n
X
2
1
′
yt yt I(yt−k ∈ B) ,
n−k
t=k+1
where B may consist of {u ∈ Rp : kuk ≤ kyt k} for t = 1, · · · , n.
– p.61
Example 8. Consider the daily returns in 2 Jan 2002 – 10 July
2008 of six stocks: Bank of America, Dell, JPMorgan, FedEx,
McDonald and American International Group.
n = 1642, p = 6

−0.227

 −0.203


 0.022
b
B=

 −0.583

 0.804

0.144
−0.093
0.031
0.550
0.348
−0.041
−0.562
0.201
0.073
−0.059
0.158
0.054
−0.068
0.436
−0.549
0.005
0.096
−0.129
−0.068
−0.012
0.668
−0.099
−0.409
−0.033
0.008
0.233
−0.012
−0.582
0.131
0.098
−0.028












Segmentation for transformed series:
CUC of Fan, Wang & Y (2008)
– p.62
−10
0
5
Daily returns of 6 stocks
2003
2004
2005
2006
2007
2008
2002
2003
2004
2005
2006
2007
2008
2002
2003
2004
2005
2006
2007
2008
2002
2003
2004
2005
2006
2007
2008
2002
2003
2004
2005
2006
2007
2008
2002
2003
2004
2005
2006
2007
2008
−10
0 5
−15
−5
5
−15 −5
5
−20
−5 5 15
−15 −5
5
2002
– p.63
CCF of squared returns
0.8
0.0
0
20
15
0
−20 −15 −10
0.8
10
15
20
0
5
0
10
15
20
0.4
0.8
Y4^2 & Y6^2
5
10
15
20
0
5
10
15
20
0.0
0.4
0.8
Y5^2 & Y6^2
0
5
10
15
20
0
5
10
15
20
Y6^2
0.8
Y6^2 & Y5^2
−5
20
0.0
0
0
15
0.4
5
Y5^2
−5
10
Y3^2 & Y6^2
Y4^2 & Y5^2
0.8
−20 −15 −10
5
0.0
0
20
20
0.8
20
0.8
10
15
0.0
15
0.4
5
10
0.4
0.8
0.4
10
0.8
15
5
Y2^2 & Y6^2
0.0
10
0.4
0.8
0
20
0.4
0.8
5
Y6^2 & Y4^2
−5
15
Y3^2 & Y5^2
0.8
0
0
0.0
−20 −15 −10
5
Y5^2 & Y4^2
−5
10
0.0
0
0.0
−20 −15 −10
0.4
0.8
0.0
0.4
0.8
0.8
0.4
20
0.4
0
0.4
0.8
0
15
0.0
−5
Y6^2 & Y3^2
−5
10
Y4^2
0.8
0
0.0
−20 −15 −10
5
0.0
0
0.0
−5
5
Y2^2 & Y5^2
0.4
0.8
20
0.4
0.8
−20 −15 −10
0.4
0.8
0.4
0.0
0
15
−20 −15 −10
Y6^2 & Y2^2
−5
10
Y5^2 & Y3^2
0.0
0
5
0.8
0
0
Y3^2 & Y4^2
0.4
−5
0.4
0.8
0.4
−5
0
0.0
−20 −15 −10
Y6^2 & Y1^2
−20 −15 −10
0
Y5^2 & Y2^2
0.0
−20 −15 −10
20
Y4^2 & Y3^2
0.8
0
15
0.0
0
0.0
−5
Y5^2 & Y1^2
10
0.4
0.8
−5
0.4
0.8
0.4
0.0
−20 −15 −10
5
Y3^2
Y4^2 & Y2^2
20
0.0
0
0.0
−20 −15 −10
15
0.4
20
10
Y2^2 & Y4^2
0.0
15
5
0.8
10
0.4
0.8
0.4
0.0
0
0
0.8
5
Y3^2 & Y2^2
−5
20
0.0
0
Y4^2 & Y1^2
15
0.4
0
10
0.4
0.8
0.0
−5
Y3^2 & Y1^2
−20 −15 −10
5
Y2^2 & Y3^2
0.4
0.8
0.4
0.0
−20 −15 −10
0.4
0
0.0
20
0.0
15
Y2^2
0.8
10
0.4
5
0.0
0.4
0
Y1^2 & Y6^2
0.0
20
0.8
15
0.4
10
Y2^2 & Y1^2
Y1^2 & Y5^2
0.0
5
0.0
0.4
0.0
0.4
0.0
0
Y1^2 & Y4^2
0.8
Y1^2 & Y3^2
0.8
Y1^2 & Y2^2
0.8
Y1^2
−20 −15 −10
−5
0
0
5
10
15
20
– p.64
CCF of residuals from fitted GARCH(1,1) for each return series
20
0.8
0.0
0.8
0.4
5
20
−20 −15 −10
−20 −15 −10
15
20
0
5
0
10
15
20
0.4
0.8
Y4 & Y6
5
10
15
20
0
5
10
15
20
0.0
0.4
0.8
Y5 & Y6
0
5
10
15
20
0
5
Y6 & Y5
−5
20
0.0
0
0
15
0.8
10
Y5
−5
10
0.4
5
0.8
15
20
Y3 & Y6
0.4
10
0.8
0
0
Y4 & Y5
0.0
−5
20
0.0
5
15
0.0
0
Y6 & Y4
0.8
−20 −15 −10
15
0.8
15
0.8
0
0
10
0.4
10
10
Y2 & Y6
0.0
5
5
Y3 & Y5
0.4
0.8
−5
0
0.0
5
Y5 & Y4
0.0
−20 −15 −10
20
0.8
0
0.4
0
15
0.4
20
0.0
−5
10
0.0
15
Y4
0.0
0
0.4
0.8
0.0
0.4
0.8
0.8
0.4
0
0.4
0.8
−5
10
0.8
20
Y6 & Y3
0.0
−20 −15 −10
5
0.0
15
−20 −15 −10
0
5
Y2 & Y5
0.4
0.8
0.0
10
0.4
0.8
−5
0.4
0.8
0.4
0
5
Y5 & Y3
0.0
−20 −15 −10
0
Y3 & Y4
0.8
0
Y6 & Y2
−5
0
0.0
−5
0.4
0.8
0.4
0.0
0
0.0
−20 −15 −10
0
Y5 & Y2
−5
20
0.4
0.8
−20 −15 −10
Y6 & Y1
15
Y4 & Y3
0.0
0
10
0.4
0.8
0
0.4
0.8
0.4
−5
Y5 & Y1
−20 −15 −10
5
Y3
−5
20
0.0
0
0.0
−20 −15 −10
15
10
15
20
15
20
Y6
0.8
20
10
Y2 & Y4
0.8
15
Y4 & Y2
0.0
−20 −15 −10
5
0.4
10
0.4
0.8
0.4
0.0
0
0
0.8
5
Y3 & Y2
−5
20
0.0
0
Y4 & Y1
15
0.0
0
10
0.4
0.8
0.0
−5
Y3 & Y1
−20 −15 −10
5
Y2 & Y3
0.4
0.8
0.4
0.0
−20 −15 −10
0.4
0
0.4
20
0.0
15
Y2
0.8
10
0.4
5
0.0
0.4
0
Y1 & Y6
0.0
20
0.8
15
0.4
10
Y2 & Y1
Y1 & Y5
0.0
5
0.0
0.4
0.0
0.4
0.0
0
Y1 & Y4
0.8
Y1 & Y3
0.8
Y1 & Y2
0.8
Y1
−20 −15 −10
−5
0
0
5
10
– p.65
CCF of squared transformed returns
0.8
0.0
0
20
15
0
−20 −15 −10
0.8
10
15
20
0
5
0
10
15
20
0.4
0.8
X4^2 & X6^2
5
10
15
20
0
5
10
15
20
0.0
0.4
0.8
X5^2 & X6^2
0
5
10
15
20
0
5
10
15
20
X6^2
0.8
X6^2 & X5^2
−5
20
0.0
0
0
15
0.4
5
X5^2
−5
10
X3^2 & X6^2
X4^2 & X5^2
0.8
−20 −15 −10
5
0.0
0
20
20
0.8
20
0.8
10
15
0.0
15
0.4
5
10
0.4
0.8
0.4
10
0.8
15
5
X2^2 & X6^2
0.0
10
0.4
0.8
0
20
0.4
0.8
5
X6^2 & X4^2
−5
15
X3^2 & X5^2
0.8
0
0
0.0
−20 −15 −10
5
X5^2 & X4^2
−5
10
0.0
0
0.0
−20 −15 −10
0.4
0.8
0.0
0.4
0.8
0.8
0.4
20
0.4
0
0.4
0.8
0
15
0.0
−5
X6^2 & X3^2
−5
10
X4^2
0.8
0
0.0
−20 −15 −10
5
0.0
0
0.0
−5
5
X2^2 & X5^2
0.4
0.8
20
0.4
0.8
−20 −15 −10
0.4
0.8
0.4
0.0
0
15
−20 −15 −10
X6^2 & X2^2
−5
10
X5^2 & X3^2
0.0
0
5
0.8
0
0
X3^2 & X4^2
0.4
−5
0.4
0.8
0.4
−5
0
0.0
−20 −15 −10
X6^2 & X1^2
−20 −15 −10
0
X5^2 & X2^2
0.0
−20 −15 −10
20
X4^2 & X3^2
0.8
0
15
0.0
0
0.0
−5
X5^2 & X1^2
10
0.4
0.8
−5
0.4
0.8
0.4
0.0
−20 −15 −10
5
X3^2
X4^2 & X2^2
20
0.0
0
0.0
−20 −15 −10
15
0.4
20
10
X2^2 & X4^2
0.0
15
5
0.8
10
0.4
0.8
0.4
0.0
0
0
0.8
5
X3^2 & X2^2
−5
20
0.0
0
X4^2 & X1^2
15
0.4
0
10
0.4
0.8
0.0
−5
X3^2 & X1^2
−20 −15 −10
5
X2^2 & X3^2
0.4
0.8
0.4
0.0
−20 −15 −10
0.4
0
0.0
20
0.0
15
X2^2
0.8
10
0.4
5
0.0
0.4
0
X1^2 & X6^2
0.0
20
0.8
15
0.4
10
X2^2 & X1^2
X1^2 & X5^2
0.0
5
0.0
0.4
0.0
0.4
0.0
0
X1^2 & X4^2
0.8
X1^2 & X3^2
0.8
X1^2 & X2^2
0.8
X1^2
−20 −15 −10
−5
0
0
5
10
15
20
– p.66
CCF of residuals from transformed return series
20
0.8
0.0
0.8
0.4
5
20
−20 −15 −10
−20 −15 −10
15
20
0
5
0
10
15
20
0.4
0.8
X4 & X6
5
10
15
20
0
5
10
15
20
0.0
0.4
0.8
X5 & X6
0
5
10
15
20
0
5
X6 & X5
−5
20
0.0
0
0
15
0.8
10
X5
−5
10
0.4
5
0.8
15
20
X3 & X6
0.4
10
0.8
0
0
X4 & X5
0.0
−5
20
0.0
5
15
0.0
0
X6 & X4
0.8
−20 −15 −10
15
0.8
15
0.8
0
0
10
0.4
10
10
X2 & X6
0.0
5
5
X3 & X5
0.4
0.8
−5
0
0.0
5
X5 & X4
0.0
−20 −15 −10
20
0.8
0
0.4
0
15
0.4
20
0.0
−5
10
0.0
15
X4
0.0
0
0.4
0.8
0.0
0.4
0.8
0.8
0.4
0
0.4
0.8
−5
10
0.8
20
X6 & X3
0.0
−20 −15 −10
5
0.0
15
−20 −15 −10
0
5
X2 & X5
0.4
0.8
0.0
10
0.4
0.8
−5
0.4
0.8
0.4
0
5
X5 & X3
0.0
−20 −15 −10
0
X3 & X4
0.8
0
X6 & X2
−5
0
0.0
−5
0.4
0.8
0.4
0.0
0
0.0
−20 −15 −10
0
X5 & X2
−5
20
0.4
0.8
−20 −15 −10
X6 & X1
15
X4 & X3
0.0
0
10
0.4
0.8
0
0.4
0.8
0.4
−5
X5 & X1
−20 −15 −10
5
X3
−5
20
0.0
0
0.0
−20 −15 −10
15
10
15
20
15
20
X6
0.8
20
10
X2 & X4
0.8
15
X4 & X2
0.0
−20 −15 −10
5
0.4
10
0.4
0.8
0.4
0.0
0
0
0.8
5
X3 & X2
−5
20
0.0
0
X4 & X1
15
0.0
0
10
0.4
0.8
0.0
−5
X3 & X1
−20 −15 −10
5
X2 & X3
0.4
0.8
0.4
0.0
−20 −15 −10
0.4
0
0.4
20
0.0
15
X2
0.8
10
0.4
5
0.0
0.4
0
X1 & X6
0.0
20
0.8
15
0.4
10
X2 & X1
X1 & X5
0.0
5
0.0
0.4
0.0
0.4
0.0
0
X1 & X4
0.8
X1 & X3
0.8
X1 & X2
0.8
X1
−20 −15 −10
−5
0
0
5
10
– p.67
Conclusions
• A new version of PCA for time series: finding a latent
segmentation via a contemporaneous linear transformation
• When the segmentation does not exist, provide effective
approximations by ignoring negligible, though significant,
correlations
P
• Why works? Wy = k Σy (k)Σy (k)′ = AWx A′ ,
tr(Wy ) =
p
XX
k i,j=1
ρyij (k)2
= tr(Wx ) =
p
XX
k i,j=1
ρxij (k)2 =
XX
k
ρxii (k)2
i
provided all components of xt are uncorrelated across all
time lags
Components of xt are predictable, due to stronger ACF!
– p.68
• When latend segmentation does not exist,
Wy =
X
use zt = Γ′y yt ,
Σy (k)Σy (k)′ = Γy DΓ′y .
k
Hence Σz (k) = Γ′y Σy (k)Γy , and therefore
Wz =
X
Σz (k)Σz (k)′ = Γ′y Wy Γy = D,
k
i.e. Wz is a diagonal matrix.
Note. Σz (k) are unlikely to be diagonal, though the
off-diagonal elements tend to be small.
– p.69